32] TO THE THEORY OF SPHERICAL COORDINATES. 215 
Section 2. Geometrical Applications. 
Consider any three axes Ax, Ay, Az, and let X, ¡a, v be the cosines of the 
inclinations of these lines to each other. 
Let A, M, N be the inclinations of the coordinate planes to each other; l, m, n 
the inclination of the axes to the coordinate planes. Suppose, besides, 
a = i -x 2 (17),. 
b = l -y\ 
C = 1 - v\ 
f = fiv — X, 
% = vX-fi, 
1) = Xfji — v , 
k = 1 — X 2 — /A — v 2 + 2X/u» (18) ; 
we have the following systems of equations : 
V (t)C) cos A = — f, V (be) sin A = (k), V (a) sin l = (k) (19). 
v (ca) cos M = — g, a/ (ca) sin M = ^ (k), V (b) sin m = sj (k) 
V(ab) cos N = — f), V (ab) sin N = V (k), V (t) sin n — v (k). 
a + vf) + /.tQ = k, (20). 
vsi + b + xg = 0, 
/¿a + xb + g = 0. 
b+vb+/if=0, (21). 
vf) 4~ b + Xf = k, 
yu-b + Xb + f = 0. 
g + vf+ fit = 0, (22). 
vq + f + xt = 0. 
yu.g -f- Xf -t- £ == k. 
be— i 2 = ka (23). 
ca— Q 2 = kb, 
ab - b 3 = , 
gb - af = kf, 
bf - bg = kg, 
fg - cb = kh, 
abc - af 2 - bg 2 - cb 2 + 2fgb = k 2 .. 
(24).