562 Mr. G. B. Jerravd's Reflections on the Resolution 



the equation for Vp would not rise above the third degree. 

 But although the eight functions, 



Vp, Vp,, 



Vp/ denoting what Vp becomes when / is changed into /', 

 will be necessarily equal to each other; and consequently if/ 

 be changed successively into i, k, I, and F into I, K, L, there 

 will arise, the index 1 remaining fixed, eight functions equal 

 to Vi ; eight to Vr ; and eight to Vl ; we must not conclude 

 that Vp/j^x, in which the index 1 is supposed not to be fixed, 

 will be capable of coinciding with one of the three functions 

 Vj, Vk> Vl. It is true that all the roots of the equation 

 P^2 + Bi P^' + . . = may be evolved separately from a single 

 expression whilst the index 1 remains fixed. But the ques- 

 tion here relates to the possibility of evolving them four at a 

 time in a certain definite order. And in effect if we examine 

 the function Vpz-^^y we shall find that 



wherein I suppose y„ (^/3y), y^(^/3y^, . . to be denoted by 



g^^ 5"y» • • ; A(^.?)' f(i(P,D> • • ^y^c^' -^S' • • ; and the accent 

 attached to the ^ and h to indicate, as before, a change of a^ 

 into a 2*. Whence it is clear that the four roots which com- 



* It may be easily shown that 



Observing that A^— -^^ + j/s + ^i^af and that consequently 

 h^ (|3 5) (xO — ^ ^'«' ^^ immediately find from the theorem (h.) 



where <r and t may have more than one set of values assigned to 

 them. Hence the same expression for P may be evolved from 



*'''e.f)»=°lasf.on,'''(a(tO = 01 



*V(8f)„ = oJ V(/?.8)(r.0=ol 



We must accordingly have 



