applications of method of least squares to the adjustment
TRANSCRIPT
Scholars' Mine Scholars' Mine
Professional Degree Theses Student Theses and Dissertations
1922
Applications of method of least squares to the adjustment of level Applications of method of least squares to the adjustment of level
circuits: Adjustment of elevations in Dawson and St. James circuits: Adjustment of elevations in Dawson and St. James
Townships, Phelps County, Missouri Townships, Phelps County, Missouri
Clarence Edward Bardsley
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Recommended Citation Recommended Citation Bardsley, Clarence Edward, "Applications of method of least squares to the adjustment of level circuits: Adjustment of elevations in Dawson and St. James Townships, Phelps County, Missouri" (1922). Professional Degree Theses. 165. https://scholarsmine.mst.edu/professional_theses/165
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APPLICATIONS OF TIlE METHOD OF LEAST SQUARES TO THE ADJUSTMENT OF LEVEL CIRCUITS.
ADJUSTMENT OF ELEVATIONS IN DAWSON
AND ST. JAMES TOWNSHIPS, PHE['PS COUNTY,
MISSOURI.
BY
CLARENCE EDWARD SOLOMAN BARDSLEY
A
THE SIS submitted to the facul~y of the
SCHOOL OF MINES AND METALLURGY OF THE UNIVERSITY OF MISSOURI iB partial fulfillment of the work required for the
D E G R E E 0 F CIVIL ENGINEER
Rolla,Mo. 1922.
ASSISTEO BY
CHARLES EDWARn COOKE
~y-,-,r~ 1d,",-~ , PrG seor ef €ivil IRaiae.ria,.
APPLICATIONS OF THE METHOD OF LEAST SQUARES TO THE ADJUSTMENT OF LEVEL CIRCUITS.
ADJUSTMENT OF . ELEVATIONS IN DAWSON AND ST. J!MES TOWNSHIPS,
PHELPS COUNTY, MISSOURI~
CON TEN T S •
INTRODUCTION. Remarks on the Introduction. • • .. . . . · . . • • • • Liscrepancies among Observations •••••••••• • Carefulness of Observation • • • • • • • • • • • • • •
• • • · . . . . . Types and Occurance of Errors • • • • Circ.umstances affectin~ Observations Princip~e of Least Squares Stated • •
•. . • • • • • • • ix·
• • · . . . . . . .On the Uniformity of Least Square Results • • • • • • • Not in General Use Among Engineers • • • • • • • • • • Precision, Subsequent Measurements Dependent on results Theory "Stated for Weigbted Observations • • • • • • • • Acknowledgments • • • • • • • • • • • • • • • • • • • • x
PART. I. THE INDEPENDENT ADJUSTMENT OF A MULTIPLE LINE.
METHOD OF LEAST SQUARES APPLIED TO THE ADJUSTMENT O~ THE LINE A VIA B, C, AND D TO E.
BASED ON THE LINE OF LEV£LS RUN 8Y THE U. S. Q£otOGteAl SURVEY IN 1908 ALONG THE FRISCO R. R.
W IT K PER IlIA In lilTS Ellie H. ARK S , - A - A·Y ST. "A" E $, AND -E- AT KNOBYIEW •
. Multiple Lines of Levels •• • • • • • • • • • • • •• 2 sta.tement and Solution for .Adjust~ent Correction ••• 3 ~eneral Principle of Lea~t B~u8res • • • • • • • • • • 4 The Weights fe>r Level Lines .. • • • • • • • • • • • •• 4 Probable Error of Wei~hted Arithmetic Meall. • • • •• 4 Intermediate Points on Level Cireuits •••
IBterBiediate junction pointsB and, C • /
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• • • • • •
• • • • • •
4
5
CON TEN T S •
PAR T I I . . TilE LEAST SQUARE ADJUSTMENT OF A LEVEL NET
BY MET BOD OF OBSERVATION EQUARIONS. APPLIED TO THE ADJUSTMENT
OF THE DAWSON-ST. JAMES TOWNSHIPS LEVEL NET,
PHELPS COUNTY, MISSOURI.
lASED ON THE A VIA 8, C, AND 0 TO E BASE.
Observed Differences of Elevation by Links • . . . . . . 7 7 7 8
8
9
Level Base Lines • • • • • • • . . . . . . · .. . . . . . On the Solution of the Problem by Least Squares Statement of the Conditions . . . . . . . . . . . Observation Equations • • • • • 0 • • • • • • • •
Computation of Jnct.Elevations on Arranged Circuit 0 . . Rule: Formation of Normal Equations from
Observation Equations 0.. 10 Solution of Normal Equations
Elimination by Addition and Subtraction .. . . . Check on Residual Values • . . . . . . . . . . . • 0 0
Summary of Residual Values 0 0 0 • • • • • • • •
- -
• 12 o 15 • 15
Calculation and Tabulation of Adjusted Jnctopoints 0 •• 15 solution of Normal Equations by the P,q,r Method 0 0 0 0 16
(0) Normal Equations 0 0 • 0 •• 0 ••• 0 • 0 • 0 16 (P) Solution of Normal Equations 0 • • 0 • 0 • 0 • 16 (S) Most Probable Differences of Elevation • • • 0 16 (T) Most Probable Elevations • • • • 0 • • • • 0 • 16 (q) Solution of Normal Equations 0 • • • • • • 0 • 17 (r) Solution of Normal Equations • • • 0 0 0 ••• 17 Adjustment Quantities • 0 0 • 0 0 0 0 0 0 0 0 17 Check on Resid uals " • • 0 • • 0 • 0 • • 0 0 • 0 0 17
- -Adjustment of Intermediate Junction points o • o 0 • 18
Adjustment of Points Land K 0 00 0 • . • • . . • 18 - -Adjustment of point P •• • . 0 • 0 • • • e · • • • 19 Adjustment of Points Nand 0 . .. • • · . . . o • • 20
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• • • • • • • 21 • • • • • • • 21
C 0 K TEN TS • Adjustml:)nt of Spur Linl:)s PQ, NR, and OSO ; ••
Adjustml:)nt of Point Q • • • • • • • • • • Paralll:)l Duplicate Line of Levels • • • • • • • • • 21
Unweighted Measureml:)nts . . . . .. . . . . . • • • • • • • 22 probable Error of a Singll:) Ml:)asureml:)nt. • • • • • • • • 22 Probable Error of n Ml:)asurements •• • • • • • Adjustment of Point~ R •• • • • • • • • • • •
• • • • 22 • • • • 22
Adjustment of point.S • • • • • • • • • • • • • • • • 23 Duplicatl:) Line of Levels • • • • . . . . . ')3 · . . . '-_. -
Summary of A.II Junction point Eleavations . . . . • ••• '24 Adjustment of Intermediate Bench Marks • • . . • • • 25
Description of Benchmarks and their Adjusted Elevations Adjustment of Benchmarks Between "A" ..and "I ". -
" " " " " " " " I and P - -
It " II " " " " " p and Q
" " " " " " " " A and D
PAR T I I I • THE LK4..ST SQtl.~RE ADJUSTMENT OF' A LEVEl. NET
BY METHOD OF OBSERVATION EQUATIONS. APPLIED TO THE ADJUSTMENT
OF THE DAWSON-ST. JAMES TOWNSHIPS LEVEL NET,
PHELPS COUNTY, MISSOURI.
BASED ON THE LINE OF LEVEl8RUN BY THE
•
· •
• • • 26 • • · 29
· • • 31 • • • 34
U. S. GEOLOGICAL SURVEY IN 1908 ALONG THE FRI8CO R. R. WITH PERMANENT BENCHMARKS,-A- AT ST. JAMES,AND
-E- AT KNOBV lEW.
computation of Elevations in Arranged Circuit •••• Closures in Arranged Circuit .. . . .
Observation Equations •••••• • • • • •• • • • Rule: Formation of Normai Equations from
Observation Eq~atJons •• statl:)me~t on the Solution of Normal Equations .. • • •
• •
· . • •
• • • •
39 39 41
41 41
Formation of Normal Equations for Level Net. • • • • •• 42 Summary of Normal Equations showing Redundant Numbl:)rs.. 44
(P) solution of Normal Equations '. • • •• • • ••• 45 .( q) solution of Normal Equations. '.. • • • ,. • •• 46 (r) solution of Normal Equations • '. • • • • • • •• 47
- -Adjusted Elevations of Junction Points ,. • • • • •• •• 47
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CON TEN T S •
PAR T I V • THE LEAST SQHARE ADmSTMENT OF A LEVEL NET
flY METHOD OF CONIHTION EQUATIONS AND
CORRELATIVES. APPLIED TO THE ADJUSTMENT
OF THE DAWSON-ST. JAMES TOWNSHIPS LEVEL NET,
PHELPS COUNTY, MISSOURI.
BASED ON THE LINE OF LEVELS RUN BY THE U. S. GEOLOGICAL SURVEY IN 190B ALONa THE FRISCO R. R.
11TH PERMAN£NT IENCHMARKS -A- AT ST. JAMES, AND wE- AT KNDIVIEI.
Tabulation': ,Linka, Observed Diff • in Elevation, Distances, ,Weights, Deri ved Correction, Corrected Diffs. in
Condi tion Equations • • • • • • • • • • • • • • • • • • • On the formation of Condition Equations
Table of Correlat~s and, Corrections to Links • • • • • • •
Elev.50 •• 50 •• 50
. .. . . . . . . 51 Rules: Formation of Normal Equations from Condition Equations 51 Normal Equations. • .• • • • • • • • • • • • • • • • • • • •• 52 Notes on the (P),(q),(r) Solution of Normal Eq~ations •••• 52
(P)Sollltion of Normal Equations. • • • • • • • • • •• 52 (q) Solution of Normal Equations •• '. • • • • • • • •• 53 (r) or Back solution of Normal Equations. • • • • • •• 53
- -Adjusted Elevations of Junction points •••••• , •••••• Abridged Method A for the solution of Normal Equations . ' •••
Normal Equations with Constant and .check 'Columns .• • • • Abridged Solution A with Notes on the precess •••••• Correlat~s with Notes on the Pcocess of forming Them •• Discussion: Substitute for the (P)' (q), (r) Method ••••
Use of Check Terms. • • • • • • • • • •• • • Abridged Method B for the 'Solution of Normal Equations • • •
Normal Equations with Constant and Check Terms ••••• ,Abridged Solution B with Notes on the Process .. • • • • • Use of ,Check Terms shown in Red.. • • • • • • • • • • •
--Back Solution with Notes on the proc~ss •••••••••
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54 55 55 55 56 56 56 57 57 57 57 57
APPLICATIONS OF T~E METHOD OF LEAST SQUARES TO THE ADJUSTMENT OF LEVEL CIRCUITS.
ADJUSTWENT OF ELEVATIONS IN DAWSON AND ST. JAMES TOWNSHIPS,
PHELPS COUNTY, MISSOURI.
INTRODI)CTION. An introduction to a thesis on least squares could be drawn
out to considerable len~th in explanation: as the purpose of this tbesis has been an investigation in applications of the method of least squares to tbe adjustment ot level circuits,it will be found in the tody of Ute rCllort considerable space has been devoted to the explanation of tbe different applications of the method,witb a vie. that these investigations may bo of use to eD~iDeers in the profession; so the intr'oductory remarks will be brief.
It is well known amOn~ engineers and otberstbat measurements made ~ith apparatus of " most any kind,chains,rods,scales,verniers, .icrometer8t~tc., and no matter ho~ careful one may be in making measurewelits,there will ~e discrepancies aWOD~ tbe o~servatioDs. If a series of benchmarks ~e COlJuected by lines of levels,~ome of wLicb are check lines forming witb tbe others. complete circuits, it is necessary to adjust tho difference in elevation so "that aU the circuits will close exactly,in order that the difference of elevation between any t"O ~encb.arks will be constant when co.puted througb two or more series of lines by the several routes • . This is illustrated b) several applic~tions of the aethod of least squares to tbe adjustment of elevations in Oa"son and st. James Townships,Pbelps County. Missouri.
It is ullderstood that observations are always Jiade as carefully as possible,tor tbe most pro~able results and miniau5 errors. Positive and negative errors of the same magnitude occur witb equal frequency; tbey are equall, prot able. Errors of increasing lIagnitude occur with decreasing · frequency. Saal1 errors occur flore frequentlYj8re lIore proba~le tbat. large ones. Very large errors seldoll occur;tbey are likely to belong in the class of mistakes rather in tbat of accidental errors. Acci~ebtal errors are systematically lIodified by tbe circumstances of the observations. The c·hief cire.umstances affeeting a set of observations are atllospber ic conditions, tbe skill of the ofiserver, andtbeprecision of tbeinstrullents.
CON TEN T S •
Adjustment by the Method of Correlates. . , Theory • • • • • • • • • • • • • • • • • • • • • • • 58
Sum of the Weighted Squares a Minimum. • • • • 58 Conditions multiplied by2C ••••• • • • • • 58 Summation of Weighted Squares and the 'Conditions58 The First Derivutive placed equal to zero ••• 58 Nomenclature of Symbols used • • • • • • • • • 59 Theoretic Formation of the Normal Equations • • 59 ,DisCllboion: On Methods for Formation of
Normal Equations .• .• ,,!)9
Application of the Theory • • . . . . . . . . • 60 Derivation of Terms for Level Net(Dawson-St.
James Townships l,Te~) •• 60
PAR TV. PROPORTIONAL ADJUSTMENT OF INTERMEnIATE BENCHMARKS ,
FROM CORRECTED EI.JEVATIONS OF AnJACENT JUNCTION POINTS. APPLIED TO THE ADJUSTMENT
OF INTERMEDIATE BENCHMARKS
IN THE DAWSON~ST.JAMES TOWNSHIPS LEVEL NET,
PHELPS COUNTY, MISSOURI.
BASEO ON THE LINE OF LEVELS RUN BY THE U. S. QEOLOQlCAL SURVEY 'N 1908 ALONG THE FRISCO R. R •
• ITH PERM~.ENT BENCHMARKS -A- AT ST. JAMES, . AND -E- AT KNOBYlew.
Diagrams showing Adjustment Corrections, Description of Benchmarks,Observed Elevations,Adjustment Corrections, and Adjusted Elevations of all Benchmarks .in the Level Net.
Adjustment of Benchmarks between "An and "I" • • • • • 63 - 66 " " " " n n " 'I and P .... • • .. - - 68 " " " " " " fI P and Q .. '. • · ..
and -
71 " " " " " " " 'I J .. • • • •
" " " " It " " B and B .. • · .. I. 73 - 76 " " " " " " " H and J • ,e I. • • -77 " It n " " " n J and M • • • • • -79 Adjustment of Points N and 0 • • • .. • • • . .. • • • •
-v-
CON TEN T S . Adjustm~nt of Benchmarks b~t,,~en "M" and "N" • • • • • 80
" " " " n " " N and R 82 • • • • • -" " " " " " " P and M • 84 • • • • " " " " " " " E and D 87 • • • • • -" " " " " " " D and G 88 • • • • • " " " " " " " A and B 92 • • • • •
" " " " " " n B and C 93 • • • • • " " " " " " n C and D 95 · • • • • n " " " " " n C and F 98 • • • • •
" " " " n " " H and F 101 • • · • -" " " " " " " F and G .• 103 • • • -" " " " " " " G and 0 104 • • · · " " " " " " " 0 and s. 106 • · •
" " " " " " " N and 0 108 · • • •
CON C L IT S ION · Main Cla8s~s of Probl~ms,M~thods of ·Solution,Appropriate
Applications. 110 Dir~ct Obs~rvations of a sing1t:~ Quantity. • .. ' . .... 110 Indirect Observations • • • • • · • · .. • • • • • • 110 Condition~d Observations · • • • • • • • • • • • • 110
Two M~thods of finding Most probabl~ Values of Unknown Quanti ties • .• 110
Obs~rvation Equations. • • • • • • • • • • • • • • 110 'Condi tion Equations. • • • • • • • • .. • • • • • • 110 Comparison and the Appropriate Application of the
Two Methods. 110 B~st Methods of Solving Normal Equations .• .. • •• • • .. 110
-B I B L lOG RAPHY .•
.Bibliography • .• . . .. . .. .. .. • • .. .. . • .. .. .. • • • • .. 111
I N D E X
Ind~x,Alphab~tical .. .. • .. · • • • • • . .. • · .. · .. .. .. 113
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APPLICATIONS OF THE METHOD OF LEAST SQ(TARES TO THE Al).JI1STMENT OF LEVEL CIRCUITS.
ADJUSTMENT OF - ELEVATIONS IN DAWSON AND ST. JAMES TOWNSHIPS~
PHELPS COUNTY~ MISSOURI.
ILL U S T RAT ION S • Folder of Plates preceeding page 1
-Plate .A. Map of Dawson and North Half of St. James Townships showing Lines of Levels,etc. • • • •• 1
Plate B.Showing Observed Elevations. • • • • • • •• 1 Plate C. Rearranged. Elevations of Points for the
Discussion in Parts I and. II • • • • • • • • • • 1 Diagram of the Circuit A, B, C, D, E,A ' •••••••••• 2 Diagram of Adjustment Corrections between Points A and D. • 5
" " "" lie " " "A and I. .18 Diagram Showing the :Adjustment of point P ••••••••• 19
" " " " " points Nand 0 ••••• 20 Plat showing Observed EI.,:vations for the Points P, Q,M, N,
R,O, and B ••••••• 21 -Diagram of Adjustment Corrections between Points.! and I.
" " " " " " " " I and P. - -
" " " " " " " " P and Q.
" " " " " " " " .A and D. Plate showing .Arranged Circuit for Adjustment of Entire
.25 .• ~29 .31 ,.34
Net from the Railroad Base by the Method of Observation Equations •••••••••••••••.• .• 40
Plate showing the Selection of Small 'Circuits for the Adjustment of the Entire Net from the Railroad Base by the Method of Condition Equations and
. Correlat i Ves • • • • • • • • • • • • • • • • • Graph for the Adjustment Corrections between.! and I.
- -Diagram Showing the Adjustment of Point P ' • .••••• Graph for the Adjustmtmt 'Corrections between I and p.
- -" " " " " "e " "P and Q •
" " " " " - " " " " "
" " " " ~vii-
" "
I and 'J. Band lI.
• • .49 • • .63 • • .66 • ,. .66 .. • .68 ,. • .71 • .. .73
ILL U S T RAT ION S . f l .' j.
Graph for th~ ,Adjustm~nt 'Corrections b~tw~~n Hand 'J • .. • 76 n n n " " " n " J and M 77 • • •
Graph showing . .Ad,j ustment of PointeN and 0 • • • • • • • • 79 Graph for th~ .Adjustm~nt Corr~ctione b~t"~en M and "N • • • 80
n n n " " n " " N and R 82 • • • -" n " " " " " " P and M 84 · • •
" " " " " n " n E and D 87 • • • -
" " " " " " " n D and G 88 • • •
" " " " tI " " " A and B 92 • • • n " n " " " " " B and C 93 .. • •
" " " " tI tI " " C and D 95 • • •
" " " " " " " n C and F 98 .. • •
" " " " " tI " " 'H and F 101 • .. • -
" n n , n " " n " F and G 103 · • • -" " n " n " " " G and 0 .. • • 104
-1.06 Graph showing th~ .' Adjustm~nt of Point "8. • • • • .. ... • •
Graph for th~ Adjustllt:mt Corr~ctions b~t",=,~n N and o. • • 108 Plat,=, Showing Figur~ .Adjustm~nt of A G~od,=,tic Quadralateral
-of th~: (P).~(q)~ (r), and Givin~in D~tail the Proc~ss
Solution of Normal Equations A.pplicabl~ to 'Level Net Problems • • • . . . • • • • • • • • • • • • '. .Finas.
- Viifi-
The principle trom which the term "leQstsq~a~es" 8ris~s is the following: In .e4s~re.ents of eq~al _re~J$Joft.the aost _robable val~es of observed q~QntJtles are those that render the s~r. of the sq~ares of the resJd~al er~ors a .JnJ.~". The errors that lIIay be corrected,however,by tbis metbod, are only those accidental errors that are governed by no laws other than those of cbance.
The lIIethod of least squares takes first rank as an aritblIIetical process and by its use unitor8ity ' is secured in tbe process of adjustaent and comparison, and the 1II0st pr'obable balanc,iDg of errol's of observations lIIay be expected.
It is believed tbat the .ethod of least squares is not in general use allong engineers and that approximate aethods are lIIore ot'ten substituted for the adJust.ellt of quantities,although far superior results.igbt be expected through the elllployment of toe 1II0re exact aethod. There are those,Do doubt,who although versed in aatheaatical science, lIIigbt find it difficult to utilize this lIethod, for lad. of concel.ltration eBIl patience to carry this process througb t~ completion.
It is recolllmeLded, however, whenever precision is delllanded and subsequent measurements are dependent upon tbe results.
In the adjustment of level circuits, tbe several lines are weighted inversely as tbeir leagths,tbis causes us to state the above principle 8sfol10.8: In obs.rvat Ions ~h.". ~neq~al . 'e.lght s are appls.d,the aost _robable valves of the observ,d q~a"tJtJes are thos, that render the s~* of the a'Jghtedsq~Qr.s of the resJdvol e~ror$ a ~'"'.v.. '
The levels in Dawson and St. James Townships were run un~er .Y superv isior1 by .essrs. E. E. Dec,ker, W. R. Denison:,and F .lJ. Bollow.
In the arrangement of this thesis, and in tb~ coaputations and construction of drawings, I wish to ac~nowled2e'tbevaluable assistalJce rendered by Major CnarlesE. Cooke,Pro'essorof Topographic Engineering.
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..",O!!!~~~¥~::::-::9J>' oo/usrd hdWNn~ ond £ on t""Dilrood lin.
;'1r,..,..,.$ itNIN:(lt. d'-,..cT- I ion tJfle .. l.silldrcuib. I
J"Mh __ I""inf ele"..!,#n.s I .. re ;, ilMdr, e/lfflll'enG&3 of ele,./,im /lnJ in retl I .nd bl_", c;los .. ",,,,,,, I /" bl_. tt¥1"k>_~ Oi"e. I /n 9re.". D/6IwJOU I:' n.ll co".,."'" "';,." .,J,,, .. ,, i" bu.Lin • .s F'Q.,Nfl. 0
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• -Ci~~",~ IJT /t!'~/s,fir.sl;ond flJd wll./e ~ h ~ I I 1-a.IS$ north is 1J1~e/ onll,B.C,DQ.$ a,BP-$ • .
J _____ ..: ____ ___ _ ___ _____ __________________ _____ __ •
&I. l. e. oft h •
No,.th.a.t po,.tlon of PHELPS Showln, atlon of L EV ELL I iii E S, BENCH IIARkS, JUHCflOH PIS, wit h ,. •• p.ct p to aactlon IlfI •• and phy.l.al
'1 •• at u ft •• , • t c 0r-'-'n DAWSON T.p. _ ~ i and vlolfllty. 'V" , T hi. II A P a I • UJ-r .how. tho ' D IRE C T ION 'n :ot"""--;i'f-T-1'-,.--"";"'::::'=~"""~,....,.---F-,.--+-..,---t-----!"-t---.--t:='T---'------7---,
which tho LEVELS •• ,.. RUN. • • • • •• • • 00 In follo.l n 3 tho dlacu •• lo off thl. THES adju.t tho COVER IIAPS ov.,. thl. BASE MAP a. t h •
a ,..
• • •
PAR T I • . - -
THE INDEPENDENT ADJUSTMENT OF A MULTIPLE LINE. METHOD OF LEAST SQUARES APPLIED T0 TEE ADJeSTMENT
SF THE LINE A VIA B~ e~ AND D TO E.
(BASEO ON THE LINE OF LEVELS RUN BY THE U. S. QEOl'QI~AL SURVEY IN 1908 ALONS THE FRISCO R. R.
WITH PERMANENT BENCHMARKS~·A· AT ST. JAMES, AND BE" AT KNOBVIEW).
THE O~TERW ' INATIO~ A~O AOJUSTMENT OF ELEVAT10NS IN
PHELPS COUNTY MISSOURI.
(rawson Township L~v~l N~t)
Adjustm~nt of lin~s 'A, B, C, I; and E,r as Multipl~ Lin~s of 1~v~I~. S~e sketch below and Plate C.
-~ULTIPL F LI~KS. By a multiple line of levels is'
meant a set of two or more lines connecting th~ same two b~nch marks by routes of differ~nt length. In ord~r to find th~ most probabl~ value for th~ diff~rence of el~vation between th~ terminals of a multinle line, it is n~c~ssary to wei~ht each constitu~nt line inversely as i ts len~th. Havina weiahted the sev~ral 1 ines , as thus explaned, the caSe becom~s identical with any case of weiahted measurements, and hence the probable error of a sinQle measurement of unit weiaht is aiven·by the formula:
the pr()bable error of any of the linea of the wei~ht "J! by the tormuls:
and the probable error of the wei~hted ~rithmetic mean by the formula:
/ ZE V 2 = 0.6745 \(Zp(n-1)
Our problem is as follows: G I .;. n two I In •• o f 1 • .,.I,·,a, ,hown I n til.
a" 0 v. f I 3 u .~ • , • h I 0 h l I v. t ". f 0 I low I n l ~ •• u It,:
A v I a B, 0, to 0, 6. 25 .1 I • I In. , + 1 1. 1 1 1 ft.
A via E to 0, 1.00.11. lin., +11.049 ft. Not.: A to E llv.n •• 13I1t of Inflnltr.
T h. • I • vat 10 n of 8. It • at A 1017.800 ft .. , and the .I.v-
atlon of B .... at E • · i074.001j ft.
Rft'aui ~ e 1 t: he mos t orO b lblf! s l e v at io'l of
the 00 " t ') , ano tha proba~l .. e rr o r o f tni 9 result.
SOL UTI 0 N
· M P pM V V2 pV2 11.111 o. 160 . 1. 77776 · +0.0 53 0.00'2803 0.00044944 11.-043 1.000 11. 04900 -0.003 0.000081 0.00008100
Ep:l:-160 _ ... _-----) 1! ~. 82676 --------11. 058
O.t.".· ln·atlon of t II I allov. quantltl •• :
A 1087.800 0 . 1098.911
--------+ 11.111
01 ff.EII • . ,. C t 0 D.
ft. ft .•
ft. • ,. a 8,
A • 1081.800 ft. E. 1074.005 ft.
1-3 .. 795 ft Dlff.EI.v A to E.
D f~o. O~ark H-1 lin. • 1098.911 ft. o f ~ o. K" 0 " • I • w 8. • III. ' . 1 0 9 8· . 8 " 9 ft.
Ob •• ~v.d .rror at D .0.062 ft.
----------Ep~O. 00053044
n=!2.
D 1098.849 ft. E. 1014.005 ft.
• 24.844 13.195
• 11.0 4 9 ft. Dlff.EI.".A via E
to D.
RESIDUALS: In oba.~.,atl · ona of unequal •• Illlt,the dlff.~.nc. b • t • I • nth e w. I lilt. d a" I ::t:h •• t I 0 •• a nan d t". 0 b •• ,. II. d "a I u • of eaoh obs.~vatlon I. a ,. •• idual. If the ob,.,..,.d "alu. I. 3~latl" tllan the .el ~ht.d a~1 th •• tic .ean, ~h. ,..aldual quanti t yl. p I u •• , and If I... t han , h. •• I :I h t • d ." I t h • • t I 0 ••• n , the ,. •• Idual quantity I, .Inua.
FI~.t o",."".d dlff.ln EI.v II ,.Il"t.d .~Ith •• tlo •• an pli
11.111 ft. 11.058
FI".t " •• i dual .""or •• 0.053
s • con doll •• r w • d d If , 1. I n E I ." II 1 1. 0 4 9 ft. Wel]hted a,.lthlletlo lI.an pI! .11.058
S.co~d ,.e.1 dual ."'0" -- O.OOg·V.
GE,\rEriAL PliIN'CIPL-E: In ob·serV.:Jtl.071S of unequ.:JI />rec~s.orn,
the mast probab!e values of the observed quantities are
those that render the sum of the ~elRhted squares of the
res. ~ d u a I err 0 r sam I r, I mum. I t I. 0 n a coo u n t 0 f t hi. P r I n _ clpl. th.t the lIethod of least .quar •• has 1I •• n so na.ed.
THE WEIGHTS P are the reciprocal's of the distances for each line. Thus:
A via B, C, to 0 6.25 rall •• ,r.clp,.ocal or wt.-0.160.
E to o 1.00.11. .. .. .. .. If • 1. ·0 0 •
N.o t e: R • c I pro 0 a I san d S qua ,. • .• can b. If.' Y oon".nl.ntly look.d up n Bar'ow'. Tables of Squar •• , Roots, E~o.
Nth. nUllb.,. oft oll.ervatlons.
CHECK: A. a oh.ok on the solution .0 ... i ] h t • dr •• I d u a Is", U st. q u • I z. ""0 •
p o. 160 1. 000
V +0.053 -0.009
pv +0.009
:~!.~~2 l:p v - o. 0 00
far, the s U II .. a t Ion
Then the probable error of the weiQhted arithmetic mean is:
r = 0.6745 IJQ. OQ~53044 =': 0.0144 ft. p a V 1. 1. 0 x 1
1087.800 · ~ 11.058 = 1098.858 ft., And the most probable value for the elevation r is:
~ = 1098.858 ~0.0144 ft.
