354 ME. W. SPOTTISWOODE ON HYPEEJACOBIAN SUEFACES AND CUE YES. 
where 6 is indeterminate. If for V there be substituted its value given by (1), these 
equations will serve to eliminate the ratios a : b : 0 in two ways ; and by this means we 
shall obtain two equations in x, y , z, t , which, being independent of a, b, will hold good 
for any particular surfaces of the pencil U, and will consequently represent a curve 
passing through all the points of contact of the surface U with the pencil V. The 
resultants, combined with the equation U=0, will determine the coordinates of all the 
points of contact ; and the particular surfaces of the pencil which actually touch U will 
be found by substituting successively the values of x:y:z:t , so determined, in the 
equation (1), and thence deducing the corresponding values of the ratio a : b. 
The two resultants may be represented by the following formulae : — 
P, Q, R, S=w, v, w, k = J((U, <p, 4))=0, 1 
a, b, c, d !> 
a', V , d, d' J 
(3) 
in which P, Q, R, S represent the four determinants which can be formed from the 
matrix (3) by the omission of each of the four columns in succession. Of these, of 
course, two only are independent; and they represent, as mentioned in the Introduction, 
the Jacobian curve of the system U, <p, \f/. The Jacobian curve presents some pecu- 
liarities at the principal points of the system, i. e. at the points where U=0, <p=0, -<p=0. 
In order to examine them, it will be convenient to transform the expressions (3) as 
follows : — 
Qs — B.y= — zw, k , u — k , u, yv 
zc , d , a d , a, yb 
d' a' 
d', a', yb' 
= yv -1 -zw, u, k = u\ J, u, k, 
yb -\-zc , a, d m<p, a , d 
yb' -\-zd, a!, d' m-ty, a', d' 
or, putting (n—m) : m= m, the above expression takes the form 
Qz— Ry=U-f-mU, u , k =A-fmUA, suppose. 
<p, a, d 
4/, a!, d' 
More generally, let 
A,, B„ Cj, F 1} G 1? H !=a, b ,c,d, 
d, V, c', d', 
(4) 
i. e. A 1? B„ . . are the determinants formed from the matrix (4) by omitting the columns 
two and two in the usual order, viz. 2, 3 ; 3, 1 ; 1, 2 ; 1, 4; 2, 4; 3, 4. Also let 
