362 MR. W. SPOTTIS W 0 ODE ON HYPER-JACOBI AN SURFACES AND CURVES. 
we shall then have for the Hyperjacobian curve the following expression: — 
P, Q, . . . =w, v, w, Jc , nl, nJ, nK, 
a , b, c, d , L, M, N, 
V, d , d', L', M', NV 
( 3 ) 
and it is not difficult to see that, by a transformation similar to that used in § 1, we 
may write 
(t : m)( P, Q, . . 0L=P o +mUP lf Q 0 +mUQ 1? (4) 
and consequently that 
{t : m){b x , ~d y , d,)( P, Q, . . .)=m(w, v, w , ^)(P 1? Q„ . . .) ; 
( 5 ) 
that is to say, the Hyperjacobian surfaces touch the given surface at the principal 
points, and that the Hyperjacobian curve has a node at those points. 
Again, a transformation similar to that employed in § 3 will give 
(t: m)AP=m{l + 2(m— 1) : (n— ljjHP^^ — 2 to)HPj : m. . . . (6) 
But it is to be observed that a similar process would have led to the relations 
(£:ra)A 00 P=m]l+2(m— 1) : (n— lj}H 00 Pi+(w— 2m)H 00 P 1 : m, . . (7) 
as well as to the corresponding relations with the suffixes 0, 1 ; 1,1 respectively. These 
show that, in the case considered before, viz. where n=2m, the Hyperjacobian surfaces 
have three-branch contact with the given surface, and consequently with one another, 
at the principal points. At the same points the Hyperjacobian curve will have a 
triple point. 
§ 5. Nature of the Contact at the Secondary Points of the System. 
We have hitherto considered the degree of the contact of the Hyperjacobian surfaces, 
and the nature of the points on the Hyperjacobian curve, at the principal points of the 
system. Suppose that, at some of the principal points, the surfaces U, <p, . . . not 
only meet, but touch one another ; and let these points be called the secondary points 
of the system. When this is the case we shall have the relations 
u : v : w : Jc 
■=a : b : c : d 
=a! :b' :d: d' 
> 
( 1 ) 
J 
Suppose now that P, Q have the same values as in § 4, and that P represents a deter- 
minant containing the first four columns ; say, let 
P, Q, B -=w, v, w, tc, nl, nJ, nK. 
( 2 ) 
