I 



* 



Oji the Resolution of Equations. 277 



h 



F 



and —0?* +2p^ - {p^ +4q^p)x'^+q^x'^ ^ =c 



or x^^~{2p^ -c)x''+{p^+4q''2^)v^-q*=0: (3) 



tlie equations (1), (2), and (3), joined to x^ -^-yx^ -^^X'}'q=Q are 

 sufficient to determine y. 



(C) 

 Next putp=0 and Tvyc becomes 



a; > 2 „ (^4 _j_4^y^^8 ^Oqy'^x'^ — q^ np^x^ (G) 



^H-y^M-2 "" p'x^ ' (H)' 



and by composition, 



but (IP) is a factor of x^^ -{y^ ~4fy^'\-^y+2p'^)x^-\-p^ 

 4qyp'^+4q^p+2q^y^)x'^q\ (G^^), andconsequendyof (G'HG'') 

 or of {4pY -2y3ja?3 — (p'* -4qyf^ ^4qY+np')x^ j if we now 

 make x^-^-yx^ -{-p^x+q — O, we also have {4p^y^ —2^^-)x^ 

 4qyp^^-\-4q-p^-np^)x^—0; n being arbitrary we assume »'* 



qyp^^ '{-4q^p^ — np' 



(G-) 



i a?' 2 --(y*-f-4g'y)a?«4-(p'''-42'y/3'3_|.4j3p/_|_222y2j^4_o4==.o, 



Jp -f c =- J we have 4qy= — a 



+ap3-f- 



42='=^^; (2) 



and o;^ — (^y*-{-4qy)x^-{-2q^y^ — q^x~*=c 



or a;'2 _(y4_^4^^)^s_^(222}/2_c^^4_^4_0: (3) 



(1), (2), and (3), joined to x^ +yx^ -^^x-^q=0 are sufficient to 



determine j9. Lasdy put q=0 in TyyT and we have 



a?^3— (y«— 4j?y3_{_2^3)j8-^^4j.4 „^/ (E^ 



and by composidon, 



x^^ -{y*—4j)y^+2p^)x^-]-p*x^+nq' nq' (E') 



W (F) is a factor of x^''~{y*—4j)y^-\-4q'yi-2j>'^)x^-i-{p*- 

 4q'yp-\.4q'''p-^2q'^y^)x* -q'\ (E"), and consequendy of (E') 

 (E") or of ~4q'yx^—{4qyp''-{-4q'^-p-\-2q'^y'')x*-\rnq'-\-q'^ : t 



Let 



+pT+2'=0. 



4q'^px* — 2* 



wri 



ting for ng' in (E') its value, x^"" -{y^ —4py' -{-4py^2p')x''.-{■ 

 iJP^-\-4q^p)x^ -qi=Q^ which compared with q^ - aq- -\-hq-TC=Oy 

 gives 4px*=-a, {\) 



4yx^=h, (2) 



