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 povi^ers of y separately 

 equal to ; a must therefore be a root of the equation formed 

 by eliminating y between the two equations thus obtained. 

 In this manner, by a very simple process, we get the following 

 equation in a, 



64 a" 4. 32 A^a* + 4 (a^^- 4 A4) a"- a^^ = 0, (2) 



which is of a cubic form. 



Equation (2) will be at once recognized as equivalent to 

 the auxiliary cubic 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. 



Xi, iih, X;i, x^, being the roots of equation (1), Lagrange 

 seeks the equation whose roots are the expressions 



OCi + X2 — X3 — Xi X3 + X4 — Xi — X-i . 



Xi-\- X3— X2 — «4 X2-\- Xi — Xi — X-^ 



X\ — p X^ — X2 ~~' X^ Xo, ~j~ ^3 — X\ — X^ 



and finds it to be 



w^+ 8a2M^ + 16(a2'-4a4)m2 _ 64A32 = 0. 



Comparing this with equation (2), we see that Aazzii. If 

 then we put 



Xi ■\-X2 — X3 — X4 = 4 Ui X3 4- a?4 — Xs_ — X2 = 4 04 



^1 + ^3 — ^-2 — Xi = 4«2 X2 + Xi — Xi — x-^ — Aa^ 



Xi + Xi - X2 — X3 = 4a3 X2 + X3 — Xi — Xi = 4«o 



and attend to the relation x^ + «2 + x^ + X4=: 0, which sub- 

 sists, inasmuch as the equation (1) wants its second term, we 

 shall find 



Xi — ai=. ^ (aji — X2) 



X2 — «! = ■§• (X2 — Xi) 

 X3 — t?i := ■2{^oX3-\-Xi) 

 Xi — «i = ^{3Xi + X,^ 



