MR. W. SPOTTISWOODE ON HYPER JACOBI AN SURFACES AND CURYES. 357 
H =u u 
w'. 
f. 
V , 
A 
w 1 , 
v i > 
vJ , 
mi. 
B 
v', 
v ! , 
W u 
n ' , 
C 
l' , 
mi, 
ri , 
D 
A, 
B, 
c, 
D, 
A =u n w' , v' , V , A, ~d x 1 
w', v t , v ! , mi, B , 
«/, vi , w/ , C, 
V , mi, n' , D, ~d t y . 
A, B, C, D, . . 
• 
n=l+2(m— 1) : (n— 1) ; 
then the required formulae will be 
• ( 2 ) 
B a y:M=b,y :«=B Z Y :w=^V:&=AV:wH=0, .... (3) 
the last of which, being a quadratic in m : tj', will determine the two directions sought. 
If the value of V given by the equation (1) be inserted in (3), we may eliminate the 
ratios a : b : c : Q in two different ways. The resultants will be the Hyperjacobian sur- 
faces, and their intersection the Hyperjacobian curve. 
The two independent results may be comprised in the formula 
■u , 
v , 
w. 
Jc, 
nH=0. 
a , 
b. 
G j 
d. 
A<p 
a', 
V, 
(j , 
d>, 
A\|/ 
a", 
b"r 
d". 
A* 
Among these, the expressions P, Q, R, S may be combined in the same manner as 
the corresponding expressions in § 1 ; and if we then write 
A, B, C, F, G, H=U, 
u , 
v , 
w, 
k , nH, 
a , 
b, 
c , 
d , A <p. 
a', 
b', 
d. 
<1 
*53 
a", 
b". 
d\ 
n; 
<1 
% 
A„ B 1} Cj, F„ G„ H,=<z , 
b, 
c , 
d. 
* 
a', 
V, 
d , 
d'. 
A*, >• • 
a", 
b", 
c", 
d". 
A* J 
we may take instead of the four expressions P, Q, R, S, the six 
A+mUAj, . . F+mUFj, (7) 
and any two of these equated to zero, or any one of them combined with 
T^To-f-mUT^O, will serve for the two equations required. From these it is easy to 
see that, in the same way as in § 1, it may be shown that the Hyperjacobian surfaces 
touch the surface U at the principal points of the system. 
