BALL-A LOGARITHMIC SLIDE-RULE 285

ON A LOGARITHMIC SLIDERULE FOR REDUCING READ- INGS OF THE BAROMETER TO SEA-LEVEL.

BY JOHN BALL, PbD., B.Sc., F.O.S., Assoc.M.1nst.C.E.

[Read May 17, 1905.1

1. Jntroduetwn. THE recent institution by the Egyptian Government of a considerable number of meteorological stations In Egypt and the Soudan has involved the performing a t Helwan Observatory of large numbers of reductions to sea-level of barometric readings for the purpose of preparing isobaric charts. These reductions being. generally based on the monthly means of observations taken thrice daily, the data are sufficiently accurate to warrant the corrections being computed with a fairly high degree of precision. A t first they were calculated by Angot's method, using the Internalional Tnbles ; but the necessary interpolations were found to be laborious, while the accuracy attained was occasionally somewhat below that required. Recourse was then had to the logarithmic tables in the same volume ; these left nothing to be desired in point of accuracy, but they involved more labour than was convenient. Attempts were then made by the author to devise some mechanical means of calculation which, whilst equally accurate in its results, should be quicker and less liable to errors of computation than the method by the logarithmic tables. These attempts were successful. A slide-rulc for each station was devised, which met all requirements, and which could easily be prepared in the observatory office. Reductions are now made with the aid of these rules in less than one-tenth of the time previously occupied, and with a very high order of accuracy.

I n the belief that it niay be useful to others who have to perform similar reductions, it is proposed in the present paper to describe the method of constructing such slide-rules, so that similar instruments can be readily made by any one to suit any given meteorological station. The instruments used at Helwan are' adapted for metric measures ; but in this paper the British system of measures will be adopted as more useful to British meteorologists. When an observed barometric height is mentioned in what follows, the reading is of course supposed to have been already reduced to 33" F. from the reading of the attached thermo- meter of the instrument, and also corrected for the effect on the height of the mercury-column of the variation of gravity with latitude and altitude.2

2. Principle of the IlIelhcl. The barometric formula employed in the International fiIetemologica1

Z = K ( l + c + a O ) (lip) - ( l + y ) ( 1+- z,z') log H 2 . . . . (l),

Tables, based on Ruhlmann's modification of Laplace's formula, is H

Communicated by permission of Capt. H. G. Lyons, Director-General of the Egyptian

a The barometric formula itself contains factors involving the variation of gravity, but Survey Department.

those are of course for the effect on the weight of the ctir-co/unin, not on the mercury.

286 BALL-A LOGARITHMIC SLIDE-RULE

where 2 =altitude of upper station, z=altitude of lower ahtion,

K = the barometric constant, R = the mean terrestrial radius.

a = the coefficient of expansion of air,

@=0.578*, where +=mean tension of aqueous vapour

0 =mean temperature of air, y=0'00259 COB 2X, where X=latitude of place, H"= the barometric pessnro at the lower station, H =the barometric pressure at the upper atation.

t ) and t ) = mean pressure of air,

Inserting numerical values for the British system, the equation becomes

and the lnterntitionrcl Tables give the values of log A, log B, log C, log D, where

log A=lOg 60368'6[1*00157 +0'00?037(8 - 32)],

log B = colog (1 - 0.3;9:), log c= log (1 +0'00259 COS SX),

using which notation, it is evident that log Z =log A + log B + log C +log D +log (log H, - log H),

which, for the purpose of sea-level reductions, may conveniently be written in the form

l ~ ~ ( l ~ g H ~ - l ~ g H ) = l o g Z - l o g A - l ~ g B - l ~ g C - l ~ ~ D . , . . (3).

Considering first the right-hand member of equation (3), it is clear that for any given place log Z, log C, and log D are constants. There are thus in this member of the equation only two variable logarithms (A and B) to be subtracted from the constant (log Z - log C - log D). Now if we assume any fixed laws of relationship between 6 and 4 on the one hand, and the observed temperature and vapour tension at the upper station on the other hand ; and if we assume (as it will be shown later me may justly assume) q to have a constant mean value for the purpose of calculating log B; it is obvious that this snbtraction can be readily performed by o simple slide-rule suitably graduated for temperature and pressure as observed at the upper station.

