solution of the equation of the fifth degree

Annals of Mathematics

Solution of the Equation of the Fifth DegreeAuthor(s): Alexander EvansSource: The Analyst, Vol. 5, No. 6 (Nov., 1878), pp. 161-166Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/2635845 .

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THE ANALYST.

Vol. V. November, 1878. No. 6.

SOLUTION OF THE EQUATION OF THE FIFTH DEGREE.

TRANSLATED FROM THE THEORY OF ELLIPTIC FUNCTIONS OF

BRIOT AND BOUQUET, SECOND EDITION.

BY ALEXANDER EVANS, ESQ., ELKTOJST, MARYLAND.

421. It was announced by Galois that, so far as n = 11, the degree of

the modular equation may be lowered by unity. M M. Hermite and Betti

have given two demonstrations of this theorem based upon different prin- ciples. Let us consider a symmetric or alternate function (va, vb) of two

roots of the modular equation, then, the same function of two other roots, and

so on as far as the last two roots, and let us designate by U a symmetrical function of these J(w+1) quantities, we easily demonstrate, by the aid of the laws of permutation before established, that when n does not surpass 11,

among the different manners of associating the roots, two by two, there ob?

tain those for which the function U only acquires n values for each value of u and by consequence satisfies an equation of the degree n.

About each one of the critical points a root va remains holomorphic and

the associated root vb acquires the n other values; the quantity (va, vb) and in consequence the function U takes n values, forming a circular system.

If this function only acquires n values in the whole extent of the plan, it is impossible that two of these values can have a common element (va,

vb); for by a suitable way, va becomes equal to the root which remains ho?

lomorphic about one of the critical points; if two values of U had a common

element (va, vb) without being identical, they would engender two circular

systems of n values each.

We may associate two roots va and vb taken at pleasure; for, about the critical point where va remains holomorphic, vb acquires the n other values.



?162?

In a single value of the function [/"two elements cannot present the same

difference of indices, because if we should describe the contour or lacing (O) [le lacet] a certain number of times one of the elements would become equal to the other. Since we have perceived that the contour (a0) causes the

function Uto acquire the same values as the loop (O), we are certain that

this function has only n values in the whole plan, the other lacings relead-

ing to the loops (a0) and (O)- 422. These considerations allow us to find easily the favorable ways. We should associate the roots by way of difference and form the product

of the J(w+1) quantites. For n = 5 if we take as first factor V? v0, the combination

(i) Cr=(F_t,o)(,l_?4)(?2_V3) is the only one in which the difference of the indices in the last factors is not the same.

By the contour (O) this function acquires the five values

Ux = (V?v1)(v2?v0)(vs?v4), (2) <{ U2 = ( V-v2) (t,,-^) K-^0),

I Us = {V?vs)(v4?v2){v0~v1), L ?4 = (V?V?) (V0?VS) {V!?V2).

After the law of permutation written in number 408 the contour (a0) re-

produces the same values in another order; from that we conclude that the function ?7 has only the five preceding values, and by consequence that it satisfies an algebraic equation between u and U of the fifth degree in TJ.

For n = 7 there are two favorable combinations,

(3) U= {V-v0)(v2?vs)(v^?v6)(v1?v5),

(4) U= (V?v0)(v2?v6)(v5?vA)(v1?vz). We obtain them in the following manner; upon the contour (a0) the law of

permutation is ( V, v5, v6, v?, vS9 vl3 v2). Let us take as first factor V?vQ, and as the second v2?va, for example

v2 ? vd; when we describe the noose (a0) the product (V? v0)(v2?vs) be?

comes (v5?vQ)( V?v^: the first product being completed, before reprodu-

cing the second, the contour (O) once run over, will contain the factor

v4?v6, which gives origin to v5?v0, and by consequence the last factor

will be vx?v5. By the noose (O) the function (3) acquires seven values: these same values reproduce themselves upon the loop (a0); whence we

conclude that the function U satisfies an algebraic equation between u and

Uf of the seventh degree in TJ. The function (4) posseses the same property. For n =311, we have also two favorable combinations



?163?

