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

First Class. 23
2~. The Norm of the sides of the triangle; and for the function N (see ~ 33)
he also suggests the names Second Staudtian, or the Norm of the angles of
the triangle.
6. Prove cos A cos B cos = sin.(34)
sin a sin b sin c
7., tan A tan i B tan = s —. (35)
NOTATIONS.-The arcs which join the vertices of a triangle to the middle
points of the opposite sides are called medians, and are denoted by ma, mb, mc,
respectively. The arcs of great circles, which are drawn from the vertices
at right angles to the opposite sides, are called the altitudes, and represented
by ha, hb, h,. The arcs which bisect the interior angles, called the interior
bisectors, are denoted by da, db, dc; and the bisectors of the exterior angles
by da', db', de'.
8. Prove cos b + cos c = 2 cos osma. (36)
9. Prove in a spherical parallelogram that the sum of the cosines of the
sides is equal to four times the product of the cosines of the halves of the
diagonals.
10. Prove that the norm of the sides of a triangle is equal to the norm of
the sides of any of its colunar triangles.
11. If ABCD be a spherical quadrilateral; and if
AB=a, CD = a'; BC= b, DA = b'; AC= c, BD = c',
and the arcs joining the middle points of a, a'; of b, b'; and c, c' = a,,, 7,
respectively, it is required to prove
cos a + b + cos = cosa' os bc cos c' 4os cos y. (37)
cos b + cos b' + cos c + cos c' = 4 cos 2 a cos ~ a' cos a. (38)
cosc c + cos a + a cos a'= 4 cos b cos b' cos. (39)
12. Prove
cos a + cos a' + cos b + cos b' + cos c + cos c' = 2 cos 2 a cos a' cos a
+ cos b cos 2 b' cos 3 + cos c cos c' cos y}. (40)
13. Prove
cosa + os a'+4 cos a cos a' cos a =cos b +cosb '+4 cosb cos b' cos3
= cos c + cos c' + 4 cos c cos i c' cos Y. (41)