The left-hand member of the equation (3), though it appears a t first sight more difficult to deal with, can be treated equally easily. For if 31 and N are any two quantities comparable in magnitude with the extreme heights of the mercury column a t any given place, and if c and d are any two other quantities comparable in magnitude with the extreme

B A L G A LOGARITHMIC SLIDE-RULE 287

corrections required to reduce the barometric height at that place to sea- level, then, the squares and higher powers of quantities such as : being negligible, we may write, without sensible error,

a1

whence

M + c c

11 +d 11 100 -=- "' BI M'

N + c c log,, - =- N N' N + d d loge - =- N N'

log. =p'

an expression which will clearly not alter in value when we replace M throughout by N. log [log (31 + C) - 10:: 31 J - I O ~ [log (31 + (2 ) - I O ~ 81 J

which is the condition required in order that i t may be possible to obtain the corrections c and d by a simple pair of suitably placed slicles, one of which bears divisions proportional to the differences between the lognrithms of dil'eretit values of the observed barometric height, and the other is graduated with the interval between the corrections c and d equal to

where M is any barometric height between the extremes observed at the station. The mean barometric reading is conveniently taken for M, but the investigation shows that the results will not vary by more than an entirely insignificant amount if any other reading between the extremes be used instead of the mean.

It will be shown later that it is possible so to locate this second pair of slides, and the first pair already mentioned, on a single rule, that by placing any given observed vapour tension (on the log B scale) against any given observed air temperature (on tlie log A scale), the correction to sea-level can be immediately read off against that point on the log H scale which corresponds to any given barometric reading H simul- taneously observed a t the station.

Thus, the slides being once made and gradnated, tlie rule will at all times and under all conditions (excloding abnormal inversions of tempera- ture and humidity in the air strata, in which cases no fixed method of pro- cedure whatever can hold) give very accurately the correction to sea-level for the barometer at the station for which it is designed. And as this is done by a single setting of the rule, without the smallest calculation or reference to tables, reductions can easily be performed at the rate of two per minute. When it is added that the-calculation of the scales and the construction of the rule in cardboard will not require more labour than the reduction of a score of observations by the International Tables, it will be apparent that the value of the rule as a time-saving machine may be

Hence we have

=log[ log(N+~)- log N ] - l ~ g [ l ~ ~ ( N + t l ) - l ~ g N] . . . ( 5 ) ,

log [log (bI+c) - log 111 - log[log ( M +d) - log 111,

2aa B A L G A LOGARITHMIC SLIDE-RULE

very considerable where a large number of reductions for the same barometer have to be performed.

It may be well, before describing the construction of the scales, to justify the approximations mentioned above for the graduation of the vapour-tension and correction scales by actual trial in the case of a high station with largely varying meteorological conditions, in which case the errors involved in the approximations will appear a t thcir maxima. We will take as an example a station 2000 ft. above the sea-level, which is higher than any meteorological station yet dealt with by the author, and assume possible ranges of air temperature from 50" to 100" F., and of pressure from 27 to 29 ins. The correction to sea-level will then range approximately from 1.8 in. to 2.1 ins.

If under these extreme conditions we investigate the error produced by considering (for the purpose of the vapour-tension scale only) the barometric pressure to be constant at its mean value instead of varying between the given limits, we find the maximum uncertainty in the resulting correction to be 0.0004 in., which is about the variation which would result from an error of O O . 1 F. in the temperature, and is there- fore quite negligible.

With regard to the second approximation (that of the correction scale) the uncertainty is even less significant. For if we substitute the given limits into the approximate equation ( 5 ) mentioned above, it becomes l0~(10~28~8-l0~37)-10~(29~1~l0~27)~l0~(l0~30~8-l0~29)~10~(l0~31~1-10~ 29),

the two membcrs of which work out to be 0.06466 and 0.06481, the difference indicating an uncertainty of only 1 in 900 in the range of correction deduced from the mean; and as the total rangc of the correction is only 0.3 in., the uncertainty in the result is only about 0.0003 in., a quantity quite negligible even in the most refined barometric measurements.

