360 Mr. 6. B. Jerrard on a Method of Tramforming Equations 

 Accordingly we shall have piiiHttii 



1 



R=— ^^@. (4S-f5T);)wlbi9rf* lofinflo « 



We conclude, therefore, since we may assume T=l, that neither 



P nor R will be of the form . „ > unless S Be of i^ariorm. 



Substituting now these expressions for P and E til t^^ (^qiiation 

 @.(0P + 2R + 3S + 4T)2=0, '.;',;,:,. . 

 the first member of which is an integral function of P, Rj>S> 3\ 

 we shall obviously be conducted to a quadratic equation in S 

 with determinate coefficients. This equation I shall represent by 

 aS2 + 2/9ST+7T2, *- '^''-'^"^ ^ ^ 



a, yS, 7 being certain rational functions of Ag, A4, . . Km -' Tne 

 expression for S will consequently be 



u 

 Unless, then, a be equal to zero, S will not take the form 



G 

 (l-l)HV ,, 



Now since, in order to obtain a, we need not consider the 

 whole development of the function g. (0P + 2Rh-3S + 4T)2, 

 but only that part of it which is aifected with S*, it is clear that 

 if we assume ^^^ 



P=;?S+y, ^' 



R=rS + r', 

 assigning to jo, p\ r, r' such values as are deducible from the 

 expressions previously found for P and R, we shall obtain a by 

 merely writing p and r for P and R respectively, 1 for S, and 

 suppressing the term 4T. Hence 



a = @ . (Op + 2r + 3*)^ ^^ »terf« ^^w- 



where 



@3 ©4 , 



Further, if we expand @. (0p+2r + 35)^ according to the 

 descending powers of 5, the expression for a will become 



©3* s^ H- 2©3;. (Op -f 2r)s + © . (Op + ^rf ; 

 • or since ©0 = w, ©2 = 0, 



{(©3)^-©6}5« + 

 2{(m-l)©3;?+(-@5)r}^ + 

 w(m-l)/ + (-©4)r2; 



