Wild€r''s Algebraic Solution. 273 



6 b^ d c2 



.r«-{--x* + (rr — t)^:^ — — =0, which reduced is given in 



most books of algebra. 



^i2_^S,^«+S3a.'4S^_^ (G) 

 Again, let us assume — — v^r . "' Tua' which 



becomes, when (H) is a factor of (G) independently of ^, 



x^-\-yx" -\-px-\-q 

 ip'^ —4qyp^-\-4q^p-\-2q^y^)x*—q* , 



x'^-\-yx^-\-px-{-q 

 (H)=0 and of course (G)=0, we have by comparison with 

 y*-\-bi/^-^ci/-\-d=0, after having divided 



(G)by:r« 4p+^ = -6, (1) 



^-iTq=c, (2) 



eliminating x-'^ from(l) and (2) by the process indicated 

 2jp2 (1) -f^ (2), we have Sp^ + ^^+^bp"" —cq=0. This 

 equation is satisfied by making 4p-{-b=0, (4), and 4q — c=0, 

 (5). The three equations, (3), (4) and (5) joined to (H), are 

 sufficient to determine y, for from (3) we have 

 x^^ -{-{d — ^p-)x^-{-{p*^4:q")x'^ — g4=0, then we obtain by 

 the second example, x'^=^lbcd), and from (H), 



jx^+px+q) _ {x^-ix+i) 

 ^ x^ x^ 



we evidently have { — 4qyp^ -t-^q^y^)x'^, identically nothing, 

 otherwise, there would be a relation between a, b and c, 

 which is not the case. 



Let us write in tyt. '2y for «/, and 2?/= —p for p, which 



changes this function to 



x^"-\-Sy*-Spy^-8qy-'2p^)x^-^{\6y^-S1py^-3'2qy^+24p^y^ 



x^-\-2yx^ 



+ {2y^-p)x+q 



and then we have, by a comparison with the function 

 y^-\-ay^-\-by^-\-cy'^-\-dy^-\-€y^'\-fy-{-g'=0., 



Vol. XVI.— No. 2. 8 



