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

90 Spherical Excess.
therefore C. C a + b
sm 2 + sinm 2 - E cos + cos
* c^ /C7 ~\ c a+b
sin ' - sin E cos 2 - cos 2
2 22 y 2
Hence tan - E cot ( C - E) = tan tan - (s- c). (376)
Similarly, from the first of Delambre's Analogies,
tan i lEtan î (C-E) = tan - - (.Etan t -(s-b). (377)
Ience, multiplying and extracting square root, &c. See Nouvelles Annales, 1856, p. 91.
EXERCISES XXV.
1. If the arc AD, drawn from A to a point D in the side BC, bisect the
area of the spherical triangle, prove that
cos AB: cos AC: sin ~ B1): sin 2 DC. (378)
7r 1r i
2. If ABC be a triangle, having a = b =, c =, prove sin = B.
3 2 3
3. If in fig., ~ 79, the arc EM be cut off equal to AB4, and MN be
drawn perpendicular to the great circle DE; then MN= E. (GUDERMANN.)
cos 1 b cos 1 c sin A
4. Prove that sin (A - E) = cs 2 b c c sin (79
cos2a
5. Prove that
cs ( - E in \ b sin ~ c + cos 1 b cos 1 c cos
cos(A - cs) = (380)
cos a
6. Prove that in a right-angled spherical triangle tan E = tan ~ a tan ~ b.
7. If a', b', c', ', B', C' denote the sides and angles of the triangle
supplemental to ABC, prove
cot e s cot s' = tan (s - a) tan ~ (s'-a'). (381)
8. In the same case, if 2E' denote the spherical excess of the polar
triangle, prove that
tan LE tan ~ E' = tan (A - E) tan (A' - E'). (382)
9. Prove that the arc joining the middle points of any two sides of a
spherical triangle is less than a quadrant.