Sir William Rowan Hamilton on Equations of the Fifth Degree. 339 



and, therefore, conducts to expressions for a, /3' + 7' + ^^ /^T^? and ^V -|- 7^8'^ + 

 8^j3^, as rational functions of a/, jo, y, r. Again, by the theory of cubic equations, 

 we may write : 



^- = e-\- K-\-\ 7' = e + 0a: + (f\ 8^ = e + 0^ + OX, (10) 



in which is a root of the equation 



02 + -1_ 1 ::: 0, (11) 



while e, *-A, and k^ -|- X^ are symmetric functions of /3^ 7'^ 8^. Making, for 

 abridgment, 



^78 rz 17, Af\ rr <, 



we have, by (10) and (11), 



/r'^ + \3 = ^^ — £3 4- Set, 



and 



/S^ + 7^ + 8^ = 3e, PY + 7^8^ -I- 8'^j3^ = 3 (e' - t) ; 



and, therefore, by (9), 



— 4a = y ; Qi^c^ — e) -zz a/"^ -\- p ; 



— 4tt='+12ae — 8i; = y^+p,r'4-y; ' 

 a* - Qa\ + 8a»7 — Se^ + 12^ = x" + J9x'^ + ya/ + r ; 



conditions which give 



a = ~i^ ; 



e = -^i^(5y^ + 8p); 



t = +^:f(10y*4-ll;>y^+9?^' + p^+12r). J 



Thus, a, e, 7/, and «, on the one hand, are rational functions of x', p, q, r; and, 

 on the other hand, x\ x", a/", a'^^, x^ may be considered as functions, although 

 not entirely rational, of a, e, rj, i. In fact, if these four last quantities (denoted 

 to help the memory by four Greek vowels) be supposed to be given, and if, by 

 extraction of a square root and a cube root, a value of k be found, which satis- 

 fies the auxiliary equation 



/ _ (^2 _ ^3 ^ 3,^) ^3 _(. ^3 _ 0, (17) 



2x2 



(12) 



(13) 

 (14) 



(15) 



(16) 



