On the Resolution of Equatims. 277 



and — a?* +2p2—(p4 4-4g2p)j;-4 4-54^-8=0 



or a?i2_(22?=^ -c)a?«+(p*+4^^i?)j;*-?''=0: (3) 



the equations (1), (2), and (3), joined to x'^-{-yx^-\-px-\-q=Q are 



sufficient to determine y. 



(C) 

 Next putp=0 and TVyc becomes 



x^^ —{y'^-{-4:qy)x^+2qy^x^ — q'^ np'x^_ (G) 

 x^-{-yx^ ■\-q p'x^ (H) 



and by composition, 



x"^ — {y'^+Aqy)x^-\-{^q^y^-\-np')x'^—q'^ np'x^ (G^O . 

 x^-{-yx'^-^p'x-]-q p'x^ (H') 



but (H') is a factor of x'^^ —{y'^—4p'y^-\-4qy+2p'^)x^-{-p'* — 

 Aqyp'^ ■^4:q^p-\-2q^y^)x'^-q^, [O"), and consequently of (G')-(G''') 

 or of {4p'y^ —2p'^)x^ — {p"^ —4qyp'^ '\-4q'^p'-{-np')x'^ 'j if we now 

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

 4qyp'^-\-4q"p^—np')x*=^0', n being arbitrary we assume p'* — 

 qyp'^-^4q^p' — np'=^0, and writing forwp' its value in (G'), we have 



and dropping the accent and comparing this equation with p* 4-ap- -{- 



6p+c=^0, we have 4qy= — a, (1) 



4q-^=b, (2) 



and x^ — {y'^-{-4qy)x'^-\-2q'y^ — q'^x~'^=c 



or x^^—{y'^+4qy)x^ + {2q^y^—c)x*—q*=0: (3) 



(1), (2), and (3), joined to x^ -{-yx'^ -{-px-\-q=0 are sufficient to 



(C) 

 determine p. Lastly put ^=0 in "TryT and we have 



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

 x^ -\-yx^ -\-px q' (F) 



and by composition, 



x'^^ -{y* — 4py^-\-2p^)x^-\-p'^x'^-\-nq' nq' (E^) 

 x^-l-yx^'-^-px-i^q' ~~9^' W^' 



but (F) is a factor of x^'^ — [y'^—4py^-\-4(^y-\-2p'^)x^-\-{p'^ — 

 4q'yp-{-4q'^p+2q'^y^)x^ -q'^ ,{¥."), and consequently of (E') - 

 (E'^) or of -4q'yx^—{4qyp^-\-4q'-p-{-2q'^y^)x^-\-nq'+q"^ : the 

 above expression becomes equal to zero vfhen x^ -{-qx^ -\-px-{-q' =^0. 

 Let us make 4q'yx^ —4q'^px* —q*=nq' and then we have, by wri- 

 ting for nj' in (E') its value, x^^ —(y'^ —4py^-\-4py-\-2p'^)x^-\- 

 {p'^+4q^p)x'^ — q'^=0, which compared with q'^ — aq^ +bq-\-c=0, 

 gives 4pa?*=-a, (1) 



4yx^=b, (2) 



