392 
[474 
474. 
ON THE GEOMETRICAL THEORY OF SOLAR ECLIPSES. 
[From the Monthly Notices of the Royal Astronomical Society, vol. xxx. (1869—1870), 
pp. 164—168.] 
The fundamental equation in a solar eclipse is, I think, most readily established as 
follows: 
Take the centre of the Earth for origin, and consider a set of axes fixed in the 
Earth and moveable with it; viz., the axis of z directed towards the North Pole; 
those of x, y, in the plane of the Equator; the axis of x directed towards the point 
longitude 0°; that of y towards the point longitude 90° W. of Greenwich. Take 
a, h, c, for the coordinates of the Moon; k for its radius (assuming it to be spherical); 
a', b', c, for the coordinates of the Sun; k' for its radius (assuming it to be spherical); 
then, writing 0 + </> = 1, the equation 
{0(x — a) + cj)(x — a)\ 2 4- {0(y — b) + (f> (y — b')} 2 4- [0(z — c) + </> (z — c')} 2 = (0k ± <f)k') 2 
is the equation of the surface of the Sun or Moon, according as 0, cf) = 1, 0 or =0, 1: 
and for any values whatever of 0, cf), it is that of a variable sphere, such that the 
whole series of spheres have a common tangent cone. Writing the equation in the form 
0 2 {(x — af 4- (y — b) 2 + (z — c) 3 — k 2 ) 
4- 20(f) \(x — a) (x — a') + (y - b) {y — b') + (z — c) (z — c') + kk'} 
+ </r {(x — a') 2 + (y — b'f + (z — c') 2 — k' 2 ) = 0, 
or, putting for shortness, 
p = a? + b' 2 + c 2 — k 2 
p' = a f * + b' 2 + c 2 - k' 2 
a — aa' + bb' + cc' + kk' 
P = ax 4-by 4- cz 
P' = a'x 4- b'y 4- cz,