(51.) 

 (52.) 



'"-^ T. Y. ( • ' (^^') 



MKTHOD OF TRANSFORMING AND RESOLVING EQUATIONS. 303 

 A + M = L (50.) 



and then the two parts, of which the expression for 1/ is com- 

 posed, will take the forms 



A/ A. , . // A. , A /// A . 



a- + A X + A a; = Po + Pi 'f! 



M' x''' + M" oT" + M'" x''"' + Miv a^'" =;/^ ^j/^ ^ 



and the expression itself will become 

 y =/(.r) =;^o+i?o+ {P\ +/>'!)* 



At the same time we see that the case to be avoided, for the 

 reason lately assigned, is the case of proportionality of j^'oi p'm 

 ' • • P'm-v ^l^o->Pii ' ■ •Pm-v ^^ ^^ therefore convenient to 

 introduce these new abbreviations, 



irrr" ^=^-» 



^ wi— 1 



and 



P'o -PPo = 9o, P\ -PPl = 9l, - P'm-2-PPrn-2 = 9ra.i', (55.) 



for thus we obtain the expressions 



p'o = ?0 +PPo^ P'x = q\+PPi, '" K„_2' 

 = Ira-^ + PPm-2^P'm-l = PPm-V 



and 



y =/(•*•) = (1 + P) iPo+Pi'^' + —Pm-i •*"'"~^) 



(56.) 



(57.) 



and we have only to take care that the w — 1 quantities, 

 qQ, q^, ... 9'^_2 shall not all vanish. Indeed, it is tacitly sup- 

 posed in (54) that P^_y does not vanish 3 but it must be ob- 

 served that Mr. Jerrard's method itself essentially sujjposes that 

 the function A' x + A" x + A"' x is not any multiple of 

 the evanescent polynome X, and therefore that at least some 

 one of the m quantities Po, Pi, "• P^_i is different from 0; 

 now the spirit of the definitional assumptions here made, and of 