INTERMEDIATE POINTS. By an Int.r •• dlah point i. "alit on. Iyln] only on a sln31. In. of I.".,., and h.n •• h a If I· n l n 0 I n flu • no. 0 nth. l. n era I a d j u • til. n t. T h ••• • a y ooour on a line whoes .nd. half. b •• n tatl 'faotorl Iy ad-
o f
justed or on a olos.d olrcult. In .Ith.r oa •• ,th. r.qulr-.d adju.tlle"t Is dlstrlbut.d uniforlll), throulhout the 11".,lIallln3 the oorr.otlon b.t· ••• n an)' two polntsdlreotly proportional to the I.n]th b.t ••• n those t.W points.
ADJUSTMENT OF INTERMEDIATE JUNCTION POINTS B & C.
Ollser·v.d valu. of D - 10 9 8 • -9 11 ft.. (Ozark H.'1. LI".) Adjusted val ue o f D - 1098.858 ft.
--------AdjYstllent • + O. 0 5 3 ft. 10 •• r at D and tap e r to 0 at A.
-4-
A. lhe dlatance A "Ia tha dl.tanc. A to B 1.00.1
B, . ; C to t h • n
o • t h •
6.25 .. 11 •• , and adjuat •• nt to
poInt 8 I. aa foll .o.a:
Adju.t •• nt t 0 8 0.008 ft.
And the IIO.t probabl. el."atlon of point 8 ia:
1074.666 0.008. 1074.658 ft.
In like .. ann.r, a. the dlatanc. A to C" 4.00.11 •• ,
tit. a d j u.t •• nt to C - --~-- " O. 0 53 - o. 034 ft. 6. 25
II n d t h • .0. t p rob a II I • • I • " a t I on o f pol n t C I • :
- 1083.400 ' - G. 034 • 10S3.366 ft. - -- - - - - -
• • • • • • • • 0 • • •••• o ••••
C H E C K 0 n • I • " a ~t : Ion 0 f 0 fro III E •
EI."atlon of 0 from E • EI,vationof E
Olff. In EI,,,.E to 0 Corr,otlon for 1.00 .. lIe E to 0
109S.849 ft. 10~"oI00!) '
.24.S44 ft. -.0.009
24.853 ft. 1074.005 --------
Mo.t probabla E I a " • of 0;'1098.858 f t
Ob •• rv.d EI. v. 0 t 0-1098.849 f t Adju.t.d E I • t. of 0-1098.858
--------Adju.t •• nt. o. 009 ft.
o • • • • • • • • • o •••• ••••• ( 0 • K. for
• •• e .e
S •• Plat. C for the .. dju.t.d " .. Iu •• for this clroult •
• g~
~.O~~~~=+~ __ ~ ________________ ~-L ____________ ~D~-L Adju.t.d Llna
10-------- 4: OM;. --- -----01 ~-------------6ZSHi----------------~
DIAGRAM SHOWING ADJUSTMENT OF POINTS SET. A .. nd O.
t I ..... 1· ' a •• n how the ,v .. lu ••
1 "".).
Fro. t •• abova dl .. ,r ... I alon3 the OIl •• r"adtr.d)1 to the dl.tanc •• fro .. II T~~·. 16" .. 3 r .. .. a I • 0 • a r v ••
In • .. r. lo •• rad proportloll .. 'I, do. n to t'll. .. d j u • t • d , ( II I .. c k .) I In,.
tit. In t • r •• d I • t. p ·o I n t • •• .. oh.ok on til ... dJu.t •• nt. ~f
If dr •• " to .cal • •
-§-
PAR T I I .
TllE LEAST SQUARE ADJUSTMENT OF A LEVEL NET
BY METHOD OF OBSERVATION EOUATIONS. APPLIED TO THE ADJUSTMENT
OF THE: DAWSON-ST. JAMES TOWNSHIP LEVEL NET~
PHELPS COUNTY, MISSOURI.
(BASED ON THE: A VIA B, C, AND D TO E BASE).
-6-
THE DETERMINATION AND ADJUSTMENT OF ELEVATIONS I N
PHELPS COUNTY MISSOURI.
(Lawson Township Level Net)
Referring to the :Cawson Townshi p Levt::l Net I:iagrams indicated on pages and , the field notes show the follow-ing resul ts:
A via L, K to I = 25.875 ft. -
I via P to M = - 107.201 ft. J to H = + 73.460 ft. M to J = + 13.695 ft. H to F = - 40.447 ft. F to G = - 80.066 ft. G via X, L to .A = -+ 166.589 ft. I to J = - 93.531 ft. A via B to H = - 45.750 ft. A via C to !i' = - 86.657 ft.
-M via N, 0 to ,.., = - 33.201 ft. I-:!
It is desired to find the most probable values for the elevations of the level net junction stations I, M, J, E, F, and ~ Opon finding these junction elevations, it is then desired to find all interme~iate and remaining elevations in this net. Referring to pageZ,it will be noted that the line of levels running from A via B, C, :C, to E has previously been adjusted; levels along the Frisco R. R. have also been adjusted:these lin~s will be considered base lines and their values will remain unchanged. The :Cawson Township net will be considered tp hinge on the O. S. Geological SurveyStaf;ions A (at St. James) and E (at Knobvi~Hd.
SOL UTI 0 N 0 F PRO B L E M BY MET HO~ 0 F lEA SoT 8 0 U ARE s.
As there are but,six unknown Bench Marks (B.Ms.) · (I, M, j,sH, F, and ~),there can be but six independe~1 unknowns ih the observation equations. As the linesAI~IM,
- -MJ, JR, HF, and FG, may .evidently be selected as the indep-enden1t' unknowns, we may write for . the most probable values
-tr--
of the correspondinQ differences of elevation:
A via L, K, to I = '25.875 + VI -I via P to M = - 107. '201 '+V2
.' M to J = + 13.6:15 + V3 J to H = + 73.460 :+ VI,. H to F = - 40.447 + V5 -F to G = - 80.066 + V6
The conditional equations involved in the se'veral closed circuits may then be avoided by wri tin ~ all the observation equations in te'Ilms of these Quanti ties. W ritin~ the reduced obsetvation eQuations directly from the dia~~am on pa'(1e1C, we have, by ,comparison with the observed values,
R ° \I t. ' VI R • • I d u a f • g'5:~6~·(wei~~l3~:\60); A Vla L,K,toI =
-I via P to M = V2 = 0.000 (wei £!h t 0.156);
M to J = 'V3 = 0.000 (wei£!ht o. 31'2) ; - 0.6:25); J to H = VI,. = 0.000 (wei!!ht H to F = 'v 5 = 0.000 (wei f,1ht 0.571);
-F to G = V6= 0.000 (wei ~ht '2.000) ; -
(wei~ht 0.222); G vi a X, I, to A = - 'V 1- V 2- V 3- V I,.-V 5-V 6=:+0. 155
A via E to H = 'V 1 +V 2 +'V 3 + VI,. , =-0.171 (wei Ith t 0. '2:14);
A via C to F = 'v 1: +V 2 + V 3: +'V 4: +V 5 = +0.'28:1 (weiltht o. 333); - V3:+'V4+'Vfj:+lV6=-0.157 (weiltht 0.1:24); M via N,O, to G=
I to J = V2+V3 =:+0. 0~ 25 (wei~ht 0.6'25). . ,q .. atlo".
To show how the above reduced observatlonAwere form-ed, a compilation of the valu'es for elevations of the several level net junction points determined from the suc6essive differences of elevation alon~ the selected circuit, A via L, K, I, P, M, j,B, F, G, X, 'D,to A 1.s shown below.
1
The closures shown above for the several circuits are extended below, as well as the calculation of the respective
wei ~hts.
-2-
Observed values for the elevations of stations alon2 the Circuit A via L, K, I, P, M, J, H, lI, ': , X, I, to A, Ietermined from successive differences of elevation:
A = 1087.800 ft. U.S. S.S.Iron Post at St. James. 2.5 .•. 875
--------(-I = 1061.925 ft. --------1 = 10.j1.9' ~5 ft.
M =
J =
it =
=
=
A =
+
+
-;..;.------- , I I I ,
--93~506 ft. 9 3 .5 31 ft. I to J. '
I
,-------- , ' ~ 0.0:25 ft. Closure. 968. 41'j ft. -------
I
73.460 ____ '_-"__ A =1087.800 ft.
1041.87) ft.---- ," ' ~ . ~:T: ----------40.447 45.121 ft.
-------- 45.750 ft.A 1001.4~2 ft. --------
, - 0.171 Closure. "" 80.066
viC!. B to B.
912l. 366 ft. , " A= 1087.800 ft ,
', ____ 1
+ 166. 589 " " ---86~368- ft. .. ' ".
" 1087.955 ft. ... 86.657 ft. A via C to F.
'" ---.-----"'~ 0.:289 ft. Closure.
" 0.155 ft Closure. "" M = 954.7:24 ft.
" , " ,, - ... ~ " ,--- . : , ' -33. 358 ft.
-3;.201 ft. M via 0 to G.
A . ROt r *. Vla . , , to ..
I via P to M
M to J -J to H
H to F F to G -G via XiI, to .A
.A via E to H -
M v iaN, 0, to G
I to J ' .A 'vi a 'C to F
= 3.21 = 1. 60
= 1.75 := 0.50 = 4.50 = 3.40 = 8.05 = 1.60
= 3.00
--6:I~~-ft. Closure •
" 0.156 1-- .. ----
= o. 31:2 It will be noted
= 0.6:25 that in each case
= 0.571 the respecti ve
= :2.000 wei ~h ts ar~ the
" " "
O. '2212 reciprocals of the = " = 0. ':294 distances.
" " = O. 1:24
" = 0.625 . . If = 0.333 -9-
We next write the normal equations,which are ari ved at as summarized in the followina rule:
'R {j L' E: 10 for 111 the . nor 111 a l . e qua t ;. I) 11 j 0 rea c;, 0 n e 0 j
t h; e u n ~ n 0 it n qua n t ;. t ;. e' s, 111 U l t ~ ply e a C hob s e r vat ;. 0 r, . e q 'U ,:1-
t;. onby t~e product oj the ~ elght oj that observatlon
and the algebralc coejf;.c;.ent oj that unknoanquant;.ty
.n that equat;.on, and add t h e results.
Bavin~ formed t~e several normal equations,t~eir solution as simultaneous equations aives the most prob~ able values of the unknown Quantities.
Then by the above rule; we have: NOR II ALE 0 U A T ION I NV 1 • (Solution f 0 ,. )
O. 160'v 1
O. ~9 4V 1'+0 .2) 4'V 2'+0. ' ~9 4V 3 +0. :2) 4V 4
O 2A~V +0 3 A"A'V '+0 ?: A ~V '+0 ~ ;z :zV '+0 ~?:7V • :"'.-1.-1 1 • -'.-I 2 ' • .-1 -' .-I 3 . ..).-'.-' 4' • .-1 .-1.1 5
NOR W ALE Q U " T ION I N lV % •
'+0. 156V 2
=+0.000000
=-0. 050!L7 4
=+0.096333
=' +0. 000000
0. '22'2'V 1:+0. ' 2' ~:2V 2 +0 . :2'22V 3+0. '22: 2V 4:+0. ''L''L- 2V 5,+0. '2:22V 6=-0.034410
o .~) 4'V 1 +0. '2) 4'V 2'+0. ' ~94V 3+0 .'2) 4'V 4 =-0.050274
0.33~V1+0.3 3 3Vi+O.333V~+0. 33 3V4+0. ~ 33V5 =+0.096333
:+0. 6:2FjV 2:+0. 6'25V 3 , =:+0. 0156~5 II') ~~-;~~j' ~-:-l - ~~~~-:~-~~l~-:~-;~I ~~-:~-~~~~-:~-~-~~:::~-~~ry=ry~ , L, . ' " , .J 1 .", " . ' , " , 4 . , . ' 0, ' , , ... .,
NORWALEOUATIONI.., V3 •
'+0 ?:]/r 'v ' • .1 L 3
O , ,- , ,~ ! r v '+0 II',r ,r>V '+0 ,rr,r,v :+0 ,rf ,rv ·+0 ,rL;r,'.2V +0 ,,~r r;.v =-0 034410 • , L, L ' L l ' • , [L.· L 2' • ,L. L , L. 3. , L L.L 4 • ' L 5' • L L 6 •
, 0 . 1 ~9 4V 1:+0. '29 4'V 2:+0. 129 4'V 3:+0. '29 4'i 4 =-0. 050~7 4
O 3AA'V ' 0 ?: A ~V '+0 ~~:z'v '+0 ~~~'V '+0 3;Z~V =; ~. 0.0963~~ • .-I.J l ' + • J .J .) 2' • -' -'.-I 3' ..) -' -' 4' . , .-'.-1 5 - -
, +0.1' ~4'V3+0. l' 2 4V4+0.1'~4;V5: +0.1'~4'V6=-0.019468
:+0. 61~5V 2:+0. 6·r;.fJV 3 ='+0. 0 156-r;.5 ~) +~~~4?V~:1~474V;:1 ~)lnv~:~~77?v::n~~71V~:G~?~7,v6::~~007807,
-1@-
NOR MAL E QUA T , 0 N 'N 'V 4.
+0.6~5V. =+0.000000 O ";, ':;'S-V +0 ,I""L ,I""L' I')V '+0 ,,:,I""'I""LV '+'" ,I"" ,r ,rv '+0 ,r ,r ,rV '+" ,r .rr,v - 0 0;:14"10 • "-"-"- 1 • "- ' 2' • 4 ' L ~ ' -J. L 'L.L.' • L,LL 5' VO LL.£ 6-- • J "'i
=-0.0.50'274
NORMAL EQUATION I NV 5 •
+0.571V 5 =+0.000000 O ,f",r:l'"'V '+0 ,I"'\""'I"'V '+0 , ""~ '"'V +O· ir"irroV +0 f"""""'V ' 0 ,(''>t')2V 0 0::'<:4410 • ,L·£,L l' • , LL>L 2' 0 ·L.L.·L, J o , ,L,LL... ..L.LL 5 + • ,.L..L 6=-' ~
O. 333Vi+0. 333V;+0.333Vi+0.3~3Vi+0.333V5 ~+0.096333 :+0. 1' ~4'V 3:+0. 1':24V • +0. 1"24'V 5+0. 1'24'V 6=-0.019468 ---------------------------------------------------------5) - ~~'r " ~ 'r'-'] ~ "1"" "7 " /h,"', . ,. -,,- , '" " '/ ' ) o"\~~,-'r "'" , .'. : .... r ' ' f , "'" r ~ - , ~ ':' h " .. ' ' ........ .' i ' , " .J. ,' : ......, ( -i ; .+ ' ' ''''': 1, . +. ' .. ,--' I, ~ -:::: .. ,-... , 1 L.I.. _.' , ' ~": f'
• • ' • ' I 1, . ' " ' :> ,' . . "") • .' ~.1. - .. 'i'" , ",- .. ' :. , ' .
NOR MAL E QUA T I 0 Ii 'N 'V 6 •
: ~2.000V6~+0~000000
0. '22.LV i+O. !222V 2+0. '222V 3:+0. '222V .:+0. '2'2:2V 5+0. :222V 6=-0.034410
___ ~ ___________ ·~~:~~~'~i~~:~:~~·~!~~:~'~~~i:~~:~!~~'~!::~:~~~~~~ 6) .. oJ, ") ./~ ,~ \I + '" -: r: ':' ,: .. ') 7 ~ r \ I . + .'" ;: .A ~ ,;,' + -, 7' :: \i + r;, ,' 1 t.:) 'I • -= _ .~ n ;~ 7. q 'y:~
. • .t .~, , ' 1 , .,' ..' , ~ ? - ' . ~ ") ,I . 1 .. , ,, . .. . ' I ) "Ii " ' . ." V ") I. . . , .
Wh.nc. fro. tit. aOllltJlon for the Nor.al Equationa, w. writ. tit. el" foltowln3 nor.al .quatlo". tllat ar. to . b. • 0 I, " • d • I .. II , t a" • 0 u • ',/ f 0 ,.. til. "a I II •• 0 f 'V 1 ,V 2' V 3 ,V. ,V 5' & V 6.
NORMAL EOUATIONS.
1):+1. 009V 1:+0. 849V 2'+0'. 849V 3:+0. 849V .:+0. 555V 5:,"0. 222V 6=:+0.011649
'2 ):+0. 849'V 1:+ 1. 6 30'V 2:"'1. 47 4V 3:+0. 8'49'V .+0.55 5'V 5"'0 . '222V 6 =· ... 0. 0!27~27 4
3):+0'. 849'V 1;+1. 474'V 2+ 1. 910'V 3:+0 • 97~'v+0. 679'V 5 ,+0. 346'v 6=+0.007806 • 4)+0. 849'V 1:+0. 849'V 2.+0.97 3V 3:+ 1. 598'V .:+0. 679V 5:"'0. 346v 6=-0.007819
5) '+0. 555V 1:+0. 555'V 2:+0. 679'V 3:+0. 679'V.+ 1. !250V 5 +0'. 346'v 6=;+0. 04!2455 6) ... O. 1222Vl:+0. '222V 2· ... 0. 346'v 3:+0. 346'v . :+0. 346'V 5:+!2. :46'V 6=-0.053878
• • • • ••• • •• • •• •• • • •• •• • ••••• Now w. ar. to ao'". til. abo" •• quatlo" •• I.ultan.ou.lr~
SOLUTION OF NORMAL EQUATIONS.
-11-
SOL UTI 0 N 0 F ~ 0 R 101, 1. L E QUA T ION S.
FI ,.at Sat.
I I I I .. I n a 11 e tV:5'
1 ) :+ 1. 0090V 1+0.8 490'V 2 +0. 8490'V 3+0. 849o"V • +0. 55 50Y 5+0~ '2'220V 6= +0.011649 i ll. 5.1 6 12 .
~ 2) U.l1~90V1: +0.9390V2~ +0.8490V:5+0. 4890V.+0. 3197V5+0.1'279V6=+0.015713 7 ). +o:-~205v~:O:09 OO'v;---------:~O:"§ 606,v::~c5:;2 3 53V~~0:0941V~;:0: 0040 64
!L) +0.8490 V 1 +1.6 30V 2 :+ 1. 47 40V 3'+0.8 490V • +0. 5550V 5 +0. ;2220V 6=:+0. 0~ 27'27 4 -0·17..113
3) /+. t)IJ9'2V 1 + 1. 137 5V 2 + 1. 47 40'V 3+0.7 509V .+0. 5~ 240V 5+0. '2670V 6=+0.0060'24 8 ) : +O:19~8'V~·~O:4}25V~---------:~O:0981V:~0~0310V~:6:0450V~:::O:0'2i250
3)+?i 8i~~XJ+1. 4740VL+1. 910'V 3 +0. 9730V. +0. 6790V 5+0 • 3460V 6 =+0.007806 4)/1.;bbbVi+1.6666v2+1.910V3+3.1369V.+1. 3~29V5+0.6792V6 =-0.015349 9 )-o:8176V~:O:19~6V~--------:'2:1~3§V::O~64~9V~:6:333~V~-:::O:O'23i§5
4) + 0 • 8 .9 0 V A + O. 8 • 9 0 V 2 + 0 • 9 7 :5 0 V 3 + 1. 59 8 V • + 0 • 6 79 0 V 5 + O. 3 .6 0 V 6 . • ,- o. 0 0 7 8 19
5)/d!7~~~~1+0.79 5 3V2+0.9730V3+0.9730Vt1.791LV5+0.4958V6 =+0.0608~6 10) +O:O.537v~:O:0537v~---------:O~6250Vti~lT~'2v~:O·:1498V~-::O~068655
5) ·+0. 5550V 1:+0. 5550V 2+0. 6790'V 3+0. 6790V o4+1. :250V 5 +0. 3460V 6 =+0.04'2455
6)/d~~~~i~~+0.4357vi+0.6790V3+0.6790V.+0.679V5+4.6039V6 =-0.1057~L 11) : +O:1193V~:o:1193V~;:--------:--------:O:5710V~:4:2579V~:O:148187-
Su ..... ,.y of fl,.at .et of equatlona,
7)+0. 5200V 1 -0.OjOOVi+0. 3600Vi+0.235~V~+O.0941V&=-0.004064 8) +0. 19 38V 1'+0. 49'25V 2:+0. 0981V 4+0.0 310V 5-0. 0 450'V 6=:+0.021250
9)-0.8176'V1-o. 19'26V 2- '2. 1i)39'V.-0. 6439V5-0. 333'2V6=+0.O'23155 10) +0.0 537v 1+0. 0537V 2:+0.6 250V .-1. 11':22V 5-0. 1498'V 6=-0.068655 11) : +0.119~Vl: +0.l19· 3V2 '+0. 5710V5-4. '2579V6='..-O'.148187
• • • • • ..... 0. , ..... • ••••
Second sat.
1111 .. lnate V 4 • "
7) '+0 . , 5~~OOV ~ -0. 0900V 2 +0. 3600V . :+0. :;: ~5:3V 5 +0. 0941V 6=-0.004064
8)/o:7111~~1: +1.8073V2' +0. 3600V.,+0.11 38'V5-0. 16 51V6=:+0.0779812 1!:;:) - 0: 191:2v~:1:89.73V~---------::o:1!2T§v~::o :~59\2V~::O:08:2046
'8) ;+0. 1938V 1:+0. 49!25'V 2:+0. 0981V . ;+0.0' 310V 5-0. 0450V 6 = +0. 021!25O'
'3) /:..~ b ~'i'v~!o. 0087v 2-0. 0981V .-0. ()29!2V 5 .... 0.01~lV6:::+0. 001050 13): +& :I567V~;:O:4838V;---------:~o:o618v~:O~060Iv~~:o~O~;~306 , '
-12-
SUMMary of second set of equatione:
1':;;)-0.ljl~V1-1. 8')7~v2 , +O.1~15V;+J.:;:5'1:2V6==""0. 082046 13)+0. 1567v 1 +0. 48 ~8V 2,+0.00 18V 5-0.0';01 V, == +0.0':;: L 300
14)-0.6~17v1-0.0067v2-4.4946V5-0.851~V6=-O.;14545 11):+0. 119 3v 1 +0. 11'1 3V 2 +0. 5710V;- 4. :2,79V 6= +0. 148187
••••• • • • • •
Tn i rd set.
14) - o. 6311 V ~ - 0.0061 V 2- 4. 4946 V ;- O. 8518 V 6. - O. 214545 11)!~!~~l'~i+0.6~17Vi+3.0:;:35V -2~.5458V=+0.784658
-------------------------j---------~---------17)+0.6250V2~1.4711V5~23. 3'176V~+O.~7011~ ;
su •• ary of Thi rd set of equation.: , '
15)-1. 3070v;+0.t~37v~+0.185'1V6=-0.054836 16):+0. 48!21V 2-1. 1131V5-0. '2714V6=-0.030'1'20 17)+0.6!250v 2-1. 4711V5-23. 3'176V6:+O.570113
• • • • • • • • • • • • • • •
Fourth set.
I I , I • , nat .V 2 •
15)-1.3070VI+0.]:237v~+ 0~1859V6=-O.054836
16)/i~~3~6~2-3.0177v - 0.7358V 6=-0.083B26 18) --------:~:8340V~:-O:5499V~::o:13866~ 16):+0. 48'21V g-l. 1131V 5- O. '2714V 6=-0.0309:20
17) 2~3~:~i~!:~:~~~Z~2:~~:~~~~'~!:;~~:~~~2~1~ 19) :+0. 0:21ov ;:+ 17. 776oV6 =-0. 47068'2,
-13-
summary of Fourth set of equation.:
18)-'2.8940V -5
19) +0. 0216V '+ 5
O. 5499V6=- 0.138~62
17.7766V 6=- 0.470682 •••• • • • • • • • • • 0 ••
Fifth set.
I I I I .. I nat eV 5 •
18)~2.8940V~- O.5499V6=- 0.13866~ )(13~·9.8148
19) /.3:'~~~~~.2~3l~~:.2~23~.2::§~:.~~'3~7~ 20) :+:2381. 185 3v 6 =-63. 201333
summary of FI fth Set of equations:
20) +2381. 1853V 6 =-63. '2013 33 ThenV 6=-O.0'2654
• 0 • • • • • • • •
Substl'tuts this va'ue fo,' V61n equation IS) and sol". for V5:
18 )-'2.8940V 5 +O.014594346 =-0.13866'2 Then -2.8940V5=-O. 153256346 And V5=+0.05'2956 subatl tut. theT,,,,lu •• found for V6 and V5 In equation 15) and ao\ 've for v2:
15)-1. 307ov 2 +O.0065506572 -0.004933786 =-0.054836 Then -1.3070v 2=-0.0564528712 And V2=+0.0431927
Su b s tit ute the val ¥.,. .. f 0 u n d for V 6 ,V 5 ,a n d V 2 In. qua t Ion 11) and .olv. for VI:
11)+0. 1193Vl+0.00515288911 +0.030237876 +0.113004666 =0.148187 Then +0.1193V 1 =-0.00020843111 And Vl=-0.001747
SUD. t I ,u ~. t h. v a J.u •• for V 6 ,V 5 ,V 2 ,a n d V 1 I n a qua t Ion 7) and .0 I v. for V 4: :<:6 0 7)-0.00090844 -0.063887~2~~+8.3t24605468 -0.002497414 =
==0. 004064. Then +0.3600V4=-0.0092313498 And V4=-0.025643
-14-
Sub.titute the In equation 1)
" a I u •• f 0 u n d fV r V 6 ,V '" , 'V 2 ,V , and V and .01". fOT' ..., 1 4 3 • .
1)-0.00176'2723 +0.0366706023 +0.843V 3-O.021770307-
Thtm ·+0. 849V 3=-0.02498667:23 And V 3=-0.02943
-0.00583188 =+0.011649.
00000 00000 00000 00000 00000
'Check these V values thru one of the equations in the first set, and we find: 4)-0.00148~203 +0.036670857 -0.02863539 -0.040377514+
+0.035957124 -0.00318284 =-0.007819 And we find that these values will dhebk 0 = 0 thyu the fourth aecimal place.
Summary of residual values. 'V 1=-0.001747 ft; V2=+0.04319~ ft; 'V3=-0.029430 ft; V4=-0.02564~ ft; V5-+O.052956 : ft; V6=-0.026540 ft.
Now we find that the most probable values for the dif-ferences of elevation are: A via L, K, to T= -~25. 875 0.0017 = - '25.8767 ft;
-I via P to M =-107. '201 + O. 04 ~2 = -107. 1578 ft; .. M to J '= + 13.695 0.0294 = :+ 13.6651) ft; -J to H =+ 73.460 -0.0256 = + 73.4344 ft; H to F -- 40.447+ 0.0530 = - 40. 3940 ft; - 80.066 o. 0~26 5 F to f"< -- - = - 80.09'25 ft. \,;l
- - -Then from A via the route L,K,I,P,M,J,H, F,G,back to A, we have the most stations. A= 1087.8000 ft.
-25.8767 r,i6;i:§!233 ft~
-197.1578
M= -9."54:7655 ft.
probable elevations
M= 954.7655 ft. +13.6656
J= 968:-'431"1: ft. +73.4344
H= i041:-8655 ft
, -19-
of the unknown
H=1041.8655 ft. -40.3940
F= 1001:-"4715 tit. -80.0925 •
G= ~9!21:3790 ft.
(0 ) N 0 R M A L E Q U A T I 0 N S . V1 Va V3 V. Va V, Absolute
Term 1) +1.009 +0.849 +0.849 +0.849 +0.555 +0.222 -0.011649 = 0 2) ~e.~~9~.630 +1.4?4 +0.849 +0.555 +0.222 -0.02?2?4 = 0 3) +~ • .a/4,9 +,1. ftJl"il +1.910 +0.9?3 +0.346 -0.00?806 = 0 4) +9. a~;i) +9. jJ~,9 +XJ. ~7Z + 1.598 +0.346 +0.00?819 = 0 5) +~ • .5S5 +.0 • .5,513 +XJ.Bff~ +0.~7~ +1.250 +0.346 -0.042455 = 0 6) +G.n2' +0.Z7-~ +0'.3:4,.6' +,0'.,3'4".6' +.0. Z~$' +2.346 +0.0538?8 = 0
(Nu.~er. ~ e low t ". red I I ... ar. redundant nu.~.r.).
(p) Solution of Normal Equations. 1 2 3 4 5 r; Absolute
Term P.1 +1.0090 +0.8490 +0.8490 +0.8490 +0. 55pO +0.2220 -0.011649 Pit (-0.991~ -0.8414 -0.9414 -0.8414 -0.5501 -0.2200 +0.011545
P3 • .. .. • +b.915~ +0.?59? +0. 134? +0.0880 +0.0352 -0.01?4?3 P4 ' - · - . . ·(-1.092~ -0. 829? -0. 14?1 -0.0961 -0.0384 +0.019082
Pa ' . · - . . . ' . . +0.5654 +0.1469 +0.1401 +0.1300 +0.016968 p, • ' . .. .- . . ·(-1.?68~ -0.2598 -0. 24?8 -0.2299 -0.030011
p, •. •.• . . • +0.825? +0.162? +0.1202 +0.015?82 • - • • ' . • (-1.2111) -0. 19?0 "';0.1456 -0.019114 P.· . . . . . . . ~
P, . . . . . . ... . . - •• +0.8694 +0.1646 -0.041682 • ·(-1.150~ -0.1893 +0.04?943 PI. 0 • - • . • - • . •. • . .". • . • • . • . • •
Pu Pu
... .. . .. . . . . 0 . 0 .. , . , _,
· . .... ... . . ... . .... . . - .. . - .. ' . . • +i.21?~ +0.058803 · · (-0.450~ -0.026514
(S)Find most probable values for the dif-ferenoes of elevation.
