33 
[106 
e as of the spinode- 
be also a stationary 
3 ; the node-with-node- 
3-couple-develope twice 
ne on such duplicate 
The tacnode-plane is 
107] 
a point of intersection 
at a consecutive point 
ie-with-node-plane ; the 
point has therefore a 
ition is a node-couple- 
le-curve. Consequently 
and thence also the 
couple-develope. 
can have a stationary 
the curve of contact 
dy a stationary plane 
that the node-couple- 
: except in the points 
touch at the tacnodes, 
e-planes are stationary 
are stationary planes 
iched at the tacnodes 
lanes of the flecnode- 
107. 
ON THE THEORY OF SKEW SURFACES. 
[From the Cambridge and Dublin Mathematical Journal, vol. vn. (1852), pp. 171—173.] 
A surface of the n th order is a surface which is met by an indeterminate line 
in n points. It follows immediately that a surface of the n th order is met by an 
indeterminate plane in a curve of the w th order. 
Consider a skew surface or the surface generated by a singly infinite series of 
lines, and let the surface be of the w th order. Any plane through a generating line 
meets the surface in the line itself and in a curve of the (n — l) th order. The 
generating line meets this curve in (n — 1) points. Of these points one, viz. that 
adjacent to the intersection of the plane with the consecutive generating line, is a 
unique point; the other (n — 2) points form a system. Each of the (n — 1) points 
are sub modo points of contact of the plane with the surface, but the proper point 
of contact is the unique point adjacent to the intersection of the plane with the 
consecutive generating line. Thus every plane through a generating line is an ordinary 
tangent plane, the point of contact being a point on the generating line. It is not 
necessary for the present purpose, but I may stop for a moment to refer to the 
known theorems that the anharmonic ratio of any four tangent planes through the 
same generating line is equal to the anharmonic ratio of their points of contact, and 
that the locus of the normals to the surface along a generating line is a hyperbolic 
paraboloid. Returning to the (n — 2) points in which, together with the point of 
contact, a generating line meets the curve of intersection of the surface and a plane 
through the generating line, these are fixed points independent of the particular plane, 
and are the points in which the generating line is intersected by other generating 
lines. There is therefore on the surface a double curve intersected in (n — 2) points 
by each generating line of the surface—a property which, though insufficient to 
determine the order of this double curve, shows that the order cannot be less than 
(n - 2). (Thus for n — 4, the above reasoning shows that the double-curve must be 
c. II. 5