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

82 Various Applications.
and from A'PA,
cos PA' sin iMf+ cos PA sin MA' = cos PM sin AA'.
Eliminating PA', we get
cos PA sin a sin iTA'
cos PB sin A'C + cos PC sin BlA' + A
sin AM
cos -Plsin AA' sin a
sin Aix
but
2no = sin AlMsin BA' sin a; 2nb = sin AJllsin CA' sin a;
2n, = sin a sin MA' sin a; 2n = sin a sin AA' sin a.
B o B
Fig. 29.
Hence, eliminating B4', CA', MXA', AA', we get
nC cos PA + nbcos PB + nPC =n cos PM.
Cor. 1.-If P be a point, such that for given multiples
1, m, n, I cos AP + m cos BP + n cos CP is constant, the locus
of P is a small circle.
Cor. 2.-If there be any number of fixed points A, B, C, &c.,
and a corresponding system of multiples 1, m, n, &c., and P any
point satisfying the condition 2 (1 cos AP) = constant, the locus
of P is a circle.
Cor. 3.-If p be the spherical radius of the circle, touching
the inscribed and escribed circles of a spherical triangle (HAiT's
circle),
tan p = tan R. (348)
DEM.-Let P be the pole of Hart's circle, I, Ia, Ib, I, the
poles of the in- and circumcircles; thus
IH= p - r, IaH= p + ra Ir I= p + rb, Ic= p + r,.