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473] ON THE CONSTRUCTION OF THE UMBRAL OR PENUMBRAL CURVE &C. 
To prove this, take x 2 + y 1 — 1 for the equation of the fixed circle, (x — a) 2 + (y — /3) 2 = 7 2 
for that of the variable circle; the foregoing law of variation being in fact such 
that a, ¡3, 7, are linear functions of a variable parameter 6; the equation of the 
common chord AB is — 2ax — 2¡3y + 1 + a 2 + /3 2 — 7 2 = 0 ; viz., this equation contains 6 
quadratically; hence the envelope of the common chord is a conic; and thence 
(reciprocating in regard to the fixed circle) the locus of the pole of AB, that is, of 
the point Q, is also a conic. 
Consider now a solid figure in which the circles are replaced by spheres; viz. 
we have a fixed sphere, and 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 the line. 
The envelope of the variable sphere is a right cone; the intersection of the cone 
with the fixed sphere is the envelope of the small circle of the sphere, say the 
circle AB, which is the intersection of the fixed sphere by the variable sphere. This 
circle AB is also the intersection of the fixed sphere by a sphere, centre Q, which 
cuts the fixed sphere at right angles; and by what precedes the locus of Q is a 
conic. Hence the penumbral curve is given as the envelope of the circle AB which 
is the intersection of the fixed sphere by a sphere which has its centre Q on a 
conic, and which cuts the fixed sphere at right angles. It is obvious that the circle 
AB always cuts at right angles the great circle which is the section of the fixed 
sphere by the axial plane, or say the axial circle. Project the whole figure stereo- 
graphically; the projection of the circle HI? is a variable circle which cuts at right 
angles the circle which is the projection of the axial circle, and which has for its 
centre the point Q' which is the projection of Q. But the locus of Q being a conic, 
the locus of its projection Q' is also a conic; and we have thus the projection of 
the penumbral curve as the envelope of a variable circle which has its centre on a 
conic, and which cuts at right angles a fixed circle. 
We may in the axial plane construct five points of the conic which is the locus 
of Q, by means of any five assumed positions of the variable circle, and somewhat 
simplify the construction by a proper choice of the five positions of the variable circle. 
This is not a convenient construction, and even if it were accomplished we should 
still have to construct the projection of the conic so obtained, in order to find, in 
the figure of the stereographic projection, the conic which is the locus of Q'. I do 
not at present perceive any direct construction for the last-mentioned conic; but 
assuming that a tolerably simple construction can be obtained, the construction of the 
projection of the penumbral curve as the envelope of the variable circle is as easy 
and rapid as possible. Probably the easiest course would be (without using the conic 
at all) to calculate numerically, for a given position of the variable sphere, the 
terrestrial latitude and longitude of the two points of intersection of the variable 
sphere by the axial circle; laying these down on the projection, we have then a 
position of the variable circle; and a small number of properly selected positions would 
give the penumbral curve with tolerable accuracy. 
I have throughout spoken of the penumbral curve, as it is in regard hereto that 
a graphical construction is most needed; but the theory is applicable, without any 
alteration, to the umbral curve.