A via L,K,to I = -25.8?5 0.0016 = - 25.8?66 ft; I via P to M =-10?201 + 0.0440 = -10?15?0 ft; lot to J =+ 13.695 0.0304 = + 13.6646 ft; J to H =+ ?3.460 0.025? = + ?3.4343 ft; H to F =- 40.44? + 0.0530 = - 40.3940 ft; F to G =- 80.066 0.0265 = - 80.0925 ft.
(T) Then from A via the route L,KLltP,M,J/H,F,G,b&Ok to A, we have the most probable elevations or · he unKnown atation$:
A = 108? 8000 ft., M = 954. ?664 ft., H = 1041.8653 ft., =_~Q~§Z22 t!~L2~i~ =_~QL~~~Q
I = 1061.9234 ft., J - 968.4310 ft.,F = 1001.4?13 ft., =~ZL!22Q ±2~Lt~t~ =~§QLQ~g§
lot = 954. ?664 ft., H = 1041.9653 ft., G = 921.3?88 ft.
-16-
(q) Solution of Normal Equations.
2 3 4 5 p, Ahsolute Term
q,. +1.6300 +1.4740 +0.8490 +0.5550 +0.2220 -0.027274 q,. qe ,..P,!7.1.43 -0.7143 . -0.7143 -0.4670 -0.1868 +0.009801 ~ ----
+O.~i5~ +0.75a7 "+0.1347 +0.0880 +0.0352 -0.Oi7473 __ ~ ___ ,. ___ o __ -, _____ -.-... .. .- , _" ......- ___ ... ____ , ...".,- ____ . _-_. _ _ _ • __ ~ ____ , _ __ , _ _
q, • • •• +1.9100 +0.9730 +0. 6790 +0.3460 -0. 007'806 q, q.. • . , • . • -0.7143 -0.7143 -0.4670 -0.1868 +0.009801 q.. q, • . • -0. 6303 -0.111~_ -0. 071? -0. 0292 +0~_97~ qs
+~5854 +0.1469 +0.1401 +0.1300 +0.016968 -_. __ ._ ._--,-----" ----:-..... _. -..-. ---_ ...... -.:
.0. • +1.5980 +0.6790 +0.3460 +0.007819
. " • . • •• . " -0.7143 -0.4670 -0.1868 +0.009801 ..... _ . .
Ql0 • .
qll • qu . , Q13 • Q14 •
.. .. ... .... . ... ... .. . _. . ... . .. ... . ..
Q:f 15· • . • . • • . •
Qie • • • . • . Q1. 7· • • • •
-0.0198 -0.0129 -0.0052 +0.002570 -0.0382 -0.0364 -0.0338 -0.004408
+0.8257 +0.1627 +0.1202 +0.015782 --,-----.-~--.. ---. -
. , • • +1.2500 +0. 3460 ~O. 04245.5 • . " . • . -0. 3053 -0. 1221 +0. 006408 • . • • • •• -0.0085 -0.0034 +0.001679 • . • • • • • -0.0347 -0.0322 -0.004205
• • • . " -O~ 0321 -0.0237 -0.003109
+0.8694 +0.1646 -0.041682 -~:--~--------I-. . . • •• +2.3460+0.053878
• •• -0.0488 +0.002563 • •• -0.0014 +0.000671
<118 •••• e_ . o ._ • • . • • • . -0.0299 -0. 00~901 •• , ••••• " -0.,0175 ~O. 002298 Qs.: e • . • • . • . • • • • . •
~o • • . • . • - • . • . • • ·0. -0.0312 +0.007890
+2.2172 +0.058803
(r) Solution of Norm~l Equations. V1 V2 \b V4 V", V6
~ +0.011545 +0.019082 -0.030011 -0.019114 +0.047943 -0.026514
r2 +0.005833 +0.001018 +0.006096 +0.003860 +0.005019 -----~ ~0.029134 -0.005089 -0.013124 -0.010434 +0.052962 -. ---1'-~ +0.021614 +0.003779 +0.006674
ra +0. 025.549+0. 0~5~ -0.030365
. ~ -0.037007 +0.043983 !
ra, -0.001600
-0.025688
AdJustment V" =-0.001' V2 =+0.0440 Va .y:-0.0304
Quantities: V4 =-0.0257 Va =+0.0530 v, =-0.02'5 Cheok ~hese V values thru one
of the equations in (O),and we find: .
L1) -0.001614400 +0.037341567 -0.025779885 -0.021809112 . +0.029393910 -0.005886108 -0.011649000 =
And we find that these values oheck 0-0 thru ~he seventh deoi.al place. . '
-17-
o j
THE DETERMINATION AND ADJUSTMENT OF ELEVATIONS IN
PHELPS COUNTY MISSOURI.
(rawson Township L~v~l Net)
ADJUSTMENT OF INTERMEDI ATE JUNC~ION POINTS L, K, P, H, O.
No1/e:(See discu •• lon on nter.edlate Point. on pa3. . )
to
Ob •• rved valu. of Adju.ted valu. of
'" 1061.925 ft.(R.publlc Rd.Llne) 1061.923
Adjuatll.ntl + 0.002 ft.Lo •• r lin. at and taper to 0.00 at A.
d I at. A. th ... A via L, I(
L • 0.5 ",lleeithen t a I • 6. 25 • I the adju.t",ent
dlatance i. a.
follo.a:
ea,and the to point L
Adjuat •• nt to L _~&.5_ " O. 002 • 0.00016 q. 6. 25
And t h • fa 0 , t pro b a b, I • e I .vatlon ° f pol n t L I , ,.
1078.427 O. 000 1078.427 f-J. --------I n I I k. • ann.r,a, the dletance A to K • 1. 75 • I I e s ,
adjust •• nt to K _1:.12 6. 25 " 0.002 • 0.000150 ft.
And the "'oet probable el.vatlon of point K I a.
0.001 • 1092.693 ft •
.A o,,,;;j ,K Adjuated Lin. I . io----/.7SMi.-----< S 10----------'-----6.2 Mi·~. -----------1
D I A Q R A.. S H 0 'w I N GAD JUS T .. E N T 0 F POI M T S BET. A and I •
Frail ih. above dlagra. It Ie .... 11' •• e" how VII. "at ••• along the Ob •• w-".d IllIe ar. lo •• ,..d proportional to tbe dl.ta"o. f,.o. A to til. Adjuat.d II" •• Tb. dl .. gra. al,o •• ,. v •• a. a 0 h • 0 k ° n til. a d j u • t •• n t 0 f t". I n t .Ir" •• d I .. t e Pointe If drawn to •• al.a horlzontallr and ".ntloallr.
-12-
Obser"ed Adjusted
yalue of yalue of
pol n t pol n t
1061.925 1061.923
ft.(Republlc R 0 ad)
Adjustment + 0.002 ft. II ne to ~e lowered at I.
Arran]ed value of point -_101]-(In lar]e ciroult fro", sU"'Matlon of consecutlye dlffs.ln eleyatlon.)
Adju.ted yalue of po nt 101
Adjust.ent
954.724 ft. (fro,," the
4.0/1,:
0.042 ft.line to •• Il t M.
6.4l1i -
'" e • t )
r a lee d
f~ ~ , ____ -'-c'"'Q"'o~t/on Line --+--- - --.-------1 ~-Io..i
I Adjusted lin e p .-0 ., "'
DI AGRAM SHOWING A D JUS T 101 E N T 0 F
POINT P.
Fro", the dla]ra .. I I n e. I n a n 9 Ie •• juet",ent follow.
;X) ' ,>
<,
Ilcco"'panyln:J ... ty proportionate
the .1",llar trlwe affect the .d
to the point P as
Relations stated ae a propol'tl ,-Q ~' ,
6. 4 0.042·0.002
----------------4. 0 (X.0.002
Sol II e
0.042 + 0.002
x + 0.002 - --------------- x 4.0 Ii. 4
(0.042·0.002 " 4. 0
"
x - -O.0()2 6. 4
And we find:
X -.0.0255 ft. (Adjust .. ent to point P.
Then the .. oet pl'otable elevat on of point P
928.552 .0.026 928.578 ft.
-19-
0.'"
.. ..
(h
m
A,. ,. an J 8 d ( I n I a r' J a SU"'IIIatl on d Iff s • I n
Ad j u S t 8 d
Ar'r'an,ad (In lar'la sUllllllatlon dlf1s. In
Ya I u a 01 pol n t II cUr'cult 1r'olll of consacut I Ya a I • Y a t Ion )
va I u a c 1 pol n t II
Adjust",an
'1alua of polnt-'Gclr'cult fro", o f con. a cut Iva ala'lation).
Adju.tad '1.lua of point G'
t
954.724 ft. (frolll tha .ast)
o. 042
921·523
921.379
ft.Llne to a t M.
ft. (froOl
ba ,..I •• d
III via N;O to
Adju.t",ent + 0.144 ft.Llna to ba Icwa,.ad a t G.
Fro .. the a c COlli pan yin l dlaJ"." by proport on .. t • G; I I n e • I n t h • • I 1ft I I a ,. t r I a n J I a ., • a affact t h a adjust .. an t of pol n. t • N • n d 0 a. folio •• :
r /"
/' A 0 JUS TilE N T OF POI NT N. ... ....
Ra a t I 0 II:' • tat a d a • a P,.opo,.t ..' on «
8. o 5 O. 144 + O. 042 • -------------
O. 9 (0.042 X)
By alJab r a :
O. 9 _x • (0.181) " ----)
8. 0 5 Sol YR l :
x O. 0 2 12 1ft ant to
1t.Adjustpoint N.
Tllan til. 1.1'I.tlcR
'" 0 • t 01
probabla
p c I n t 0' / ' _~" "
j ' ' . '
875.092 ft.
. ·IT' '.: "' I .... , '
I ~ ;-<. I~ ;',, :
01 ' .~.~ ~ M
"
.•.. ~,
". ~ .t:' -. '. ' .
t · ., "
".
., ~o DIAGRAM SHOWING
ADJUSTMENT OF POINTS Nand O.
C on.sfrucfion L /ne,
!.------·a . .3Mi:-. -----..,
/'
.. '
G
1o---------------8·05I'1i.-- - ·- --- --------I ADJUSTMENT OF POINT O. 8.050.144 + 0.042
In tha sa.s .annsr' •• aboya:----·--------~----3. 3 ( .y + o. 0 4 2 )
Solv n3: '.0.0342 ft.Tlla adjust.ant to point o.
Than tn. III0.t pr'obabla .Iavatlon of point
0.034 -969.702 ft.
-20-
G)
THE DETERIIINATION AND ADJUSTMENT OF EL "EVAT'ONS IN
PHELPS COUNTY MISSOURI.
(Cawson Township Level Net) -
Adjustment of lines p~ ani NH as Parallel Luplicate Lines of Le:vels, and the adjust'Tlent of t!lt!
line asa ~s a Suplicate Line of Levels. See sketch below and the three Plates of Caw son Township.
9!J7.Z4-() R t86.,g.~.tJ 9-'/'2..5/61 l.....-- B.M. o4J ", ;S.M. I .-- ~--r- -,- -, --;- -,-1-"-- --,---,- .... -. --r- ..,. -,---,- - 1"'"1
I 1 , I I I I Iii I • I 'I S69:~ d..:::a 758 I
~---~-.--+---I.---~--.:-- +--- --~---;----~-J- -~---~--~--- --: -~- .~ -~'-!--~--J r; I I 1 : I I (t I I : ,96g : I I I
I, I I I I, I;; I I •• ~ • I
[----~----+---1t- · -:~---~--·~- -- ---:---~---I- - ' ,----;---- ---r-- -~---~~ ----:---:---~---I
__ .. : ___ i ____ ~ __ ~ ___ ~. __ ~ ______ : ____ : ___ L _ i~tr. __ : __ L _ _ __ ~ ___ ~ __ ~ __ __ ~ __ -J --J-__ I I : : 1 ~<15 I :: !:: ~ I I: " I : I I' 1
I I I~ I : I I I 11 I' I I 1 I I: I J t---r--r-?- t~.s~ ~ ----t--- --1- --r-- i --- ---;---r--r-- ----1----r---~--I---i---r--r-J ..----:---~---li--,---l' !, - ---- i~--i----:---:--- ---:--T--:--~----T---~--- ;--,
~---:----~--- ---,:---J---L------~---~-J ~J ____ L __ : _______ ~ ____ L __ J __ ---~---~---:'--1 ' I I ,--" . I I I I I' I, I
~: I': : r :MI~ I ,I I:; :': I I . L.: I :. t I , "r t I ' . . .' I r----f----r--'? ---;----r--l---- ---~ ---:---1---- 9'--r- -~---:- -- ---~---:--~--- -----!---i----,---i
: ~ :: I I I:: :: u>: ; • ~ : ::: : :t3 : I
Adjusted values are as ~ 1--p • follows M •
N • I) •
928.578 954.7 66 87~.092 9 6 9.702
ft. .. .. ..
It is required to find the most prcbable elevations of the points Q, R, andS.
ADJUSTMENT OF POI NT Q.
Paralle~ Duplicate Lines o~ Le}els: By a parallel duplicate line of levels is meant a line that is run twice over the same route with equal care and in the same direction.Levels are ~ometimes run in this · manner when there are two instru-
-ments on the work,as was the case when the line PG was run, also the line NR.
Ob.erved .. .. e I e v • t Ion .. .. of poInt P .. .... Q.
Dlf1.renoe In elevation •
Ob •• rved elev«tlon .. .. .. .. of .. point P • .. .. . t •
Dlffere"oe In elevation •
-21-
928.552 1 t • 997.242 .. 1, t 'n.tru.ent. -------+68.69 0 .. .. ..
928.552 1 t. 2nd ItI,tru •• nt. 997.2H .. .. .. .. .------+68.685 .. .. .. ..
{IN ;\'ETGHTEL l!EH SU k ElrfENTS. If U,e d1,jjerence of elev
at1,on oj tmo stat;ons 1, ' S measured a number oj t1,mes 1,n
the same manner,over tAe ' same lenght of l1,ne. arid under
'such cond.t1,ons that tf,e d1,fferent deterrr •• nat.ons may be
reearded as oj equal <l:e1"ght. t fl en the ar.th met.c me<ln of
the 'several re'sults '/, ' s tile most pro b able value of tins
d1,fje r ence of elevat1,on. The probaDle error of a sin)le ... asure"'ent I. ll"en Dy the forllula:
The proDable error of the arlthlletlo .. ean 01 n me •• ure_ ",ent. lIy tha for .. ula:
ADJUSTMENT OF POINT Q.
Ob.erved V.lue.
h8.630 {to 68.685 ft.
2)13Z:'375
v
+0.00:25 -0.0025 -6:6050
v2-
0.00000 ,S25 0.00000625 0.00001250 =L:vz,
z = 68. h875 ft. C hac k n = ~2. (No. 0 fOb s. )
r.= 0.6745 + = 0.0017 ft.
Then the most probable v~l u e for the difference in elevation = 68.6875~ 0.0017 ft.
And the most probable elevation of the point Q is as
follows: Adjuet~~ ¥alue of p~int g ~ 328 .•• ~78 ft.
Adiusted LlfLln elev~P to "" 68 688
Most probable elev. point Q= _~~Z~3§§ ft.
O, b.er"ed • •
-ACJUSTMEN T OF POINT R.
e ley a t Ion • •
01 point N· It f- . R.
875.250 866. H~
Dlfferanoe In elevation .. - 8.867
1 t • • 1. t
• w
O ••• ,."ed ele"atlon o 1 pol n tN. 875.250 ft.
• •
• • • w • • R· 866.293 • 2nd In.tru.ent.
Difference In ele"atlon • - 8.957 • • • •
-22-
ADJUSTMEMT OF POINT R.
The line N to R is a parallel duplicate line of levels.
O~ •• r ... d ... Iu ••
8.867 ft. 8.')57 ft. -----
2z = 17.8'24
z = 8.')12 ft.
r.= 0.6745
v - 0.045 of. 0.045
0.000 C h • c k
0.0020 25 0.002025 --------0.004050 =~V2
n=2. (No. of Obs. )
of. 0.0304 ft.
Then the most probable value for the iifference in elevation = 8.jl"~.: 0·.0304 ft.
And the most probable elevation of the point R is as follows:
875.092 ft. Adjusted value of point N = Adjusted Ciff.in elev.N to R = - 8.')12
Most probable elev.point R = 866:180 ft.
'"
ACJUSTMENT OF POI NT S. CUDlicate Lines of Levels! By a 'duplicate line
of levels is meant a line that is run twice over the same route with equal care, but in onposite iirections.
L.ttln] a r.pr ••• nt the dl.cr.p.ncy ~.t ••• n tn. ,. •• u It obtaln.d from tn. for •• rd I In. ~nd th.t obt.lned from tne r .... r •• lin., •• h .... :
d v = of. ---
1 2 and
d v 2 =
2
Su~.tltutln~ th •• e ... Iu •• in the followln] equatIon .nd r.pl.clnl r 1 with r, for the c ••• of dupllc.t. lin •• , •• ha". the proba~l •• rror of a .'n~l. ",.t.r.'n.tlon (forward or r .... r •• ) by. line of the ""3th 1:
d 2 d 2 '2d 2 (of. ___ ) + ( _ )==+---_
'2 '2 4 d 2
2:~a+ /7.2
Sabsti tuting:
" L ~ o. 6745 y---~~----~: ~Z~2_ y. -n- ___ d ~-1-2(n - 1) 1. 4142136
In the ca •• of duplloa; ••• asur.lII.nts.(n - 1)=(2-:1)= 1. T h • n :
ri.= 0.47f)9 va: =.: 0.471)9 d.
And for the error of the arithmetic mean in duplicate level in~:
r = ",
':0.47i)] d = ----------
VT"
",DJUSTMENT bF POINT S.
Ob,.rv.d .I .... atlon of point 0 • 9119.758 ft(Froa South) "" "" • •• S. 942.516 B
Dlff.r.no. In .I.vatlon -- 27.242(!ioln~ North)
Ob,.rv.d .I .... atlon of point S. 942.516 ft. ." "" • ". o· 969.743 .(FrOIll North)
Dlff.r.no. In el.vatlon - • . 27.227(aoln3 South)
Ob •• r .... d Val u ••
27.242 ft. Then~ r L =':O.47f)j x 0.015 + 0.0071 = '27.227 ft. And:
+ . 0.015 =.: 0.0051 r",=-0.3373 x
------d = 0.015 ft. and 1/2 d = 0.0075 ft.
Then the most probable value for the difference in elevation = 27. '242 - 0.0075 =27. 2495.:!O.0051 ft.
And the most probable elevation of the point S is as follows:
ft. ft.
Adjusted value of point 0 = 969.702 ft. (On Circuit) Adjusted Tiff.in elev.O to S = -27. :250 ft. -------
Most probable elev.point S = '942.452 ft. - - -----SUMMARY OF THE ELEVATIONS OF ALL O~ THE JUNCTION POINTS.
A" 1 () fl 7.f\ :) ~1) ft. '3 ~ :)1 .~ 7 9\) ft. " .. i\ 1 ;; .()9?() ft. B " 1 3 7 .4 • Ii 'i 3· 0 ft. H", 0 4 1 • rl Ii 'i 'i ft. o.. ·~ ~J _ 7 () ? () t t ·. c " 1 !) <i ~ . ~ Ii "(I ft. ,.. 1 n Ii 1 • -1 :> ~ ~ ft. P.. . '~ 2 8. Ii 7 B 0 ft.
" J oq f\ _(l ~ B:) ft. J OI)8.4)li.to Ov" '? 9 1.?lllip ft. ,. 1 () 7 4. I) 01) 0 ft. K· 10 q 2. "i '1 ~ () ft. . R" 8 ~ 15. 1 8 cr-h ft.
F • 1001.471 'i ft. L" 1078.4270 ft. ~ .. 01? 4~20 ft. -t,!. 9. ';i •• 7 1\ ') Ii ft. . .
------~---~------~-----------------------~-.--------------.
THE DETERMINATION AND ADJUSTMENT OF ELEVATIONS IN
PHELPS COUNTY MISSOURI.
(Dawson Townshin Level Net) ADJUSTMENT o F I N T E R M E 0 I ATE TEMPORARY 0.0020 ft.
BEN C H MAR K S BETWEEN THE POINTS ~. I -:-- -- - - ---_ ... .. -
A ( a t St • .)a",as)AND I (a t M I I a •
School). . ' ._ .- - .-- .' - ... _.- '- - .- . . - ----_. -.. -~-
W
f ~ \,2 __ ,. r" '-'--. --... --.--....... -----.---.- . _____ . ____ .E..-" :,.t r . f'. R
0. .. ~ .. r . c(
o o
o
a
o It'>
o
..
....
o .. H .... ....
01 AGRAM TO SCALE SHOWING ADJUSTMENT OF ALL
POINTS BETWEEN A AND I. (See dlaJ~am on paJe )
o • 0 0 1 I)
0.0010
0.0005
0.0000
Tne adjustment 01 these Intermediate Sanch Mark. can be effected by .cal in 3 the adju.t.ant correction. from the above dlaJralll,and subtractln3 aacn corraction from tne Ob •• ,.ved Elevation a. it appears In the Level Not e s ; 0 r, I f . des ire d , t h. a d j u 8 t m • n too r r e c t Ion sea n b e
•
•
•
computed by the fo I 10winJ formu la for tni ••• ctlon(1 to 1)1
FORMULA: Olst.A to Intar.Pt. Adjustm 'ent to Int.rmediata Point
z ____________ ~ _____ _
6. 25
I 'n" ]t :n I sea •• , ltth. A d J u • till. n t Cor rae t ion 8 are t 0 b a subt"acted from the Observed Elevatlons,as stated abov ••
MERAMEC SPRINGS AND REO BIRO OUADRANGLES. - -
PHELPS COU NTY. ST. JAMES AND DAWSON TOWNSHIPS.
The elevations in the followin~ list are b~sed on a precise-level line run in 1908 by E.L.McNair from the United States Army En~ineers'bench mark at Labadie Mo., the accepted elevation of which i s 472.572 feet.
The levelin~ under adjustment was run in the Fall of 1921 by the Vocational Department of the Missouri School of Mines and Metallurgy from the Dntted Btates ~olof!ical Survey bench maiDks at St. James and Knobview, the accepted el~va tions 0 f which are 1087.800 feet and 1074.005 feet respectfvely . .
-25-
MERAMEC SPRINGS QUAD~ANQLE.
( L at I t u d e '5 7· " 5 ' - '5 8 0; I 0 II :I i t u de 9 1 • '5 0 ' - 9 1 • " 5 ' )
(St Ja ..... Town .... lp)
LI NE FROM ST.JAMES NORTHWEST VIA JEFFERSON ROAD TO INTERSECTION WITH REPUBLIC ROAD, THENCE NORTH VIA REPU8LIC R 0 A D TOP H E L P S -G .A S CON 'A DEC 0 U N T Y LIN E •
Elevation Adjuatlll.nt Adjust.d No. D"cI"lptloll By F I ., d Book COl",.ectloll EI .... atloll
A. St •. Jam~s, corn~r of feet fee t J e et
M~r am~c ani 'Nashin~ton St,'s. , abo u t 70 ft. north of cen. t~ 'r lin~ of Frisco R. R. track, 25 ft. from do·~,neT of Comm~:~ci al Hot~l,in an ,~n ~ betw~en cross walk and siie walk; iron post stamp~d "1098". (s •• Bull.tln 1568 p a :I • 10. ) 1087.800 0.000 1087.800
RED BIRD QUADRANGLE.
L. , ( L a tit u d. '58· - '5 8 • 1 5 'j lon ' :I I t u d. 9 1 • '50 ' - 9 1 • " 5 ' )
St.James,0.5 miles north of,on Jefferson ~O~ at intersection of Parker Lane road, concrete culvert, north end wall,top of; :chistiled. squaTe.
K. st. James,!. 75 miles north of,on Jefferson road, at intersection of Republic road,on south side of road, 10" black jack tree, in base of;copper nail and washer ,
1078. 4'27
109:2.694
..;26-
0.000 1078.4''27
- 0.001 1092.693
P. B. M.
No. O.sorlptlon
1. st. James, 2. '25 miles north of, oJ: Republic road,Jefferson road,0.5 rui I es no rth 0 f, cuI vert, wes tend, sandstone slab, top of;painted
EI.Yatlon CO", By Fld.Bk.
fee it, 'fi e e t
Ad;" E I • 'y •
feet
square. ------------------------~ 1034.345 -0.001 1034.344
2. Jefferson road,l.O mile north of on 3epublic road,T road east,at Cox's residence,100 ft. east of, sandstone boulder in N. E.Cor. o f T road; topcorner brolen off,
painted white. ------------------ 1024.715 -0.001 1024.714
3. Jefferson road, 1.5 miles north of, Robinson creek,O.75 miles sout~ of, Fitzenreider's residence,lOO ft. west of,wa~on gate post,in base of; copper nail and washer. --------- 338.469 -0.001 998.468
4. Robinson Creek,0.25 miles south of, on Republic road,on east side of road, 10" post oak tree near Kenndy's ffiCiil box, in base of;nCiil. 9'23.575 -0.001 923.574
5. Jefferson road,2.25 miles north of, on Republic road, Robinson Cr~ek, west bank of,24" white oak tree, i~ root of;cop~er nail and washer. 917.604 -0.001 917.603
(Dawson Township)
6. Eo binson Creek,0. :25 miles north of, 15 ft. west of road, sandstone boulder on St. James-Dawson Twp. Line,oppdsite wire fence ~orner, point is on south hillside, 300 ft. south of Wooftchoppe~'s cabin; painted cross on top of boulder. 1022.334 .... 0.001 10.22.393
-..27-
E I • , a t Ion No. O •• cI"lp.tlon. By Fld.Bk. COl".
7. Robinson Gr1eek,0.75 miles nOTth feet feet
of,on Republic roarn,Township line,0.5 miles north of,at north end of rid~e, 30 ft. west of road, sandstone boulder, top of; chisel-
ed square. -------------------- 1039.423 -0.001
8. Robimson Creek, 1.0 mile north of, on Republic road,at foot o~ hill, concrete cuI ver t aero ss ro ad, W.
end wall, top of; chiseled Sg.--- 934.325 -0.00'2
9. Robinson CJr.eek, 1. 25 miles north of,Miles Sohool Souse, 1.0 mile south of,T road west,Denter of roads; road eldvation.--------- 976.005 ~0.002
10. Milds Bchool House,0.5 mil~sS. of,on Republic road,Marrs Cemetary,400 ~t.north of, east side of road,satidstone boulder at rail fence corner,top of; chisel-
Ad j d. E I • v. feet
1039.4'22
934.323
976.003
ed square. -------------------- 1059.077 -0.002 1059.075
11. 'St.James,6.0 miles north of,on Republic road,Sa:fe, ~ 2. 5 miles E.
of,Miles School, 25 ~t.W~of,S.W. Cor. S. E. c;tr.of the S. W. Qtr. Sec. 20,T.39N.,R.6w.,60 ft.S.E.of, concrete well curb, top of; bronze tab~et stamped" "
I. St.James,6.0 miles north of,on Republic road,·Safe,2.5 milesE. of,cross roads at Miles School, northeast corner of school li-oqse, li!I).estone wheel guard,top
-0.00'2
o f; painted s q~artl.-------------l061. 9'25 -0'.00'2 1061. 9'23
-2S-
ADJUSTMENT OF INTERMEDIATE TEMPORARY
BETWEEN THE POINTS1 Cat MI I •• Sohool) AND lie road on •• 11 •• outh of County Line.)
C'I •
0 ...
• • 2 o • c .... """1 LC"I
• I"-.. E .00 - ..
• 0 C'I
... -
.. It"
l"-. ... Ad j u. t. d
14
.. • .. "" 0 0 (\I "" 0
C'I N " f<'\
L I n • . 16 17
8 ENe H MARKS
PCon R.pub_
.. • 0 0 It" 0
I<' • 18 P 13 O"r
~~-------T------~------~------~--r-----~-------r----~ 1:.5
I
01 AGRAM TO SCALE SHOWING ADJUSTMENT OF ALL POI N T S BET WEE N 1 AND ( S e. d I a l r a '"
:: ' ."
t)
" P •
on p
Th. adjust",.nt of th ••• Int.r •• dlat. b.nch marks can be .ff.ct.d by
on
1.5""
<" ... -1
,.. .. ,-,
/ I'
scalln] th. adjust ... nt cor-
\,!':
--. f:~ . ,
r.ctions fro .. t .h. abo". dialralll,
< ' ...
and addln] .ach oorr.ctlon to th. ob •• ry.d .I.yatlon a. It app.ars In th. l.y.1 not •• ;or,lf d •• lr.d,t". adju.t •• nt corr.otlon. can b. co.put.d by th. followln] forllluia for thl •• ectlon (I to p).
FORMULA:
"" " ' R "< e' A (~ '+- 1
S E
",
........... " '-..
. ... 'C
(\I
0 . 0
• c 0 .. 0
• .. .. 0
(.)
.. c
• II
• " ... -<
0.028 x DI.t, I to Int.r.Pt. Adj. Cor. to In t.r. Pt. aC------- ___________________ _ -0.002)
4. 0
Add th ••• corr.ctlons to th. ob •• ry.d Yalu •••
El.Yatlon No. D •• crlptlon. By Fld.Bk. C,or.
