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

Third Class. 47
21. Prove that the sines of the perpendiculars from the orthocentre on
the sides of a spherical triangle are proportional to the secants of the opposite angles.
22. If the base BC of a spherical triangle be given in position and magnitude, and the sum of the sides AB, AC be given in magnitude, prove that
the locus of the intersection of the bisector of the external vertical with a
great circle perpendicular to AB at B is a great circle.
23-30. Prove the following analogies due to Breitschneider. See CRELLE'S
Journal, Band. xiii., Seite 145:sin E. cos (A - E) sin s. sin (s- a)
I ~. n --- -. - 2 (208)
sin 3 A cos ~ a
cos 2 E. sin (A - E) cos s. cos (s-a)2. 2- - ---— 2 ---- - OS — 2__,=. (209)
sin 2 A cos a
sin E. sin (A-E) sin(s-b) sin (s -c)
30. -_= 2 — - --------. 2 2U (210)
cos 2A cos a
cos E. cos (A-E) C cos ( s-b c)c (21)
cos A cosa a
cos (B- E) cos E) sin s.cos ~ (s - a)
50.. = 2.-2 __ — _-. (212)
sin A sin ~ a
sin' (B-E) sin~(C-E) cos~s.sin (s-a)213
60. -f -- _ _ (213>
sin A sin a
7 cos(B — E) sin (C-E) sin(s-b) cos(s-) (214)
7. 2 — --- - - -- - - - -- - - -- 2 --------; -- _ ----------, ~(2 14 )
cos ~A sin a
8. sin(B-E) cos(G-BE) cos~(s-b) sin(s-c)
COS 2 Z 51n 2 a
cosS A sin 1a(215
These analogies are all inferences from Delambre's. For example, 1~ is
obtained by subtracting both sides of the equation
cos ~ (B + C) cos ~ (b + c)
n -c from unity.
sin 'A cos a