361] 
ON QUARTIC CURVES. 
469 
of intersection by the tangent is = 0 or else = 2; and there is at least one portion 
of the oval for which the number of intersections is = 0; for otherwise the oval would 
be concave at every point, which is impossible. Hence there is a tangent which does 
not meet the oval (except at the point of contact), and we may in the immediate 
neighbourhood of the tangent draw a line which does not meet the oval at all. 
Precisely the same considerations apply to the case of an oval which is part of 
a spherical quartic, the tangent being of course a great circle; and the conclusion 
arrived at is that there exists a great circle which does not meet the oval at all; 
that is, the oval lies wholly in one hemisphere. 
I remark that the demonstration would, as it ought to do, fail, if we attempted 
to apply it to an oval portion of a spherical sextic; the tangent circle meets the oval 
in a number of points which is =0, 2, or 4; and the number cannot be for every 
tangent circle whatever = 2; but there is nothing to prevent it from being for every 
tangent circle whatever = 2 or 4. Hence we cannot, for every spherical sextic, obtain 
a tangent circle not meeting the oval except at the point of contact; and consequently 
we do not obtain in the immediate neighbourhood of the tangent a circle which does 
not meet the oval at all. And in fact such circle does not in every case exist; that 
is, the oval portion of a spherical sextic does not in every case lie in a hemisphere. 
It has been shown that the oval portion of a spherical quartic lies in a hemi 
sphere ; but we have to consider the case where the quartic consists of two or more 
ovals. To fix the ideas, let A, A' be a pair of opposite ovals, and B, B' another pair 
of opposite ovals, components of the same spherical quartic. If there exists a tangent 
circle of A which does not meet B, then there exists in the immediate neighbourhood 
of the tangent circle a circle which does not meet either A or B; and we may assume 
that A and B lie on the same side of this circle; for if B were on the side opposite 
to A, then B' would be on the same side with A ; and we have only, instead of B, 
to consider the opposite oval B\ Hence we may consider that the ovals A and B lie 
on the same side of the circle; that is, we have a spherical quartic consisting of or 
comprising the ovals A and B in the same hemisphere : the two ovals are, it is clear, 
external each to the other. 
But every tangent of A may meet B in two points; consider the whole spherical 
figure, and suppose that the tangent (or say, the tangent circle) of A, A' meets the 
ovals B, B' in the points K, L and the opposite points K', L': then considering the 
tangent circle as moving round A, A' until it returns to its original position, the 
points K, L, K', L' are always four distinct points; and K and some one (say L) of 
the two points L, L' will describe the same oval, say the oval B; while the opposite 
points K\ L' will describe the opposite oval B'. We have here the oval A included 
in the oval B (and of course the opposite oval A' included in the opposite oval B'). 
But the oval B, qua portion of a spherical quartic, lies wholly in one hemisphere; 
hence the two ovals A, B lie wholly in one hemisphere. It is easy to see that there 
is not in this case any other portion of the spherical quartic, but that the two ovals 
A, B are the entire curve.