106i~?~ 25 -6~662 1. (See desc ri ption on p ",ge )
12. MilesSchool,0.75 miles north of,on Republic road,W.L.Mart-in's h 0 us e, 150 ft. S. F. of, on top of hill,on east side of road, 12" post oak tree,in base of;copper nail and washer.--- 102~.857
13. Miles School, 1. '2 miles north of,on Republic road, Bourbeuse River,0.75 miles south of ford, at dim cross road,southeast corner of, sandstone boulder;
+0.003
painted sguare.--------______ 9'24. '251 :+0.006
.. 29-
Ad j d. E I • v •
1(~ll:j23
1023.8f)0
Elevation Ii 0 • D.scrlptlon • . By Fld.Bk. Cor.
feet feet
14. Miles School. 1.75 miles north of,on Republic road, Eourbeuse Ri ver, O. ':2 miles sou th of, farm house, 600 ft. east of,on east side of road, 15" walnut tree, in base of;copper nail . and washer. 873.567 :.0.010
15. ~ ash School House.0.75 miles south of,Bourbeuse River,at ford, 100 ft. south of,east bank of river, 30" sycamore tree, in base of; copper nail ani washer. 8~J. jj8 '+0.014
15a.Bourbeuse River, at ford,on Republic road, center of river; road elevation. --------.-:---- 834.000 :+0.014
16. Bourbeuse River, 0.25 miles N. of,on Republic road,at T.road east,northeast corner of,in base of fence post;copper nail
and washer. -----------------
17. Bourbeuse River,0.75 miles ~. of, Wash School, 400 ft. S.W.of, -J.W.Grossner's house, 300 ft. S.E. of,8 ft. west of three mail boxes,on west side of road,14" sealey bark hickory tree,in root
840. 106 :+0.016
of; copper nail and washer. --- 947.596 :+O.Olj
18. 'hash -School,O. 5 miles north of, on Republic road, Little Bourbeuse creek,at ford,onsouth bank,west side of road,lO" 8caleybark hickory tree,in base of; copper nail and washer. ------- 835.9'';.7 :+0.0'23
-30-
Ad j d. E I • v. feet
873. 577
840.012
834.014
840. ILL
947.615
835. j50
N ().
-P.
O,scl'lptlon. EI."atlon By Fld.Bk.
feet Wash School, 1.0 miles north of, Little Bourbeuse Creek, 0.5 Mi. north of,Harrison School,O.5 Mi. south of,Phelps-G9scon9de County Line,1.0 mile south of. Austin E. Breuer's hou se ,:200 ft. west of, at T.road east,west side of Rep-ublic road, ::2 ft. inside of fence, 10" post oak tree, in base of;
nail and washer. --------
COl' • feet
Ad j d. E I • " • feet
copper , __ .0.
I
. j : ~8 . 578 r---------~._~------~----------~~------~~
Adjust.d
j 28. 55' ~+0. o~6
.. ... 'C
C'I
0
0
..
+
c .. 0
c-... E .. .. • . .. " .. 0_ 0
,,0
c
C> C>
C>
.... '" C
o:: '.::
---
C>
C>
Lin ••
"" IX
o I
0·"
{" ...... -., .'Ii 4-
(,... ° 1
t oo. ~se I
r v "._ d ~· ! -I J.l --fl __ <: , .. _ ___ . __ J _ '" . '1') ~
c:'"l + .> c: +~
0.... ,- , ....
.. LinR _ .,J r'
<>
" t , I.. e
c ...
~.~ ~~j
--- .~ . (\.~ .
_ c- ...
2 ,-.
<> o
...
DIAGRAM TO SCALE SHOWING THE ADJUSTMENT OF ALL POINTS
BETWEEN P ANO G.
I t w I lb. 1' . ' '" ... b • I' • d t hat t h. I In. P toO I s a .pUl' lin. of 1.".1. In OUl' .ch .... of adjust •• nt. Th. lin. p to Q was I'ul\ as a pal'all.1 dupllcat. lin. of 1.".I',and the .o.t probabl. "alu •• fol' the .I.yatlon. of the point. on, u C h a I In .. I. t h. a". I' a 3. 0 f t h. ..". I' a I d. t. r. I nat lon, • Th. aao"e dla3ra ... how. a (red) In. of l.y.l. run by a Brandl. Inatru.erit and a (alu.) I In. of l.y.l. I'un by a GUl'I.y In.tru •• nt:a. th ••• In.tru •• "t. w.r. of equal r.fln_ •• " t, • qua I w • I .. h t • a. a p p I I • d to to tn. • 0 r k don. ' by. a c II In.tru •• "t. Th.(lr •• ") II"e I'.pr ••• "t. the ay.raJ' of tn. I " • t I' u .... " t a I d • t • I' • In.a t I 0 " • r In · t h e I. y. In. tad j u • till. " t ju.t pr.o.edlnl, It .a. found that the point P wa. 0.026 ft. low. A. the .pur lin. P to Q hlill.' directly on the point P,the" tne whole lin .. P to Q .1 I hay. to ae ral •• d 0.P26ft. I n tile d I a a I' a • t II I. a d j u • t •• "t I. ." 0 w " b Y t h. (a I a c k ) -1 In • .•
-3~-
Th. adjustlll.nt Is a constant of 0.026 ft. which Is to b. added to .ach of the ave"a~. inst,.u ... ntal obs.,."ed " a I u e. bet 10 •• n Pan dO. T his I. a p P •• r-. n t fro :~ the d I a ~ r a ••
FORMULA:
Adj.Cor.to Int.r.Pt.-Av.Obs Valu. + 0.026.
No. O •• cr-Iptlon.
-P. (See descri ption on paf1e 31 )
19. TNash School, 1. '25 miles north of, Little EourbeuseCreek,O.75 ~i. north of, HarrisonSchool,O. :25 miles South of,~helps-~asconade County 1ine,O.75 miles south of, Steve Lorts' house,70 ft.north of, T. road west(Mail road to Hif.lh jate in Maries Count7),no~thwest corner of roads, 'L4" oak stump in fence corner, top of; ten penny. nail.-----------------------Brandis
!20. Harrison School House, on St.
Gu rlev AveraE!e
James-Republic road,west side of School EuildinE!,north concrete door s~ep.top of; chisel-ed square. Brandis
"
Gurley AVeralte
,,-
'21. Harri son School House, O. 25 Mi. north of,on Republic road, - -Phelps-Gascoiade County Line, 0.'25 miles south of,pri'vate T.road west,southwest corner of,16" bl~zed black oak tree; bent nail in base of. ----- Brandis
Gurley AVera;1e
-32-
EI.vatlon Br Fld.Bk.
feet
j~8. 55' ~
96'2.732 96:2. 726
-96,"2:729
965.674 j65.658
-96'5:666
959.594 959.578
-9:59:586
Cor. feet
~o. G~I)
'+0.0'26
:+0.0'26
Ad j d. E I • v. fee t
9'28. 578
96'2.755
959. 61!~
No. Oe,crlptlon.
-
Elevation By FI d. Bk. Co r.
feet feet
Ad j d.
E I • v. feet
Q. Harrison School House,0.50 miles north of,on Republic road,intersectionSt.JamesRep~blic road and Hi~h GateRed Bird road,0.5 miles so~th
- -of,Phelps-Gasconade County Line,'28 ft. south of,center of road, 30 ft. Wt::lst <?f, 5 ft. insidt::l wire fenct::l,in line with center of road north into -GasconadeCo~nty, dim T.rosd west,50 ft. south of;iron post stamped "997 ".-----Erandis '
Gurl ey 997.242 997.'237 997.240'+0.0:26
-997.266 P. B. M.
1hlS co~pletes the deSCrtptlon and adJustment oj the l~ne
of level ,s fro71l 'St.}ames north12est v~a Jefferson Road to In
tersectlon ~~th Republ~c Road. t~ence north v~aRehubl~cRoad
to Pnelps-Gasconade CountyLIne.
MERAMEC S PRINGS OUADRANGLE.
(Latitude 37°45'-3So;lon3itude 91°30'-91°45'
( S t.
LINE FROM ST. JAMES EAST VIA OZARK HIGHWAY TO PHELPS-
CRAWFORD COUNTY LINE,SAID POINT ONE WILE NORTHEAST O.F KNOB
VIEW RAILROAD DEPOT.
ADJUSTMENT OF INTERMEDI ATE BENCH MARKS ALONG THE ROUTE
A VIA BAND C TO D.(A is the P.B.M. at St.,)a .. e. and 0 ia the
T.B.M. at the nterllectlon of the Ozark HI]hway and the Pnelps-
crawford , County Line.)
BELOW IS 01 AGRAM TO SCALE SHOWING THE ADJUSTMENT OF ALL
NTERMEDI ATE POINTS ALONG THE OZARK HI GHWAY BETWEEN"A"AT ST.
JAMES ANO"D"AT THE COUNTY LINE NEAR KNOBVIEw.
T h a affacted Jram, and a I e vat Ion bookajor,
adjustment of the.e Intar ... di at. ban en by aeal inJ the adjustmant corrections .ubtractlnJ each correction from the aa It appear. In the level note. and f de.lred,the adjuat ... nt corrections
putad by the f ° I I 0"" I n J for II u 1 a for t h a a act i on
FORMULA:
A D ,.I • 0 F. I N T E R • PT.· - - - - - - - - - - - - - - - - - - - )( O. 0 Ii 3 • 6. 2 5
( See D I a J ram 0 n pal e
Elevations pre v lou a I y
on this .ect l on adju.ted,aa follow.
A B C D
• 1087.8000 1074.6580 1083.3 6 60 1098.8580
f t , " • •
.. ... o o o
,
A. 22
::I( •
o 0 o ..
o 0
I , -:;1" ,
B 2~ 25 27 28 Adjusted
• ••• •• • o o ..
-34-
o It>
if I
C Lin e. • • o It>
o C'< . .. ..
..
t L o W E R
",arka c.n b. from the dl.observed description can ba eolOl-A toO:
/1T / fT : I
" I I I ,' IT,' · I ! I ,
I , I: : i :! I i I i I I i I 11 1
: I : I , : :! 10-I ' I. I ~ Ie I ~ I • 1, . I'
il : I I I I
I I I I
31 0
• It> .... It>
I.I ... I 11('\
i", 10 I • 1-0 r
I
No. D.scrlptlon. EI.watlon By FI d. Bk.
10~~~~00 A. (See description on pa ge 2 6 )
22. St.James,0.4 miles east of ton St. James-Cuba road(Ozark Hi~hway), Frisco R. R., 50 ft north of, tu rn in road to no rth; 500 ft. W. o~,concrete culvert across road, north end wall, top of; chiseled
square. ---------------------- 1065.363
RED BIRD QUADRANGLE.
Ad j- d. Cor. EI.".
0!666 1081~~60 P.B.M.
-0.003 1065.360
(Latl tude 38-~~8-15'; IO~3Itu~, 91-30'-91-'5') Qt J 1 0 °1 i.) ~. Ja~es .L O wnS .. l p ) B. u • ames, . m 1 e , . 1'., . 0.,
'''3 L. •
on Ozark giQhway,T.ro ad east (intersection of Ozark Hi~hway and Red Bird road),400 ft.S.of, Cr. F.A.Scott's house,300 ft w. of, on W. side of road, 10" elm tree,in root of;copper nail
and washer. ------------------ 1074.666
st. James, 1. ,5 miles N. E. o f, on Ozark Hi~hway,0.5 miles east of Red Bird road intersection,on N.side of road, 12" post oak tree,in base of; cop per nail
and washer. ----------------- 1062.669
'24. St. James, 1. 75 miles northeast of,on Ozark Highway, concrete culvert across road,south end
''"'5 L., •
wall, top of;chiseled trian~le. 1034.437
St.James, '2.0 miles N.E.of,on Ozark RighwaY,on top of hill, center of road; road elevation. 1054.446
-35-
-0.009 1074. 658
-0.013 1062.656
-0.015 1034.422
-0.017 1054.4'29
No. O •• crlptlon.
26. St.James, 2.5 miles northeast of,on Ozark Hi~hway, FriendShip School, 1.5 miles west of,concrete culvert(errected 8-13-20 by T.Hall),north end wall,toD
E I • vat I 0 'n: By FI d. Bk.
feet Cor. feet
of; chiseled square. --------- 1010.057 -0.021
'27. Friendship School, 1.25 miles west of,on Ozark HighwaY,at T. road N.to Hinkle P.O.,Corru~ated iron culvert,rock on east end of culvert,top of;chiseled
triangle. ------------------- 1018.92~ -0.023
'28. Fri~dshi p School, O. 5 miles W. of,on Ozark Hi~hway,Gerber's Store,600 feet W.of,concrete culvert,north end wall, top of; chiseled trian~le. ---------- 1062.045 -0.030
C. Friendshio School,'on Ozark Highway, southwest corner of concrete porch of school house, top of;chiseled triangle, oainted "1083.3"------------- 1083.400 -0.034
'C 1. F r I • n If • II IpS 0 1\ 0 0 I ,on 0 z a ,. k
Highway, 50 ft.S.W. of,in School Ylird,10 ft. from fence corner, 30 ft.N.of center of Ozark Highway, 30 ft.E.of center of T. road north;iron post stamped
"1083". --------------------- 1083.458 -0.034
29. FriendshipSchool,0.25 miles E. of,private T.road N.to Louis ~all's residence,75 ft.W.of, concrete culvert,S.end wall, top of;chiseled square. ----- 1037.041 -0~036
-36-
Ad rd. E I • v. fee t
1010.036
1018. )03
1062.015
1083.366
1083.424 P.B.M.
1037.005
No. O.scI"lptlon.
~O. Friendship School,O.9 miles E. of, on Ozark HiQhway,PhelpsCrawford County Lin~,1.1 miles west of,concrete culv~rt,south end wall, top of;chiseled
EleYatlon 8y Fld.8k.
feet COl" • feet
Ad j d. E I • Y •
feet
square. -,--------------------- 1042.297 -0.044 104'2.'253
31. J riendship School, 1. 5 miles east of,on Ozark HighwaY,PhelpsCrawford County Line,0.5 miles west of,Knobview,O.5 mi les north of across field, concrete culvert under Ozark Highway,north end wall, top of;chiseled square.- 1028.412 -0.049 1028.363
ID. Friendship School, 2.0 miles east of,on Ozark Hi~hway,at cross roads on Phelps-Crawford County Line,southwest corner of roads, 14" hickory tr~e,in base of; copper nail and washer. ------------- 10J8.911 -0.053 10)8.858
Ti/.1,'S completes the descrlpt-:-otn and adJustment of the
~'/,ne of levels from St. James east V1.a Ozark ll,.gi~1tay to the
P;~elps- 'Crau:jord 'County Luu .• sa,.d 1>olr~t o .. e rule northeast oj
Knobv~e1t Ra1.lroad ·£e/>ot.
P. A. R T . 1 I I .
THE LEAST SQUARE ADJUSTMENT OF A LEVEL NET
BY METHOD OF OBSERVATION EQUATIONS.
APPLIED TO THE ADJUSTMENT
OF THE
DAWSON-ST. JAMES TOWNSHIPS LEVEL NET,
PHELPS COUNTY,
MISSOURI.
(BASED ON THe LINE OF LEVELS RUN BY THE U. S. GEOLOGICAL SURVEY IN 1908 ALONG THE FRISCO R. R. WITH PERM~NENT BENCHUARKS,cAw AT ST. JAMES, AND
WE- AT KNOBVIEW).
-38-
Showing ~lev8tions and closures when level circuits are arranged as shown in plat on page40.
It = 1087.800 - 25.875
(St.~ames ) Adjusted value.
I = 1061~925 (Miles School) -107.201
C = 1083.221 + 15.511
M = 954.724 (Royal) + 13.695
D = 1098.732 D ~ 1098.579 in arr.Cir.
J = 968.419 + 73.460
H = 1041.879 + 32.608
B =, 1074.487 + 8.734
C =, 1083 .. 221, - 82.223
F = 1000.998 - 80.066
G = 920.932 +177.647
D = 1098.579 - 24.844
+ C. J 53 Closure at C.
(Red Bird and Jakes Prairie Rds.)
(Red Bird Rd.and Ozark Highway.) H = 1041.879
- 40.44~ (Friendship ~chool ) _._--F = 1001.432 F = 1000.998 in arr.Cir.
+ 0 . 424 Cl osure at F. (Jakes Prairie and Tribby Rds.)
(Ozark Highway and County Line.)
E = 1073.735 (Knobview Depot.) E = 1074.005 (n ") Adjusted value • .
- C. 270
I = 1061.925 - 93.531
J = 968 . 394
Closure . At Vncbvi e~ .
J = 968.419 in above arranged circuit. - 0 . 025 Closure on circuit above . At
M = 954.724 - 33.201
G = 921.523" G = , 920.932 in above arra~gedcireuit.
Note: The values given on this page are not adjusted values, but only an arrangement to
f~cilitate the ad-- justmeat that i~ .J •
to be made on the following pages by
the theory ef Least Sqaares.
+ 0. 5"91 Closure on circuit above.At G. A =, 1087.800
- 13. 134 B =1074.,668
L B = 1@74 ... 487 ia sieve arraagedeire1li:t. + 0.179 Closure on circuit above.At B.
-39-
910 01 919 35' 9130' , 38°10' __ ~ __________ +-________________ ~ ____________ ~~ __________ -T~3~8°10'
- "*
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a.6 •• U .ra ~ ec Sp rin ge
91 401
10$7.880 8. S.
W N 91
-40-
S 35'
Ouadra nll i.
B I p
Observation Equations.
(A to I) +\11. .......... e , .... -. ....... -....................... = +0.000 (weight 0.160) (I to M) ".-... +V2 ...... -............................ -• .•• = +0.000 (weight 0.156) (M to J) · .... -..... + ~ ... ' .. " .............. " ................... = +0.000 (weight 0.312) (J to B) · ................. + ~ ...... ' ............... ' ......... = +0.000 (weight 0.625) (H to B) ........................ - ...... +. 'Vs .... ' ........................ = +0.000 (weight 0.294) (B to C) ....... ~ .........•. +Vs ............ ' ..... = +0.000 (weight 0.333) (C to F) •.•....•........•.• ~ ... +~ .............. = +0.000 (weight 0.333)
~~ to g~ .... ". " ................... -. ' ....................... +~ ..... = +0.000 ~weight 2.000~ to ......... -.... ................. ' ...... +V9= +0.000 weight 0.222 ~D to J~ -V1 -V2 -Va -V4 -Vs -ik -\; -i;{, -~ = -0.270 f weigh t 1. 000 ? ' I to · .. +V2 + ~ " ............................... -......... .. =, -0.025 . weight 0.625 (A to B) +V1 +V2 +Vs +v;.. +V5 ........... -.. ' ... . = +0.179 (weight 1.000) (H to F) ........ -..................... +Vs +~ +\; .. ............ = +0.434 (weight 0.5'71) (M to G)" ........ .. '. : +~ +V4 +Vs +'" +\; +~ = +0.591 (weight 0.124) (C to . D) ........ ' ... .................... . <It . . ....... +~ +~ +~ = ,+0.153 (weight 0 .. 445)
.. Formati.on of the Normal Equations •
Normal Equations are formed "in the following manner -from the Observation Equations:
RlJLE: 10 form the no~malequat, ,,o1ls 101" e'ack one of the un
known .quant: ~t .'Jes, 7Ilult'J.ply 'each o/}'se1"vat'f.on ' equat·'!.on by th~.p1"od-
9f ,tbewe$ght of ,that o~ser'Va~$onand ,the alge~~~$c coetf;c~ent
of that unknown quant'lity .Ut ·that ·equat:.on, and add ·theresfllt,s.
Having formed the several normal equations, 'their solution as simple simultaneous equations gives the most probable values of the 'unknown quantities •
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ha •• tb' •• r ••• try.
-41-
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G
te
Dte
O ••• '''I.d
fil
ff.,
.no
. I'
n .1
...
..
1=
-2
5.8
75
M
=
-ie
7.2
01
J
= +
1
3.6
95
H
=.
+73
.460
B
=
+ 3
2.6
08
C
=
+ 8
.73
4
F =,
-8
2.2
23
G
,=
-8
0.0
66
D
=
+
177.
647
E
= -
24
.84
4
,Aaj
ustli
re11
t Q
uaD
titi
est'.
-0.0
43
72
2
+0.
3473
24
V 1 =
-0
.02
96
VIS
=+
0.17
36
-0.4
49
00
0
+0.
060
~~2
~= -
0.0
54
4
~ =
+0.
0607
+0
.135
Qo~
Va =
+0.
0354
V,
=+
0.13
60
__
__
__
_
+0
.00
44
61
~=
+0
.02
31
~=+0.0045
-0.6
65
58
1
+0.
6655
76
~ =~.0~37
-0.6
65
58
+
0.6
65,
58
, A
dT
u •
t •
d
E x
o.
I I. n
t 0
h •
0 k
•
o 0
, r
eo
t I
0 1\
fA
~, t
n ~
oe
Ad
JU
S te
d
E leva
t ,. 0
11
0 f
.J un
c t
,. 0
n P
o t,
n t
· s.
:':'0
.029
6 ~ -
25
, .. S04
6 ft
. In
D
aw
son
TO
rl'n
sh
"p
,Ph
elp
s
Co
un
ty.
-0.0
54
4
=-1
07
.2,5
54
II
! ~
lo8~
8oo
fi'r
h =
1
04
1.8
55
5
ft.,
+
0.0
35
4
=~ 1
3.7
30
4
II
-8~.9046
+ 9
2.78~?
~
+0
0231
:;+
7
3.4
83
1
II
• ---
'1-,
+0:
1737
=+
3
2.7
81
7"
J =
10;
61',
695
4
/I
J:;
=
+0.
0607
=+
8
.79
47
"
-10
'7_
85
54
+
0.13
60
=-
82
.08
70
" k
=
9~4
.?~
04
/I
Ie
=
+0
.00
45
=~
~8e.
e6:\
.5
" +
1
9. ?~
\ -0
.04
37
=
+1
77
.60
33
" I
== 9
68
. C,
'!0
4
/I
f [t
=
-0.0
35
7,
=~
24
.87
97
"
+'7
3_
48
g1
10
'iL2
. 6 s
s:,~
11
'l
"~
+
8.
79
4?
,l
ti ~
~
«:I
~\j
\()
108c
,.4~99
11
• •
ti ~
-88
.08
?0
\~
......
-.
0-
11
" ...
l.O
Ol.
.S
4!<.
9
-8
0.0
62
.5
II .
L =
1041
.6F
,~r;
, "
IG =
9~
:Z. ~8
14
II
(.!)
:::
::;
l"
t--I 0
'I
' 0
-l{
) l>
;)
t--<:>
~
oo
c •
.. •
l>;)
"<
t 'l
" 0
'\ 5~
l'"
0 C
...
... II
" <-
:\ ~
I
PAR T I V
THE LEAST S~UARE AD.JUSTMENT OF A LEVEL NET BY METHOD OF CONDITION EQUATIONS
AND CORRELATIVES.
APPLIED TO THE ADJUSTMENT
OF THE
DAWSON-ST. JAMES TOWNSHIPS LEVEL NET,
PHELPS COUNTY,
MISSOURI.
(BASED ON THE LINE OF LEVELS RUN BY THE U. S. GEOLOGICAL SURVEY IN 1908 ALONG THE FRISCO R. R.
WITH PER.ANENT BEN~HUARKS wAw AT ST. ~AUES,
AND wE- AT KNOBVleW).
-48-
91°30' 38°10'--+-------------~------------------------------------______ _+----------------~8°10'
I
I GAS CON A DEC 0 U N T Y. ! T--J;£. -H E -~ P -:.~:~ (; - U N -T-;-. ~::- -.Ll
• f ij ~. ;; :1.
;>11 ,> I
" ,..... ""<", . ...1 I
i"i ';'1 I
D SON I p: I:
MAR I E S I
C 0 U N T Y. I I
--- - -- -~------I
D ILL 0 N I
:S
TOWNSHIP
...... ~ <\2 1
~ i-,
o
o
o
38°GO'~~------------~--~~~~~~~======~------~~~~----~-----------~8°eo· 91 40' ,-.~_ ............. l8t. James 91 0 30'
U. S.P .. B.Il. 1087.800
-49-
Least Square Adjustment of Level Net.
Link Observed From To Ditterenoe
A to B B" C C" 0 0" E C" }P
F! G D" G
- 13.134 + 8.734 + 15.511 - 24.'844 -82.223 - 80.086 -177.647
}P" H + 40.447 B" H - 32.608 H" J 73.460 A" I - 25.875 I" J - 93.531 J" W - 13.695 I" M -107.201 G if M" + 33. 201
Name
a b o d e t , g h i j k 1 m n p
Miles Distanoe
D.
1.0 3.0 2.25 1.0 3.0 0.5 4.5 1.75 3.4 1.6 6.25 1.6 3.21 6.4 8.05
P-.eight _.1_
D
1.000 0.333 0.444 1.000 (j.333 2 .. 000 0.222 0.571 0.294 0.625 0.160 0.625 0.312 0.156 0.12'4
Derived Correotion
-0.0306 +0.0808 -0.0569 -0.0354 +0.1387 +0.(1047 +0.0453 +0.0633 -0.1731 -0.0236 -0.0294 +0.0076 ·0.0353 '-0.0543 +0.1568
Con d i t ion E qua t ion s •
1) 0 = 0.062 - ' d - c b a 2) 0 = + 0.153 + g - f e + c 3) 0 = - 0.434 + e + h i + b
4) 0 = -0.157 h + f + p m - j
5) 0 = + 0.204 + a + i + j - 1 k 6) 0 = - 0.025 + m n + 1
correote'd Ditterenoe
- 13.1846 + 8.7948 + 15.4541 - 24.8794 - 82.0863 - 80.0813 -177.6017 + 40.5103 - 32.7811 - 73.4836 - 25.9044 - 93.5234 - 13.7303 -107.2553 + 33.3578
Each complete circuit furnishes the condition that the sum of its adjusted differences of elevation shall equal z&ro when given the proper signs as if run continuously around the circuit1clockwise or counter-clockwise.
Number of Conditions = (No.of lines) - (No. of Jnct.~oints) + 1. " " " " = + 15 10 + 1 = 6.
The minus signs applied to the small case letters in the above condition equations result from changing the direction of the arrows (when red arrow goes in opposite direction from the black arrows in in diagram on preceeding page) so as to be continuous around each circuit. It is not necessary that all the circuits be traversed in the same direction in a given problem1but forelearness the work will appear more cODsi8tant if all the eireuitsare traversed in one direetiOb,eitbereloekwise or counter-elockwise.
I 'JI ~
I
TABL
E OF
O
OR
REL
ATE
S.
Pro
du
ot
of
Os
wit
h Oorr~lates.
+.3
535
+.1
806
+.5~
63 +
.1~4
8 +.
O~71 +
.0~4
8 L
ines
~p
1 -1
a b o d e r g h i j k 1 11
1 n p
.10
.3
0
.22
5
.10
.3
0
.05
-1
-1.
.. 1 • • · . •
• .4
5
.17
5
••
.84
• •
• •
.18
.8
25
••
• •
.18
.3
2'1
••
••
.84
.8
05
••
2 • •
• •
+.1
• •
-1
-1
+1 • •
. . . · . • •
• • • •
• •
• •
3 4
5 +
1
6 ~ o o o
Sum
(Sum
x ~p)Lines
• •
+1
• • • •
+1 • •
• •
+1 ~1
• •
• •
; .. • •
• • • •
• • · . • •
• • • •
+1
• •
-1
• •
-1
• • • •
-1
• •
+1
• •
• •
• •
• • • •
• •
+1
+
1 -1
-:-
1
• • • • • •
• • · . •
• •
• • •
• • • •
• •
• •
• •
• •
+1
+1
-1
· .
-1 o o
+1
o o o -1
o o -1
+
1
-.3
53
5
• • •
• ••
• •••
• • •
•••
+.0
47
1
• ••
••
-.3
53
5 .•
••••.•
+.5
56
3 ....
.....
• •••
• • •
•••
-.3
53
5
t,1
00
6
•••••
• •••
• • ••
••
• ••
••
-.3
53
5
• • •
• ••
• •
•••
• •••
• • ••
••
......
• •••
••
-.1
00
8
+.5
56
3
• ••
••
• ••
••
• ••
••
• ••
••
-.1
00
6
•••••
+.1
94
8
• ••
••
• •••
• • •
•••
+.
10
06
•••••
• ••
••
• ••
••
." ...
. • ••
••
• ••
••
· ....
· .. " ..
+.5
56
3
-.1
94
8
-.5
56
3
•••••
• ••
••
• •••
• +
.04
71
• ••
••
• ••
••
'0' ••
• • ••
••
. • II ••
.... . . .
• ••
••
• •••
• • •
•••
• •••
• • •
•••
· ....
• ••
••
• •• I
.'
• •••
• • •
•••
• ••
••
• •••
• • ••
••
-.1
94
8
+.0
47
1
•••••
•••••
-.0
47
1
•••••
•••••
-.0
47
1
+.0
84
8
-.1
94
8
•••••
+.0
84
8
•••••
• ••••
-.0
84
8
+.1
94
8
• ••
••
• ••
••
NO
R M
AL
E
Q
UA
T
IO
N
S •
-.3
08
4
-.O
a0
8
+.2
02
8
+.0
60
8
-.2
52
9 -.
05
69
-.
35
35
-.
