112 
their signs. In order that this condition should be fulfilled, 
we must have, in the transformed equation, the sums of the 
terms containing the even and odd powers of y separately 
equal to 0; @ must therefore be a root of the equation formed 
by eliminating y between the two equations thus obtained. 
In this manner, by avery simple process, we get the following 
equation in a, 
64a°+ 324,0'+4(a.—4a,)@’—aZ = 0, (2) 
which is of a cubic form. 
Equation (2) will be at once recognized as equivalent to 
the auxiliary cubie arrived at in Lagrange’s, and indeed in 
every other known method of solving the biquadratic equation. 
Nor is it difficult to shew why the roots of Lagrange’s auxiliary 
cubic are thus related to the different values of the quantity a, 
by which the roots of the equation (1) are diminished in the 
method here presented. 
Ly Vay Lz, X4, being the roots of equation (1), Lagrange 
seeks the equation whose roots are the expressions 
& + Ly — U3 — Wy Lz + Ly — Ly — 2p. 
X, ++ 13 — 2 — Uy @y + %y— X% — Us 
& + Uy — %— 23 y+ %3 — Xj — WM 
and finds it to be 
u® + Saou’ + 16(a.°—44,)u? — 644,? = 0. 
Comparing this with equation (2), we see thatda=u. If 
then we put 
YQ +a — %3— &y = 4a, t3+%4,—2%,—Mm=4a% 
@ + 23 — %,— A= 4a, Ly + %y — &% — 4; = 4a; 
2, + %y — %— 1; = 4a; La + %3 — 1, — %y = 404 
and attend to the relation a, + % + 23; + a;=0, which sub- 
sists, inasmuch as the equation (1) wants its second term, we 
shall find 
xv, — a = $(a@ — ®) 
Lg — Gj = 3 (%2 — 2) 
Xs — a, = $(3434+-%) 
Vy — A, = ¥(3%44+23) 
