of Algebraic Equations of the Fifth Degree, 567 



will denote a function of that value of P, the index of which 

 is/(ab)(cd)... 



37. It is evident from the equations (s.) and (t.) that we may 

 have 



e„,/(ab)(cd) . = yK + ^'""^H + ''"S + ''''^S + ''^^. )ft!^)ft?).. y (^,) 



or 



0«,/(ab)(cd). .= y(^. + i''^^ + »2«^^ + .3«^^ _^ ,4«^^ )(ab)(cd).. (^ ^^^ 



if e„,/(ah)(cd). = »" (G)"«,/)ft!')(c.?). . 



wherein ^ and >; are not independent of (^.!^)(^.^). •• This 

 theorem is remarkable not only for its hypothetical character, 

 but also for being composed of two branches. 



38. If we suppose the operation denoted by (j^y) Q^,^^ •• 

 to take the particular form Ta b^ ^a b^, there will result from 

 the first branch, 



^n,f= V^^- + '^"•'■/^ + '^''^y "^ •^"''s + '''•''^^ I. (Vj.) 



if 0„^y=.?10'„^^ J 



Further, if, observing that P//,3sWyS\ = P^, and consequently 

 0„,/(-/j3)(yS) = 0w,/5 we suppose ('a b^ ('cd^ .. to become 



if e.,/=^(0'„,^)(^a)(yS) 

 Lastly, since l^f(cc(i)(yi) = I/) the same branch will give 



if 0„,/=^(0'.,/)(«.^)(yO J 



And analogous results will be obtainable from the other 

 branch (w.). 



Now we see that (vj.) cannot generally appl^' to ®"„y (since 

 the condition B"„ y:-=: »^@'^y> cannot be satisfied without in- 

 ducing certain relations among a-j, x^^ . . x^^) ; but that it will 



* See the equation (u.). 



