348] 
ON THE THEORY OF INVOLUTION. 
299 
10. It may be proper to remark conversely that given the equation Q = 0, if 
in this equation we write a = (a + ka') — ka', ... so that Q becomes a function of 
a + ka, ... a', ... k, then the equation Q = 0 will be satisfied irrespectively of the values 
of a',... k by a plexus of equations involving only the coefficients a 4- ka', ... and which 
is in fact the very plexus (equivalent therefore to two relations) which gives the 
conditions in order that the equation W — 0 may have a threefold root. 
11. Thirdly, suppose that the functions U, V are such that there exists a range 
U+kV=0 having a pair of twofold points. If Aq be the proper value of k, then we 
have U +k l V=(x — ay) 2 (x — /3y) 2 ©, and therefore U= — k 1 V + (x — ay) 2 (x — ¡3y) 2 ©. Hence 
forming the Jacobian of U, V, we have for the determination of the critic centres the 
equation 
j KV, 8 x .(x-ay) 2 (x-ßy) 2 & 
! 8 y V, 8 y . (x - ay) 2 (x - ßyf © 
= 0, 
which is of the form 
(x — ay) (x — ßy) il = 0 ; 
or, we have x — ay = 0, or x — /3y = 0, a pair of critic centres; and for each of these 
the corresponding value of Aq given by the equation —k : 1 = 8 X U : 8 X V is k = k lt so 
that k = k x is a twofold root of the equation in k. 
12. By the like considerations as for the threefold root (observing that if the 
equation W = 0 has a pair of twofold roots we must have between the coefficients of 
W a plexus equivalent to two relations, and of the order 2 (n — 2) (n — 3)), we see 
that the condition for the existence of a range U + kV= 0 having a pair of twofold 
points is of the form P = 0, where P is a function of the degree 2 (n — 2) (n — 3) as 
regards the coefficients of U, and of the same degree as regards the coefficients of V; 
and conversely that, given the equation P — 0, we may find the plexus. 
13. The equation □ = 0 will be satisfied if R = 0, or if Q = 0, or if P = 0; and 
in no other cases. To prove this, suppose that x — ay =0 is the critic centre corre 
sponding to a twofold root Av of the equation in k. We have U= — k x V+(x — ay) 2 ®, 
and thence the equation for the critic centres is 
8 X V, 8 x (x-ay) 2 <d 
8 y V, 8 y (x — ay) 2 © 
= 0, 
which is an equation of the form (x - ay) 0 = 0; and where, corresponding to the 
root x — ay = 0, the equation — k : \ —8 X U : 8 X V gives k = k lt Since k 1 is a twofold 
root, there must be another critic centre also giving the value Aq of k. This new 
critic centre may be either x — ay= 0 (the same as the first mentioned critic centre) 
or it may be a distinct critic centre x — ¡3y = 0. In the former case 
8 X V, 8 X (x — ay) 2 © I 
ByV, 8 y (x - ay) 2 © 
38—2