280 Wilder's Algebraic Solution. 



\-q 



— for y 111 (G), 



x^-\-px-\-q 

 here, writing 3/'=- — for y in (G), we have 



y'i - 4py'^ -{-Aqy' -\-2p2 — 



a;' 



x' 



And since p, q and a; are independent, we may make any 

 three hypotheses we choose ; accordingly, comparing 



y4_4^y2+4^y/+2p2_-^l±-^^+^_xS with 



a function y'" -\-by'^ -\-cy'-\-d=tO, y' being the same in both 

 functions, we have 4p= - b 

 4q=c and 



the same as obtained before. 



The fourth proposition gives the same result, perhaps more 

 satisfactorily. Continuing to denote by x the function (p{bcd), 

 we shall then have, 



y'=- (x^-\-px-\-q) , 



y'= —(a^x^-{-apx-{-q), 



a^x^ 

 y'=z — {a6x^ J^a ^px-\-q) , 



y'— — {a^x^-\-a^px-\-q) 



—-^ ; let us transpose the 



second members, and it requires no great skill to see that 

 the continued product of the four factors will be, 



y'i — 4py'^+4qy'-{-2p^ — (p* - Aqy'p^ -\-4q^p-{-2q^y'^) 



_ 



q^ 

 +— - a;* =0 ; but by the hypothesis, 



(p^-\-Aq^p) (/4 

 y— 4p3/'4-4(/?/'+2p2- — 4.— _a;4=o, conse- 



— 4qy'p^+2q^y'" 

 qaently, ^ =0 ; hence, the product of the 



four factors is y''^-\-by'^-^cy'-{-d=0. 



Let us now take y^-{-by^-\-cy'^-\-dy-]'e=0, for the given 



(A') 

 equation. Here w=5, and if we make m=5, the function 7^77 



will be proper to resolve this equation. 



