Second Cafe of the Cubick equation x % —qx—r. 915 
Invejligation of the third exprejfion of the value of the 
root x. 
1 1. The third expreflion for the value of x, or the 
laft of the two mentioned in Art. 8. to wit,^ 3 |^+^j 
sl 3 or \J 3 | tH + \j 3 \ T~ s > ma y be Gained 
as follows. 
Since a*+h } is -r, it follows that h % will he —r-a‘\ 
But a 3 is lhewn in Art. 5. to be ~-r + \] 
4 27 
Therefore 
r-a i is ■= r— \g + \j Confequently b l 
is - WF? and b is = P tw 
{ r / 
rr 
q 3 
— 
2 1 ’ 
1 2 \ 
4 
*> is - fpli-ff- t,< m (p'> ttin g". 
as before, \] 3 [i+s+\j 3 Q- E. I. 
Synthetick demonflration of the truth of the f aid third ex- 
preffon, 
1 2. Here again we may demonftrate fynthetically, 
that this expreflion is equal to the true value of x in the 
equation x i — qx = r, by fubftituting it for x in the left- 
hand fide of the faid equation. For, if we make that 
fubftitution, we Ihall find that the value of x % - qx, 
5X2 thence 
