s * 
A New Method of Resolving certain Equations. 89 
To resolve a general Equation of the fourth degree. 
Let the given equation be, a4 +4dan* +662? +4cx-+-d=0. 
wt +-4amex* +4emx =1—m.x‘ —6bmz? —dm, 
Add (4a°m? +20)x% io aa fn both sides ; 
2 oe 
a*-+4amz°+(4a*m? +2-) 2? --dcme+—5=1 —mxt+ 
: c c? 
(4a?m?+2° —6bm)x* +75 — 
Bo: ets “ya =1—m.x4 2 whee —Gbm42-).0? or 
bac ~~ oo 
Rsdice the lastmember andl at —m ohn dn). 
Then VImmx f= — an=20" m* — Some mc \) 
Also a? + 2ame-+ S/T mia? +S Oo ie (B) 
Now (A)  4a*m? —6ba*m+2ac=o 2a? —2a?m Xf 2c? 
—2da*m, Put 2a2m=2z. 
Then 2? —3bz+ 2ac=f 2a? —2z Xf 2c? —dz. 
Expanding; z* —6bz? Li ae | ea — 12abez+-4a?c? = 
c? —2a?dz—2c?z-+dz?,, 
Transpose and divide by z; he +(4ac+9b? —d)z— 
2abc+-2a?d+2c?=0. (C). 
Then (B) ax? iteteves —a?m. x? +-4/c?—da* m. 
ax*-+-zx-+c = s 02 fet —dz = (D) 
Vou, IX.—No, 1. 12 
