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

Spherical Triangles. 9
13. If a variable sphere touch three planes, the locus of its centre is a
right line.
14. Describe a sphere, passing through four points which are not coplanar.
15. If A, B, C, D be four points on a great circle, prove that
sin BC. sin AD + sin CA. sin BD + sin AB. sin CD = 0.
16. In the same case, prove that
sin BC. cos AD + sin CA cos BD + sin AB cos CD = 0.
SECTION II.-SPHERICAL TaIANGLES.
11. DEF. X.-The figure formed by the shorter arcs joining
three points on the surface of a sphere, no two of which are
diametrically opposite, is called a SPHERICAL TRIANGLE.
Two points on he surface of a sphere can be joined by two distinct arcs,
which together make a great circle. Hence, when the points are not diametrically opposite, these arcs are unequal, and it follows from the definition
that each side of a spherical triangle is less than a semicircle.
If ABC be the triangle, O the centre of the sphere, the
planes OAB, OBC, O CA form a solid angle O - ABC
c
Fig. 5.
(Erc. XI., Def. II.), whose face angles AOB, BOC, COA
are measured by the sides of the spherical triangle ABC,