03
54
+
.45
57
+
.13
67
+
.09
42
+
. 0
04
7
+.1
00
6
+.0
45
3
+.3
81
5
+.0
63
3
-.5
09
2
-.1
73
1
-.1
47
7 -.
02
38
-.
04
71
-.
02
94
+
.04
77
+
. C
076
-.1
10
0
-.0
35
8
-.0
84
8
-.0
54
3
+.1
94
8
+.1
56
8
The
norma~
equ
atio
ns
may
be
wri
tten
by
insp
ecti
on
as
foll
ows
owin
g to
th
e si
mp
lici
ty o
f th
e co
nd
itio
n e
qu
atio
ns.
To
fo
rm
the
Nor
mal
E
quat
ions
: 1
et.
S
qu
are
each
q
uan
tity
in
'oo
lum
n
1 le
ft
han
d
sid
e
tab
le
of
oo
rrela
tes
an
d
mu
ltip
ly
by
it
s 1
/10
p
,an
d
add
th
e resu
lts:
+0
.72
60
; 2f
t,d.
Mu
ltip
ly
eao
h
qu
an
tity
iq
o
olu
mn
1
by
q
uan
titi
es
in
oo
lum
n
2 o
n
sall
ie
lin
e
mu
ltip
ly
tiy
their
1
/10
p
, an
d
ad
d resu
lts-
-0.2
25
0;
3d
•
Mu
ltip
ly eao
h
qu
an
tity
in
o
olu
mn
1
by
q
uan
titi
es
in
oo
lum
n
3 o
n
sam
e li
ne m
ult
iply
b
y th
eir
1
/10
p
, an
d
ad
d resu
lts:
-0.8
00
0;
4t1
\,.
Mu
ltip
ly
eao
h
qu
an
tity
in
o
olu
mn
1
by
q
uan
titi
es
in
oo
lum
n
4 o
n
sam
e .
lin
e
mu
ltip
ly
by
th
eir
1
/10
p
, an
d
ad
d
resu
lts=
+
0.0
00
0:
5;~lh
Mu
ltip
ly
eao
h
qu
an
tity
iq
o
olu
mn
1
by
q
uan
titi
es
in
oo
lum
n
5 o
n
sam
e "
lin
e
mu
ltip
ly
tiy
their
1
/10
p
, an
d
ad
d resu
ltsa
-0.1
00
0;
~~.~
Mu
ltip
ly
eao
h
qu
an
tity
in
, o
olu
mn
1
by
q
uan
titi
es
in
oo
lum
n
8 o
n
sam
e li
ne"u
ltip
lyb
y th
eir
1
10
p
, an
d
ad
d re
su
lts=
+
0.0
00
0 •
. ~al.~ives
the fir
st
no
rmal
eq
uati
on
. 7
lt,b
;"
Sq
uare
eao
h
qu
an
tity
in
o
olu
an
2
an
d
mu
ltip
ly
by
it
s
1/1
0
p,
and
ad
d
resu
lta=
+1
.02
50
: "tMi
i~'
Mu
ltip
ly
eao
h
qu
an
tity
in
o
olu
.an
2
by
eao
h
ad
jao
en
t q
uan
tity
in
tu
rn
in
the
oth
er
oo
lum
ns
by
t.h:air
1
/10
p
ae
ab
ov
e
an
d
ad
d resu
lts·
r;\:B
,., "
.ro
oeed
as
.bo
v-a
u
nti
l all
·· te
rae
hav
e
been
m
ult
ipli
ed
an
d
t.h
e '6
th
':bera
al
eq
ua\i
on
h
ae
be.
'!!
0Q
.ipi.
,t..e
d.
' '
a b o d e f g h i j k 1 m
n p
NOR ),{ A L E QUA T ION S •
1 :2 ~ 4 5 {: ( q) SUll' Cbeck +0.7250 -0.~250 -0.3000 +0.0000 -0.1000 +0.0000 -0.062 +0.1000 +0.0380 -D-.-2~+1.0250 -0.3000 -0.0500 +0.0000 +0.0000 +0.153+0.4800 +0.6030 -0.-3000 -0.-3000 1+1.1150 -0.1750 -0.3400 +0.0000 -0.434 +o.oeoo -0.4340 +0.-0000 -0.-0560 -o-.1~l+1.5100 -0.1600 -0.3210 -0.157 +0.8040 +0.6470 -0-.-1000 +O-.eOOO -0-.-34-00 -Q.--lOOOl :1! 3850_ -0.1600 +0.204 +0.6250 +0.8290 +0.0000 +O-.-Geoo +o.eooo -0-.--3"21-0 -o.lbOO i~J_.1210 -0.025 +0.6400 +0.612-.Q Note: Sum and C~eck oolumns will not be used in the following aolution of the above normal equations; but will be used later in a subsequent solution of normal equationa.The (q) is the absolute term.
(P) Solution of Normal Equations. 1 :2 4 Ii I) ABSOLUTE
TER II & +0.7250 -0.2250 -0.3000 +0.0000 -0.1000 +O.OOCO -0.0620000 & (-1.3793) +0.3103 +0.4138 +0. 00 CO +0.1379 -0.0000 +0.0855166
l' • • • • +0.9552 -0.3931 -0.0500 -0.0310 +0.0000 +0.1337614 +0.0325 -0.0000 -0.1400348 P. • • · • (-1. 0469) +0.4115 +0.0523
f§ • • • • • • • • • +0.8291 -0.1956 -0.3942 ·+0.0000 -0.4046128 +0.4754 -0.0000 +0.4880035 Ft • • • .' . • • • .(-1.2061) +0.2359
-
'Pr • • • • • • • • • • • • • +1.4613 -0.2546 -0.3210 -0.2454525 (-0.6843) +0.1742 +0.2197 +0.1679631 18 • • • • • • • · . . . . .
-
& • • • • • • • • • • • • • • • • • • +1.1384 -0.2159 -0.0353133 It. • • • • • • • • • • • · • • • • • ·(-0.8784) +0.1896 +0.0310192 -:r11 • • • .. • • • • • • • .. • • • • • • • • • • +1.0096 -O.O856~13 I12 • • • • • • • • • • • • • • • • • • • • • • (-0.9905) +C.0848079
Notes on The P,G,R Solution of Normal Eauations.
Opposite flagella of this thesis will be found the least. SQuare figure adjustment of a Geodetic Guadralateral on the folded plate. The problem cn the plate was taken from the Topographic Instructions of the United States Geological Survey page 61 to page 71,for the instruction of students in my class in Geod~tic Computations. On comparison with the Geological Survey publication in the description of the method for solving normal equations, it will be found that the encluded plate goes much more into the details of the method.
As the ~,~,R, solution of normal equations used here in this level net adjustment is identical with that used in the adjustment of the triangulation scheme,by referring to .tbe plate opposite pagel18, it wilJ be found very easy to follow the process and steps in tb~ give!). computation and also in other computations in tbis thesis where the P,G,B solution of normal equations has been employed.
-52-
(q) Solution of Normal Eouations. 2 1; • 'i 6 ABSOLUTE
ql ++.O~50 -0.3000 -0.0500 +0.0000 +0.0000 +0.15~~OO q2 -O~209§ __ :2~2221 __ ~2!2QQ2 __ :Q!2312 __ ~2~2222 __ :Q~Q~2!2§§
+0.9552 -0.3931 -0.0500 -0.0310 +C.OOOO +0.1337614 -------------~-----------------.......;-'-----qa •••• • +1.1150 -0.1750 -0.3400 +0.0000 -0.4340000
q~ ••••• -0.124'1 +0.0000 -0.0414 +0.00(0 -0.0256556 q, -0.1618 -0.0206 -0.0128 +0.0000 +0.0550428
q. • • q, • • qs • • q,
+0.8291 -0.1956 -0, 3942 +0.0000 -0.4048128 ...-..---------------------------+1.5100 -0.1600 -0.3210 -0.1570000 -0.0000 +0.0000 -0.0000 -0.0000000 -0.0026 -0.0018 +0.0000 +0.0069957
• • • •• • • '-0.0461 -0.0930 +0.0000 -0.0954482 +1.4613 -0.2546 -0.3210 -0.2454525 ------------------------
q iD· • • • • • •. ' . . . +1.3850 -0.1600 +0.204C((0 qu' • -0.0138 +0.0000 -0.0085498 q 11· • . ••• . • -0.0010 +0.0000 +0.0043472
• • • • . -0.1874 +0. COOO '-o .1923529 q 23 •
q if· • • • • .•••• -0.0444 -0.0559 -0.0427578 +1.1384 -0.2159 -0.0353133 ~-------~-----------.---. q 15 •
q •• . . . . . .
q 27· • • • • • • • q.. . . . . . . . . q... . . . . . .. . . . . q.,. • . . . • • . . . . . . . . . .
+1.1210 -0.025((00 -0.0000 +0.0000000 -0.0000 -0.0000000 -0.0000 +0.0000000 -0.0705 -0.0539259 -0 .• 0409 -0.0066954 ---------~----- ----+1.0096 -0.0856213
(r) Solution of Normal Equations. 1 :? 3 4 f5 6
r1 +0.0855166 -0.1400348 +0.4880035 +0.1679631 +0.0310192 +0.0848079 r2 +0.0000000 -0.0000000 +0.0000000 +0.0186323 +0!2~~2Z96 ra +0.0064949 +0.0015307 +0.0223908 ~2!00~~~~ +0.0470988 r4 +0.0000000 +0.0101880 +0.0459533 +0.H~48000 r, +0.2302166 +0.2289370 +0.5563476 r. +0.0312227 :0:1006209 Some authorities on Least --------r, +0.3534508 Square, call Iter) Solution of Normal
Equations" the'l "B~'Ck Solution".
Summary of Junction Elevations in Dawson Township, Phelps County,Missouri~
A = 1087.8000 -----> 1087.8000 - 13.1646 - 25.9044 --------- . -----
B = 1074.6354 + 8.7948 --,.......--..-.-
C = 10B3.4302 + 15.4541 --~
I = 961.8956
D = '1098.8843 ---> 1098.8843 - 24.8794 -177.6017 ------ - .. ---
E = 1074.0049 G = 921.2826 Cheok + 80.0613 --F = 1001.3439 ----->
+ 82.0863 1001.3439 + 40.5103 -----.-
C = 1083.4302 H = 1041.8542 ~-> 1041.8542 Cheok + 32.7811 - 73.4836
B = 10;~'.6353 J = 968.3706 Cheok - 13.7303
954.6403 <,--- M = 95~~6403 +107.2553 - 33.3578
A D JUS TED
A = 1087.8000 B = 1074.6354 C = 1083.4302 D = 1098.8843 E = 1074.0050
I = 1061.8956 Cheok
E LEV A T ION S •
F = 1001.3439 G = 921.2829 8 = 1041.8542 I = 1061.8956 J = 968.3706
M = 954.6403.
G = 921.2825 Cheok
Points K, L, N, 0, P, are adjusted as intermediate points and take corrections proportionate to their distances from adjacent junction points.
ADJUSTED ELEVATIONS. K = N = L = ° =
p = Points Q, R, S, are at the ends of spur lines and will take the same corrections as P, N, O,respectively.
ADJUSTED ELEVATIONS. Q= R= S=
A b
rid
g e
d·
Met
:h 0
d
A.
N.
0 R
11
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LE
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UA
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e co
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3)
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o
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, wh
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r n
ot,
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5100
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10
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70
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that,
whe
n th
e ab
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ged
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rm
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use
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s (5
) •
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50
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00
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90
ju
st
to
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left
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e co
eff
icie
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m
ust
(8)
.•••.••
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25
0
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be
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ad
dow
n an
d to
th
e ri
gh
tjio
e:
L 1
ine(2
);
add
colu
mn
2 w
ith
li
ne(
2)=
+ ~030
cr p
\\ $0
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-~
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luti
on
o
f N
orm
al
Eq
uat
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s by
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ed
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E
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ck.
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mal
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atio
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y (I
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=
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A ••
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lao
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efe
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mad
e ab
ov
e b
etw
een
th
is p~ge
and
the
pag
e p
reo
eed
-in
g
th1
S.)
.
The
m
etho
d g
iven
ab
ove
is
a su
bst
itu
te f
or
the
(P),
(q),
(r)
So
luti
on
o
f N
orm
al
Eq
uat
ion
sj
all
o
ther
st
eps
in
the
pro
cess
o
f le
vel
n
et a
djust~
men
t ar
e id
en
tical
wit
h
tho
se
giv
en
in P
art
of
this
T
hes
is,
and
wil
l n
ot
be
giv
en a
gai
n
her
e.
Not
e:
the
I (C
heck
m
ark)
at
the
ends
o
f ro
ws
(II)
,(II
I),(
IV),
(V),
an
d(V
I),
are
wri
tten
to
in
dic
ate
th
at
the
I o
f C
olum
n an
d Ro
w
are
exac
tly
sati
sfie
d.
Una
void
able
dis
crep
anci
es
in
the
last
fig
ure
o
f th
e Check~term,due
to re
m
ain
der
s,sh
ou
ld b
e re
mov
ed
by
arb
itra
irly
cor~ctiDg
the
check~term
bef
ore
p
rece
sain
g w
ith
th
e n
ext
step
in
th
e eli
min
ati
on
;th
is i
s b
est
done
by
d
raw
in
g a
li
ne
thru
th
e er
ron
eou
s fi
gu
re
and
wri
tin
g
the
co
rrect
one
abov
e it
.' A
B R
ID G
E D
M
ET
HO
D
B
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OB
)l
A L
E
QU
A
T I
ON
S
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1 2
34
n
6 N
l,
1 +
0.7
25
0
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0
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ote
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1.0
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0
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00
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. 0
50
0
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+
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03
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It
mu
st
be
rem
e'm
ber
ed
that
the (
Ch
eok
E)
3 •
• •
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1.1
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0
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o
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o
f th
e
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i t
4,
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00
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10
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70
+
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47
0
an
eq
uati
on
,wh
eth
er
wri
tten
g:
no~
:: :h
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wh
en
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rid
ged
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rm
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th
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' 6
• •
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ust
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e re
ad
, do
wn
and'
toet
~~er
ight
. S
olu
tio
n
of
No
ra.l
E
qu
ati
on
s b
y
Ab
rid
g;a
eMe
't&ga
~ o
f ch
eck
o
po
sit
e
lin
e
(14
).
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1 2
3 4
5 6
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2. Pr9c~ss
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1)
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rmal
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ADJUSTMENT BY THE METHOD OF CORRELATES. THE 0 R Y •
The final adjusted values of the observed quantities must satisfy the conditions of the problem in hand, and must be the most probable values,acoording to the Theory of Least Squares, that will satisfy all the conditions. Therefore,the sum of the weighted squares of the corrections(residuals,v) must be a minimum.
Then: U =W:J.ff +v&. b2 + 1't3 c2 + Nc. d!- ••••• . + V+1 rz! = a minimum
which ~ust be satisfied simultaneously with the following conditions. Multiplying the condition equations on page 50 in suc.cession by the factors, -2~, -2~, -203, etc., respectively,
+2Ot Ch. + 2Ct d + 20t c +2Ct b +2Ct a = 0 -2G< ~ -2~ g + 2~ f+2(1 e -2(1 c = 0 +2Ce ce -203 e 2<1 h +2Cb i -203 b = 0 .. . .. . ... .. . ... .. . .. . ... ... . -20; qs -20; i -2Cs j +20; 1 +2Cs k -20; a = 0
(2)
Adding these equations to (1) and collecting the coefficients of the separate a, b, c~ etc., we have the requirement that,
?il.tf + 2a(Ct - Os ) +
Vi2 b2 + 2b (Ot - 03 ) + (3)
VI5 c2 2c(~ + Ct ) +
. . . . . . . . . + + 2(Ct.Q1-CeQ2+Ce~+ ••• ) = a minimum.
For the minimum, the first derivative of this expression with respect to each a, b, c, etc., must b~ placed equal to zero. Therefore,
dU = 2W;t a + 2(Ct - Cs) = 0, dividing by .2 and solving for a, da 1
dU - 2W:l b + 2 (~ - Cb )
db
a = - (-Ct. + Cs), W1
= 0, dividing by 2 and ~olving for b,
b =. ! (-OJ. + Ce ), Yi2
-58-
dU 2~ Y -2((1 c,.) dividing = + = 0, by 2 and solving for c, dc
1 (-0,. ~) c = - +
~
dU = . . . . . . . . = 0, etc. dz
Where W1, Vi2, ~, etc.= ~ are the weights respectively, d= distances U = ~ weighted residuals squared, desired to be a minimum, a, b, c, etc. = corrections for the links, c,., Cl2, Cs, etc.= the adjustment factors solved for in the
normal equations. ~, CJ2, ~, etc.=constants of observed closure respectively
in the several circuits.
~~ = the differential of U with respect to a, or the first derivative.
The constant 2 comes from the differentiation of the expressions in each equation equated to O.
substituting these values of the a, b, c, etc. in the condition equations on page 50, and combining the coefficients of ~, Ck, Cs, etc.,we obtain the Normal Equahons: Which partake of the following form:
aa , [w]~
[a:]~ [~] OJ. w
+
+
+
[a:]~ + r~]Ca bb
(b:] Ca Iw)~ +
[bc]~ w + [~]Cs w
+ . . . . + qi = 0
+ + ~ = 0 (4)
+ . . . . + Cb = 0
If the foregoing substitutions and and like Cs combined.,we arrive at the normal equations as appear on the top of page 52.
In forming the normal equations, it may be found somewhat simpler to follow the rules of thumb formulated from the foregoing theory. which will be found on page 5~ for the formation of normal equations from the condition equations, and on pages 10 apd 41,by a similar demonstration, for the formation of normal equations from observation equati~ns.
-59-
APPLICATION OF THE THEORY. Least Squares computation consists in rendering the val~e
of U in the following expression a minimum~
o = a2 + .333b2 + • 444c2 + d2 + .333e2 + 2£2 + .222f + • 571h2
+ .294i2 + .. 625j2 + .16Ir + .62512 + • 312nf + .156rr + .124J!.
Condition Terms. -2Ct. [-.062- d - c - b - a] -2~ [+ .153 + g - f - e + c] -2<1 [-. 434 + e + h - i + b]
-2G.. [-. 157 - h + f + p - ill - j]
-2Cs [+.204 + a + i + j - 1 - kJ
-205 [-.025 + m - n + 1]
The coefficients of the above co~dition terms is taken arbitrarily as -2 in order to simplify the expressions for the "V" corrections (ie. a, b, c, etc.) for the links.
To make the above expression for U a minimum requires that its differentials with respect to a, b, c, etc. be each equated to zero, as follows:
2 a + 2Ct. - 2Ce = 0, then
I
a = - Ct. + Os ........ = 1 times (-Ct. + as )
.666b + 2Ct. - 2Ce = 0, then b = - 3Ct. + 3<1 ••••• = 3 "( -Ct. + <1 )
.S8Sc + 2Ct. 2~ = 0, then c = -2.25Ct. + 2.25~ .= 2.25 II (-Ct. + ~ )
2d + 2Ct. = 0, then d=-~ .. , ............ =l
.666e + 2<il - 2Ce :: 0, then e- = - 3~ + 3<1 ........ = 3
"
" ( -Ci + Q, )
-60-
These expressions when simplified and with the Cs transposed to the right-band side of the equations, will then appear as extended to the left, and form the correlates as they are tabulated on page 51.
4f + 2~ - 2(4 = 0, then f = - .5~ + .5~ .......... -.
.444g - 2~ = 0, then g = + 4.5~ .......................
L142h - 2~ + 2(4 = 0, then h = +L75~ - 1..75(4 •••
.. 588i + 2~ - 20s = 0, then i = - 3 .. 4(1 + 3 .4~ ••••
1.250j + 2(4 - 20s = 0, then j = - 1.6(4 + l.6Qs .......
.. 32k + 20s = 0, then k = - .. 625~ " ....... " .......... -
1.2501 + 2~ - 2~ = 0, then 1 = - 1.6Qs + 1.;6~ .......
.. 624m + 2(4 - 2~ = 0, then m = - .. 321(4 + .321~ ...
.. 312n + 2Qs = 0, then
.248p - 2(4 = 0, then
= 0.5
= 4.5
= 1..75
= 3.4
= 1.6
= 6.25
= 1.6
= 0.321
p = + .805(4 .............. = 8.05
-61-
times (~ + ~)
" (~ )
n ( +(1 - ~)
II (~ + Cs)
" (-(4 if- Cs)
"
11 (~ + ~)
fI (~ + ~) \
fI +~
n +(4
PAR TV. - -PROPORTIONAL AD·JUSTMENT OF INTERMEDIATE BENCHMARKS
FROM CORRECTED ELEVATIONS OF ADJACENT ' .JUNCTION PoINTS.
APPLIED TO THE ADJUSTMENT
OF
INTERMEDIATE BENCHMARKS
IN THE
DAWSON-ST. JAMES TOWNSHIPS LEVEL NET~
PHELPS eOUNTY~
MISSOURI.
(BA'E~ ON THE LINE O~ LEVELS ~UN 8Y THE U. S. GEOLOaJeAL SU~VEY IN 1908 ALOMG THE FRISCO R. R.
11TH ~E •• AN£NT BENCHMARKS -A- AT ST. dAMES, AN. -E- AT KNOBVIEI).
-62~
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Merimec -Sprines ~uadrangle. (Latitude 37 45'-38;10ngitude 91 30'-91 45 1 )
(St. James Township)
Line from st. James,northwest via Jefferson road to the intersection of Republic road, -thence north via Republic roqd, 9. 5 miles to the Phel p s-Gasconade Coun·ty Line.
A. St. Jalnes, corner of Meramec and Wasl1ington Sts.,about 70 ft.R. of,center line of Frisco R.R. track ,25 ft. from corner of Commercial Hotel, in angle between cro ss walk and side Walk; iron post stamped "1098 n. ----------- 1087.800 0.000 1087.800
\ i - .
L. st. James, 0.5 miles N.W. of, on Jefferson road at intersection of Parker Lane road, concrete . cu1vert,north end wall, top of; chi se1ed square. -------- ------ 1078.427 -0.002 1078.425
K. st • J arne s , 1. 75 mil e s N. W .0 f , on Jefferson road at intersection of Republic road,on south side of road, 10 n bl ack jack tree, in base of;copper nail and washer. 1092.694 -0.008 1092.686
1 St.Ja.mes,2.25 miles N.of,on Republic road, Jefferson road,O.5 miles N.of,culvert,west end of, sandstone slab, top of;painted square. ----------------------- 1034.345 -0.010 1034.335
2 Jefferson road,l.O mile N.of,on Republic road,T road east at Coxts house,lOO ft. E. of, sand stone boUlder in northeast corner of T road;top · corner broken off and painted white. -------- 1024.715 -0.013 1024.702
3 Jefferson road,1.5 miles north of, R6binsons Creek,O.75 miles south o£,Fitzenriederts residence,100 feet west of,wagon gate post,in base of;copper nail and washer. 998.469 -0.016
-64-
998.453
4 Robinson's Creek,O.25 miles south of,on Republic road,on east side of road,lO If post oak tree,near Kenndyts mail box, in base of tree; nail. ____ ~ ___ ~_---------------w-- 923.575 -0.018
5 Jefferson road, 2.25 miles north of, on Republic road,Robinsons Creek, west bank of,24" white oak tree, in root of;C01)per nail end washer. 917.604 -0.019
(Dawson Township)
6 Robinsons Creek,O.25 miles north 9f, 15 feet west of road,sandstone boulder on 'St. James-Dew son Township line, opposite wire fence corner on east side of road, point is on south hill side 300 feet south of wood chopper's cabin;painted cross on top of
923.557
91? .585
boUlder. ------------------------·· ··-1022.394 -0.020 1022.374
7 Robinsons Creek,075 miles north of, on Republic road,township line,0.5 miles north of,at north end of ridge, 30 feet west o~ raad,se~dstone boulder, top of;chiseled square. ---------- 1039.423 -0.021 1039.402
8 Robinsons Creek,l.O mile north of, on Republic road,at foot of hill, concrete cUlvert · across road,west end wall, top of;chiseled square.- 934.325 -0.023
9 Robinsons Creek,1.25 miles north of, Miles School House,l.O mile south of, T road west,center of roads;road
934.302
elevation. ------------------------ 976.005 -0.024 975.981
10 Miles School House, 0.5 miles south of, on Republic roeA,Marrs Cemetary,400 feet north of,east side of road, sand stone boulder at rail fence corner, top of; chiseled square. -----------1059.0117 -0.025 1059.052
It St.James,6.0 ·miles N.of,on Republic Rd.,Safe,2.5 miles E.of,cross Rds.at Miles School ,N.E. corner of School ~ouse,limestone wheel' guard, top of; pa.inted square. ---... -------------- 1061.925 -0.030 1061.895
-65-
Kdjustment of peint P en Republie Road 0.5 miles south of Ha.rris.a Seh(!)ol, and the determination of intermediate poiats between I and P only. Et~~ .• - ~"1.'~" '~ •• r.
• I. 1"1.0". A41 •••
---,..------0 •• 29' C.r ..
.ALe.LAYt •• IF •• £ •• R. RICY' •• FOR P~." ,.
I + .0064 = .2 .. 4 .0380 6.4
I = .0360 x 2 .. 4 6.4
.OOQ4
I = .0135 - .0064 I = , •••• ,,' •• 4 te •••• r •• 4
.J.~&tl, •••
E'.~ • • 954.124. Arr •• ,.d • •• 954.131. AdJ •• t.d
---II1II--"--+ •••• ,. Cere
Elev P = 928.5520 ObserTed - 0 .. 0071 Cor.
Elev P = ~28.5449 Adjusted.
.... . -., . . . . .. e · . ....... ..... .....
.. .. • .. •
t-t
• •• .... .. (I
••
poiat at distance d. . .......
Correetion e. for any interil8aiate " from P aDd betweeD pl!>ililts I and P.
DEI '.'AI"" .F A& •• 8'IEI' FIRI'LA •
• ,0225 4.0 ',,' "",.",... 0 b C
8
- .00'71 = d
•. 0225 d C = ..:...;..----- +
~ 4.0 .0071 t. I
• £ R •
"l '1""4 -.. .. ...
COl .. •
Ii
.. .. ... • •
A D .. ' • (Wi) '1""4 • • • or
•
• o
., .qt 'I""'f
1C\ .. • ..
() . .....
E •
":'66-
I
•
L " • E to ~ 'I""'f ....
IC\ e ... -• • fit fit
POl ... e ' .. .e
• '""'" .... e • • ...
Thea C=(-.®05625d-.~"1)
.. e o • e
~ • IC\
• •
.. .... • • • •
e e .• •
I. st.,Jam.es,6.0 m.iles north of,on Republic road,6afe,2.5 miles east of,Miles 6choOl,25 feet west of, S.W.Cor.6.E.Q.tr.of-- the ~.W.~tr.,Sec.20,T 39 N.,R.6 W., 60 feet southeast of,concrete well curb, top of;bronze tablet. 1059.528 -0.030. 1059.4:98
12 Miles 6chool,0.75 miles north of, on Republic road,W.L. Martints house,150 feet southeast of,on top of hill,on east side of road, 12" post oak tree, in base of; copper nail and washer. ---~----- 1023.857 -0.025 1023.832
13 Miles 8choo1,1.2 miles north of, on Republic road,Bourbeuse River, 0.75 miles south of ford,at dim cross road,southeast corner of; painted square on sandstone boUlder. ------------------------ 924.251 -0.023
14 Miles School ,1. 75 miles north of, on Republic road,Bourbeuse River, 0.2 miles south of, fa..rm house, 600 f ·eet east of, on east side of road, lBn waJ.nut tree,in base of;copper nail snd Washer. ---~------------ 873.567 -0.020
"}:0 Wash School House,0.75 miles south of,Bourbeuse river ford,lOO ft.S. of,east bank of river,30" sycamore tree,in base of;copper nail and
924.228
873.547
washer. ------------------------- 839.998 -0.017 '839.981
T5a Bourbeuse River,at ford on Republic road, center of creek;road elevation. Road Elevation. -------------- ..... -- 834.000 -0.017 833.985
16 Bourbeuse river,0.25 miles north of, on Republic road,at T road east,N.E. corner of,in base of fence post; copper nail and washer. --------- 840.106 .0.016
l~ Bourbeuse Riv.,0.75 miles N.of,Wash 6ch.,400 ft.S.W.of,J.W.Crossnerts house,300 ft.S.E. of,B ft.W.of three mail boxes,on W.side of road,14" scaley bark hickory tree,in root of; copper nail and washer. --.. ------ 947.596 -0.015
-6~-
840.090
947.583
Adjustment of intermediate bench marks between the points P and Q on the Republic Road near the Phelps-Gasconade County Line •
• ~ w~" b. r •••• b.r.d that th ••••• P to Q I. a .pur "n. of •••• ,. 1ft oar .oh ••• of adju.t •• nt. Th. I'n. P to 0 .ae run a. a parafl •• dupllo.t. lfn. of ' ••• '.~a.d tb •• o.t probabl. yal ••• for tb •• '.Yatlo •• of the po'nt. on .uoh a II •• I. the ay.ra,. of tho •• Y.~al dat.r.lnct'one. Th. dra_ src. bal0. abo •• Ifne. of •••• ,. run by Brandt. and Qur'.y I •• tru •• nta:a. tb ••• Inatru •• nt ••• r. of .qual r.ff •••• t, .qua ••• 'Ibt ••• applJad to tb. work dona by .aoh In.tru.ant. Th. Ir ••• Itna r.pr ••• nt. tb. ayerasa cf tb. Inatru •• nt~l d.t.r.f •• tlon •• In tbe I.y.a ft.t .dJu.t ••• t pr.o •• dlftl, It ••• fouad tb.t the pofat P ••• 0.0071 ft~ hl.h. Ae the .pur lIn. p to Q bfn ••• dlr.otly on tb. polntP,tb.n the wbot. lin. p to 0 .fl ...... to b. Icwer.d 0.0071 ft.'ft tbo dfafr •• tho adJu.ted •••• I •• h •• n In blaok.
