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

116 Small Circles on the Sphere.
of half their common tangents, and denoting for shortness by
12 the sine of half the common tangent of the circles s, 82, the
condition is
0, 122, 132, 14
-2 - — 2
2Fi O, 0 3, 24
-2 — 2, -2 = o (411)
3l, 32, 0, 34
2 1-=2 2
41, 422, 43, 0
which, expanded, is equal to the product of the four factors
12.34 ~ 23.14 + 31.24 = 0. (412)
This theorem was first published in the Proceedings of the
Royal Irish Academy, in a Paper by the author " On the Equations of Circles," in the year 1866.
110. If 8s, 82, 83, 84 be a system of four great circles, and
81s, 82, 83, 8s4 four other circles (great or small), then
11', 12', 13', 14'
21', 22', 23', 24'
=0. (413)
31', 32', 33', 34'
41', 42', 43', 44'
This is proved like Frobenius's theorem by multiplying the
two determinants (xl, y2, z3, c r4), (x1', Y2', %'3 cos r4), the first
of which vanishes; since rl, r2, r3, r4, being the spherical radii of
great circles, are each equal to a quadrant, and their cosines
vanish.
111. If the second system in ~ 110 be great circles, and coincide with the first, we get, since the mutual power of two great