3. Calculation of the Scales. The calculation of the scales is easily performed with the aid of the

International Metemologienl Tables and a book of logarithms. It is of course premised that the latitude and altitude of the station are known, and also that the usual assumptions are made with regard to the fall of temperature, pressure, and humidity with increase of altitude.

The method of calculation will be best shown by an example, and for this purpose the case of Khartoum will be taken. Khartoum is situated in latitude 15" 36' N., and its altitude is 1233.6 ft. above sea- level. In accordance with the mean results of numerous observations in tropical countries, we may assume that an increase of altitude of 1000 ft. is accompanied by a fall in the air temperature of 3O.1 F.; hence the mean temperature of the air between Khartoum and sea-level may be taken as constantly greater than the air temperature a t Khartoum itself by 1O.9 F. On similar assumptions the mean pressure of the air may be taken as 0.6 in. higher than the pressure a t Khartoum itself, while the mean vapour tension may be taken as 1.07 times that observed a t Khartoum.

The approximate limits of oir temperature a t the place are 50" and 100" F., those of pressure being 28.0 ins. and 29-2 ins., while from a

BALL-A LOGARITHMIC SLIDE-RULE 289

rough preliminary computation the sea-level correction may vary from 1.15 in. to 1.35 in. These are only rough values to indicate the extent of the scales required.

A convenient linear unit is 0.1 in. for each 0.001 of logarithmic difference; this gives scales sufficiently open for the correction to be easily read to 0.001 in. while the instrument is kept within a total length of 15 ins. Needless to say, the same linear unit must be adhered to for all the four scales of any one rule.

Temperature Scale.-Let t be the observed air temperature a t Khartoum, 0 the mean temperature of the air bctween the station and the sea-level. Then, as mentioned above, we shall take 6 = 1 + 1".0 F. It will be sufficient to fix the divisions of the scale for each 10" F. of temperature interval, proportioning the individual degrees afterwards. We then, with the aid of the Intewmtionnl Tables, make the calculation as follows :- t = 500 R no Rn' m0 1 nno

The vapour tension varies from 0 to 1 in.

It is now necessary to fix on a unit for the scales.

_ _ - - - 8 =51.9 6 i . 9 Si .9 i i . 9 101.9

differences - - '00841 .00825 '00809 '00795 '00780 log A = 4.79872 4'80713 4.81538 4.82347 4'83142 4.83922

scale-lengths (inches) = '811 '825 '809 '795 T80

The space in inches between the 50" and 60" marks on the scale will thus be 0.841, and so on.

Yapour-teiuion Scale.-Assuming the mean pressure of the air to be constantly equal to 29.0 ins., we calculate the scale-lengths for differences of 0.5 in. of vnponr tension thus :-

f=vapour tension at place = 0 0 5 1'0 0'51 1.07 +=mean vnpour tension =

log B (from tables) = o 0.00307 0.00609 ditferences - - 0.00307 0.00302 scale-lengths (inches) - .307 .302

in a i r=jx1*07 } -

Barometer Scale.-This can be calculated to get the divisions for each

H= baronieter reading = 28.0 28.4 28.8 29.2 log H differences - scale-lengths (inches) = '616 *607 '599

(h-reetion Scale.-This is conveniently calculated for each 0.05 in., from 1-15 in. to 1-35 in., in the following form, where H = a n y barometric reading, say 28.5 ins., and H, Thus H can be assumed constant for this calculation; the same result for the correction-scale would be arrived a t by taking 28 ins. or 29 ins. for H in place of 28.5 ins. I t is convenient to use 6- or 7-figure logarithms for H and H,, though 5-figure tables are sufficient in the remainder of the process.

correction = 1.15 1 *20 1 '25 1 '30 1 3.5 HO

0.4 in., as follows :-

= 1.14716 1'45332 1'45939 1.46538 0.00616 0.00607 0.00599 -

H +- the sea-level correction.

