470 
ON QUARTIC CURVES. 
[361 
Cambridge, December 15, 1864. 
Reverting to the case where we have in one hemisphere the two ovals A, B 
external to each other, the spherical quartic may comprise as part of itself another 
oval C. The ovals A and B, qua ovals external to each other, have a common tangent 
circle (a double tangent of the spherical quartic) which cannot meet the oval C (for 
if it did we should have six points of intersection); hence in the immediate neighbour 
hood thereof we have a circle not meeting any one of the ovals A, B, G. We may 
consider A, B, G as lying on the same side of this circle; for if B were on the 
opposite side to A, then B' would be on the same side ; and so if G be on the opposite 
side, then G' will be on the same side; that is, we have the three ovals A, B, C 
external to each other, and in the same hemisphere. 
There may be a fourth oval, D, and it would be shown in a similar manner that 
we have then the four ovals A, B, G, D external to each other and in the same 
hemisphere. But there cannot be a fifth oval, E; the proof is precisely the same as 
for the theorem in piano; viz. taking within each of the five ovals a point, and 
through these points drawing a conic, the conic would meet each oval in two points, 
and therefore the plane quartic in ten points, which is impossible. 
Passing from the sphere to the plane, the foregoing investigation shows that every 
plane quartic without nodes or cusps is either a finite curve, or else the projection 
of a finite curve, of one of the following forms: 
1. a single oval. 
2. two ovals external to each other. 
3. two ovals, one inside the other. 
4. three ovals external to each other. 
5. 6. four ovals external to each other. 
The last case has been called (5, 6) for the sake of the following subdivision, viz.: 
5. the four ovals are so situate as to be intersected, each in two points, by the 
same ellipse. 
6. they are so situate as not to be intersected by any one ellipse whatever—the 
distinction being similar to that which exists between four points, which may be either 
such as to have passing through them as well ellipses as hyperbolas, or else to have 
passing through them hyperbolas only. 
I remark that the limitation of the theorem to the case of a quartic curve without 
nodes or cusps is necessary, at any rate as regards the nodes. We may in fact find 
a quartic curve having a single node which is met by every line in at least two real 
points, and which is therefore not the projection of any finite curve ; for if we imagine 
two hyperbolas so situate that each branch of the one cuts each branch of the other, 
then it may be seen that there exists a quartic curve approaching everywhere very 
nearly to the system of two hyperbolas, but having, instead of the four nodes of the 
system, only a single node, which is such that every line meets it in at least two 
points.