282 



Wilder'' s Algebraic Solution. 



kino- (D)=0, then (C)=0 ; treating p as unknown, and conn- 

 paring with p3+6p-l-c=0 ; we have 

 3x3j/=-6, (1), 



X^+3C^1J^=C, (2), 



p=-ix'-\-!/x), (D). 

 From (1) and (2) we have a; '^ -cx^- — =0-, 



from (1) then ?/ = — ^^ ; ^^ese two equations joined to (D), 



determine p. 



Let us take for the last example 



There are several functions, resulting from different values 

 of wi, equally proper to resolve this equation. The one, in 

 which the function is most easily calculated, is that in which 

 m=2. Our function is then, 



a;io+S„T« + S,a;« + S,^* + S3^+S_^ (A') 



' (Bj* 



p, and 



x^ -{-yx*+px^+qx^-{-rx-{-s 

 ting 83,84, 



y3_Spy-Oq 



y ~p 



Calculating 83,8,, etc., and afterwards writmg —Y" ^^^' 



for q, we shall have. 



^ 



1 +5?/' U 



— y^ 



+6py^ I 1 -dp'^y-- 

 ^io_p:,^4.Sqy >—a^6_}_36r?/2 y-^x*-^3psy f ^x- 



4.3p2 1 ^^ -^^pqy +2^« J 



-f24r j -72sj/ 



_4g'2 ! 

 _3pr J 



-S'(A') 



, , , (y'-p) 3 , {y' -^py — ^9) 



a;2 -f- ro; 4- s (B') 



2 * ' 6 



Now, if we make (B')=:0, (A') will also be equal to noth- 

 ing ; and the five independents x, p, r, q, and 5, allowing 

 as many separate hypotheses ; we therefore make y, being 



(A') 

 the same in 7g-y, and y^-\'hy^-\-cy^'\-dy''-{-ey+f:czO, 



