390 
[473 
473. 
ON THE GRAPHICAL CONSTRUCTION OF THE UMBRAL OR 
PENUMBRAL CURVE AT ANY INSTANT DURING A SOLAR 
ECLIPSE. 
[From the Monthly Notices of the Royal Astronomical Society, vol. xxx. (1869—1870), 
pp. 162—164.] 
The curve in question, say the penumbral curve, is the intersection of a sphere 
by a right cone,—I wish to show that the stereographic projection of this curve may 
be constructed as the envelope of a variable circle, having its centre on a given conic, 
and cutting at right angles a fixed circle; this fixed circle being in fact the projection 
of the circle which is the section of the sphere by the plane through the centre and 
the axis of the cone, or say by the axial plane. The construction thus arrived at is 
Mr Casey’s construction for a bicircular quartic; and it would not be difficult to show 
that the stereographic projection of the penumbral curve is in fact a bicircular quartic. 
The construction depends on the remark that a right cone is the envelone of a 
variable sphere, having its centre on a given line and its radius proportional to the 
distance of the centre from a given point on this line; and on the following theorem 
of plane geometry: 
Imagine a fixed circle, and a variable circle having its centre on a given line 
and its radius proportional to the distance of the centre from a given point on the 
line (or, what is the same thing, the variable circle always touches a given line); then 
the locus of the pole in regard to the fixed circle, of the common chord of the two 
circles (or, what is the same thing, the locus of the centre of a new variable circle 
which cuts the fixed circle at right angles in the points where it is met by the 
first-mentioned variable circle) is a conic. 
To fix the ideas, say that P is the centre of the first variable circle; AB its 
common chord with the fixed circle; Q the centre of the circle which cuts the fixed 
circle at right angles in the points A and P; then the locus of Q is a conic.