THEIR APPLICATION TO THE SOLUTION OF EQUATIONS. 19] 
includes, as particular cases, the quadrinomials of De 
Moivre and Euler, and the ordinary binomial. 
23. Thus comparing De Moivre’s equation 
2 —5av+5a2r—-2Q2y=0 . . . (12) 
with (10), we see that 
P=a, Q=0, R=-a’, and E=-2y, 
“, P=-R, and S=P?+R=0. 
Now when Q@ and S vanish, the condition (11) is satisfied, 
and 
B, B,=P, B, B3+P383=9, Bi ;+8,63=90, and 
Bi + 68+ B3+ 4P(B; 5 — Bz Bs) =— E.* 
The second and third of these equations are satisfied by 
8,=0, and the fourth becomes 
Bi + B=— E, 
which, combined with the first, gives 
Bi=-YE+ VIE PPayt Ve— ai, 
Gi=-4E- VEEP- Poy - Vop—a; 
and the roots of (12) are the five values of the expression 
2 
(1)? Vy + Vep— a+ (1)? Vy Vor = a, o 
a(1)3 
V y+ VP = 7h 
24. Next, when P vanishes, S=R, and (11) becomes 
(QRE + Q!+R*)?=0, or 
QRE-+ Q*-+ R?=0; 
Euler’s criterion. In this case (Art. 21) 
Bi B.=0, B, K3+R3P;=Q, BiB: +P2As=R, 
and 8} + P2+ Ps=— EH. 
of the quintic, but they are implicitly given. The value of v, which may 
be deduced from Mr. Cockle’s equations, will be found to differ in form 
from that given in the text (Art. 21), a circumstance arising from the dif- 
ference in our methods of eliminating v and of combining the eliminants. 
* It is to be noticed that these equations are employed in order to avoid 
the vanishing fractions which would arise in attempting to deduce the roots- 
immediately from the formule given in Art, 21. 
(1)? Vy + Vop= ai + 
