352 ME. W. SPOTTISWOODE ON HYPER .J ACOBIAN SUEPACES AND CURVES. 
Then, when the number of surfaces is equal to that of the variables, the determinant 
u , 
v , 
w , 
k 
a , 
b , 
c , 
d 
a ' , 
V , 
c\ 
d' 
a", 
b", 
c". 
d" 
is called the Jacobian of U, <p, and is usually designated by the formula 
(3) 
If, however, the number of surfaces be less by unity than that of the variables, the 
first differential coefficients (1) will form the matrix 
u , v , w, k J 
a , b , c , d (4) 
a', V, d, d' J 
from which there may he derived four determinants, whereof, however, two only are 
independent. Any two of these will represent surfaces, which may be called Jacobian 
(more strictly Hypojacobian) surfaces ; and their intersection, the Jacobian curve of 
the system. The pair of surfaces, or their curve of intersection, may conveniently be 
designated by the formula 
J((U, *>,40) (5) 
Again, if AU, A'U, . . be any derivatives of U beyond the first, we may, by means of 
them, form from the system of surfaces U, cp, %|/, . . the following matrix 
u , v , w , k , AU, A'U, . . J 
a , b , c , d, A <p , A '<p , . . 
a', V, d, d', A 4/, A'4, . . 
: : : : : : : : j 
in which it is to be understood that the number of surfaces is such that the number of 
columns exceeds that of the lines by unity. From this matrix two independent deter- 
minants may be formed ; and the surfaces which they represent may, in conformity 
with previous nomenclature, be called the Hyperjacobian surfaces , and their intersection 
the Hyperjacobian curve of the system. These will be designated by the formula 
J ((U, <p, 4/, . . )) (7) 
The principal properties of the Jacobian surface (2), and of the Jacobian curve (5) 
are known; and the object of the present paper is to investigate some of the properties 
of the Hyperjacobian surfaces and curves. 
The surfaces and curves (5) are in themselves independent of any particular mode of 
origination ; but they are here considered as arising out of the contact of the surface U 
