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

50 Solution of Spherical Triangles.
The equation (221) proves that if 2 (c + a) be greater than
90~, c is less than a, for the product of the quantities under
the radical must be +. The sign is + or - in (222), according
as a is less or greater than 90~. If the given parts c and a be
each 90~, the angle A is 90~, and b is indeterminate. It is also
evident, from the formulae (218)-(220), that c must lie between
a and -r - a, in order that the values of cos b, cos B, and sin A
may be numerically less than unity.
EXAMPLEGiven c = 37~ 40' 20", a = 37~ 40' 12"; find b, A, B.
Type of the Calculation.
c + a = 75~ 20' 32", (C + a) = 37~ 40' 16".
c-a= O 0 8, i(c-a)= 0 0 4.
1 tan (c + a) = 1'8876649, I sin (c + a) = 1i9856305,
I tan i (c - a) = 5-2876348; 1 sin (c - a) = 5-5886648;.. I tan } b = 3-5876498..~. I tan (45 + i A) = 2-3000150,
I tan aB- 3-8015172.
Hence b = 0~ 26' 37"'2, A = 89~ 25' 37", B = 0 43' 33".
EXERCISES.-X.
1. Given e = 63~ 55' 43", a = 120~ 10' 0"; find b,, B.
2., c=54 20 0, a= 36 27 0;,,
3.,, c=87 11 39-8, a= 86 40 0;,,
52. SECOND CAsE.-Being given c, A, to calculate a, b, B.
The unknown parts are found thus:sin a = sin c sin A, equation (108). (224)
tan b = tan c cos A,,, (109). (225)
cot B= cos c tanA,,, (112). (226)