H =28'50

~ 2 9 . 6 5 29.70 29.75 29.80 29.85 = 1-4720247 1.4727564 1.4734870 1'4742163 1'4749443 ;:: 2 = 1'4548449 1'1548149 1'4548149 1'4548449 1'4548449

log H,-log H = 0.0171798 0'0179115 0'0186421 0'0193714 0.0200994 log (log Ha - log H) = g.23502 z.25313 3.27019 2.28716 5.30318 ditferences - - 0.01811 0.01736 0'01667 0.01602 scale-lengths (inches) = 1'811 1 7 3 6 1.667 1.602

290 BALL-A LOUABITHMIC SLIDE-RULE

Calculation far Lotalising the Scales.-In order that the four scales may berplaced correctly in relation to each other, in making the rule, it is necessary to calculate the coincident points for some one set of conditions. Let us suppose the barometer reading is 29.0 ins., the temporature of the air a t the station 80" F., and the vapour tension 0.50 in., and calculate the correction by the logarithmic tables in the ordinary way, thus :-

log A=4*82347 log B=0'00307 log C = 0.00096 log D = 0*00003

log 2=3.09017

- 4'82753 4.82753

log (log HO- log H)=2.26264 .'. log Ho-log H=0'01831

log H=146240 log H,=1'48071

whence H, = 30.249 ins. and the correction = H, - €1 = 1.249 in.

The scales must therefore be so placed that when 1.249 on the correction scale is opposite 29.000 on the baromcter scale, 80" on the temperature scale shall be agirist 0.50 on the vapour-tension scale. The only other necessary condition is that the dircction in which the scales run ( i .e . to right or left) must be considered. If the fixed part of the rule contains the temperature and barometer scales, both graduated from left to right, then the sliding portion must have the correction scale graduated in the same, and the vapour-tension scale graduated in the reverse, direction. The calculation of localisation automatically does away with the necessity for any further consideration of log Z, log C, and log D in constructing the rule.

4. Construction of the Bide.

The instriiment is conveniently made of stout Bristol board. One piece about 15 ins. by 5 ins, is taken for the fixed scales, and another piece about 1 in, wide by 15 ins. long for the moving portion. The scales are best drawn in Indian ink after first marking out in pencil.

Fixed Pmtion of the IMe.-A horizontal line is drawn lengthwise down the middle of the card. Starting about 13 in. from the left-hand edge, this line is graduated in divisions of 10" each from the scale-lengths corn- puted ; thus from 50" to 60" will be 0.841 in., from 60" to 70" will be 0.825 in., and so on. The single degrees can readily be put in by sub- division.

The barometer scale will conveniently commence about 14 in. to the right of the 100" division, on the same line, and mill run from left to right like the temperature scale. The data already calculated give the intervals between the 28.4 and 28.8 marks, and 80 on ; the single tenths of an inch can then be filled in by sub-division of the larger spaces.

Sliding Pmtion of the Rule.-The correction scale can be a t once put on the upper edge of the strip of card, beginning conveniently about 7 ins. from the left-hand end, and the graduations running from left to right. As in the case of the fixed scales, the computed data give the large divisions, and the smaller ones can be made by sub-division.

BALL-A LOQARITHMIC SLIDE-RULE

292 BALL-A LOGARITHMIC SLIDE-RULE

The sliding portion must also bear the vapour-tension scale, whose position is, however, governed by the localising calculation above made. Placing 1.249 on the correction scale against 29.00 on the barometer scale, we can a t once mark off, against 80" on the temperature scale, the division corresponding to 0.5 in. of observed vapour tension. Now the vapour-tension scale must, as remarked above, run in the opposite direction to the others ; we therefore place the 0.0 mark 0.307 in. to the riyht of the 0-5 mark, and the 1.0 division 0.308 in. to the left. The remaining tenths of an inch can now be filled in by sub-division, and the rule is complete.

