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

138 Polyhedra.
Now'n being the first staudtian of the solid angle D- ABC,
we have 4n2=
1, cos ab, cos ac
cos ba, 1, cos bc.
cos ca, cos bc, 1
Therefore 64Rn2 =
O, a, b, c 0, a2, b2, c2
a, 2, 2 cosab, 2cosac 1 a2, 2a2, 2abcosab, 2accosac
b, 2 cosba, 2, 2 cos bc a22 b2, 2bacosba, 2b2, 2bccosbc
c, 2 cos a, 2 cosb, 2 2, 2 a cos a, 2 cb cos b, 2c2
Hence 64a2b2C2n2.R2 =
0, a2, b2, C2
a2, 0) c2, b/2
= - 1 (aa')4+ 24 (aba'b'). (442)
b2, c'2, O, a'2
C2, b'2, a'2 0
Cor.-24 VR = { 2s (aba'b') - S (aa')4.
127. The Isosceles Tetrahedron.-(NEUIBERG.)
DEF. XXXIX. —An isosceles tetrahedron is one whose opposite
edges are equal.
From the definition it follows at once (Euc. I., viI. xxxmi.)
that the four faces are equal, and that the sum of the plane
angles forming each trihedral angle is equal to two right
angles.
128. If we suppose BC= AD =a, A C= DB=p, AB=DC=y;
then denoting by a, b, c the angles of the triangle ABC, they
are also the face angles of the trihedral angle D - ABC; and