372 
ON THE THEORY OF ELIMINATION. 
[59 
Suppose that there are three sets of equations, or g = h — k + l: we have here three 
systems, ii, Cl', Cl", of which Cl contains h vertical and h — k + l horizontal rows, 11' 
contains h vertical and k horizontal rows, Cl" contains l vertical and k horizontal rows. 
From any l of the k horizontal rows of Cl" form a determinant, and call this Q"; from 
the k — l supplementary horizontal rows of Cl', choosing the vertical rows at pleasure, 
form a determinant, and call this Q'; from the h — k + l supplementary vertical rows 
of il form a determinant, and call this Q: then Q" divides Q', this quotient divides 
Q, and we have V = Q-h (Q' Q"). 
Suppose that there are four sets of equations, or g = h — k + l— m: we have here four 
systems, Cl, Cl', Cl", and Cl'", of which fi contains h vertical and h — k + l — m horizontal 
rows, Cl' contains h vertical and k horizontal rows, Cl" contains l vertical and k horizontal 
rows, and Cl'" contains l vertical and m horizontal rows. From any m of the l vertical 
rows of Cl'" form a determinant, and call this Q'"; from the l — m supplementary 
vertical rows of Cl", choosing the horizontal rows at pleasure, form a determinant, and 
call this Q"; from the k — l + m supplementary horizontal rows of Cl', choosing the 
vertical rows at pleasure, form a determinant, and call this Q'; from the h — k + l — m 
supplementary vertical rows of £1 form a determinant, and call this Q: then Q'" 
divides Q", this quotient divides Q', this quotient divides Q, and V = Q -f- {Q' -4- (Q" 4- Q'")}. 
The mode of proceeding is obvious. 
It is clear, that if all the coefficients a, /3,... be considered of the order unity, 
V is of the order h — 2k + 31 — &c. 
What has preceded constitutes the theory of elimination alluded to in my memoir 
“ On the Theory of Involution in Geometry,” Journal, vol. n. p. 52—61, [40]. And thus 
the problem of eliminating any number of variables x, y ... from the same number of 
equations U=0, V—0,... (where U, V,... are homogeneous functions of any orders 
whatever) is completely solved ; though, as before remarked, I am not in possession of 
any method of arriving at once at the final result in its most simplified form ; my 
process, on the contrary, leads me to a result encumbered by an extraneous factor, 
which is only got rid of by a number of successive divisions less by two than the 
number of variables to be eliminated. 
To illustrate the preceding method, consider the three equations of the second order, 
U — a x- + b y 1 + c z 1 + l yz + m zx + n xy = 0, 
V = a' x 2 + b'y 2 4- c' z 2 + V yz + m' zx + n' xy = 0, 
IF = a"x 2 + b"y 2 + cV + l"yz + m"zx + n”xy = 0. 
Here, to eliminate the fifteen quantities x A , y 4 , z 4 , y 2 z, z z x, a?y, yz 3 , zx?, xy 3 , y 2 z 2 , z 2 x 3 , 
x 2 y 2 , x 2 yz, y 2 zx, z 2 xy, we have the eighteen equations 
x 2 U = 0, 
x 2 V = 0, 
x 2 W = 0, 
y 2 U = 0, 
y 2 V =0, 
y 2 W = 0, 
z 2 U = 0, 
z 2 V = 0, 
z 2 W=0, 
yzU =0, 
yzV =0, 
yzW = 0, 
zx U =0, 
zx V = 0, 
zxW = 0, 
xyU =0, 
xy V =0, 
xy W = 0,