344 PHILOSOPHICAL TRANSACTIONS. [ANNO 1798. 



2m — 1 2m — 2 o 



izy m _ x + <lm. 2 3 



+ p 



+ (2m — 2) . £Z 



+ r 



4- 2mz 2m ~ 1 



-f- (2m — 1 ) . hz 2 ™-* 



+ (2m — S^rz^- 4 

 + &c. 



>v = 0. 



Assume y = v 2 ; and let the co-efficients of the terms of the former equation be 

 1, b, c, d } &c. and of the latter a, b, c, d, &c. then the equations become 

 y» -J. ^y— « _L. C y m ~* + c^™- 3 -f &c. = 0, 



AT/"*- 1 -f- B^ m " J -f C2/ m - 3 -f &C. = 0. 



These equations have a common measure of the form y ± z, where z is expressed 

 in terms of z and known quantities; and this common measure may be found, by 

 dividing, as in prop. J, till y is exterminated, and making the last remainder 

 equal to 0. 



Now, the first remainder is ((ca — Z>b) . A -f- b 2 — ca) y""* + ((c?a — be) . a + 

 bc — da) y m ~ s -J- ((eA — Z>d) . a + bd — ea) y m ~* + &c ; or, by substitution, 

 a yn-« _|_ fiy m ~ i -+- yy m ~* + &c. and, in a, z rises to 6 dimensions; in (3, to 8 dimen- 

 sions; and, in y, to 10 dimensions, &c. The 2d remainder is ((ex — Bj3) . x + 

 (|3 2 — ay) . a) y»-3 -J- ((Da — By) . a -f- (j3y — aS) . a) z/ m_4 -f &c. ; or, by substitu- 

 tion, wA 2 2/ m-s -|- nA 2 y m ~ 4 -f- &c. and, dividing by a 2 , the dimensions of z in m, are 

 15; in n, 17> &c. Let ?r, x, f, <r, t, &c. be the dimensions of z in the first term 

 of the 1st, 2d, 3d, 4th, 5th, &c. remainders; then ?r = 6, x = 15, f = 2x — tt + 4, 

 <r = 2^ — * + 4, T = 2<r — ? -f- 4, &c. the increment of the m — 1 term of this 

 series is Am + 1, and therefore the m —■ 1 term itself is 1m . (m — l) + m, or m . 

 (2m — l). Now, in the m — 1 remainder, y does not appear, and in that 

 remainder z rises to m . (2m — l) dimensions; if then this remainder be made 

 equal to nothing, and a value of z determined, the last divisor, y + z, where z is 

 some function of z, is known; and this is a common measure of the two equations 

 y m + by" 1 -' -f cj/" 1 - 4 -f- &c. = 0, and Ay m ~ l + By m ~* + cy m ~ 3 + &c. as O; conse- 

 quently, y + z = 0; and 3/ = + z; hence + \Zy, ort>, = ± V ± z; therefore, 

 by the solution of an equation of m . (2m — l) dimensions, 2 roots, z ± y' ± z > 

 of the original equation, are discovered. 



Cor. 1 . Since 2 roots of the proposed equation are z -\- v, and z — v , a: 2 — 2za: 

 4- z 2 — f 2 = O is a quadratic factor of that equation. 



Cor. 2. In the same manner that the 2 equations y m + by™" 1 + cy 1 *-* + & c « = °> 

 and Ay™ -1 -f bj/'" -2 + cy m ~ 3 -f- &c. = O, are reduced to one, may any 2 equations 

 be reduced to one, and one of the unknown quantities exterminated; also the con- 

 clusion will be obtained in the lowest terms. 



Prop. 3. Every equation has as many roots, of the form a ± */ ± t>\ as it has 

 dimensions. — Case 1. Every equation of an odd number of dimensions has, at least, 

 one possible root; and it may therefore be depressed to an equation of an even 

 number of dimensions. — Case 2. If the equation be of 2m dimensions, and m be 



