556 
NOTE ON BEZOUT’S METHOD OF ELIMINATION. 
[381 
and thence 
A — P, 
B =Q-Px, 
C —R — Qx, 
D = S — Rx, = — Rx. 
Let a be an arbitrary quantity and write 
Dz = \ U , U' 
\ aa 3 + ba 2 4 ca 4 d', a!a? 4 b'a 2 + c'a 4 d' 
we have it is clear 
□ = Aa 3 4 Ba. 2 + Ca + D, 
= a 3 P 4 a 2 (Q — Px) 4- a(R — Qx), = Rx, 
= (a 3 — a 2 x) P + (ar — ax) Q + (a — x) R, 
and thence 
—5— = a 2 P + a.Q + R. 
a — x 
The equations P = 0, Q = 0, R = 0 are respectively quadratic equations in x, the 
equations which are used in Bezout’s method of elimination; and representing them by 
P = Lx 2 + Mx + N , =0, 
Q = L'x 2 + M'x + N', =0, 
R = L''x 2 4 M"x 4 N", =0, 
we have 
L , 
M , 
N 
L', 
M', 
N' 
L", 
M", 
N" 
as the equation resulting from the elimination of x from the equations U — 0, U' = 0. 
The foregoing investigation shows that the functions P, Q, R are obtained as the 
coefficients of a 2 , a, 1 in the development of 
1 
a — x 
U , U' 
aa 3 4 ba: 2 4 ca 4 d, a'a 3 4 b'a 2 4 ca 4 d' 
or more generally, taking U, U' to be any two functions of the order n, that the n 
functions P, Q, R, &c. each of the order n — 1 are obtained as the coefficients of 
a. n ~i, a n ~ 2 , ... a, 1 in the development of 
1 
a — x 
U, U' 
u a , u: 
where U a , Ud are what U, U' become when x is replaced therein by a: and we 
have thus a simple cl posteriori verification of the form in which, several years ago, 
I presented Bezout’s Method of Elimination. 
2, Stone Buildings, W.G., March 5, 1863.