393 
474] ON THE GEOMETRICAL THEORY OF SOLAR ECLIPSES, 
the equation is 
6 2 (x 2 + y 2 + z 2 -2P+p) 
+ 20<f> (x 2 4- y 2 + z 2 — P — P' + a) 
+ (fy (x 2 + y 2 + z 2 — 2 P’ + p) — 0, 
and the equation of the envelope consequently is 
(x 2 + y 2 + z 2 — 2P + p) (x 2 + y 2 + z 2 — 2P' + p) — (x 2 + y 2 + z 2 — P — P' + <r) 2 = 0, 
that is 
(x 2 + y 2 + z 2 ) (p + p — 2a) — (P — Py — 2(p—a)P—2(p — a)P' + pp — a 2 = 0, 
which is the equation of the cone in question. 
Observe that one sphere of the series is a point, viz., taking first the upper signs 
if we have 6k + (file = 0, that is 
k' . — k 
6 
k'-k’ 
then the sphere in question is the point the coordinates whereof are 
x = 
k'a — ka' 
k'-k 
y 
k'b - kb' 
k’-k 
z = 
k'c — kc' 
k'-k ’ 
which point is the vertex of the cone: it hence appears that, taking the upper signs, 
the cone is the umbral cone, having its vertex on this side of the Moon; and 
similarly taking the lower signs, then if we have 6k — <j>k' = 0, that is 
0 = 
k' 
k 
k' + k’ r k' + k’ 
then the variable sphere will be the point the coordinates of which are 
k’a + ka' k'b + kb' k'c + kc 
k' + k 
k' + k 
k' + k 
which point is the vertex of the cone; viz. the cone is here the penumbral cone 
having its vertex between the Sun and Moon. 
Taking as unity the Earth’s equatorial radius, if p, p are the parallaxes, k, k 
the angular semi-diameters of the Moon and Sun respectively, then the distances are 
and the radii are Sm K , Sm —, respectively; hence, if h, h' are the hour- 
sin p' smp smp snip 
angles west from Greenwich, A, A' the N.P.D.’s of the Moon and Sun respectively, 
we have 
a = 
6 = 
c = 
k = 
smp 
1 
sinp 
1 
smp 
sin « 
sinp 
sin A cos h, 
sin A sin h, 
cos A , 
a = 
b' = 
c = 
sm p 
1 
sin p 
1 
sm p 
sin K 
sin p 
> sin A' cos h', 
, sin A' sin h', 
cos A', 
., _ sm k 
№ — t“ 
C. VII. 
50