298 
ON THE THEORY OF INVOLUTION. 
[348 
Now in general 
Disc*. PQ = Disc*. P . Disc*. Q. [Result. (P, Q)] 2 , 
and hence in the present case 
Disc*. (U + kV) = Disc*. (U' + kV').(U' + kV 0 J, 
where U 0 '+ kV 0 ' is what U' + kV' becomes on writing therein x = ay and neglecting 
the factor y 1l ~ l which then presents itself. We see therefore that in this case the 
equation k has a twofold root k = — U 0 ' -e F 0 '; a result which might also have been 
obtained from the consideration of the Jacobian. 
7. The condition in order that the ranges TJ =0, V= 0 may have a common 
point is 
Result. (U, F) = 0, 
say P = 0. P is of the degree n in regard to the coefficients of U, V respectively. 
8. Secondly, suppose that the functions U, V are such that there exists a range 
TJ + kV — 0 having a threefold point. If k x be the proper value of k, then we have 
U + k 1 V= (x— ay) 3 ©, and therefore TJ = — & a F + (x — ay) 3 0. Hence forming the Jacobian 
of TJ, V, the equation for the determination of the critic centres will be 
which is of the form 
KV, K'(%- ay) 3 0 
h y v, K- (>- ay) 3 © 
(x — y.y) 2 fi = 0 ; 
or we have (x — ay) 2 = 0, a twofold critic centre. The corresponding value of k given 
by the equation — k : 1 = S X U : 8 X V is k = k lf and we have thus k = K as a twofold 
root of the equation in k. 
9. But if the range W=TJ-\-kV=0 has a threefold point, or what is the same 
thing, if the equation W = 0 has a threefold root ; then we must have between the 
coefficients of W a plexus of equations equivalent to two relations. Such plexus is 
known to be of the order 3 (n — 2). This comes to saying that if the coefficients of 
W are assumed to be of the form a + kti + la", ... and if between the several 
equations of the plexus we eliminate k, we obtain for l an equation Q = 0 of the degree 
3 (n — 2). The equation in question would be of the form Funct. (a + la", a',..)= 0. 
Hence Q is of the degree 3(w —2) in the coefficients (a,...) of U. And in a similar 
manner Q is of the degree S(n — 2) in the coefficients (a',...) of F. And omitting 
altogether the terms in l, or taking the coefficients of W to be a + ka\ ... if from 
the equations of the plexus we eliminate k, we find an equation Q = 0, where Q is a 
function of the degree 3 (n — 2) as regards the coefficients of U, and of the same 
degree as regards the coefficients of F We have thus found the form of the condition 
Q = () which expresses that there may be a range TJ+kV— 0 having a threefold point.