310
[688
688.
GEOMETRICAL CONSIDERATIONS ON A SOLAR ECLIPSE.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xv. (1878),
pp. 340—347.]
I consider, from a geometrical point of view, the phenomena of a solar eclipse
over the earth generally; attending at present only to the penumbral cone, the
vertex of which I denote by V. It is convenient to regard the earth as fixed, and
the sun and moon as moving each of them with its proper motion, and also with
the diurnal motion. The penumbral cone meets the earth’s surface in a curve which
may be called the penumbral curve; viz. when the cone is not completely traversed
by the earth’s surface, (that is, when only some of the generating lines of the cone
meet the earth’s surface), the penumbral curve is a single (convex or hour-glass
shaped) oval; separated, as afterwards mentioned, into two parts, one of them lying
away from the sun, and having no astronomical significance; but when the cone is
completely traversed by the earth’s surface, then the penumbral curve consists of two
separate (convex) ovals; one of them lying away from the sun and having no
astronomical significance, the other lying towards the sun. The intermediate case is
when the cone just traverses the earth’s surface, or is touched internally by the
earth’s surface; the penumbral curve is then a figure of eight, one portion of which
lies away from the sun, and has no astronomical significance: there is another limiting
case when the cone is touched externally by the earth’s surface, the penumbral curve
being then a mere point.
It is necessary to consider on the earth’s surface a curve which may for shortness
be termed the horizon; viz. this is the curve of contact of the cone, vertex V,
circumscribed about the earth; it is a small circle nearly coincident with the great
circle, which is the intersection by a plane through the centre of the earth at right
angles to the line from this point to the centre of the sun.
Regarding V as a point in the heavens, capable of being viewed notwithstanding
the interposition of the moon; the horizon, as above defined, is the curve separating
