ME. W. SPOTTISWOODE ON H YPEE J ACOBIAN SEEPAGES AND CUEVES. 361 
Similarly, at the points in question, P being of the form indicated above, 
(*„ ^) 2 P«=0, 
d x cbPo : m=c) x P : m=mV l u : t, 
d/bP 0 : m=B y P : m=mP 1 v : t, 
B z c^P 0 : m=d 2 P : m=7nP 1 w : t, 
B 2 P 0 : m=B^P : m=mP^ : t. 
Hence, finally, the expression sought =(p 1 -\-u— l) 2 (u— l) _2 Pt X 
u l w' v' , 
w' u' ml+mv.t, 
v' u' Wj 
V-\-mu:t m'-j-mw:# w'+mw:^ Tc x -\-mk\t^ 
=(Pi+w— l) 2 (w— l) _2 (m+w— 1) 2 P^ -2 X m 1s w', «/, w', 
«/, «i, u v\ 
v', u', w 1? w', 
u', v', w', 
=(Pi+w-l) 2 (w-l)- 2 (m+w-l) 2 PJH 0 . 
That is to say, at the principal points the Hessian of P vanishes with that of U. 
§ 4. Hyperjacobian Surfaces and Curve of a four-fold Pencil. 
Consider as before the surface U, and the four-fold pencil 
V=acpH-h\j/+c^4-dw-(-eg=0 (1) 
If we now form the conditions for three-branch contact between U and V, we 
shall be able to eliminate the ratios a : b : . . . in two different ways, and thus deduce 
as before the Hyperjacobian curve. In order to form the expressions required, it 
will be convenient to write 
A =w 12 A 00 — 2OTS7A 01 + OT- 2 A n ,-j 
H =W 2 — 2JWV +Kst 2 , 
A <p=W 2 — +N®- 2 , (2) 
A^=LV 2 -2MVs7 +NW 2 , 
3 p 2 
