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

96 Spherical Excess.
the small circle, will have a constant area, namely, 2Er2. If
the side AB be given, the small circle BB1 described with A as
pole, and spherical radius c, cuts Lexell's circle in the points
B,,, each of the two triangles AB C, AB, C will have an area
= 2Er2.
In order that the problem may be possible, Lexell's circle
must meet the circle BB1; or, what is the same thing, the
angle PA'C' equal to - - E, must be sufficiently large, the
minimum of - - E, or the maximum of E, corresponding to the
case where the small circles touch each other. Then the points
A, B, P, A' are on the same great cirele, and the triangle BC'A'
is a diametral triangle;.. C' = '+ B;
but A'= r - A, and C'= 7r - C.
Hence A = B + C, and the required triangle is diametral,
a being the diameter.
Cor.-If AB be greater than A C', the circle BB, must intersect Lexell's circle, and there will be neither a maximum nor a
minimum; but if AB be greater than A C', AB + A C will be
greater than AA ', or b + c greater than 7r. HIence, if b + c > w,
there will be neither a maximum nor a minimum.
Trigonometrical Solution (NEUBERGG'S).
1~. We have, by Cagnoli's formulae (351), (352),
sin (A - E) = sin E cot b cot i c.
If cot bcotlc> 1, or b+c>180,
sin E may have any value, and then sin (A - E) may be found,
and the triangle is possible. Hence there is neither a maximum
nor a minimum.