ME. W. SPOTTISWOODE ON HYPEEJACOBIAN STJEFACES AND CUEVES. 353 
with a surface passing through the intersection of <p and ; or, say, with one or more 
particular surfaces of the singly infinite or one-fold pencil a<p+b^=V, where a and b 
are constants. For the equations expressing the condition of contact between U and Y 
will enable us to eliminate the ratio a : b in two ways, and give rise to the curve 
J ((U, (f>, ^))=0. These equations, combined with U = 0, will suffice to determine the 
coordinates of the points of contact ; and if the values so determined be substituted in 
the equation V = 0, the values of a : b, that is, the particular surfaces of the pencil for 
which such contact obtains, will be found. 
Similarly, the Jacobian (2) may be considered as arising out of the contact of the 
surface U with the doubly infinite or two-fold pencil, Y=a<p + b4/+c^. 
Again, by selecting suitable derivatives of U, <p, . . for the terms AU, A'U, . . 
the Hyperjacobians (6) and (7) may be considered as arising from contact of higher degrees 
than common (or two-branch) contact between U and some of the surfaces of the pencil 
Y=a<p+b\|/+ . . And we shall in each such case have three equations, viz. U=0, and 
the equations of the curve, which will give the values of the coordinates of the points 
of contact. These values, substituted in the equation for Y, will determine one of the 
ratios a : b : . and thereby a pencil (whose multiplicity is less by unity than that of 
the given pencil), for which the contact obtains. 
The properties here considered are those which appertain to the points, if any, 
through which all the surfaces pass, or, as they may be termed, the principal points of 
the system; and consist mainly in the nature of the contact of the Hyperjacobian 
surfaces with the surface U, and the multiplicity of the Hyperjacobian curve at the 
points in question. 
The present investigation extends to the cases of two-branch contact of the given 
surface with a one-fold and with a two-fold pencil, and of three-branch contact with a 
four-fold pencil. In the latter case, notice is also made of some properties appertaining 
to the points, if any, where all the surfaces touch one another, or, as they may be 
termed, the secondary points of the system. In particular, it is shown that, in the case 
of common, or two-branch contact and a one-fold pencil, the Jacobian curve has a 
double point at the principal points; while in the case of three-branch contact and a 
four-fold pencil, the Hyperjacobian curve has a triple point at the same points. 
§ 2. The Jacobian Surfaces and Curve of a onefold pencil. 
Consider a surface U=0 of the degree n, and two other surfaces <p = 0, \f/=0, each 
of the degree m, where m is in general different from n ; also the one-fold pencil of 
surfaces 
Y=a<p-f bif^O, (1) 
where a and b are constants. If U and Y have common, or two-branch contact, we shall 
have, beside (1), the following conditions, viz. 
d x V=0u, c^V = Ov, B 2 Y —6w, B f V=^, 
3 e 2 
( 2 ) 