E •• y. P • 928_5520 Ob •• ryed • p. 928.5449 AdJust.d
- 0.0071 Cor •.
Correction c for any intermediate point between P and Q: C = average observed elev minus 0.0071.
T I
e O 0 ,..,
t " 0 .,0 0
()'\)
o <0
o o
o
Elo •• o • 997.2400 Ob.aryad • o· 997.2329 AdJ.et.d
- 0.0071 Cor.
~.,
o IX>
o o
o
,. " " . " ' . .,
J1 I' ,.
Go .... " 1'
,~
• I
1 "- A V E R : a E I
J r
o B Sr E R V E 0 .. I , L I It E .
p
... .... o o • o
• .0 -0 ... ::'0
0;; It.,,. '. "., (f .
IC'\
'" .• o
0 t<,. L 0 0 0 • E
R I
A D .I l:I S T
•
I ' ... 0 I"- (0
0 0 0 0 . . 0 0
! I . I
L o W E R
L. Nt; ~...I'.----
Gurley 0 r.a • •
I ... 0 ... <0 o <:> o 0
o co
C\ I"-
• q
Ie' N 0 0
t IC'
N 0 0
• 0
• ... 8RAPH FOR THE ADoJUSTUEIT OF .IUERIEDI' AfE POJIT8 8ETWEEN P AltII
-68-
. -
o.
18, Wash School,0.5 miles north of, on Republic road,Little Bourbeuse Creek, 8.t ford, on south banJ.c,on west side of road,lO" sca1 ey bark hickory tree, in base of; copper nail cmd washer. -- -- - 835.927 -0. (l)l0
P. Wash 8chool,1.0 miles north of, Little Bourbeuse Creek,0.5 miles north of,Harrison Schoo1,0.5 miles south of , Phelps-Gasconade County Line ,1.0mile south of ,Austin E. Bruer's house,200 feet west of, at T road east from Republic roe.d, on west side·of road,2 feet inside of fence,lO" post oak tree, in lJa.se of; copper nail and washer. --------------------- - -- 928.552 -0.007
19 Wash Schoo1,1.25 north of,Litt1e Bourbeu se Creek ,0.75 miles north o~,Harrison Sc11.00l,0.25 miles sOuXh of,Phelps-Gasconade County Line,0.75 miles south of,Steve Lort's house, 70 feet north of,T road west(Mail road to High Gate), nortr.!.west corner of roa.ds,24" oak stunlP in fence corner, top of; ten penny nail. ----------------- 9£2.732 Bra.ndis,
962.726 Gurley. - ' .. -----
Average 962.729 -0.007
20 Harrison School Ho use, on st. Jame sRepublic road,west side of School BUilding,north concrete door step, -top of; chiseled square. -------- 9£ '5."674 Brandis,
965. 658 Gurley. _ ... --_ .... --.. Average 965.666 -0.007
21 Harrison School House,0.25 miles north of,on Republic road,PhelpsGasconade County Line, 0.25 miles south of ,private--·T road west, south west corner of,16" blazed black oak tree ;bent nail in base of. -- 959.594 Brandis,
959.578 Gurley. _ ' ___ ' __ "iiir
Average 959.586 -0.007
-69-
835.917
928.545
962.722
965.659
959.579
Q. ira.rrison School Ho use,0.5 miles north of,on Republic road, intersection of st. James-Republic road and High Gate-Red Bird ro ad, 0.5 miles south of,Phelps-Gasconade County Line,28 feet south of,center of road,30 f eet west of, 5 feet inside wire fence, in line with center of road north into Gasconade County,dim T road west, 50 feet south of;iron post stamped "997". ------------------
P.B .. Me 997.242 Brandis, 997.237 Gurley.
Average 997.240 -0.007 997.223
-70-
Adjustment of intermediate temporary bench marks between point I at Miles School of Republic Road and Point J two miles due east on the Red Bird Road.
DER~.ATJON OF AOdUSTMENT FORMULA. El.~.I.IO'1.92~e Ob •• rYe
• l.lG'1 •• 9~4 A4'u.t. Ef.Y.d.968~394& Ob •• r •• • d.96a~3704 AdJu.t.
- 0 .. 0236 Cor.
Correction c. for any intermediate point at distance d. from J and between I and J:
~
: c
r----- - . ...
CI
I • • '!II -~ ... • ::'i'f
C - .0236 d =
.0060 1.6
C .0060 d
.0236 = +
.... 0
C =
C =
· I L o wE R •
1.6
( .00375 d
( .00375 d
E R V E 0
co \0 C'oI o • •
A D ~ 0 8 TED
32 33
~ 1ft
~ ~
-i • rot 0
+ .0236
~ .®236
_L I N E .
: C'oI 0
• 0
L Ii It E . 34
0 ~ •
411
)
)
'~AP. Foa T"~ ~D~uaTIE'T OF .tln EIUEB . ,U : E Pt.ITS
-71-
\0 on (\I
0
• 0
J
0 0
• 0
BEtweel I lKO - d4
Red Bird ~uadrangle. (Latitude 38 -38 15' ;longitude 91 30 1 -91 45 1 )
(Dawson Township)
Li ne f:co!il Miles School House on Republic Road, east via l'oad 2.0 miles, to point on Red Bird Road, two miles north of St.JD~es-Dawson Township Line.
I. 8t.JarJ.es,6.0 miles north of,on Republic Road, Safe,2.5 miles east of, cross roads at Miles 8chool,northeast corner of school house, s and s tone wheel guard,top of;painted square.--- 1061.925 -0.030 1061.895
32 Miles 8chool,0.25 miles east of, : on north and south center line . and between sections 20 and 29, R.6 W.,T.39 N., at di m T road north, center of road;road eleva 1052.867 -0.029 1054.838
33 Miles School, 0.75 miles east of,Rinehart's house,200 feet north of,at cross roads,southwest corner of,sandstone boul-ber, top of;dob o~. paint. ------ 1045.151 -0.027 1045.124
34 Miles School, 1.2 miles ea st of J
via crooked road through woods, Red Bird Road,O.4 miles west of, on south side of road,1 6 1f post oak tree,in root of;copper nail and washer. ------------------- 1037.012 -0.025 1036.987
J. Robinsons Branch, 2.5 mil es north of,St.James-Dawson Townsh ip Line, 30 m.iles north of, on top of hill J
Red Bird Road at T road west to Miles School,northwest corner of roads,15" white oak tree,in root of;copper nail and washer. ---- 968.394 -0.0~4
-7·'2-
968.370
Adjustment of intermediate temporary bench marks between point B at the intersection of Ozark Highway and Red Bird Road, and point H at the intersection of Jakes Prairie and Red Bird Roads.
DERIVATION OF A D JUS T M EN T
E'ev.B • 1074.6660 Ob.er.
FOR U U LA.
• B· 1074.6352 Adjus.
0.030 '8 Cor.
I\)
J, ~ . " ,CI] o .
<.) --- - - r t B ~ __ d __ >1 H
~----'---3.4:-----> ,
Elev.H = 1042.0580 eb •• r. • H. 1041.8535 Adju ••
0.2045 Cor.
Correction c. for any intermediate point at distance d. from B and between Band H.
and
simplifying:
c = -(0.051088 d + 0.0308)
and finally
c = -0.051088
co o 1("1
o
B
• eo -0
lIIO
'" :;; 0 .
35
0 ta
• co
d
C\
A
- 0.03(8).
'I ~
~ 'i'-
S '0
0 0 co
IC\ '0 ... 1("1 Q\ " ... co <> . C> • •
D J U S T 36 37 38
0 ta 0 .0 '" In
• • . ... ... ...
~
E 0
c = 0.173? d + 0.0308 ---3:-4---
" "
J ~ · '" \
\. L 0
• E ~
0 R ... IC"t ~ 0
<> ... GO . C\I
La 0- ..... ... . K'I '" ... . 1("1 ... . ... · .
L I N E . 39 40 41 42 H
0 0 IC\ 0 0
<> ca ..... 0 ... • •
C10I C10I (\I "" K"I
8RAPMFOR THE ADJUSTMENT OF INTERIlEO-IAtE POIIiTS ' BETWEEN B
-73-
A NO H.
35 '
36
37
38
Red Bird QMadrangle. (Latitude 38 -~8 l5 1 ;longitude 91 30' -91 45 1 )
(St. James Township)
Line from intersection of Ozark Highway and Red Bird Road,which is 1.0 mile northeast of st.James, thence north via Red Bird Road,to Royal F.O.,and Phelps-Gasconade County Line.
st.James,l.O mile northeast of, ' on Ozark Higbway,T road east, {intersection Ozark Highway and Red Bird Road),400 feet south of, Dr.E.A.Scottts house,300 feet west of,on west side of road,lO" elm tree, in root of;copper nail and washer. -------------------
st.James, 1.5 miles north of, intersection of Red Bird road and Ozark Highway, O. 5 miles north of,N.E.Johnson's house, 75 feet eabt of,on east side of road,20 feet . inside of fence,12" hickory tree,in root of;copper nail and washer. ----
Flag Spring Scho.o1, 0.25 miles east of, Britton's Store,0.25 miles west of, on Red Bird road, turn in road to east,30 feet south of, sandstone culvert across road,west end Wall,top of;chiseled square. -----------
Flag Spring School,0.5 miles east of,Britton's Store,in front of, center of road at turn to north;road elevation.--
Flag Spring School,0.75 miles northeast of,St.James,2.5 miles north of,at Y road to north,in center of Y,lO" post oak tree, ,in base of; copper nail and
1074.666
1069.326
1029.621 .
1039.320
1023.293
-0.031 1074.635
-0.057 1069.269
-0.084 1029.537
-0.096 1039.224
-0.118 1023.175
39 Flag Spring SchoOl, 1.25 mile SCi
northea.st of, Britton!l::.' Store, O.75 :-mi1es north of, dim. cross roads, southeast corner of, 10" post oa.k tree, in base 0:8.; copper nail and w.asher. --------- 1016.'~ 547 -0.134 1016.413
40 Brittons Store,l.25 miles : north of; st. James 3.5 miles north of) on': Red Bird Road, at ford across Robinson's Branch, 50 f:eet south of;projecting . Cotton Rock Sta.ne ·:; in bluff, on west side ;~' of road; chiseled square.--------------- 952.626 -0.159 952.467 (Bench Mark obliterated by dyna.~iteJ
41 Robinson t s Bra.'1ch, 025 mil e s north of, on Red Bird Road, at dim cross roads,tile culvert on .·~ we st side of road, south end waJ.l, top of; chiseled squa,re.---------1025.576 -0.172 1025.404
(Dawson Township)
42 Robinso,n's Branch, 0.5 miles
H' •
H.
north of, on Red Bird' Road, on st. James-Dawson Township Line, on east side of. ·- road, opposite T.-Snow's Mail Box,#12A,gate post, inc'hue of; copper nail and washer.-------- - ---------------- 1054.905 -0.185 1054.720
Robinson's Brru1ch, 0.9 miles north of, on Red Bird Road,at intersection of Rea.: Bird and· Jakes :Prairie roads;(Y road to northeast),F.Berries house, 300 feet south of, southeas,t c.~rner ~ of roads'i14" po sto~ tree, root of; c.opper nail and washer;. ----- -1042.947 -0.205 1042.742
Robinson~s Branch, O~ miles .. north of J on ,Red Bird r oad, at intersection of Red Bird and Jakes Prairie roads, (J road northeast) J F.Berries house, 230 feet south of; crotch in'.Y roads, opposite east end of "agon gate, five feet south of P.B~M. fence; iron poststamped "1042 "----1042. 058 -0.205 1041.853
Adjustment of intermediate temporary bench marks between point E at intersection of Jakes Prairie and Red Bird Roads, and point . J on Red Bird Road two miles nortb of St'i;,James-Dawson Township Line.
])ERIP'ATION OF AlJ){fSTMENT FORAllJLA. &b •• rv. ~.= .. ~ .- f E I • " • H. 104 2 • 05 II 0 j - .= .:=,,=
• H-l&41.853' AdJ •• t·N ~
--------- c ~ Cor. -- 0.204, ~ 0\
H~ _~ lJ ....... ..... - /.6' >
E1.v.J - 968.5980 Ob •• rv. • J - 968.3704 AdJuot.
- 0.2276 Cor.
Correction c. for and between Hand
any ipt~rmediate point at distance d.from H J.
and
TQen:
C = -(O.0144~?5 d + 0.2045) and C = (-O.01443?5 d
~ - .. ....
CI:'I .... 0 eN
•
H
• •• '-0 ... " ••
B S .E R V E 0 0 ... .
K'I ... ... N
A 8 JUS TEO 43
o C\
• o
L I N E .
: ... CII
liN e • 44
o ... .; ...
1 l 0 W e R
.. ..
\0 ,.. e.a N
J
o \0
• ...
0.2045) •
Adjustment of intermediate bench marks between point J on Red Bird Road two miles north of St :·James- Dawson Twp .Line ,and point M 500 feet north of Royal Post Office.
The adjustment of these intermediate benoh Marks can be effected by scaling the adjustment corrections from tme large diagram below and subtracting eaoh correction from the observ-ed ilevati0n as it + _ appears in the level notes;or~if desired,the adjust- . - ment oorrections can be com-puted by the foll- i _ _ _ _ _ owing formula for this section,Jto M. ~
I\:)
~ I' I
J.J, ... ...... -- -3.2 / _ ..... -"; ,j, M DERIVATION OF ADJUSTMENT FORMULA.
It.v.d- 968.598 O~ •• rv.d. • J- 968.3704AdJ.at.d. E'ev.U- 954.9830 O~a.ry.d.
• u. 9~4.7304 AdJuat.d. - 0.227' Cor. - 0.1726 Cor.
Correction c. for .any intermediate point at distance d.from M and between the points J and M.
- 0 B S
O - .0550 d + ln26 - -8:21-- .'(
C = (0.017134 d + 0.1726)
I r: R V Finally E 0 0=(-0.017 L , 134 d -0.1726 }
N E .
I ;.~-.
L 0
• : E (\I r- IC\ N L
: R ... r- 0- ... CD 0 . ... co 0- 0- 0 • IC\ \0 . 01 (\I 1"4 ... 0- r- ei
• . . • ... E r- .-• R ... ... . • .
J A 0 J U S T E & L t N E •.
M 4,15 ,46 .4'1 48 4~ 5. 51
<10 co 0 0 • IC'I pC • 0 ...
0- '" . eo ... • ·01 .... 0 IN . • • • • • - . . N N'I
-0 co ... pC pC 01 '" ::II
eRAPH FeR THE ADJU8TIIEIIT OF I NTERMEO<f,lTE POtllT8 BEf.EEI oJ AMD M.
-7~-
43 Robinson's Eranch, 1.4 miles I
north of,Township line,O~ miles nort}1 of, on Red Eird road, on top of hill, at T road west, north-west corner of, 12" white oak tree base of; copper nail and washer.---------~------------- 1036.078 -0.212
44 Robinson's Eranch,2.0 miles north of, Dawson Township Line 1.5 miles north of, on Red Eird Road, Eikman's house, 50 feet east of, on west side of road at gate, s~~dstone boulder, top of; chiseled square.------------ 996.226 -0.220
J. Robinson's Eranch 2.5 miles north of, Dawson st.James Townshi~ Line, 2.0 miles north of, on top of hill, at T road west to Mile School, northwest corner of ~oads,15" white oak' tree,root of; copper nail and washer ----- 968.598 -0.228
45 Robinson's Eranmh, 2.75 miles north of, on Red Bird Road, on top of hill, N.D.Putnamls house, 40" southeast of, on west side of road, five feet inside of yard fence, 12" waJ.nut tree, root of; copper nail and washer.--------- 967.970 -0.224
46 Robinson's Eranch, 3.5 miles north of, intersection of Red Eird and Jakes Prairie Road, 2.6 miles north of, ford at Clearcreek, aa miles south of, on Red Eird Road, s~~dstone boUlder at side of road, top of painted square. ----------------------- .. - 975.188 -0.211
47 Clear Creek, O. 5 miles south of, on Red Eird Road,intersection of Red Eird Road with Jakes prairie Road, 2.8:_.miles north of, 18 "Black Oak treeat side of road, in T1)O t of; copper nail and washeJ:S94.860 -0.208
1035.866
996.006
968.370
967.746
974.977
894.652
Adjnstment of point N on Red Bird Roa.d 0.9 miles north of Royal P.O., and point (;) on Jakes Prairie road 0.25 miles south of Phelps-Gasconade County Line. Nand 0 are intermediate JUDction points on the line II via Nand () to G.
M Q
via N So ' to G.
'" c-I .... .... . 0:>
M • .0
- 0 - • :::110
N Elev.M =
-~.-. -----------
- 8. '5"-
o .... --_ .. _--...,. Jr
G Elev. M = 954.9030 Observe
n M = ~§~~Z§Q~ Adjust.
. Elev.N = - 0.1726 Cor.at M.
Elev.O =
Elev.G = Elev.G =
;t:-
N 0 0-. C>
, Q oJ \~, C . ... <::::. ": '... .. Izet Y = Cor.to O;. then: C> " '
Let x = oor.to N,then: x - .1'726 _ 0.9 --O~0690- - 8:~5-
x= LQ§2Q_~_QL~ + .1726 8.05 '
Y - .1'726 _ 3.3 --O~0790- - 8~05-
Y = Lg§~~;~53~2 + .1726
Y = -(.0283 +.1726)
x= -(.OQ'77:1" .1726)
Then:
Then: -: : Y = - 0.2009 Corl to : o o. to
0:> eD o ... . 0
-0.1803 Cor.to 0 N. Elev.O=969.7360 Arranged x =
Elev.N=875.0710 Arrang d :-_~:;1803 Cor.
=.Q~g~Q§! Cor. w 0=969.5351 Adjusted.
,...,. r--~- ~
n N=864.890'7 Adjustu ========
A 0 JUS T E & o K'I . K'I
==== == ==
l , N E • G
'" co . eo
IRAPHFOR THE ADJUSTMENT OF 'NTERMeDIATE JUNCTION POINTS N
AND 6.
-79-
Adjustment of intermediate bench marks between point M 500 feet north of Royal Post Office, and point N 11(}O feet south of Red Bird Road Ford across Big Bourbeuse River.
The adjustment of these intermediate bench marks can be effected by scaling the adjustment corrections from the large diagram below and subtract1ng each correction from the observed elevation as it ap~ears in the level notes· or, if desired, the adjustment correct1ons can be computed by the following formula for this section of the net,M to N.
DERIVATION OF ADJUSTMENT FORMULA.
---~ .. j ...-''''' .
~ - -- -<0
<.u
~~~:::-...! lN EI ••• U • 954.9030 Db •• rv.
• M. 9~4.7304 ~djuit-Eltv. N • 87~.2;OO Db.trv.
• N. 874.8907 AdJu.t.
- 0' .1726 Cor. ,
Correction c. for any intermediate point at distance d~ from M and between points M and N.
and c = Q~I§§7_~ + 0.1726 u.Q
c = -(0.20744 d + 0.1726) or
... .....
>0 r-c-a C\I r- • ... 0
• o
A D J U S T E52 M • eo 0--0 ..... -. • :EO 0
C = (-0. 20744 d
<:)
l t N E • N
. o
GRAPH FOR THE ADJUSTMENT OF INTERMEDIATE POINTS BETWEEN M AND N.
-sc-
0.1 '726).
48 Royal Post Office, 1.5 miles south Of, on Red Bird Road, Clear Creek,bridge over,south east end of . concrete brid~e, elevation on hand rail.------ 872.690 -0.200
49 Royal Post Office, 1.2 miles south of, Clear Creek bridge, 0. ;:, n iles north of, T road east, lO(!1" south of, concrete culvert across road, center of; road elevation. ------------------- 924.914 -0.194
50 Royal Post Office, 1.0 mile south of, at T road northwest, 24" Black Oak tree on west side of road, in base of; copper nail and washer.------- 983.208 -0.191
51 Royal Post Office, 0.25 miles south of, on Red Bird Road,on east side of road, 30" White Oak tree, in root of; copper nail and washer.------- 985.210 -0.178
Mt Royal Post Office, on Red Bird Road,Steen's Store, north end of concrete porch, one foot out from Building, six inches from north end of porch;pronze P. B.M. tablet stamped "961!' ------ .. --- 960.631 -0.173
M. Roy-ail Post Office, 500 feet north of,Royal School House,lOOO feet north of,at T road west to Republic Road,soutPwest corner of, telephone pole,in base of; copper nail and washer.------- 954.903 -0.173
52 Royal Pqs.:-t Office,(l.6 miles north of,on Red Bird Road,Huntsman's House, 200 feet north west of,on east side of road, 16" Red Oak Tree, in root of; copper nail and washer.--------935.969 -0.274
-81-
872.490
924.720
983.017
985.032
960.458
954.730
935.695
Adjustment of intermediate bench marks betweel'l the points Nand R OD. the Red Bird Road near the Phelps-Gasconade County Line.
!I
l o IV E R
I • . !IEI .. V~ .N.87~. 2500 Obser.
• ~ . , . N.874.8907 Adjus. ! I --------~ I - 0.3593 Cor.
)IE.e".RaS66.3380 Obser. ~I • R-s65.9787 "djus.
~I :-'0:;;;3 Cor.
j :correction c. fer any ,Iintermediate lJoint ' Ibetween N and R~ is:
!II ()
8 r a II d .. '.
OBSERV E D LINE.
... ~ I~. G" r 1., \ ~ 1'1
l.V., Inetru. t
II •
I N to R i8 a spur line of I levels in the level net l I ad.justment soheme. The line ~
N to R was run as a parallel duplioate line of lev- E els,then the most pr0bable R values for the elevations • of the points on such a line is I the avera~e of the several deter- I ~ minations;The diagram below sbo.s : lines of levele run by Gurley and iL
I. Brandie Instruments; as these CrJ ~ instruments were of equal re- I ~ I~ finement,equal weight was app- I ~ ~ lied to the work done by eaoh I '
I ~ instrument. The green line rep- I I resents the averag e of the ins-
I I Ie = averaged observed
i I elevatio~ minus 0.35931 : 'I
trumental determinations. In the level net adjustment preceedinR., it was found that the point N was 0.3593 feet high. As the spur line N to R hinges direotly on the point N,then the whole line, N to R will have to be lowered 0.3593 feet. In the dia~ram, the adjusted line of levels ~ is shown below in blaok • , I
. 1
! I I
i I ! I
feet. I
I I I I I I
~~' -= -:-=' -'::::::-=-~~' .:::: ' ..:.. . . - . - ~~ - - - - - - - - - - - - - -- __ I~ ~-- .. -.---.. -.--... - .... - .. _.-_ .... _.- 0.5· '-'·'·"-" ..... . ..... -..... --. -..... -. - ... ----... - .. ...........)-~ A 0 JUS TED lIN E • R
' ~3 GRAPH FOR THE ADJUSTMENT OF INTERMEDIATE POINTS BETIEEN NAND R.
-82-
N. ~oyal Post Office,1.25 miles north of,on Red Bird Road, at T road east to Jakes Prairie Road, 50 u feet south of T road,on east side of road,14" Red Oak Tree, in base 'of ; copper nail and washer.-------------------------875.250
53 Royal Post Office,1.45 miles north of,on Red Bird Road,Ford across Big Bourbuese River,125 feet southwest of,center of road,30 feet west of,main channel of River,80 feet so uth of,leening 12" slippery Elm Tree in south bank of River,inbase of; ten penny nail.------~------------- 812.023
R: Royal Post Office,1.75 miles north of,on Red Bird Road,Red Bird Post Office,2.00 miles south of,intersection Red Bird - st. James and Red Bird - High Gate roads,1.50 miles south of,Ford across Big Bourbuese River,O.3 miles north of, large Red Barn on east side of road,lOO feet southwest of,center of road, 25 feet west of, (Primary Tr averse Station#87),wor.m fence,4 feet inside of,in lot;Iron Post set on Phelps Gasconade County Line ·, P.B. M. stamped "866!'-------------------866.338
-83-
-0.359 874.891
-0.359 811.664
-0.359 865.979
'Adjustment of intermediate bench marks between point P on Republic Road 0.5 miles south of Harrison School, and point M on Red Bird Road 500 feet north of Royal Post Office.
DERIVATION OF ADJUSTMENT FORMULA.
EI·y.p·928.7~10 o •• tl-:':":"':'- - "':}- -'----- j E'·.,.t.I-954.9030 0 ••• • P-g28.-5449 Ad", t • "-954.7304 Adj. -------- ~ . ~ --- -----
Cor. . ' - O. 1 8 6 1" c +---d ~ I\) Cor.. - O. 1 72 6
I i p"f 'M ~- -- 2,4-= ------
Correction c. for any intermediate point at distance d.from M and between P and I.
.... \0 CD .... •
0
• •• -0 -. :II.
GRAPH 'QR THE
-l 0 W E R
C - 0.1726 = __ ~ __ --~:5I§~---- 2.4
C = -(0.005625 d + 0.1726)
and finally
C = (-0.005625 d 0.1726).
o B S E R V E 0
... eN co .... •
0
A D J a S TED 54
IC\ ,.. .. co
l- J N E •
'" co ... eo ,.. ,.. .... .... . • co 0
L " , 55 Ijb
• IS .... ... • i ... ...
E •
co '0 ,.. .... co
0 O't
• ... Ae.tUSTI£NT OF ""TER8ED1,.,'E PfUtTS
-84-
-
'0 C'I
L ,.. 0 .... • co E R
I
0 .... •
eN
8EtIE,EIt P AND M.
Red Bird ~uadrangle. (Latitude 38 -3815 1 ;10ngitude 91 30'-91 45')
(Dawson To'wl1ship)
Line from paint on Red Bird Road 500 feet north of Royal Post Office, thence west 2.5 lliiles via road to point on Republic Road 1.0 mile north of Wash Schaal.
M Royal Post Office,500 feet north of ,Royal School House ,1000 feet north of,at T road west to Republic Road, southwest corner of, telephone pOle,in base of;cop-per nail and washer. ------------ 954.903 -0.173 954.730
57 Royal Post Office,0.5 miles west of,at foot of hil1,at private road southwest to farm 11ouse,011 north side of road,infence corner,twin white oak tree,in base of;copper nail and washer. ----- 863.080 -0.176 862.904
56 Bourbeuse River Ford I at mouth of Little Bourbeuse,middle of f ord; road elevation. ---------------- 813.000 -0.178 812.822
55 Royal Post Office, Ito- west of, S.T. Mitchel's house,GOO feet east of,Bourbeuse River ford, 700 feet west of,on north side of road,at corner of woods,20n white oak tree,in root of;copper nail and washer. ----------- ---- 831.792 -0.179 831.613
54 Royal Post Office,1.65 miles west of,Bourbeuse River ford, 0.75 miles west of,Republic Rd., 0.75 miles east of,Phelps-Gasconade County Line,l.O mile south of, south side of road,at T road south,12 n white oak tree,in base of;copper nail and washer. ----- 924~692 -0.182 924.510
-B9-
P. Wash School,l.O mile north of, Little Bourbeuse Creek,0.5 miles north of,Harrison School, 0.5 miles south of,Phelps-Gasconade County Line,l.O mile south of,Austin E.Breur's house 200 feet west of,at T road east from Republic Road,on west side of road,2 feet inside of fence, 10" post oclc tree,in base of; copper nail ~,d washer.-------- 928.731 -0.186
-86- ·
928.545
Adjustment of intermediate beneh marks between point E at Knobview Depot~and point D at the intersection ef tbe Ozark Highway and the Phelps-Crawford Line.
Cor. •
DERIVATION OF ADJUSTMENT FORMULA.
0.000
E
Adjuet.
ft. Et.v~D· 1098.8499 Observe
• D- 1098.8847 Adjust. ---------to 0.0357 Cor.
Correction c. for any intermediate point at distanee d.from E, and between the points E and D.
• .0 -0 - .
C = d a:-035?- -1::-(5-
Then:-
C = (+0.0357 d). o o
E~~ ______ ~A~D~J~~U~S~T~E~D~ __ ~L~I~N~E~~.----~n
I R o -.... A
I o S
E
( I 1 II
2'
.. - !)
GRAPH FOR THE ADJUSTMENT OF INTERMEDtATE POINTS BETWEEN E AND D.
c
'" ..., o • o
Adjustment of iDtermediate beDch marks between point D at the intersection of the Ozark Hi!!hway and the Phelps-Crawford County Line,and point G at the intersection of Jakes Prairie and Tribby roads,O.5 miles south of Hinkle P.O. .
DERIVATION OF AO~USTKENT FORMULA. Ele".O-1098.849 Oh.er.
• 0-1&ge.8847Adj •• t. E le".6-921.2020 Ob.er".
• G-921.2814 Adjust.
• O.0357Cor. + 0.0794 Cor.
D .(-- ----- 4-.$------- > G -~ ~
~ - -----d --.. --- - i ~ ~ 7 _ _____ _ __ c _ _ __ _ ~
. · r Correction c. for any intermediate poin't at distance ' d.from point D and between points D and G.
c=(+O.OOe?11 d + O.035?)
