[731 
732] 
97 
97 
67 
! 51 
88 
52 
7 
53 
70 
54 
21 
55 
16 
56 
63 
57 
48 
58 
92 
59 
47 
60 
82 
61 
44 
62 
52 
63 
35 
64 
59 
65 
8 
66 
80 
67 
24 
68 
46 
69 
72 
70 
41 
71 
22 
72 
26 
73 
66 
74 
78 
75 
4 
76 
40 
77 
12 
78 
23 
79 
36 
80 
69 
81 
11 
82 
13 
83 
33 
84 
39 
85 
2 
86 
20 
87 
6 
88 
60 
89 
18 
90 
83 
91 
54 
92 
55 
93 
65 
94 
68 
95 
732. 
A THEOREM IN SPHERICAL TRIGONOMETRY. 
[From the Proceedings of the London Mathematical Society, vol. XL (1880), pp. 48—50. 
Read January 8, 1880.] 
In a spherical triangle, where a, b, c are the sides, and A, B, G the opposite 
angles, we have 
— tan \c tan | a tan \b sin (A — B) = tan ^b sin A — tan £a sin B, 
tan \c {1 — tantan §b cos (A — B)} = tan cos A + tan \a cos B; 
which are both included in the form 
, . . tan-^c — tan hb (cos A +i sin^l) 
tan la (cosB — %smB) = x + tan iHeoiA +isin A) ■ 
For the first of the two identities: from 
cos A + cos B cos G 
COS CL — • T) • j 
sm B sm (J 
7 cos B + cos A cos G 
cos b = ;— , 
sm A sm G 
we deduce 
1 /cos A cos B\ cos G /cos B cos^l\ 
cos a - cos i = gp 
1 \ (sin 2 A - sin 2B) cos G sin (A. — B) 
- sirTO sin A sin B sin G sin A sin B 
sin (A - B) 
sin G sin A sin B 
(cos (A + B) + cos G] 
sin (J. — B) 
sin G 
(cos c — 1); 
C. XI. 
13