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

148 Applications of Spherical Trigonometry.
therefore E= S 1+ ( a)2+ ( - b)2+ (s )
4r- + 12r2
Hence 2Er2= + 2 +2.
Cor.-The area of the spherical triangle is equal to the area
of the plane triangle, if we omit terms of the second degree. 1
in-.
r
134. If n denote the number of seconds in the spherical
excess, A the area of the spherical triangle on the surface of
the earth in square feet; then log n = log A - 9-3267737.(GENERAL ROY.)
DEM.-We have 2E= 0626; A = 206265 Now the
mean length of a degree = 365155 feet. Thus - = 365155;
180
substituting the value of r from this equation in the value of A,
and taking logarithms, we get
log n = log A - 9'3267737. (464)
EXERCISES.-XXXV.
1. The angles subtended by the sides of a spherical triangle at the pole of
its circumcircle are respectively double of the corresponding angles of its
chordal triangle.
2. Prove Legendre's theorem from either of the formulae for sin A,
cos, A, tan ~ A, respectively, in terms of the sides.
3. If the radius of the earth be 4000 miles, what is the area of a spherical
triangle whose spherical excess is 1~.
4. If A", B", C" be the chordal angles of the polar triangle of ABC,
prove
cos A" = sin ~ A cos (s - a), &c. (465)
5. If A'BC be the colunar of ABG; prove that the cosines of the angles
of its chordal triangle are respectively equal to
cos a cos, sinbsin (C-E), sin c sin (B - E) (466)