480 
A “ SMITHS PRIZE ” PAPER ; SOLUTIONS. 
[534 
5. Shew that a cubic surface has at most four conical 'points; and a quartic 
surface at most sixteen conical points. 
If a cubic surface has two conical points, then the line joining these has with the 
surface two intersections at each of the conical points, and therefore lies wholly in 
the surface. Hence, for a cubic surface with three conical points A, B, C, the lines 
AB, BC, GA lie wholly in the surface, and these three lines form the complete section 
of the surface by the plane ABC: it is clear that there cannot be in this plane a 
fourth conical point: but there may be, not in this plane, a fourth conical point D. 
Suppose that this is so, there cannot be a fifth conical point E\ for if there were, 
the line DE would lie wholly in the surface, and would therefore meet the plane ABC 
at some point in the section of the surface by this plane; that is, at some point in 
one of the lines AB, AG, BG; say at a point in AB: but then the lines AB, DE 
would intersect, or the four conical points A, B, D, E would lie in a plane. Hence 
there cannot be any fifth conical point E. 
For a quartic surface; suppose this has k conical points, and let any one of these 
be made the vertex of a cone circumscribing the surface; each generating line is a 
tangent of the surface; and considering any section by a plane through the vertex, 
and observing that from a double point of a quartic curve we may draw six tangents 
to the curve, it appears that the order of the cone is = 6. It is easy to see that 
the lines from the vertex to the remaining (&—1) conical points are each of them a 
double line of the cone, and that the cone has not any other double lines; the cone 
is therefore a cone of the order 6, with (k — 1) double lines. A proper cone of the 
order 6 has at most 10 double lines, but the cone need not be a proper one; it 
may, in fact, break up into 6 planes, and in this case the double lines are the 
15 lines of intersections of the several pairs of planes. Hence k — 1 is = 15 at most: 
or k is = 16 at most. 
6. Find the differential equation of the parallel surfaces of an ellipsoid. 
Let (x, y, z) be the coordinates of a point on the ellipsoid -f ~ ~ = 1; 
CL 0“ G 
(X, Y, Z) the coordinates of a point on the normal at a distance =k from the first- 
mentioned point. We have 
X — x_Y— y_Z — z 
, = p suppose ; 
x y z 
a, 2 6 2 c 2 
that is, 
and thence 
Moreover 
a 2 X b-Y GZ