537] solutions of a smith’s prize paper for 1871. 507 
are, it is clear, z — 6w = 0, x — 6-y — 0, where 6 is a variable parameter; and corre 
sponding hereto in the plane we have the line x' — 6y' = 0, viz. this is any line 
through the common intersection of the four conics. 
8. If U, V are binary functions of the form {a, &,...) {x, y) m with arbitrary 
coefficients, and if the equations U = 0, V= 0 have a common root, show how this can 
be determined in terms of the derived functions of the Resultant in regard to the 
coefficients of either function. 
Show what residts in regard to the common root can be obtained when the coefficients 
are not all of them arbitrary but (1) each or either of the functions depends in any 
manner whatever on a set of arbitrary coefficients not entering into the other function, 
(2) the two functions depend in any manner whatever on one and the same set of 
arbitrary coefficients. 
How is the theory modified when, instead of the two equations, there is a single 
equation U = 0 having a double root ? 
Suppose 
U = (a, b,...) {x, y) m a x m + ^ bx m ~ l y + «See, 
V = (a', b', ...) (x, y) m ' (= a'x m ' + ^ bx m '~ 1 y + «Sec.j . 
Then if R is the resultant, the equation R = 0 is the relation which must exist 
between the coefficients (a, &,...) and {a’, b',...) in order that the equations U= 0 and 
V = 0 may have a common root (that is, in order that the functions U, V may have 
a common factor x — ay). Imagine the relation subsisting, and that x, y are the values 
belonging to the common root, or (what is the same thing) that we have x — ay — 0; 
we have then simultaneously U= 0, V— 0, R = 0. Now suppose the coefficients a, b,... 
to be infinitesimally varied in such manner that U, V have still a common root; say 
the new values are a + ha, b + 8b,... : this implies between 8a, 8b, ... the relation 
dR 
da 
8a + ^8b+... = 0. 
db 
But the common factor x — ay is a factor of the unaltered equation V = 0; and the 
values of (x, y) are thus unaltered, viz. the equation U = 0 is satisfied with the 
original values of (x, y); so that we have 
or, what is the same thing, 
dU 
da 
8a + 
dU 
db 
8b + ... =0, 
x m 8a + mx m ~ 1 y8b + ... = 0, 
an equation which must agree with the former one, that is, we have 
mx m ~fi : «See. = 
dR 
da 
dR 
db 
&c., 
64—2