32] TO THE THEORY OF SPHERICAL COORDINATES. 217 
We have besides, by projecting on the line AO', the equation 
cos 3 = a!a + ¡3'b + y'c (40), 
or the analogous one 
cos 8 = a'a 4- /3'b + 7'c (41). 
From either of which we deduce 
cos 8 = aa' + bb' + cc' + A (be' + b'c) 4- p (ca' + c'a) + v (ah' + a'b). (42), 
k cos 8 = mu' + i)/3/3' + C77 4- f (/3y' + /3'7) + g (70' + 7V) + ft (a/3 + a'/3) (43) ; 
which may otherwise be written 
cosS= W(t, H, q) (44), 
k cos S = IF (t, t, q) (45); 
or again, observing the equations which connect the quantities t, t, 
o. W (t, t', q) 
° 0S 3{W(t, t, q) W(t, q)j (46)> 
C0S 8 = 7fF (t, t, 5) W (t', t', q)) (47) - 
forms which, though more complicated, have certain advantages; for instance, we derive 
immediately from them the new equations 
Bln8 VIW(t, t, q)W{H, t, q)) (48) > 
. ^{kW(TT\ tt', q)} 
SinS 4{W(t, t, i)F(r', t', q)) <49) - 
written more simply thus 
sin 3 = W (tt', tt', q) (50), 
\ ! k sin 8 = F{ W( tt', tt', q)} (51); 
to these we may join 
£ W (t, t', q) 
V{TF (tt',tf, q)} 
VA-cotS-- (53). 
y{TF(tt, tt', q)} 
Section 3. On Spherical Coordinates. 
Consider the points X, Y, Z, on the surface of a sphere, as the intersections oi 
the three axes of the preceding section, with a sphere having its centre in the origin. 
It is evident that X, p, v are the cosines of the sides of the spherical triangle XIZ, 
c. 28