32] 
TO THE THEORY OE SPHERICAL COORDINATES. 
219 
or 
cos 8 = 
\/Tc {cot 8 = 
W (w, w', q) 
f\W (co, w, q) W(to', &)', q)} ‘ 
\J {W ( coco', coco', q)} 
Vi^(ft), q) W(co', ft)', q)} ’ 
W(co, ft)', q) 
V{ TL( ft)ft/, &>&>', q)} 
(62). 
Equation of a great Circle. 
Let the conjoint coordinate systems of the pole be 
e — a, b, c (63), 
e = a, ¡3, 7 (64), 
then, expressing that the distance of any point P in the locus from the pole is equal 
to 90°, we have immediately the equations 
W(p, e, q) = 0 (65), 
If (ft), e, q) = 0 (66), 
which may otherwise be written in the forms 
a% + brj +c% = 0 (67), 
ax + f3y + 7^=0 (68), 
or the equation of a great circle is linear in either coordinate system. Conversely, 
any linear equation belongs to a great circle. 
Suppose the equation given in the form 
A^ + B v + C^=0 (69); 
or by an equation between cosine coordinate ratios:—the sine system for the pole is 
given by 
e=A, B, C (70), 
and the cosine system by 
e = A + vB + gC, vA -f B + \C, gA + A,B + C (71). 
Suppose the circle given by an equation between sine coordinates, or in the form 
A* + By + C* = 0 (72), 
the cosine system of coordinates for the pole is given by 
e = A, B, C (73), 
and the sine system by 
e = flA + |)B + gC, f)A + hB + fC, gA + fB + cC 
....(74). 
28—2