A Aero Method of Resolving certain Equations, 89 



To resolve a general Equation of the fourth degree. 



Let the given equation be, a;*+4ax^4-66x2+4ca3+c?=K>' 

 x*-|-4amx^+4cmx =1 — m.x^ — Gima;^ — </m, 



c c^ 



Add (402^2 -{-2-)x2 + --, to both sides; 



a;*-f-4amx=» + (4a3m2 4-2-) x2+4cmx+^ = l -m.a;*-f 



(4a^m^4-2- — 6^m)x2 + -^— am. 

 c c 



—^ — dm» 



a^ 



Assnmethelastmember =^{y\—m,x^'\-'\/^ — dm)^» 

 Then v'T-^X V— -^'^^=2«'^' — 26m+ (A) 

 Also x='+2amx+-=\/l— m.x2 4-^£_^f7m (B) 



Now (A) 4a*m2 -66a^m +2gc=:\/ 2a^ — Sa^wX-Zac^ 



— 2da^m, Put 2a ^m =5'. 



Then 2r2_3^^^2ac = v/2a2 _5;x-v/2c2— Jz. 



Expanding; sr" -6fe2;3-}_(4Gc+962)5;2_i2a5c2:4-4a2c8=; 



4a2c2 — 2a2f?2;-2c22r+t^z2. 



Transpose and divide by sr ; s^ —662:2 _^ (4ac4-96 ^ — c?)s — - 



12a6c+2a2d4-2c2=0. (C). 



Then (B) ax'''i-2a'^mx-{-c = \/a^ —a^m.x^-\-\/c'^-da^tn, 

 ax^+zx+c =^a^—z .x'^-{-\/c^ — c?z (D) 

 2 g 



Vol. IX — No. 1. 12 