• eo 0 lr\ 0 IC\ IC\ IC'I 0 0 0 11:'10 -0 IC\ C\I IC\ ,... (\I -0 0 IC\ 0 c:.. lei . . • . =- 0 ... ... r-I (II (\I 1(\ 1(\ .... .... .... D 606162 63 64 65 66 6'7686
oJ u s T E D L I N E
R (\I (\I
-oA 0 0 -0 I
0 0 \C) \OS
D. • 0 °E
0I :: 7· 1 G. ,
GRAPH FOR 'HE AOJUStMENT OF INTERMEDIATE
POINTS BETWEEN 0 AND B.
-88-
Red Bird ~uadrangle. (Latitude 38-3815';longitude 9130'-9145')
(St.James Township)
Line from Knobview Depot,east 0.5 miles via Frisco Railro8~d to Phelps-Crawford County Line, thence north via Pheps-Crawford County Line road,3.5 miles to T road west, to Jakes Prairie road, thence west via T roe.d,1.5 miles to Jakes Prairie road at a point 0.5 miles south of Hinkle Po st Office.
E. Knobvie\l!}' Depot, 90 feet e8,st of, 45 feet south of center of track, 12 feet east of R.R. crossing, 2 feet north of R.R.Property line, iron po st st81n.ped "108411. ------ -- 1074.005 0.000 1074.005
58 Knobview Depot,O.5 miles east of, via R.R.,concrete serr~phore base, southwest corner, top of;signal 949 on Phelps-Crawford County lineipainted square. ------------ 1081.484 0.018 1081.502
D. Friendship School, 2.0 miles east of, on Ozark Highway,at cross roads on Phe1p s-Crawford County Line, SOt:.. thwest corner of road s, 14" hickory tree, in base ofjcopper nail and washer.1098.849 0.036 1098.885
59 Knobview Depot,1.5 miles north of, 011 Phelps-Crawford County line r02,d, Ozark Highway,0.5 miles north of, Cucaro10's house,75 feet northeast of,on west side of road,in fence 1ine,12" locust tree,in base of; copper nail and washer. --------- 1071.411 0.040 1071.451
(Dawson Township)
60 Knobview Depot,2.25 miles north of, on Phe1ps- Cr.awford County line road, Ozark Highway,1.25 miles north of, drain to northwest,lOO feet south of, on west side of road,20 n white oak tree,in base of; copper nail and washer. ------------------------- 1064.348 0.048 1064.396
-89-
61 ,Knobview Depot,2.5 miles north 'of, on Phelps-Crawford County Line Road,O::;ark Highway,1.5 miles north of,on top of hill, opposite nail box on east side of road, center of road;road elevation. ------------------- 1087.589
62 Knobview Depot,2.75 miles north of, on Phelps-Cravlford County Line Road,Ozark Highway,1.75 miles north of, fe.rm house, 350 feet northwest of,on rocl< projecting out of ground in center of road,15 fE.:et northwest of fence corner,top -of rock; painted square. --- ... ---------- 1012.794
63 Ozark Highway, 2. 25 miles north of, on Phelps-Crawford County Line Road,at dim cross roads, on top of hi1l,15 feet southwest of roa.ds,on stump; painted square. -------------- 1071.623
64 Ozark Higbway,2.65 miles north of,on Phelps-Crawford County Line Road,Hinkle P.O.,225 miles southeast of,on top of 11ill,in old abandoned field, west side of road,conglomerate boulder,top of;painted square. ----------- 1040.245
65 Ozark Highway,3.00 miles north of,on Phelps-Crawford County Line road,right aneled turn in road east,75 feet south of,T road west to Hinkle P.O.,just opposite,on east side of road, sandstone boulder,top of; painted square. -------------- 994.501
66 Phelps-Crawford County Line, 0.5 miles west of,on Tribby road,Hinkle P.O.,1.5 miles south east of,30 feet east of drain, on south sj.de of road, sandstone
LboUlder,top of;painted square. 994.905
-90-
0.050 1087.639
0.053 1012.847
0.057 1071.680
0.060 1040.305
0.065 994.566
67 Phelps-Crawford County Line, 1.0 mile west of,on Tribby road, Hinkle P.O.,l.O mile southeast of, at dim cro ss roads,lOfeet south of center of roads, send-stone boulder projecting out of ground,top of;painted square.-- 950.010
68 Phelps-Crawford County Line, 1.25 miles west of.on Tribby road , southeast corner, S. W. Q,tr. of·-· the S.E. Q,tr. Sec. 25, T. 39N., R.6 W.,at private road north, mail box on north side of road, center of road just opposite; road elevation. --------------- 949.773
G. Hinkle Post Office,Q.5 miles south of~ intersection Jal{es Prairie and Tribby roads,(Troad east)(sign board 10 miles to Cuba), east side of Jakes Prairie road,30 feet no~th of road intersection,8 f1 walnut tree, in base of;copper nail and washer. ----------------------- 921.202
-91-
0.074 950.084
0.076 949.849
0.079 921.281
~djustment of intermediate temporary bench marks between point A- at St ~James and point Bat the intersection of Red Bird Road and Ozark Highway.
OERIYATtON OF ADJUSTUENT FORMULA.
EI.y. A - 1081.800 Cor.- 0.000
i c
I
I .. () ., o <i
E
E' ••• 8 ,. 1074.66'0 o~ •• rY. . • B. l074.63~2 Adj._t.
0·. Q::5 0 8 Cor.
Correction c. for any intermediate point at distance d. from A and between points A and B.
c = .0308
"" = v
C =
a 1.0
.0308 d
1.0
0.0308 d.
'"" ./ ... ell 1"4 CIt
• CIt
A 0 J U -s ·T ' E ' 0 . .
9 ... •
CIt
:;"92-
/
/"
' L t. ' N E •
OCI CIt
K'I CIt . CIt
CIt CIt
• ...
Adjnstment of intermediate bench marks between point B at the intersection of Red Bird Road and the Ozark Hil!hway,and point C on the Ozark Highway at Friendship School.
The adjustment of these intermediate bench marks can be effected by scaling the adjustment corrections from the large diagram below and combining each correotion algebraioally w1th the observed elevation as 1t appears in the level field notes; or t ~t desired, the adjustment corrections for the intermediate p01nts can be computed by the following formula for this section of the level net,B to c.
DERIVATION OF ADJUSTMENT FORMULA. E'.y.S • 1074.6660 Observe Eley.e • 1083.4« 0 0 Ob •• rv.
• S· 1074.6352 Adjust. • e. 1083.4299 Adjust. -.-:_-----0.0308 Cor .. .. 0.0299 Cot ..
§--~--- ~t' -B gL S ---- - ",C I - ~. I --d---~::: '-'--- ~ ~--- - ---- --- -- ~ ; ~ 8.0----- ~
Correction c.for any intermediate point at distance d.from C and between the points Band C.
and O.Q299
c = -(Or020233 d - 0.0299) then C = (+0.0299 - 0.020233 d).
C> C>
C> <:> <> <:>
R B I - j
IA 0 .. U S T E t • I I 1 :1 ~ LINE. I l I +
c C\I A 0-
I 1 .1
,I I I
I i ' I I I ,
L __ _ _ , _ _ L _ _ _ __ _ _ , _ _ ..J. _ - - - -
13 69 '70 '71 • eo <> In 0
- 0 '" COol 0 ... • • • • ::IIIt(\ N c-I c-I
'72 '73
IC\
• pol
N
• pol
C> 0-
+ t. I "'e- ~ N <:>
I • I .. _ J ___ _ '--
'74 G C1
o c IC\
• o
o . C!>
GRAPH FOR THE ADJUSTMENT OF INTERME~ . ATE
PO.NTS BET.EEN B AHD C.
-93-
A.
22
B.
69
70
'71
M~RA.EC SPRINGS QUADRANGLE.
(LatItude ~7·45~ -~8·; lo.sftude 91·30~ -91-'5')
, .'
LINE FROM ST. JAMES EAST VlA OZARK KrGHIAY TO PHELPSCRA~FORO COUNTY LIHE# SAI8 POtNT ONE MILE NORTHEAST OF KNOB-vIew RAILROAD DEPOT.
Oe.crlptlon.
(See description on pag~ 64: )
St. James 0.4 miles east of, on St.James-Ouba road(Ozark Highwayl, Frisco R.R •. ,50ft. north of, turn in road to north, 500 ft. west of, concrete culvert, acrossroad,north end wall, top of;chiseled square. ----------
Observed ElevatIon
1e8'7.8oo
1065.363 RED BIRD QUADRANGLE.
COR.
0.000
-0.0124
(Latitude 38·-38°15~; longltude 91°30' -91 0 45')
St.James,1.0 miles N •. E. of, Ozark Highway, T road east (intersection of Ozark Highway and Red Bird road},400 ft. S.of, Dr.E.~.Scott's house, 3G8 ft. w. of, on W. side of road, 10 ft elm tree, in root of;copper nail and washer. -------------------- +074.666 St.James,1.5 miles ~.E. of, on Ozark Bighway,O.5 miles east of Red Bird road intersection, on ~ N.side ofroad,~2" post oak tree, in base of; copper nail and
-0.031
washer. ------------------------... - 1062.669 -0.C21 St. James,1.75 miles northeast of, on Ozark Hi~bway, concrete culvert aeross road, south end wall, top of; chiseled triangle. , St. -i •• es~, .9.2 miles N.E. of, on Ozark Higbway, on top of hill~ center of road; road elevation.
1034.437
1954.446
-94-
-0.015
-0. Ole
AdJu.ted Elevation
1087.800
1065.351
1074.635
1062.648
1034.422
1054.436
Adjustment af intermediate bench marks between point C on the Ozark Highway at Friendship School, and paint D at the intersection of the Ozark Hi~hway and the Phelps-Crawford County Line.
The adjustment of these intermediate temporary bench marks can be effected byscalind the adjustment correotions from the lar~e diagram below ana combinind each correction accordind to - its s1gn with the observed elevation as it appears in the level f1eld notes; or~if desired, the adjustment correotions oan be Gomputed by the following formula for this section of the lavel net,C to D.
E tev.C • C
DER.fAT A D JUS T • E T
o N 0 F
FOR M U LA •
• 1~83.4000 O~.erY.
· ;~!;~;;~t(:~'~~~ _ u~tJID ~- -- 2.2 S - .?
Eley.D • 10~i.9110 Ob.erv. • o. 1098.8847 Adjust.
------~--0.0263 Cor.
Correction c. for any intermediate point at distance d. from C and between the points C and D.
and
c = -(O.~24978 d - O~0299) then C = (+0.0299 - 0.024978 d).
I"o o o
L 1<"1
<:> ,\'" 0, \C \. \ CII
E <:> R •
r---T---~~~----~--------~~~~-I I R ADJUSTED I LINE. A I
0.. I 0.. l 0 I ~ S ~~~ I o E s~ I • /' ()~ I I • :.... -1 _______ .L _____ 1-,- __ _
Sl 75 76 77 D
• • 0 1£\ tc\ CC\ - co ON ... I"-... . . . • :::E 0 0 rot ...
9RAPH FOR THE ADJ~STWENT OF tN.ERMEDtATE
POINTS BETWEEN C ANa D.
-95-
73
74
st. James, 2.5 miles northeast of, on Ozark Righway, Friendship School, 1.5 miles west of,concrete culvert(errected 8-13-20 by T. RaIl), north end waIl, top of; chiseled square.--------------1010.05?
Friendship School, 1.25 miles west of, on OzarkBi~hway, at T. road N. to Binkle P.O., Corrugated iron culvert, rock on east end of culvert, top of; chiseled trian~le. ---------------------- 1018.926
FrieDdship School, 0.5 miles W. of, on Ozark BiQhway, Gerber's Store, 600 feet W. of, concrete culvert, north end wall, top of; chiseled trian~le. -------------- 1062.045
C. Friendship Bchool, on Ozark iji~hway, southwest cornerof concrete porch of school house t top of; chiseled trian~le, ~aiDtea "1083.3". -------------- 1083.400
G1- :Frienli "Semoolf "on .021ark Highway, !jOfeet S.W. of, in School Yard, 10 ft. from fence corner,30 ft. N. of center of Ozark Hi~hway, 30 ft. E. of center of T. road
75
76
north; iron post stamped "1E)83". l083.458
Friendship8chool, 0.25 miles E. of! private T. road N. to Louis Ball's residence, ~5 ft. W. of, concrete culvert, S. end wall top of; chiseledsguare.---------:J:037.041
Friendship Sebool, (!,.9miles E. of, on ozark Bi~hway, PhelpsCrawford County 'Line, 1.1 miles west of, concrete culvert, south end end.wall, top of; ehjseled S~u8re .. - .... ---.-~------... ------------1042.29'7
-96-
0.000 1010.057
+0.005 1018.931
+0.020 1062.065
+0.030 1083.430
+0.030 1083.488
+0.024 1037.065
+0.001 1042.298
77 Friendship School, 1.5 mile~ east of, on Ozark Highway, Phelps Crawford County Line, 6~~ milee ~ west of, Knowview, O.~ miles north of across field, concrete culvert under Ozark Highway, north end wall, top of; chiseled square.-----1028.412 -0.,014
D. Friendship School, 2.0 miles east of, aD Ozark Highway, at cross roads on Phelps - ~rawford county Line, southwest corner of roads, 14" bickory tree, in base of; copper .nail ana washer.--------------------------- 1098.911 -0.026
1028.398
1098.885
Th:,s cor.pletes the descr.?,ptHJ1I. and adJustment of the b~"~ oj levels from St. )1l1r.es east v?,a OzarkS"gb:ay to the Pile ~;:s.=C,.a:IR:j ord Count J' L; r.le~ 'sa:.d ,/>oHIi one lIn l e nort hefZ'st of K •• tv'/,ewR~llroad Depot.
-9~-
Adjnstmellt of intermediate beneh marks between paint e an the Ozark Hi2bway at Friendship School,and poist F on the Jakes Prairie Road 1.0 mile s cutii of Hinkle Post Offiee •
.3.0,-- .. · _._._- ' -' , - > F __ "~~ ' ____ f _____ I
c
~ER'VATION OF AOJUaTMENT FORMULA.
Et.v.e • 1083.4aoo Ob •• rv. • e. 1083.4299 Adjust. - ........ _-- ... -
+ Ih02'99 Cor.
Correction e. for any intermediate and between the points C and ,.
- .
Etev.F • loti.1770 eb~.rY. • F. 1001.3429 AdJuet.
---------+ 0.1659 Cor. point at distance d. from C
8RAPH FOR THe ADJUSTMENT ~F fNTERMeolATE POINTS BETIEEN C AND F •
• • 0 0 0 0 0 -0 IC'I IC'I C) 0 .- . • • • • 20 , 0 ... C'I on ~ ' . 78 79 89 F - , . ;.;. ~'
: ADdUSTED LI ME. • 0 001 IC'I R
" 0 0 A co 0 I 0\ 0
°Oe 0 CII s Algebraic eR" r-4 E 0\
eo IC'I
Solution. 100 ... C ,.. ~LQ299 = d .. -0: 360-- -3:-0- , I 0.1360 a c = + 0.0299 IN --
---~:~- - r: •
-.
(+0.04533 0 .. (299). " C = d' +
-98-
Red Bird ~uadrangle. (Latitude 38-38 l5';10r.gitude 91 30'-91 45')
( St. J <?me s Town ship)
Line fr.om. Friendship School on OZCI.rk Highway, in center of section 11,T. 38 N., R.6 W., north via road 3.0 miles to paint on St. James-Jakes Prairie road,l.O mile south of Hinkle Post Office,at a T road south east.
c. Friendship School, onOzark highway,in center of sec.ll,T.38 l'r., R.6 W.,southwest corner of concrete porch of School House,top of;chiseled triangle .. ---------- 1083.400 0.030 1083.430
C'.Friendship Schoo1,on Ozark Highway,in center of sec.ll,T.38 N., R.6 W. J 50 feet southwest of, in school yard,lO feet from fence corner,30 feet north of center of Ozark Highway,30 feet east of center of T road north;iron post P. B. M. stamped.--------------~-------- 1083.458 0.030 1083.488 "1083". .
78 Friendship Schoo1,O.5 miles north of,Hinkle P.O.,3,5 mi~~s south of, T road west,east side of road,8" black oak tree,in root of;copper nail arid Washer. -------------- 1018.238 0.052 1018.290
(Dawson Townshi,)
79 Friendship Schoo1,1.5 miles north of, Hinkle P. 0.,2.5 miles south of, on township line between St. James and Dawson Towr.ships,bridge on township line road over south fork of Clear Creek,northwest corner of bridge; co~per nail and w~asher. ------- 951.418
80 Friendship School,2 .. 0mi1es north of,Hinkle P.O.,2.0 miles south of, east side of road,inrock ledge; chiseled triangle.-------------- 945.415
-99-
0.098 951.516
0.120 945.535
F. Friendsh ip School, 3.0 miles north of,Hink1e P.O.,l.O mile southwest of,on .Jakes Prairie road,at T road southeast(road to Friendship Sch.) southeast corner of roada,15" white oak tree, in base of; copper nail ~~d washer.------ 1001.177
· -100-
0.166 1001.343
Adjustment of intermediate bench marks between point B at the iD.ersection of Red Bird a,nd .Jakes Prairie roads,and point F on Jakes Prairie Road 1.0 miles south ' ~ of Hinkle P.O. and 3.0 miles north of Friendship School.
DERIVATION OF ADJUSTMENT FORMULA.
E'ey.H • ID42.0580 ~b.erY.d. • H· 1041.8535 Adju.t.d.
E 'ey.F- 1801.6110 Oll •• r".d. • Fa 1001.3429 Adju.ted. -----_.-- ---------
0 .. 2045 Cor. 0.2681 Cor.
E:~~= - ~:=--~-=:f n ~ ~ ~J~ H ~ - I. 7.s----~ F
Correction c.for any intermediate point at distance d.from H and between the points Hand F.
a .. o
. C\I ' ..
H • HI .0 -0 ... . .0
c - O~2045 = d --5:0636-- -I:?5
C = -(0.036343 d + 0.2045)
C = (-0.036343 d
A 0 J U S T 81
IC'I
a
E 0
0.2045).
L o • E R
L I
L I
N E •
o o IC\ C\I
N E 82
IC\ eN ,. ...
.
GRAPH FOR tHe AD.I'IIS,"eWT OF 1I1TERME91A~E
P&UUS . BETWEEN 8 A Itt F.
-101-
• IC\ ... . 0'4
Red Bird ~uadrangle. (Latitude 38 -38 15 t ;longitude 91 30'-91 45 1 )
(Da,yson Tovmship)
Line fro~ intersection of Red Bird and Jakes Prairie roa.ds, thence 110rthet'.st 2~ 75 mile s via Jake s Prairie road to Collins School and Hin.1<:le P.O;, thence north 4.5 miles via JeJce s Prairie road to Phelps-Gasconade County Line, 2.0 miles west of Jakes Prairie Post Office.
H Robinsons Bral1ch,0.9 miles north of, on Red Bird Road,at intersection of Red Bird and Jakes Prairie roads, (Y road }Tortl:east), F .Berrie shouse, 230 feet south of,in crotch in Y roads,o~posite east end of wagon gate,5 feet south of fence;iron P.B.M. po st stamped "1042 fl. ----- •• -------- 1042.058 -0.205 1041.853
81 Intersection Red Bird and Jakes Prairie roads, 0.75 miles northeast of, on Jakes Prairie road,me.in fork of Clear Creek, O. 5 miles southwest of, via way of road, H.Eikr:laJl 1 e house,lOO feet east ofzon eBst side of' road, 14 It po st OC'l.Jc tree, in base of; copper nail and washer. ------;.. 964.540 -0.231
82 Intersection of Red Bird and Jakes Prairie road.s, 1.25 Llil es northea.st of,on Jakes Prairie road,Hinlc1e P.O., 1.5 miles southwest of,Clec.r Creck Ford, (Main fork), 50 feet east of, at angl e in road to north, southeast corner of road,in fence li#e.20"hickory treo, in base of; cOJ?:per nail and washer. ---------- 91G.09l -0.250
F Friendship School, 3.0 miles north of, Hinl<:leP.O.,1.0 mile southwest of on Jakes Prairie road,at T road southeast(road to Friendship Sch.), southeast corner of ' roads,15 "whi te oak tree,in base of;copper nail
964.309
915.841
and washer. ------------- ... -------- 1001.611 -0.268 1001.343
· .... 102-
Adjustment of intermediate temporary bench marks between point F on Jakes Prairie Road 1.0 Mile south of Hinkle P.O. and 3.0 miles north of Friendship School, and point G 0.5 miles south of Hinkle P. O. at the intersection of .Jakes Prairie and Tribby roads.
Deriva~ion of Formula for ~he Adjustmen~ of Intermediate Points.
Elev. F • 1001.6110 Ob.erv.d. Elev. G • 921.5450 Ob.erv.d. • F. 1001.3429 Adjus·ted. • Q. 921.2814 AdJustsd.
--.---~-~ 0.2681 Cor. - 0 .. 2636 Cor.
t I\J
~ ())
t .. -- ~ - 4' -::.-.::::,.:~--~ -I\) . I Il) .
~ C ---- -d ..
F ~ 1 . ~~---.-- ---~-o~.s--... -.. ---~~ G
o 0045 d C = -~O:5--- + 0.2636
C = -(0.009 d + 0.2636)
C = (-0.009 d - 0.2636)
F
o 8 S E R V EO . _.L .1..1'1 .. <:_ •
. ~ J=~!:e:~ ~. ;nL~::, ~~:t- ~ ': ~ level field notes show .w that no intermediate o~ points were establisbed ~
• in ~his sec~ion of the ne~.
A 0 JUS TEO liN E • G
0.00 MI. 0.50 III.
GRAPH SHOWING AD~USTIIENT OF PotNTS
F AND Q.
-103-
Adjustment of intermediate bencb marks between point G, 0.5 miles south of 'Hinkle P.O. at the intersection of Jakes Prairie and Tribby roads, and point () on Jakes Prairie Road 0.25 .iles south of the Phelps-Gasconade CountyLine.
The adjustment of these intermediate temporary benoh marks oan be effected by scaling the adjustment corrections from the large diagram below ana subtraeting each oorreotion from the observed elevation as it appears in the field level notes;or,if desiredtthe adjustment corrections can be 00.puted by the follow1ng formula for this section of the leyel net,G to O.
~f-' ~ - 1t~ __ ----j-~ Co c ---d.--- I\l t I ~
. ~ ' ~ . ~
G ~~.7S ) 0 DERIVATION OF ADJUSTMENT FORMULA.
Elev.G . 921.!H50 Ob •• ryed. E 'e.,. 0 • 969.1580 Ob •• rved. • G = 921.2814 Adjusted. • 0 . 969.5351 Adj.sted.
-------- --------- 0.2636 Cor. - 0.2229 Cor.
Correction c. for any intermediate point at distance d. froll and between G and o.
and
c = -(0.008568 d + 0.2229) or C = (-0.008568 d - 0.2229).
{J R ~ if. if V Ii' fl r--- -.-- _ . ~ . - L f ~. ~ .- - -- ._-. - ---.
I ' . .
0 C> 0 0 '" 0 C> '" «I • 01 • r- :;: '0 N
"" IC1 • • IC"\ L N Col N N N N N 0 CN N '" . . • . . w ~
. . co 0 0 co 0 E
0 0 o
R
A D J U S T E D l f N E . G 83 85 86 87 88 89 90 e • 8J: .'" '" '" co 0 0 co co
-r- C<I \0 0 K'I 0 0 Ie'I C> - . • • • . . :.. • IC"\ IC"\ N C<I .... 0 0
GRAPH FOR THE ADJgSTMENT Of INTERMEDIATE
POINTS BETWEEN G AIm O.
-104-
0
• G ~inkle P.0.,O.5 miles south of,
i nter se et ion Jalce s Prairie and T:~i b1Jy ro ad s, (T rop.d e a st) ( Si:':Sll board 10 rlli1es to C1..10 a), east sid.e Jake s Prail'ie 1'oad ,30 foet :,lOr Jcl1
of 1~ 03Jl in terscctiol1,8 ''i''lalnut t:re e Ii;.;. 'base; C01)lJo1' nail and y,~:;.sher.-------------------- ·. · .... -- 921.545 -0.264 921.281
83 ColI ins SellOoI, 800 f eot ~1ort:l of, Hink1 e store 8.11(.: P.O., 40 f eet southeast of,40 feet west of cen t e r of ~oad ,in lin e with t angent of road to northeast; iron post P. B. U. sta.:-::ped fl1039!' --------- ~ ,~-- - ... - - 1038.938 -0.258 1038.680
84 Collins Sch oo1,800 feet n orth of, Hinl:1~ Sto l'e a.l1d p. O. , southeast corner of porch, top of floor; copJ?er nail Gnc. washer. ----- .. -- 1040.617 -0.258 1040.359
85 Hinkle PO.,O.S miles north of, 0~1 Jak es Prairie ' road, sou th,!;7e st corner sec.13,R.6W;T.39 N.,350 f eet north of,on w~st side of road, opposite private T road east to vrhi te house ,16" black jack tree, in base of;copper nail and vlasher. ----- .. --------- 984.9 64 -0.254 984.710
86 Hinkle P.0.,1.2 miles north of, on Jakes Prairie r oad, a t pr i vate T road east,southeast corner of road,lOIl post oa.~ tree,in iJase of; copper nail e,.nd Ylasher. -------- 1057.863 -0.249 1057.614
87 Hinkle P.O.,1.75 miles north of, Oak Grove School,2.0 miles south of J old de serted farm houseol1 vre st side of road,125 feet southeast of, 011 top of hill, e a st side o'f road, 10" black jack tree,in root of; copper nail and vras her.--------- 1086.932 -0.244 1086.688
. Adjostment of Permanent Bench Mark S on Jakes Prairie Road at the Phelps-Gasconade Connty Line. The adjustment is to be made between the points 0 and S with the line tied back at O.
This section,O to s, is a spur line to the level net; it was run as a duplicate li~e of levels, that is:the line was run over the same rout~ twice~ but in opposite directions:i.e. o to S and from S back to o. ~ee page of this thesis for a discussion on duplioate lines of levels.
A N A L Y TIC A 0 JUS T N E N T o F POI N T S.
EleY.O - 969.7580 ,baerv.'n olrolilt a vIa 0 and N to M.
• o. 969.5;~1 Adjusted. _._------ 0.2229 Cor.
EleY.O- 969.74;0 Ob.erv. on .,ur return.
• 0- 969.53;1 AdJuet.
jF~:F~-1i, v s 0
- 0.2079 Cor.
<- - -O.S --> NoW for the adjustment of S.
Let X • the correotlon de.lfed.
The n:
and X & 0.0150 + 0.2079 ---2--
and:
x • -(0.0075 + 0.2079)
and fInally: X. - 0.2154 ft.the correctfon to be applIed to the obseryed eleYatlon of s.
Then: Elev l S = 942.5160 Observed. n _ O~ 2154 Correction. If s -= 942:3006 Ad j usted.
0\ ('II
IN ('\I
• C>
A {}
• • 0 -0 ... . ::110
o 8 S-ERV-EO-
0 oJ U S T
f t 0 w E R
E
... fC'I .... '" C>
D
S
fC'I
'" . o
L I NE.
l I M E
0\ .... C>
'" . C>
. 01
o fC'I
• CI>
DtAGRAM FeR THE GRAPHtC ADJUSTleN' OF POtHT S.
-106-
. 88
89
90
O.
s.
o.
(lak Grove 8cll0 01,1.5 mil e s south of ,on Jakes Prairie road,dim T road west, 200 feet north of,farm house on east side of road,200 ft west of,i~ fence corner, east side of road,14" red oak tree,in base of; copper nail and wash.er. ------r-Oak Grove S'cho 01,0.5 mil e s south of,on 3akes Prairie road,W. E. Wright's house,200 feet north of, at T road east and private road west thru gate, top of hi1l,center of roads; road elevation. ------ ..
Oak Grove School,on Jakes Prairie road,Dawson Township,Phelps-Gasconade· County Line,O.75 miles S. of,in front of School Rouse,125 feet west of, east side of road, 14" scaley bark hickory tree,in root of;copper nail and washer.--
Oak Grove School, in Dawson Twp., 0.5 miles north of,T road east, 50 feet south of,T road west to Red Bird road,20 feet north of, northvY'est corner of roads,lO" post oak tree,in base of;copper nail and washer. ----------------
Oak Grove School t O.75 miles north of,T road east on County Line, Jakes Prairie P.O.,2.0 miles west of,at private T road west on County Line, . sewell t shouse, 60 ft southwest of, Phe:bp·s-Gascol1.ade County Line,lO feet north of,north
99~. 372 -0.238
967.072 -0.230
900 .. 143 -0.226
969.758 -0.223
east corner of roads;iron post P. B. M. stamped "942 ft • ------------------- 942.516 -0.215
Oak Grove School,0.5 miles north of,T road east,50 feet south of, T.road west to Red Bird road,20 ft north of,northwest corner ·of roads, 10" post oak tree,Ul base of; copper nail and washer. ---------
-10'-
969.743 -0.208
999.134
966 .. 842
969.535
9420301
969.535
Adjustment of intermediate temporary bench marks between point N on the Red Bird Road 1.9 miles north of Royal P.O.,and point o on the Jakes Prairie Road 0.25 miles soutb of tbe PbelpsGasconade County Line.
