258 



Sir William R. Hamilton on the Argument of Abel, 



In fact, the conditions just mentioned give, in the first place, expressions for 

 a, j8, 7, 8, a', as functions of the five roots a;,, x^, x^, x^, x^, which functions are 

 rational and integral and homogeneous of the fifth dimension ; they give, next, 

 expressions for ^', 7', 8', a", as functions of the tenth dimension ; for (3", 7", a'", 

 of the thirtieth ; and for a^^, of the sixtieth dimension. And Mr. Murphy has 

 rightly remarked that this function a^'' may be put under the form 



a^^ = kk^. a/. A3*, a/, a/, a/. Bj*. . b/. c,*. . c^. d/. . d/. e,*. . e/, (13) 



in which A; is a numerical constant, and 



Ai = X^—X^ + w {X^ — X^) + «.* {X:,—X^), 

 A2 = ^3-^2 + «'(-^5— -^2) + "'''(^5— ^4)> 



A3 = ^4— a;^ + «. (ajj— 0^5) + «.* (x^— 0^3), 

 A4 = iFj — 0^3 -f- w (a;^ — a;3) + w" {x^— x^, 



K = ^3— ^^4+ («'+ «') (^5—^2) ; J 



these six being the only linear factors of V -r- which do not involve x^. But the 

 expressions (14) give, by (7), 



\\^J AiA2A3A4= {x/+X3* + a;/ + x/— (x, + ir,) {x^-\-oi;,)Y 



+ { (^2— a^s)' + («2— ^4) (^5—^3) } { (^3-^4)' + (^2—^3) (^5—^4) } ; (16) 



and the expressions (15) give 



(14) 



(15) 



1+' 



A5 As = (^^3— ^4)'+ (^2— ^5) (^3-^4)-(^2— ^5)' ; 



(17) 



the part of a^'', which is of highest dimension relatively to x^, is therefore of the 

 form 



Na;«(K* + a?3* + a^/ + a^/-(x, + ar,)(^3+^4)P 



+ {(a'2— 3^5)' +(^2— -^4) (^5—^3)} {(■«^3--^4)'+(^2— ^3)(^5— ^4)}) 



X {(^3-^4)' + (^2-^J (^3--^4) - (^2-^5)'}^ (18) 



N being a numerical coefficient ; and consequently the coefficients, in a''^, of the 



