ME. W. SPOTTISWOODE ON HYPEE JACOBIAN SUEFACES AND CUEVES. 359 
But, writing it in full, the first term of this 
=nH+(4-n)H, u, v, w= -T+(4-n)HT 1 = (4-n)HT 1 , 
since T is supposed to vanish. Hence 
AT 0 =-| -4+n+2(w-m) : (w-lJfHT, 
= {-3w+8 + 2m-2+2w-2m}HT 1 : (n- 1) 
=— HTj. 
• (14) 
So that, substituting in (12), we finally find 
(t: m)AT=2m)l+(r-l) : (rc-l^Hl^-HT, 
• ( 15 ) 
=m{l + 2(r-l) : (w-l)}HT 1 +(w-2w)HT l : m. 
If the degree of U be double that of <p, -<p, . . , i. e. if n=2m, the last term of this 
expression will vanish, and m = l; and the equation (15) will be identical with the 
equation for determining the two branches of three-point contact of T with U at a 
principal point of the system. But (15) has been formed on the supposition that tb : tb' 
satisfies the equation AV=n5H, viz. the equation for determining the two branches of 
thre-epoint contact of V with U, at the same point. Hence, in the case where n=2m, 
and at the principal points of the system, the branches of three-point contact of T and U 
coincide with those of V with U. 
It is moreover clear that a similar process may be applied to the functions Qz— Ry, . ., 
since they are all of the form A-j-mUA!, . ., and that similar results will be obtained. 
And a transformation similar to that adopted in § 1 will show that the sam& theorem 
holds good for the surfaces P, Q, B, S, as for the surface T. 
It follows also, as in § 1, that the parabolic points of P, Q, R, S, T coincide with 
those of U ; and also that when m=n, the parabolic points of the Hyperjacobian 
surfaces generally coincide with the principal points of the system. 
It is perhaps worth while to calculate the Hessian of the Hyperjacobians at the 
principal points of the system. And it will be observed that the following calculation 
applies to all functions which can be expressed in the form (t : m)P=P 0 +mUP 1 . In 
the first place, forming the Hessian (say H 0 ) of the left-hand side of this equation, and 
writing down only the first lines of the determinants, we find 
H 0 (P t : m)={t : m ) 4 d?P, d x ^P, dAP, dAP+d x P : t 
={p:t){t:myb: P, dA p > ^A p , d*P 
=Kp— i) _1 A^) 4 ^P, SAP, BAP 
— : w) 4 H 0 P. 
MDCCCLXXVII. 
5 F 