The adjustment of thes~ intermediate temporary bench can be effected by scaling the adjustment corrections from the large diagram below and subtracting each correction from the observed elevation as it appears in the level field note book;or,if desired, the adjustment corrections can be oomputed by the following formula for this section,N to O.
o b 8 • .. '=-'=-=~.:::.=- jOb 8 ~ E I • v. N- 8 7' • 09:5 0 ~J tv E t • " • 0.969. 7 ~ 8 0
~ N Adj. 1\.l d ~ Adj.
E I. v. N· ~:~:~::: JL:-------::;-,--:;<f------:-lb E Ie v. o. :~::::~~ Cor. "'- 0.202:5 +-----2. -----~ Cor. __ 0.2229
OERIYATlON OF ADJUSTMENT FORMULA.
Correction c. for any intermediate point at distance d. from N and between the points Nand O.
N • .0 -0 -. 20
C -0.2023 d --0:0206-- = -2:4-
C = QLQgQ~_~ + 0.2023 2.4
C = -(0.008583 d + 0.2023)
C = (-0.008583 d 0.2023).
A D " U S T E 0 L I N E . 91 92 93
<> 0 0 r- t(\ 0-
• • '. 0 .-I .-I
GRAPH FOR TEE ADJUSTMENT OF INTER-MEDIATE POINTS BETWEEN N ANB o.
-ltJS-
(}
<:>
• • '"
Red Bird ~uadrangle. (Latitude 38 -38 l5';10ngitude 91 30 1 -9145 1 )
(Dawson Township)
Line from point on Jakes Prairie Road 0.25 miles south of County Line,thence west two-one half miles to point on Red Bird Road 0.5 miles south of Phelps-Gasconade County Line and 1.25 miles north of Royal P.O.
o Oak Grove School in Dawson Twp, 0.5 miles north of,T road east, 50 feet south of,T road west to Red Bird road,20 feet north of, northwest corner of roads, 10 " post oak tree,in base of;copper nail and washer.--------------- 969.758 -0.223
93 Oak Grove Church, BOO feet north 'east of,J.B.Cope1and's house,
BOO feet west of,in bottom of branch to north, north sidS of road,24" sycamDre tree,in root of; copper nail and washer. ---- 834.lB2 -0.219
92 Oak Grove Church,0.6 miles west of,T road south on line betv~en secs.2 and 3 produced north, 30 feet west, concrete cUlvert across road,north end wall, top of; paL~ted square.---------------- 895.903 -0.214
91 Oak Grove Church,l.2 miles west of, Red Bird Road,0.55 miles east of, Spurgeon's house,300 feet south-west of,on north side of road,twin trunk walnut tree, in root of; copper nail and washer. ------- 863.954 -0.209
N - Royal P. 0.,1.25 miles north of, on Red Bird Road,Oak Grove Church,
-1.85 miles west of,Phelps-Gasconade County Line,0.5 miles south of,Red Bird Ford across Bourbeuse River,llOO feet south of,at T road east to Jakes Prairie road,50 feet south of T road,on east side of road,14" red oak tree,
969.535
833.963
895.689
863.745
in base of;copper nail and washer. 875.093 -0.202 874.891
-109-
CON C L U S ION •
In conclusion, a brief outline will be ~iven coverin~ the main classes of problems which have been considered and the methods of solution and their appropriate applications to the problem at nand. Direct Observations of a Single Quantity.
AdJustment. Take the mean or the weighted mean of the observed quantities. Indirect Observations.
AdJustment. Write the observation equations and from them the normal equations; the solution of the latter ~ives the unknown Quantities themselves or the corrections to their assumed approximate values. The number of the observations equations will be the same as that of the observations; the
, number of the normal equations will equal that of the unknown quantities, which must always be less than that of the observations. Conditioned Observations.
id,ustment. Write the condition , equations in their ~eneral form and then in their simple form involving the corrections. From them form the normal equations, the same in nu~ber as the conditions. The solution of the normal equatiGDs gives a set of factors, called correlates, one for aaeh ciondition equation, from which the desired corrections to the observed quantities are determined.
In summarizing 'what has been stated above,there are in ~eneral two methods of finding the most probable values of the unknown Quantities in cases involvin~ conditioned quantities.
In the first method the cond~t~onal equations are avoided ( or illlminated ) by impressing their significance on the observation equat~ons,which reduces the problem to the cases as considered in Parts (II) and (III).
In the second method the observat~on equations are illiminated by impressing their significance on the conditional equations,when the solution !!lay be effected by the method of eorrelatives as considered in Part (IV).
The first method is the most direct in elementary problems, say with not ,more than three residual llnknowns,hut the second method by usimg conditional equations instead of observation equations and . completing the solution by the method of correlatives in connection with the results obtained from the solution of .the normal equations, greatly reduces the work of computation in the case of complicated problems with four or more unknown quantities to be solved. L The best methods of solv~ngDormal eguations,as I hav1
found them, are those given in Part (IV) pages 55, 56, and 57. -110-
APPLICATIONS OF THE METHOD OF LEAST SQUARES TO THE ADJUSTMENT OF LEVEL CIRCUITS.
ADJUSTMENT O~ ELEVATIONS IN DAWSON
AND ST. JAMES TOWNSHIPS,
PHELPS COUNTY,
MI$SOURI.
-B I'B L lOG RAP H Y,.
-11-1-
BIB L lOG RAP H Y •
References on the Method of Least Squa~es. Manual of Spherical and Pract..icalAstronoay.,Chauvenet.
.J.F. Lippincott Co., Philadelphia, 1885. Approx.imate Deterllination of Pronatlle Error" Appendix NO.13,Report
for 1890,U. S.CoastandGeodet,ic Sorvey. Theory of Errors and Method of Least Squares, Johnson •
• John \tiley & Sons, New York, 1893. Practical Astronomy as Applied to Geodesy and Navigation,Doolittle,
,John Wiley I'r Sons, New York., 1893. Precise Surveying and Geodosy,MerriJlan.John Wiley & SOIlS,
New York, 1899. Adjustllent of OtJservatioDs,Wrigbt anti Hayford. D. Van Nost.rand & Co,
New York, 1904.- 1907. Text Fook 011 Geodesy aouLeast Squares, Crandall.
John W.iley & Sons., New York, 1907. flandtJ ucb der VerllessuhgsktWde, I. Metzler, Stuttgart, 1910.11y Jordan. Ausgleichuugsreclmung. By Belliert. Tullner.,Leipzig ,1907.
Method 01' Least Squares, Merrillall, Wiley, New York, 1913. Method ot Least Squares, Comstock. Ginn, Boston, 1895. Metb.<Ml of Least Squares, Fartlett,Foston, 191~.
Iftu).ory of Errors and Least Squares,Weld. Macllillan,New York,191fi. Theory of Errors and Method of Least Squares, Johnson.Wiley 1892. Combinations of Observations, Srunt, Callbrid~e Uuiversity Press,1917.
Af ,licatioll of Least Squares to the Adjust.ent of Triangulation, Special Pub lication No. 28, tl. S.C.& G.Survey,1911).
Practical Least Squares, O.M. Lelar14l, McGraw-Hill Sook Co., New York, 1921.
GeOdetic Surveying and the Adjustment of Observations (Method of Least Squares)" E. L.Ingram, McGraw-Rill Book Co." New York, 1911.
Adjustment of Trianguiation,E.M.Douglas,Scbool of Mines Quarterly .. No.4 Vol.14 Colullbia Col1ege,New York.
Topographic Instructions of the U. S.Geological Survey., Wa.shington, 1918 Topographic, Trigonolletr ic., and . Geogetic Surveying,lUlson.
John ltiley & Sons, New York, 191'2.
Of the above treatme ou the sub jec·t of Least Sq~ares, the . last five mentioned were consulted as well · as the book furt.her up iI. · the list ·. by Welti. Tbebookb,y :Leland .is reCO.lll.llelltied as an exceptionally good book on tile subje.ct..
-112-
APPLIOATIONS OF THE METHOD OF LEAST SQUARES TO THE !f1.JUSTMENT OF LEVEL CIRCUITS.
ADJUSTMENT OF ELEVATIONS IN DAWSON AND ST. JAMES' TOWNSHIPS,
PHELPS COUNTY, MrSSOmH.
I N D EX. Abridg~d M~thod A for Solutton of Normal Equations • • • • • 55
" "Solution A with Nottls on the Process ••••••• 55 Abridg~d Method B for Solution of Normal Equations.. • • • 57
" "Solution B with Notes on the Proc~ss ••• • • • • 57 Acknowledg~ment.s • • • • • • • • • .• • • • • • • • • • • •• x
- -Adjustment of Int~rmediat~ Junction Points • • • • ••••• 18
" " " pointaL and K • • • • • • • • • • • • • • • • 18 19
.79,20 .~. ." " " " " " " " "
- -" Point P • • • • • • • • • • • • • • • • • • • " points N " Point Q
" " R
" " S
and ° • • • • • • • • • • • • • • •
• • •• .. . . • • • • • • • • • • •
• • • • .. . • • • • • • • • • • • • 21 • • • • • • • • • •
• • • • • • • • • •
22 23
Adjustment of Spur Lines PQ, NR, and asa • • • • • • • • • • 21 Adjustment of Interm~diateBench Marks • • • • • • • • • .2~,63
" " " Benchmarks Between A and I ••••••••• 26,53 -
" " " " " " I and P ••••••••• 29,56
" " " " " If
" " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " It " " " " " " " " " " " " " " " " " " " " " " " " tt " " " " " " " " " .- " " " "
" " " " " " "
A and D -
I and J .B and H
-Hand J -J and N
M and N Nand R -P and M E and D
-D and G C and F Hand F
-F and G -
• ••••• 34,92,93,95 · . . . . . . . . . 71 • •••••• • • • • 73 • • • • • • • • • • 76 • . • . . • · · • · 71 • • • .. . . • • •
• • •
· ••••• • 80
• • • • • •• 82
· • • • • • • 84 • ~ • • • • • 87 · . . .. • • • • • • 88
• • • • • • • • • • 98 • • • • • ••• .... 101 • ••••••••• 103
G and 0 ••• • • • • • • .104 .106 .1:08
o and S • • • • • • . .. . 'N and 0 ,. .. . . • • .. . ..
-113-
I N D E X A
Adjustment by the Method of Correlates • • • • • • • • •• 58 Adjusted Elevations of Junction POints(Summary) •••• 47,54 Application of the TheorY(M~thod of Correlates). • • • •• 60
B Back Solution(r Solution) of Normal Equations16,47,52,53,56,57 B~nch Marks(See Adjustment of Benchmarks •••••••• 63,25 , ,
Base Lines for L~vel Nels ••••••••••• 1,6 ,38,49,62 &:sat Methods for Solving Normal Equations •••••••• 110
C
Calculation and Tabulation of Adjusted Junction POints15,47,54 Carefullness of Obs~rvation. • • • • • • • • • • • • • • • ix Check Terms • • • • • • • • • • • • • • • • •.• • • • • • 57 'Check on Residual Values • • • • • • • . . • • • • • • • 15,17 Circumstances Aff~cting Observations . . . . • • • · . • • ix Closures in Arranged Circuits • • • • • • • • • • • • •• 39 Computation of Junction Elevations on Arranged Circuit. 9,39 Comparison and Application of the Two Methods ofL.S.. • 110 Condition Equations •••••••••••••••• §0,58,110 Correlates{Correlatives) and Corrections •••••••• 51, 56 Corrected Differences in Elevation. • • • • • • • • • •• 50
D Dawson Township ••• • • • • • • • • • • • • • 1, 6, 38,49, 62 D~rivation of Terms (Theory) for Level Net. • • • • • • •• 60 Description of Bench Marks and Their Adjusted El~vs26,29,31,34 Diagrams (S~~ Illustrations) •••••••••••••• vii,viii Differences,Observed • • • • • • • • • • • • • • • • •• 7,50 Dirrect Observations of a 'Single Quantity. • • • • • •• 110 Discrepancies Among Observations • • • • • • • • Discussion: Methods of Solving Normal Equations
• • .. . • • • • • • .. .
ix 56 59 Discussion": Methods for Formation of Normal Equations
Duplicat~,Parallel Duplicat~,'Lines of ~v~ls ••••• E
• 21,23
Elevations, Most Probable Difference of .. . - • • • • • • • 16
" " Most Probable • • • • • , • • • • • • • • • • • 16
" " Description of B.M.s and .AdJ.Elevs .26,29,31,34,54 , -114-
I K D E X
Elevations, Computation of • • • • • • • • • · . . . . . . " ", Difference in • • • • • • • • · . . . . . . .
Ellimination by addition and Subtraction • • • • • • • • •
39 50 12
Equations,Normal,Solution of. • • • • • • ,41,52,55,57,59,110 " " , Condi tional. 50 • . • • • • • • • • • • • • • • • •
" " ,Observation. .41 • • · • · · • • • • • · • • • • • •
" " ,Normal, Formation of 41,51 • • • • • • • · • • • • · Errors, Types and Occurrance of • • • • • • • • • • • Error, Probable error of a Single Measurement ••••
" ,probable Error of N Measurements • • • • • •• F
· . · . • •
ix 22 22
Figures (See Illustrations • • • • • • • • • • • • • • • • vii I
Formation of Normal Eqs.from Observation Eqs ••••• 10,41,42 ~ormation of Normal Eqs. from Condition Eqs. • • • • • •• 51 Forrr.ation of Normal Equations, Theoretical. • • • • • • •• 59
G
General Use of Theory among· Engineers. · . • • • • • • • • ix General Principle of Least Squares • • • • • Graphs (See Illustrations) • • • • • • • • •
I
• •
· . Illustrations • • • • • • • · . • • • · . . . .
· . . . . 4
· . . . • vii
· . . • • vii Indirect Observations • • • • • • • • • • • • • • • • 110
5 Intermediate Junction points. · . . . • • • • • • • • • • Intermediate Point~ on Level Circuits • • • • • • • • •• 4 Interrr.ediate Bench Marks (See Adjustmett of Benchmarks) •• Index
J
Junction points and Intermediate Points. • · · · • · • 5,15,18 Junction Points, Summary of • • • · · • • • • • • • • .24,47,54
L
Least ~quares,Principle of · • • .. • • • • · • • · • • x, 4,60 " " " ., Results of • • • • • • • • • • • • • • • •• x " " ", Use Among Engineers • • • • • • • • • • •• . x
Levels,Multiple Lines •• • • • • • • • • • • •• • • •• '2 " , Weights f or the Links • •• • • • • • • • • • •• 4
Level Bas~Lines • ••• • • • • • • • • • • • • • • • •• 7
-115-
I N D E X
Levels, Parallel Duplicate Lines of • · • • • • · . • • • • 21 Duplicate Lines of .. • . • .. • • • • • • . • • • • 23 " ,
Links • . . . . . . . . • • • • • • • · . · . . · . • • • • 50
Main Classes of Problems Mean,Weighted Arithmetic Measurements,Unweighted • Methods of solution • • • Most Probable Elevations Multiple Lines of Levels
M
• • • • • • • • • • • • • • • · . . . . . . . . . . . . . . · . . . . . . . . . . . . . . • • • • • • • • • • • • • • •
• • • • • • • • • • • • • • • · . . . . . . . . . . . . . . N
• .110 · . 4
• • 22 • .110 • • 16 • •
Nets,Diagrams of Dawson Township Net(See Illustrational) vii Normal Equations from Condition Equations ••••••••• 51
" "
" "Solution of1416,17,41,45,47,52,54,55,57.59,110 " "from Observation Equations ••••• 10,41, 42
o Observed Differences of Elevations by Links • e.. • • • • • 7 Observation Equations • • • • • • • • • • • • • • • • 8,41,110 On the Uniformity of Least Square Results ••••••••• ix On the Solution of Problems by Least Squares . . . . . . . 7 On the formation of Conditional Equations
p • • . . . . . • • 51
Parallel Duplicate Lines of Levels. • • • • • • • • • • • • 22 Plates (See Illustrations) ••••••••••••••••• vii Precision, Subsequent Measurements Dependent on Results Principle of Least Squares. • • • • • • • • Probable Error of Weight~d . .Ari thmetic Mean.
• • .. . • • • • • •
.. . • •
• •
x x
4 Probable E~eor of a Single Measurement. • • • ••• • • • •• '22 Probable Error of N Measurements. • • • • • • • • • • • • • 22 Probable Values of Unknown Quantities ••••••••••• 110 Problems, Solution of by Method of Least Squares • •• • •• 7 (P)Solution of Normal Equat~ons ••• •••••• • 16,45,52,56
Q Quantities,Summary of Residual Values ••••••••••• 15
" ", Check on Residuals • • •• • • • • • • • • • • • 17 .-116-
I N D E X Quantiti~s, Summary of junction El~vations •••• Quality of Obs~rvations,Car~fulln~ss in Obs~rving (q) solution of Normal Equations ••••.•••••
• •• 24,47
R R~marks on th~ Introduction .• ••• R~siduals (v), Ch~ck on ••••••
. .
. .
• • • • ix 17,46,52,56
• • • • • • • • • • ix . . . . . . . . • • 15
R~sid ual val u~s • • • • • • • • • • • • • • • • • • • •• 15 Rul~: Formation of Normal Equations from Obs~rvation Egs .10,41 Rul~-: Format1on of Normal Equations from tCondi tion Eqs ... 51 (r) or Back Solution of Normal Equations •••• 17,47,5'2,5~,57
S Singl~ M~asur~m~nt,probabl~ Error of • • • • • • • • • •• 22 Solution of Normal Equations •••• 12,16,41,46,52,56,110,Finas Stat~m~nt and Solution for Adjustm~nt Corr~ction • • • •• 3 Stat~m~nt of Conditions • • • • • • • • • • • • • • • • • 8 Subs~qu~nt M~asur~m~nts D~p~nd~nt on R~sults,pr~cision •• x Subtraction. Elimination by Addition,~tc. • • • • • • • •• 12 Summary of R~sidial Valu~s • • • • • • • • • • • • • • •• 15 Sum of W~ight~dSquar~s a Minimum. • • • • • • • • • • •• 58 Summation(l:)of W~ight~dSquar~s and th~ 'Conditions • • •• 58 Symbols.Nom~nclatur~ of 'Symbols us~d in Th~ory • • • • •• 59
T Tabulation:. • • • • • • • • • • • • • • • • • • • • .• •• 50 Tabl~ of Corr~18t~s and Corr~ctions to Links • • • • • •• 51 Th~ory Stat~d for W~ight~d Obs~rvations. • • • • • • • •• x Th~ory-: Application of the Calculus. • • • • • • • • • 58,59 Two M~thods of Findin~ Most Probabl~ Values of Unknown Quant110 Typ~s and Occuranc~s of Errors • • • • • • • • • • • ••• ix
U Uniformity ofL~astSqu8r~ R~sults • • • • • • • • • • •• x Unw~ight~d M~asur~m~nts. • • • • • • • • • • • • • • • •• 22 Unknown Quantiti~s,Two M~thod8 of Finding ••••••••• 110 U8~ of the Theory of L~ast Squ8r~sAmong Engin~ers • • •• x Us~ of Check T~rm8 ••••••••••••••••••• ~~,57
-117-
.~ N D E X V
Values, Check on the Residuals • • • • • • • • • • • • Values, Summary of Residuals ••••••••••••• v,Residuals ••••••••••••••••••••••
W
· . • 15 • • • 15 • • • 15
Weighted Observations, Theory of ••••••••••••• x,58 Weights for Level Lines, 1/ d. • • • • • • • • • • • • • • • 4,50 Weighted Arithmetic Mean, probable Error of • • .• • • • • •.• 4
... 118-
( 25611 )
:TIC COMPUTATION SQUARE ADJUSTMENT Rules for sides
stand outside anq!e: sid8S bQth reqd tow(lrcis WzqJe's vertex: Ilff'f-hand sid(J read-, r/ql7t hand sid! next cmd read T. Consider DicJr,tffen - 6/8 f 1/9
:U' JDick
(e) Sine Eauation
htJrica} exaMS goff"" Tram 'tJlimiM!y computatiOl1.
. second Groupe h.aV~~i 'J'1S cho!ltliJdin {I)e~ whMtht! diff.~or r is min~. h4n th" sign fW11t7JnS 3(71118.
Nohl thot th(J dlff. fbr rf'or a'?9/&lI o~(Jr 80' is minu:t.
1
1 J
Sides .... h fOb
r -&/ot'ID -~/J'r·11
-1 -abr'b -:om r"/o -Ok r 11
Anq/8 Sine I)/ff.fbrl.' 8:J"-.<H3 - ~:16 9.9S.7Jf6JftJ 1'022 ~-18J7 3.lJ130J:JO r~.6 "t5'i2-~7.0"" S,85107.3f na9
l' 8.IJfjJ/572B
.. 0"-09' 14. '16 ~"''t 1'2+.8 't5'-,3B'-.:3+. :9<: 9.8!5.y.()fS76 "'2lJ.e 815 . 23 -07. '2 9.8980787 -2.
t - S,(!JS15906 t r 9.661:572B
Error-ISO
I £:s::t/:;; 1.33
r2.12 1'278
-a.S) 2.97
1O.7T
Fronf and back angles Front ang/M T Back (l'7!l/es-
I~l" 'SIne - -$ - 9.997"fo6<f.Z - t~ - 9.IJ130#02 - ~15-:::; ~O7'7 9,RAil5B21
- -23 - 9.B09Lr520 - -6f ~ 8.8IYrasT5 - - .:o~07=
s.ee/lJ821
fD ll. { 0--""'-=->.="'-....... 2 ... -.209"="'~ } { .... .,...... -249'1>f.206'1o-206"b-.a20"IJ f:0206b
divided fy JOO, S use o'?)':3 of'4 angle equaTion
, problMl IMve oiJt ((I)
ians
ComlJirM like trJrms alg8btYrical!y; thM
0~/.BOt227·hf.022°b-.452"1ot2.y.B'Jo-2CJ931I1'.228""~"t2(){J'1.-:02UIl
of Condifkm Table I of'lI Cor;reJoj8S I ! J! 1 J J 70 t'/o -Ollt-t,.-IJ.'t0/r (from aJ iii (k) (JJ m) In)
__ A~/"u
o .,."", -DJzt~Jz "brO" (f'rom bJ 1 - f'lrff'l fIIr(,,) f' .... lh)if'lrW l+az7.'iI--n.DA'i -G!,/i9 t,'9.fln7 d 1
~o tllo ""'ld,,,-Ib.,.O,,, (f'romoJ '10 +/ f'/ 1ta2+61+1l275 -0.589 l+a740- "'0.¥~6 -I . L __ I"fo I-f It'll t-Q'052Hl275r-£l063 I 1-/.3/59 -1-1.897 -I
-1 Title Pagt' 6 Contt'nts 2 Introduction 4 Front Platt's in Foldt'r
118 Body of Tht'sis 1 Rear " Plate
132 Pages Total.
Thesis C. E. S. Bardslt'y., 'Candidate for the Degree
of iCi;vi l En~ineer.
GEODETIC COMPUTATION LEAST SQUARE ADJUSTMENT
T'!)'lor2.:--______ .lBrowning
Rules Tor sides ofr ~
D~3~ ____________ ~OE~
Male Eduations
L -St"/=No. of'anq/e equations L=No. of'lines /nrYqurYJ =6. _ S=No. of'stations =l:j. Then (j-LffJ=3 aryle equaTions
L -25 f.J =No. of'sin8 equations l (J-(2X4Jt3=/sine equations J ~
StaTion Side Observed Ana/B Cor.f'orAnq/e CorrYJctedSaAna/e
leo' 00' ~.(J4 2.74 leo' 00' -0 .SO
Dick -"bt'/3 +0" 09'-/+.16 -0.90 +0' 09'-1328
179''"59' --s8.29 +I. +:'J ISO' -00'-00. 78
Stand oufside anq/t:' sid8S both rtlad fow(I('(is t a(lqlt:'3 vflriex, Ilff'f-hand sidfl rtlad-,r/qllt hand ~ sid! next and rtladt: Consider DicH,tfftln -a/s f '13
~U' .:JOic/r
~i~ (e) Sine EQuation
Sides
Fronf and back ang/fJ3 Front angJ~ 1-Back a'?!}/t:.S -
l£Jk
-3,,'1'-"
f---r----.-[
+- - /Of'lo ~5'-12-:37.0+ 5.8510731 f20S t2J5 - I He -0 !J.lXJI0777
This second 9rtJUPfi h.aye~i '/7$ 8din (/)e)(, when the ~or I" is min~, hen the sign
I"8/7I(7l17swm~
/VoftJ thol' the diff. fbr I"/'or a'?9'M o~er SO·isminU3.
1 J
""bf '/3 %f'/o
-Ok -t""
-t 8.&515728 9.tJaI6821
+O"-OS' 14. '15 9~"t."I
~ 38'-:3Lt::90 s.6!540575 96 . Z3 -07. 52 S139807tj7 t - Me/590B
9,8615726 Error 180
+2+.8 r20.8 -2.0 t
-0.91 - -23 - 9 .809+520 -2.97 - -81 9.8l5+0515
ITU71 ~ - - 9.S8B076t' !J.tJtJl!582J
r ;r,::;'''s / 1 ""- a~8 L ____ =============----==:+======v:::'" X::"::' :::::":::::'==------1 Con use a'7Y.:3 of'4 angltJ equations This probll!lm l(JUvtJ out (tf)
{o)Normol Equations k Im(J
m ua
y. Z I~m m ,,,,, I
I 2 3 1
~. Ab30Jute Torms
arr03 , umTTl in I
1 £guotions of Condifion
{f')o--3.'t5-·IO+''.2..- 0fJ.t-tc:'lat Ok (from al (9) 0- r"t.70"'lo", ': _,;:' N (t"rom bJ rh) 0- t2.7~"'lo t'lo ~" ""t ~. (;
Table of' 2 Co~, GI'-'.,' ov / 2 ..
(vI (k) (mJ (nJ .... ' _.m.,; yoluo I... '£J Ifaz75!-<t063 1-0$89t3007 I~
1-0.58.9 +0740- Iro.~26 1-006.3 ' -1-1.697
t6.000 -2.000 +2.000 fa8'<7. 3.'<50 ..... >n t L--__ --, 11Q.063 1tQ~ to -lrl.Z:
-2.000 t6.000 1'2.000 0.863 • t4.700 ..... --1'2.000 1'2.000 t6.o00 tOJ22 t2.740 ___ ---to.$'<7 -0.863 to. 122 TO.5/7792 -1.800 - -t
(PJ Solution of' Normal Equafions
I I I I 2 3 ~ ~ PI n;.ooo 1-2.000 11"2.000 ItO.9+7 -3.'<50 P:. (-0.165; 1ta333 1-0.333 1-0.1578 fO.575
~ 1+5.:333 1-t2.667 1-ll547~ '3.550 J'+ .f) JFi7.Z;: 'r0.500 ItOJ028 -0.886
P6 ItLt:OOO -0.0800 1'2//5 -P. "- '(-0250) 1-0.0200) -0.529
f7 'rTa310B() -O~'t r3.007
(5) Anq/e Corrections
:k: .U ... " 1M OS
~ , ",
..J. rom or cv-ror In
Fl"'Orn I or vretJr "
ItO.5B9 -C I u 11"025'# to./. -11'0.86+
t-o.589 -al . -1-1.217 Ita.338
0275 1-0275
1-0063 1-G58.9 1:066 -1-a58B
!t02Zi
1 (q) 'Add ".:7' -~
2. qJ 11'6.o0G +2l)OQ 1-0.863 '4." 00
+0.667 1+1I .31S~ . ISG If5.333 r2.6/ 73 't3.550
ao t/ It( . -t2.740 --0 QIo -0.8/ 1-( ~7 rl. /50
(r)
11 I Ll--!:f-~I"~o .. S~7~S-F.·~ 0.:,=,tl!=-'6-j~_ -'7': a;~:~;.o~.9o--+l..!. t"""".3 .. to=7----j L-==: '" a47S 1'0.308 ,- ,~,
n. ·alge .,0.294 1-0.589 r. 0021 ·a~ f'b . Q275
"_ I I
,or Anal" of FJk Anall! oflJrownina MoM of Dicit 1
~ -Cor "0· - /.271 _~D" --0254- Cor '/3 -"1'0 160"'- fjopypbovt! 2.0rrtlGfjrr::/n;!,hg:frr;e:r;Pjg~~/ons l _______________________________________________________________ -'
r+-"'Cor"'--"'IO!!..-"-:-1'';;a-ii47'2~6'--+'t-'Cor~..!·/!..J »'--:'-/.7"'27f,1.f7;---jIt!-'ICor=-..!D~:o_-_-~ u.,5~8'86H 'I ~~'s'1:t,n 6!/u~'i:;S. s ' J 0:7 J Jr' - 0.845 -/.4t71 0.Jt-26 I- l
3 OOA'
AnqllJ at Dick Anq/t) at fjrownino Ana/I! af Tov/or Cor~~ = -0746 -Ca- 3
/, s 1'1.217. k20r'k - tf!..275 rCor '/:I = - 0.160 -t Ca- '" = TO.96't tCor '% - -Q083
0.908 T2J8I t 0212 I-
DIRECTIONS FO/low mr;jorOP8r:oTlons dITJhab8fical!J and the
d6tal/" num6Jficaljy . Computation ty ;u... YEAGEJ{,.
YocatiOflfl/ SttdtJnt
MISSOURI SCHOOL OF MINBS AND METALLURGY