^iR William Rowan Hamilton on Equations of the Fifth Degree. 361 



5 "^l \J^2 'T" '^0 "V "^d I "^eji 

 % (^2 "^0 r ^c '^2 1 "^d ^e "T" -^e "^dji 

 ^ y^c '^d I •'^2 "^s ~T~ "^e "^2 ~r '^'' "^c-V' 

 ^ (^JT^ .Tj -{- Xg Xi -j- JTj J^c "T~ Xq X.^Ji 



|(rr/ + ^/ + a^/ + a:2*)a;,; 



(137) 



which are to be combined with the other parts of y, derived, in like manner, 

 through X, from the other terms of w, and to be submitted to the processes in- 

 dicated by the foi'mulae (132), in order to deduce the values (133) of n', and 

 n"„ and thence, by (131) and (98), the relation (134) between n, and Mj, which 

 conducts, by (130), to the expression (135). For example, the first and last of 

 the five parts (137) of y, contribute nothing to either of the two quotients 

 (132), because those parts are symmetric relatively to x^ x^, Xe', but the second 

 part (137) contributes 



— I (^i + ^i ^d + ^2 a;/ + a-/ -f xj" + J-/ Xc 4- X, Xc^ + x,'), 

 to the quotient 



* crfe ^ edc 



(•^2 — "^d) (-^e — •*"c) 



and 



+ I (-^2' + -^2' ■^■e + .^2 ^e" + ^e' + ^c' + ^c" X'i + X^ X^ + ^/), 



to the quotient 



Ycde ^dci 



(138) 



(139) 

 (140) 



(141) 



\«*2 ^e) \X(. XfiJ 



this second part (137) of y contributes therefore, by (132), 



— I (•^2' + ^i ^3 + -^2 ^i + ^f + ^' + ■^4' ^5 + ^* ^t + -^a'), (142) 



to the quotient n'j, and the same quantity with its sign changed to the quotient 

 n", : and the other parts of the same two quotients are determined in a similar 

 manner. 



32. The two other factors (119) may respectively be expressed as follows : 



^ 125 (1 - e^) {x., - X,) {x^ - X,) M2, (143) 



and 



- 125 (0 - 0") {x^ - X,) {x, - X,) u, ; 



(144} 



VOL. XIX, 



3 A 



