MB. W. SPOTTISWOODE ON HYPEEJACOBIAN SUEEACES AND CURVES. 355 
A, B, C, F, G, H=U, H ,v, w, k, ^ 
cp, a , b , c , d, y (5) 
v|/, a', V, c’, d’, J 
i.e. A, B, . . are the determinants formed from the matrix (5), by retaining the first 
column (on that account marked by the asterisk *), and omitting the four others, two 
and two, in the same order as before. We may then form the following system : — - 
(Qz— By) : m=A+mA,, (P£ — S#) : m=F-\-mF ] , 1 
(Bx—Fz) : m=B+mB 15 (Qt—Sy) : m=G+mG 1 , ^ . . . (6) 
(Fy — Qx) : m=C +mC l5 (R£— Sz ) : j 
any two of which may be regarded as the equations of the curve in question. 
By means of these equations it may easily be shown that at the principal points each 
of the surfaces (6), or say each of the surfaces 
x, y, z, t=Q 1 ^ 
P, Q, R, S J 
touches, or has two-branch contact with U. For, on differentiating the first equation 
of (6), we obtain 
B,.(Qz— Ry) : m=A E A+mMA 1 +TOUd. c Aj. 
But, on reference to (5), it is clear that at the points in question B v , B,, B,)A=0, 
or more generally 
(d x , B„ B„ B,)(A, B, C, F, G, H) = 0 (8) 
Hence 
B x (Qz— R?/) : u='dy(Qz— By) : v=’d z (Qz—By) : w='b t {Q,z— By) : ; 
in other words, the surface Qz — Ry touches U at the principal points of the system. 
Still more generally, we may write, for the whole group, the following formula, viz. 
(3„ d„ 3,)(jr, y, z, t) 1 ^ 
|P, Q, R,s| J 
=mm ( u , v, w , ^)(A n B 15 C n F„ G 1} H,), 
which expresses the fact that, at the principal points of the system, each of the surfaces 
(7) has a two-branch contact with U. It remains to show that the same is the case 
with each of the surfaces P, Q, R, S. Since each of the expressions for P, Q, R, S 
vanishes at the points under consideration, we have 
B 2 (Rr— Pz )=^B 2 R— zd x F =mmB,M, 
~d x (Py — Qx)=yb x F —x’d x Q=mmC l u. 
