ON THE DETERMINATION OF THE 
418 
[476 
orbit-plane is z — 0. We have therefore merely to transform the equations of the ray 
to the new axes by writing for x, y, z, the values 
ax' + ay' + a" z\ 
/3x' + fi'y'+13"z', 
you' + ry'y' + y "z\ 
and then putting z = 0, we find x', y', the coordinates in the orbit-plane of its inter 
sections with the ray. 
47. The equations thus become, 
a x + a y' — A — Rf = 0, 
fix' + fi'y' — B — Rg = 0, 
yx' + y'y' — G — Rh. = 0, 
or, what is the same thing, we have 
x' : y' : R : 1 
= 
1 
1 
-1 
- 1 
a, 
a', 
f, 
A 
a, 
f, 
A 
a, 
a', 
f, A 
a, 
a', f, 
A 
13, 
fi', 
& 
B 
fi, 
/3', g, 
B 
fi, 
fi', 
g, B 
/3, 
B 
Y> 
y, 
h, 
C 
y, 
y, h, 
C 
y, 
y, 
h, G 
y, 
y, h, 
G 
= 
a', 
f, 
A 
: — 
a, f, 
A 
: — 
A, 
a, a.' 
f, 
a, a' 
fi', 
g> 
B 
/3, g, 
B 
B, 
fi, fi' 
g> 
fi, fi' 
y, 
h, 
G 
y, h, 
c 
G, 
y, y 
h, 
y, y 
In these formulae we have identically 
fiy ~ fi'y, 7 a ' ~ a &' ~ a 'fi = a "> I 3 "’ y"’ 
and if we write moreover 
a, b, c, —Cg — Bh, Ah—Of, Bi—Ag, 
(whence identically af + bg + ch = 0, and where (a, b, c, f, g, h) are the “ six coordinates ” 
of the ray), then we have the very simple formulae 
x’ : y : R : 1 
= (a, b, c$V, fi', y') : - (a, b, c$a, fi, 7) : (A, B, O^a", fi", y") : (f, g, h^a", fi", y"), 
or omitting (as not required for the present purpose) one of the proportional terms, we 
have 
x' : y’ : 1 = (a, b, c$V, fi', y) : -(a, b, c][a, fi, 7) : (f, g, h^a", fi", y"), 
which are the required expressions for the coordinates.