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

Lexell's Theorem. 97
2~. If cot b.cotlc=l, or b+c=-r,
we have A - E = E, and the triangle becomes a lune formed
by the circle C'AC, and the greater circle tangential to
Lexell's circle in C'.
3~. If cot} b.cotc < 1, or b + c < 7, sin E is no longer
arbitrary. In order that sin (4 - E) < 1, sin E must be
<tan b tan-c. The maximum of E corresponds to sin E
= tan b. tan c, and then
A-E=, or A=B +C.
2'
EXERCISES.-XXYI.
1. Construct a lune equal in area to a given triangle (make use of Lexell's
circle).
2. Construct by means of Lexell's theorem a triangle ABIC' equal in
area to a given triangle ABC, and having two given sides bl, cl.
3. Construct on the side BO of a given triangle ABC an equivalent
isosceles triangle.
4. Convert a triangle ABC into an equivalent isosceles triangle, having a
common angle A.
5. Transform a spherical polygon ABCDE into an equivalent spherical
triangle.
[Employ Lexell's circle in the same manner as parallel lines are employed
in the corresponding question in Plane Geometry.]
6. Being given a spherical polygon ABCDE, if the sides be produced in
the same sense, and with each vertex as pole, an arc of a great circle be
described, limited by the sides of the corresponding exterior angle of the
polygon, prove that the total figure thus formed is equal in area to a hemisphere.-(NEUBERG. )
7. Being given A and E, prove that, if a is a minimum, b = c.
sn sin b sin o sin A
We have sin E = "si b sin
COS 2
Hence, from the required condition, sin ~b sin c is a maximum; but
H