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

132 Polyhedra.
volume of the parallelopiped is equal to the product of the base
and altitude = bc sin a. AD1; but A1) = a. sin A 0D = a. sin aod;. vol. = abc sin a. sin aod = 2abc. n, (433)
n being the first staudtian of the solid angle O - ABC
= abc \/1 - cos2a - cos2/ - cos2y + 2 cos a cos 3 cos y.
(434)
A
Fig. 51.
Cor. 1.-The volume of the tetrahedron O - ABC = abecn.
(435)
Cor. 2.-To find the volume of a tetrahedron in terms of its
six edges. Let BC a', CA = b', AB = c'; then we have
b2+ - a'2 + a2 - b' a2 + b2- -2
sa= 2bc 2 cos = 2c= ab
= abc v/ 1 - cos2a - cos2f - cos3y + 2 cos a cos p cos y.
Hence 144 F2 = Sa2a'2(b2 + b'2 2 + ' + ' a2 - a'2)
- a2b'2 '2- b2 '2a'2- 2a'2b'2- a2 b2 2. (436)
Cor, 3.-If T~, Tb, T, be the areas of the three faces OBC,
OCA, OAB of the tetrahedron O - ABC, and N the second
staudtian of the solid angle O-ABC,
V2 = T T,bTcN. (437)