If now any given vapour tension be set against any given air tempera- ture, both being observed at Khartoum, the correction to sea-level can be at once read off against the simultaneously observed barometer reading. I t is convenient to gum a pair of cardboard guides for the slide near the ends of the fixed portion of the rule. Drawings of the fixed portion and the slide are shown separately in Figs. 1 and 2.

By Rule. By Tables. ---

16 seconds minutes

5. Speed of Torking.

By Rule.

in. 1.248

1.208

1.263

In order to test the cconomy of time rcsulting from tlie use of the rule under ordinary working conditions, sea-level corrections were taken out for three sets of conditions chosen at random, the reductions being performed in each case by the rule and by an experienced computer using the logarithmic tables. The results are shown below :-

By Tables. -- in. 1.248

1.208

1.263

Air

90 I 75

Vapour Tension.

-- in. 0.30

0.80

0.60

Barometer Reading.

-- in.

28.10

28.70

2910

The rule thus yields results of equal precision with those of the tables ; but while the reductions with the rule occupied less than a minute and a half, those by the logarithmic tables consumed 1 6 minutes.

DISCUSSION.

The CHAIRXAN (Capt. D. WILSON-BARKER) thought that the thanks of the Society should be given to Dr. Ball for his paper, as all those who had much to do with making a considerable number of barometer reductions would appreciate any help that would tend to lighten their labours and difficulties.

Mr. W. MARRIOTT mid that he was glad to see the slide rule which Dr. Ball had designed for reducing the reading of the barometer to sea level. Anything that tended to lessen the time and labour in the reduction of the observatione wm always welcome. He (Mr. Marriott) gathered that the time occupied in reducing an observation by Dr. Ball's slide rule waa half a minute, but he himself had brought before the Society in 1875 a Table which combined three corrections,

DISCUSSION-A LOGARITHMIC SLIDE-RULE 293

viz. index error, temperature, and altitude, by the use of which a greater saving of time was effected t h m by using Dr. Ball's slide rule. This combined Table was used by the Society's observers, and was also set out in the Hints to Meteorological Observers. When the Royal Meteorological Society started their Second Order stiltions in 1875, he found great difficulty in getting simple tables for reduciug the barometer readings. The various treatises on Meteorology only gave the corrections for two sea-level pressures, viz. 27 ins. mid 30 ins. ; it was - consequently necessary to go through some calculations before the correction for any intermediate reading could be obtained. He subsequently calculated and printed in the Hints the corrections for pressures a t 28 ins., 29 ins., and 31 ins. from 10 to 1000 feet above sea level at temperatures from 20" to 80". As a change of 0.6 in. in the pressure produces the same ainount of variation in the correction for altitude as an alteration of 10" in the temperature of the air, it was possible by taking the mean height of the barometer to combine the correction for temperature with that for altitude, and when the index error is the same throughout the scale, as in the Fortin baro- meter, it could also be included. By the use of such a combined Table the three corrections could be made by one operation, a i d the liability to error thereby greatly reduced. The time occupied ill applying these three corrections was rather less than that taken by Dr. Ball in applying one correction only.

