A treatise on spherical trigonometry, and its application to geodesy and astronomy, with numerous examples. By John Casey.

114 Small Circles on the Sphere.
triangle, and r,, r2, &c., their spherical radii; then, multiplying
the determinants
0, xI, y2, Y 1, cosr' 0, X1 yt, s1, - cos r'
0, X2, Y2, Z2, COS 72 0, X2', 2 ', y- COS 2'
0, X Y3, 3, COS r3, O, X3'/. Z3/ - COS r3
0, X4, Y4, COS r4 0, x4', z4', -cos r4'
0, *X5> Y5, Zg5 COS r5 0, 51 y5/, z5', -cos r5
the proposition is evident.4
107. If the angle of intersection of the circles s, st be
denoted by af/, it follows at once, from equation (13), that
the mutual power (a/3) is equal to sin ra. sin rp cos aB. By
this substitution, equation (405) is transformed into
cos 11', cos 12', cos 13', cos 14', cos 15'
cos 21', cos 22', cos 23', cos 24', cos 25'
cos 31', cos 32', cos 33', cos 34', cos 35' = 0. (406)
cos 41', cos 42', cos 43', cos 44', cos 45'
cos 1', cos 52', cos 53', cos54', cos 55'
108. If the second system of circles coincide with the first
we have, for any system of five circles on the sphere,
1, cos 12, cos 13, cos 14, cos 15
cos 21, 1, cos 23, cos 24, cos 25
cos 31, cos 32, 1, c, cos 34 35 =0 (407)
cos 41, cos 42, cos 43, 1, cos 45
cos 51, cos 52, cos 53, cos 54, 1
* This theorem is the fundamental one in a Memoir by HIERR G. FROBENIUS, "Anwendungen der Determinantentheorie auf die Geometrie des
Maasses." CRELLE'S Journal, Band 79, pp. 185-245, for the year 1875. It
is also given in the Philosophical Transactions, vol. 177, part 2, for the year
1886, in a Memoir by R. LACHLIN, B.A., "On Systems of Circles and
Spheres."