ON THE GEOMETRICAL THEORY OF SOLAR ECLIPSES. 
395 
474] 
It appears to me not impossible but that the foregoing form of equation, 
(x 2 + if + z 2 ) (p + p — 2a) — (P — P') 2 — 2 {p — er) P — 2 (p — a) P' + pp' — ar = 0, 
for the umbral or penumbral cone might present some advantage in reference to the 
calculation of the phenomena of an eclipse over the Earth generally: but in order to 
obtain in the most simple manner the equation of the same cone referred to a set 
of principal axes, I proceed as follows: 
Writing 
Then, if 
a = be' — b'c, 
b = ca' — c'a, 
c = ab' — a'b, 
(and therefore 
f = a — a, 
g=b - b', 
h = c — c, 
af + bg + ch = 0). 
(bh — eg) x + (cf — ah) y + (ag — bf) z 
Va 2 + b 2 + c 2 Vf 2 + g 2 + h 2 
a# + by + cz 
Va 2 + b 2 + c 2 ’ 
£x + gy + h z 
Vf 2 +g 2 + h 2 ’ 
X, F, Z, will be coordinates referring to a new set of rectangular axes; viz., the 
origin is, as before, at the centre of the Earth, the axis of Z is parallel to the line 
joining the centres of the Sun and Moon; the axis of X cuts at right angles the 
last-mentioned line; and the axis of F is perpendicular to the plane of the other 
two axes; or, what is the same thing, to the plane through the centres of the Earth, 
Sun, and Moon. 
The coordinates of the vertex of the cone are therefore X Q , F 0 , Z 0 , where these 
denote what the foregoing values of X, Y, Z, become on substituting therein for x, y, z, 
the values 
k'a + ka' k'b + kb' k'c + lcc 
V+ k ’ k' + k ’ k' + k ’ 
and the equation of the cone therefore is 
where 
(Z - X 0 ) 2 + (F- F d ) 2 = tan 2 X(Z- Z 0 ) 2 , 
sin A, = 
k' + k 
G ’ 
50—2