22 
[490 
490. 
ON A PROBLEM OF ELIMINATION. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xi. (1871), 
pp. 99—101.] 
I WRITE 
P = (a, y, zf , Q = (a', ...Jx, y, zf, 
U = {a, y, z) m , V=(b, y, z) n , 
and I seek for the form of the relation between the coefficients (a, ...), (a, ...), 
(a,...), (bin order that there may exist in the pencil 
XP + pQ = 0 
a curve passing through two of the intersections of the curves U = 0, V = 0. 
The ratio X : p may be determined so as that the curve XP + pQ = 0 shall pass 
through one of the intersections of the curves U = 0, V = 0; or, what is the same 
thing, so as that the three curves shall have a common point; the condition for 
this is 
Resit. (XP + pQ, TJ, V) = 0, 
a condition of the form 
(Xa + pa', ...) mn (a,...) kn (b, ...) km = 0 ; 
or, what is the same thing, 
(a, ..., a', ...) mn (a, ...) kn (b, ...) km (X, p) mn = 0, 
which, for shortness, may be written 
(A, .. $X, p) mn = 0.