(5) U= (V?v0) (v10?v5) (vs?v6) (v9?v7) (v2?vt) (v4?vs\

(6) ?7= (V?vQ) (v10?v9) (vs?v5) (v7?vz) (v6?vt) (v2?v4), which we obtain in the same manner.

Upon the contour (a0) the law of permutations is (V, vl7 v6, v4, vs, v9, v2,v8,v1,v5,v10).

We will take as first factor V?v0, and as second v10?va. For n= 13 there is no favorable combination.

423. From the depression of the degree of the modular equation for n = 5, M. Hermite has deduced a method for the resolution of the equation of the fifth degree by elliptic functions (Comptes rendus, t. XVIII).

Let us form the equation of the fifth degree in U. The values of U be?

ing finite for all finite values of u, the coefficient of U5 is equal to unity. For u = co the six values of v are infinite, one of the degree 5, the others of the degree \ (number 408): the five values of U are infinite and of the

degree %?-; the sum of the negative orders of the function U being equal to

27, the equation is of the 27th degree in relation to u (number 135). For values of u exceedingly small the approximate values of v are

2 _1 _2_r* V= ?2~2u5, v0 = 2~%5, vt = v0e 5 , v2 = v0e

and consequently those of U are,

(7) U0 = 2*6*?* Ux = UQe% U2 -= U0e

We have put ? = -^; putting in the same way 0 = -^ = (? ? ?0)

X($i?f 4) (?2??%)>tne coefficients of the equation in f being entire polynomes us (number 401), those of the equation in 0 will possess the same

property; the values of 0 only becoming infinite for u = 0, this equation is of the form

u*P*P+^\*\cb+b9tf+opul*+ ? ? ?) @5~p+(a5+b5u8+c5u1?+ .. .)=0.

Eeplacing 0 by U-^u15 and multiplying by uz, we obtain the equation

u8(/V9) U* +Tu8i^)+1%p+bpu*+...)U^+u%a5 +?,*+ ..) = 0.

For very small values of u, the values of U being very small of the degree

f, and forming a circular system, we have /?0 = 9, a5 = ? 265l, 8((3P? 9)

+15p> 3, and in consequence ftp = or > 10?2p; we will take {tp= 10

?2p. The equation sought for is then of the form



?164?

J?=4 (8) U* + 2 u8-p(ap+bpus+cpul&)U^+u%a5+b5u9+cy'+d5u24) = 0.

p=i If we consider the function (TJ) = [(F)? (v0)] [(vx)?(v4)][(v2)?(v3)],

relative to the variable ux = 1-r-u, we have

since the product of the roots of the modular equation (number 411) is

equal to ? m-6.

So the equation (8) ought not to change when we replace in it u by 1-4-w, and Z7by U-t-u6. But, by this substitution the equation becomes

U5+P2\i?-* {ap+bpu-*+cpu-16)U5-p+u%d5+cy+b5^ p=i

and we hence conclude that it contains no term in TJ*, and that we have 62 = c2 = 0, c3 = 0, 63 = a3, c4 = a4, d5 = a5, c5 = 65. The equation reduces to

(9) J f/H^^^+^^+^^+^+y+a^D' \ +<a6 +b6u*+b5u"+a6u*) = ?-

For w = 1, the root t?0 of the modular equation is equal to +1, and the five

others to ?1 (number 406); the five values of ZJreduce themselves to zero; the polynomes in u, coefficients of the diverse powers of TJ in the equation

(9) ought to become null for u = 1; and we deduce a2 = 0, a3 = 0, 64 = ?2a4, 66 = ?a5, which reduces the equation to the simple form

(10) Z75+a4u4(l? u*)2TJ+a5u*{l?u*)2(l+u*) = 0.

We know the coefficient a6, it remains to determine the coefficient a4; we follow the road which has been pointed out in number 411 for the cal-

culation of the modular equation. To a real, positive, and very small value of u, corresponds a value of p

of the form p = si, s being positive and very great, and in consequence a

value of q real, positive, and very small.

By developing <p(/>) in series and restricting oneself to the first two terms

there results

u = 2?j*(l?a), v0 = 2 V?"(l - g?), ! _1_ 2jn 1. __2wi

vx = 22g40(e 5? q5e 5

), i JL. **? X _4jrt

v2 = 22g40(e 5

?q5e 5), i _L_ _4ffi J, 4wt

v3 =22g40(6 5

?q5e 5

),

vA =2"2g40(e 6?g6e5),



?163?