[Dr. J. BALL, in a letter to the Secretary subsequently, said that he desired to point out that hie slide-rule method waa not designed to compete with that of Mr. RIarriott's tables in the ordinary cases, where only a moderate degree of accuracy was required, In such cases no method could possibly be simpler or more couvenient in use than the tables given in Hints lo Neteoroloyicnl Observer& But in north-east Africa the problem of sea-level correction presented itself in a somewhat peculiar aspect. The ordinary non-periodic variations of pressure there were of a mnch smaller order t h m those in Europenn countries, and con- sequently in the reduction of reading3 it was necessary to go to a very high degree of precision, in order to compare them properly. When tlie greatest possible precision was reiluired in the reduced readings, the tables of hlr. BIarriott were insufficient. The nssumption that the change produced in the correction by a rise of 0.6 in. in the baronieter wiu equal to that produced by a fall of 10" in the air-temperature, though no doubt a very close apprositnotion in most cases, was not strictly accurate. It was further essential, for great accuracj-, to take account of the varying amount of aqueous vapour in the air, and of the effect of vari;ition of gravity on the weight of the air-column considered, both of which, being sinall effects, were neglected in the tables of BIr. RIarriott. As an instance, he had calculated the correction to sea-level for a barometer-reading of 29 ins. at Merowe (lat. 18" 29', altitude 876 feet), the air-temperature and vapour-tension at the station being respectively 50" and 0.4 in. The tables of hIr. Marriott gave the value as 0.919 in., while rigorous calculation by the logarithmic tables, with the usual assumptions of temperature-gradient, etc., gave 0.938 in. ; and this latter value of course agreed with that found by the slide- rule. He thought Rlr. Marriott would admit that in cases such as this, true variations of pressure might, i f very small, be entirely masked by the small errors inherent in the method of formation of his tables ; and in order to obtain a sufficiently accurate reduction in such cases, it had hitherto been necessary to employ either the method of M. Angot, or the logarithmic method, both of which were embodied in the Iwternatioml Tables. The new slide-rule was founded on the logarithmic method, and while immensely more rapid in use than the Inter- national Tubles, i t gave results of equal accuracy with them. Thus, although his rule was not designed to replace the useful tables of Jlr. Marriott in ordinary cnses, he trusted that it would find a field of considerable usefulness in cases where the utmost precision was required in barometric reductions to sea-level.]

Y

294 RAINFALL IN FIJI, 1904

aainfallinFiji, 1904. Lat.

16'-38' S., long. 178O.37' E. ; height above sea level, 71 ft. ; distance froni sea, 1 mile :-

The following is the rainfall taken at Delanasau, Bun, Fiji, in 1904.

Rainfull. No. of Ruiny JIux. Fall Days. In 1 Day.

1001.

January. February Narch . April . May . June . July . August. September October November December

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ins. 22.52 7'26 14.13 10.10 3 '40 1.13 0 '34 3'03 4'01 2.41 4.48 10.76

28 21 20 19 11 9 4 8 11

5 21

L

Total . . . 83.56 164

ins. 4.92 1.68 3.31 2.19 0 * 6 i 0.37 0.28 1.06 1.40 1'63 2.26 3.39

4'92 -

The average fall in 3 4 years waa 93-49 ins. The greatest yearly f d l was 159-51 ins. in 1871, and the least 52-55 ins. in 1903. The year 1877 was our worst season of drought here, and a year of extraordinary fall elsewhere in the group. In the first three months of that year there fell here 62.38 ins., in the subsequent nine months only 18.15 ins., and i n the five months ending December 31 only 5.29 ins. fell. In contrast to this I may mention the phenonienal rainfull in that year, 1877, in South Taviuni in particular. Mr. James Newall, living at &am Walu near Vuna, two and a-hnlf miles from the sea, and 594 feet above sea level, aeut ine the following particulars of the rain- fall :-In 1876 there fell there 243.07 ins. on 236 days ; and in 1877, 251'57 ino. on 228 days. Constant, strong Southerly winds caused all this, foods and droughts in the two instances respectively, Vuna being exposed to the wind of€ the sea, while here the same Southerly wind had to cross Vnnua Levu, and so became dry.

I n the sun's rays I registered 172" on December 21, 1877, the extreme liniits of the instrument,-a black bulb therniometer,-and on December 24, 1904, I observed 156"-5. In December 1877 the salt r a t e r in a tidal creek here registered 97" five feet below the surface. Yet, i n contrast to all this, the lowest temperature in the shade at night occured here on September 14, 1877, namely 56'.6 ; the mnxirnum in the shade, also R record, 98.7, occurred on January 6, 1878. Yet one more record for that year was the hygronieter readings on Noveniber 18, 1877, viz. dry bulb, 93O-6, wet bulb, 7 4 O - 8 ; difference 18O.8; the percentage of moistnre in the air being only 3?.-R. L. HOLJIER, F.R.Blet.Soc.