83 < Mr. waring on the General 
other quantity whatever, and there will refult an equa- 
tion of the formula of a quadratic from which the other 
b or a may be found, whence from the equation ( p = 
— ) p may be deduced, and confequently the refolution 
of the cubic required. 
In the fame manner for p may be affumed any quan- 
tity whatever, and in the equation qj= a}p + b l p z for b 
fubftitute its value — , or for a its value — , and there 
3 «P sty 
• P 3 P 3 
refult the equations qj= a l p + , and qj= — — + ^ 3 />% 
which have the formula of a quadratic, from which 
may be deduced the refolution of the cubic r quired. 
2. Let the refolution affirmed be x -a 'p + by P z + 
iC-yp 3 ; exterminate the irrational quantities, and there 
refults the equation x z -( 2 b z + /^ac)px - 4 (a z bp+bc z p 2 ) x- 
a ! p + + 2 a z c z p z - <\,ab z cp z - o ; fuppofe p - 1 , and 
the given equation x‘ i +qx z ~rx + s=o, let the correfpon- 
dent terms of the given and refulting equations be re- 
fpedtively made equal to each other, and there refult 
the three equations ib z + 4 ac — —q, 4 b (a z + c z ) = r, and 
^-e 4 +c 4 -- 2 a z c z + 4 ab z c -—s; reduce thefe equations in- 
to one, fo that the unknown quantities a and c may be 
exterminated, and there refults the equation 4 b^ + iqb* 
7. % 'l 
+ (^-j) b z ~~-o of the formula of a cubic, from which 
. the 
