X. On Ilyperjacobian Surfaces and Curves. 
By William Spottiswoode, M.A., Treas. B.S. 
Keceived April 23, — Bead May 17, 1877. 
§ 1. Introductory. 
In a paper published in the ‘ Mathematische Annalen ’ (vol. iii. p. 459), Brill has 
discussed the question of curves having three-point contact with a doubly infinite pencil 
of curves; and in particular he has investigated some of the properties of the curve 
passing through all the points of contact with the individual curves of the pencil. In 
the same Journal (vol. x. p. 221) Keey of Kiel has applied a method similar to that of 
Brill with partial success to the question of curves having four-point contact with a 
triply infinite pencil. Some formulas, however, given in my paper “ On the Sextactic 
Points of a Plane Curve” (Phil. Trans. 1865, p. 657) have proved to be directly appli- 
cable to both questions. An application of them to Brill’s problem will be found in 
the ‘ Comptes Bendus’ for 1876 (2nd semestre, p. 627), and a solution of Krey’s 
problem in the ‘Proceedings of the London Mathematical Society for the same year 
(vol. viii. p. 29). 
The present subject was in the first instance suggested by the foregoing papers ; and 
from one point of view it may be regarded as an attempt to extend the question to the 
case of surfaces ; viz. to determine a curve which shall pass through the points of 
contact of a given surface U with certain surfaces belonging to a pencil V, and to inves- 
tigate some of its properties. From a slightly different point of view, however, it may 
be considered as an extension of two ideas, viz. first, that of the Jacobian surface, or 
locus of the points whose polar planes with regard to four surfaces meet in a point; 
and secondly, that of the Jacobian curve, or locus of points whose polar planes with 
regard to three surfaces have a right line in common. More particularly, commencing 
with the facts, first, that if a surface of the form a<p -j-b^+c^ touch a surface U, the 
point of contact is a point on the Jacobian, and secondly, that if a surface of the form 
a<p-f b\J/ touch a surface U, the point of contact is a point on the Jacobian curve, I have 
endeavoured to extend them to higher degrees of contact. 
Consider, then, a surface U=0 of the degree n, and other surfaces <p = 0, \p=0, . . 
all of the degree m, where m is in general different from n. Let the first differential 
coefficients of U, <p, \|/, . . be thus expressed : 
(d» <b) U =u, v, w, k, 1 
P. a„ a.) *=a. I 
(d x , cb) 4 , =a!, V , d, d', | 
: : : : : J 
3 e 
MDCCCLXXVII. 
