356 ME. W. SPOTTISWOODE ON HTPEEJACOBIAN ST7EFACES AND CUEYES. 
But on multiplying the first of these equations by z and the second by y, and subtracting, 
we obtain 
oc(xbjL -f yb x Q -f zc^R) — (x 2 + ?/ 2 -f £ 2 )d JP = —yC^u. 
But, since ^P+^Q+2:R=0, this reduces itself to 
d x P : u=mm(z¥> l —yC l ) : (x 2 -\-y 2 +z 2 ). 
By a similar process applied to the other equations, and by writing x 2 -\-y 2 -\-z 2 =r 2 , we 
should obtain 
c^P : w=^P : v=d z P : w==b t P : Jc=mm(zB l — 2 /CJ : rA 
c^Q : w=c^Q : v=~d g Q : w=~d t Q : Ic=mm(xC 1 — zAJ : r 2 , l . . . (10) 
Hence each of the surfaces P, Q, R, S has two-branch contact with U at the principal 
points of the system. In other words, each of the Jacobian surfaces J((U, <p, \p))=0 
touches the surface U, and consequently they touch one another at the principal points 
of the system. At the same points the Jacobian curve therefore has a node, and along 
each of its two branches the contact of the Jacobian surfaces is three-pointic. 
It should here be noticed that if all the surfaces are of the same degree, i. e. if n=m , 
then m=0, and consequently 
(3„ 9,)(P, Q, B,S)=0; (11) 
so that, in this case, the Hessian of each Jacobian surface vanishes at the principal 
points ; in other words, the principal points are parabolic points on the Jacobians. 
§ 3. The Hyjaerjacobian Surfaces and Curve of a twofold Pencil. 
Consider the surface U=0, as before; the three surfaces <p, -ty, and the pencil 
V=a<p+b'4'+c;£=0 (1) 
Now, it is well known that if two surfaces touch one another, the curve of intersection 
has a double point at the point of contact, and that along each of the branches the 
contact is three-pointic. The formulae for determining the directions of these two 
branches are as follows. Adopting the notation of my memoir, “ On the Contact of 
Surfaces” (Phil. Trans. 1872, p. 259), in which a, j3, y, S, a', /3', y', B', are arbitrary 
constants, and 
ax +j3 y -\-yz +$£ =©-, 
<x!x -J- ft y + y = ?«j ' , 
A — a'ny — ctz&', 
B=j3'sy — /3zzr', 
C =y'zcr — yzj', 
I) = h'vr — cW , 