(5) U= (V?vQ)(v10?v5)(vz?v6)(v9? v7)(v2? vx)(vA?vs\

(6) U = (V?v0) {vt 0?v9) (vs?v5) (v7?vz) (v6?vx) (v2? vA)y which we obtain in the same manner.

Upon the contour (a0) the law of permutations is(V, vl9 v6, vA, vSf v9, v2^8,v7,r6,t?10).

We will take as first factor V? v0, and as second vx 0?va. For n = 13 there is no favorable combination.

423. From the depression of the degree of the modular equation for n = 5, M. Hermite has deduced a method for the resolution of the equation of the fifth degree by elliptic functions (Comptes rendus, t. XVIII).

Let us form the equation of the fifth degree in U. The values of U be?

ing finite for all finite values of u, the coefficient of U5 is equal to unity. For u = <x> the six values of v are infinite, one of the degree 5, the others of the degree ? (number 408): the five values of U are infinite and of the

degree ^-; the sum of the negative orders of the function U being equal to

27, the equation is of the 27th degree in relation to u (number 135). For values of u exceedingly small the approximate values of v are

We have put f = -^; putting in the same way 0 = -^ = (? ? f 0)

X(?x??4) (?2?f 3)*tne coefficients of the equation in ? being entire polynomes u* (number 401), those of the equation in 0 will possess the same

property; the values of 0 only becoming infinite for u = 0, this equation is of the form

gfl jj=4 ?0 u Ho0*+i; u^p(ap+bpu*+epul?+ ...)05-p+(a5+b5u?+c5u1?+...)=0.

Replacing 0 by U-t-uu and multiplying by uz, we obtain the equation

m80V9) U5 +TuSi^)+1%p+by+...) U^+u%a5 +a5U>+..) = 0.

For very small values of u, the values of U being very small of the degree f , and forming a circular system, we have ^0 = 9, a5 = ? 265f, 8(/?p? 9) +15p> 3, and in consequence fip = or > 10?2p; we will take j9p = 10

?2p. The equation sought for is then of the form



-166-

? = 2*3*(1?q+2qt?Ztf+4tf),

v0 = 2*^(1?q^+2q$?3q$+4qt). We find also for the first combination

tf=i2^(i-^y+/), whence

?7 = ffl-7*, a5 = 0, ?, ?-.aTttj^, a6=2?7< [l-(kM)].

We deduce the second combination from the first in replacing i by ?t.

The following is extracted from Todhunter's Theory of Equations, sec- tions 340, 341.

"The general equation of the fifth degree can always be reduced to one of the following forms:

"x5 + px + q = 0; x5+px2+q = 0; xt+pxP+q = 0; x*+px*+q = 0.

[Due to Mr. Jerrard or to E. S. Bring. Quarterly Journal of Mathemat-

ics, Vol. VI.] "Mr. Jerrard considered that the algebraical solution of equations of the

5th degree could be effected; his proposed method formed the subject of an

enquiry by Sir W. R. Hamilton in the Reports of the British Association, Vol. VI. Most mathematicians admit that Abel has demonstrated the

impossibility of the algebraical solution of equations of a higher degree than

the fourth. An abstract of Sir W. R. Hamilton's exposition of AbeVs ar-

gument will be found in the Quarterly Journal of Maihematies, Vol. V.

A simpler demonstration due to Wantzel will be found in Serretfs Cours a"

Algebre Superieure" [See Analyst,p. 65-70, Vol. IV.] "AnEssay on

the resolution of algebraical equations by the late Judge Hargreave has

been recently published: the results arrived at are to some extent at vari-

ance with those of Abel and Sir W. R. HamtitonP

TO DRA W A CIRCLE TANGENT TO THREE GIVEN CIRCLES.

BY ISAAC H. TURRELL, CINCINNATI, OHIO.

This problem is a celebrated one in the history of pure geometry, When or by whom it was first proposed I am unable to ascertain; but

the best mathematicians of modern times have not thought it unworthy



solution of the equation of the fifth degree

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