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



-&z=. w^ -\- u? — w' — tt), 



c — io" — u? -\- u? ~ w, V (30) 



D n w'' — iii^ — ai^ + w. 



But also, by (22) and (21), 



ticdea — ^taicdef * bcdea — t abode ) (, " / 



making then 



^1 abed —— T^abcd) (."^/ 



the twenty-four values of the function t^ will be those of the function t which 

 arise from arranging in all possible ways the four indices 2, 3, 4, 5 ; that is, they 

 are the fifth powers of the twenty-four expressions (24) . . . (29). It is required, 

 therefore, to develope these fifth powers, and to examine into their composition. 

 15. For this purpose it is convenient first to consider those parts of any one 

 such power, which are common to the three other powers, of the same group, 

 (24) or (25), &c., and, therefore, to introduce the consideration of six new func- 

 tions, determined by the following definition : 



VaJc ^^ 5 (T^iabc ~\~ "^aieb "T '^bcia "T '^cbai) » (*^«^) 



which gives, for example, 



V3,, = (- 5a)^ -f 60 (- 5a)^ BCD/378 



+ 10 {(— 5ay + 2bcdj378} (b^/3^ -f cV + d'?') ■ (34) 



+ 5 (- 5a) (b^^ + cy + X)V + 6B^c'i8V + 6c^DVg^+ 6D^B^g^^) ; J 



this being (as is evident on inspection) the part common to the four functions 

 ^23455 T3254, T^523' ^5433, Or to tho fifth powcrs of the four expressions in the group 

 (24). By changing /3, 7, 2, first to 7, 8, ^, and afterwards to 8, /3, 7, the ex- 

 pression (34) for V345 will be changed successively to those for v^^g and v^3j, 

 which, therefore, it is unnecessary to write ; and \^^„ v^y, v^3^, may be formed, 

 respectively, from v.^^, V453, V53^ by interchanging 7 and 8. Or, after substitut- 

 ing in (34) for /3", 7-, 8^, their values (10), and writing 17 for j&y8, it will only 

 be necessary to multiply a: by 6, and X by 6% wherever they occur, in order to 

 change V345 to v^^.,; and to repeat this process, in order to change v^53 to \^: 

 while V345, V453, V534, will be changed, respectively, to V354, Vj„, v^g^, by inter- 

 changing 6 and 0', or k and X. 



