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

Spherical 2fean Centres. 81
26. In a quadrantal triangle, of which the side c is the quadrant, prove
that
cotR=sin(C-E), cotRA = sin(B-E), cot RB=sin(A-E),
cot Rc = sin E. (344)
27. If the angle A of a triangle remains constant, and also the perimeter,
the envelope of the side BC is a small circle.
28. If the angle A remains constant, and also b + c - a, the envelope of
BC is a small circle.
29-31. Construct and resolve a spherical triangle, being given
1~. A, a, b+ c; 2~. A, a, r; 3~. A, a, R.
tan (A - P-) tan (B - E) tan (C- BE)
32. Prove cos2=tan( ) +tan (B - ) +tan C-B) (345
33. Find the simplest formulae for r, ra, rb, rc; -R, RA, RB, Rc, for a
diametral triangle; that is, for a triangle for which
c a b
C = A + B, or sin2 - = sin2 -+ sin2.
2 2 2
34. If a spherical quadrilateral be such that it is inscribed in a small
circle of radius R, and circumscribed to another of radius r, prove that if 8
be the distance between the poles,
sin (R + r + 8) sin (R + r-8) sin (R —r + ) sin (R- r- a) = sin4r cos4R.
(STEINER.) (346)
SECTION IV.-SPHERICAL MEAN CENTRES.
DEF. XXVI.-If the triangular co-ordinates of a point M with
respect to a triangle ABC be na, nb, n,, we have seen (Ex. xxI., 8)
that the arcs AMK, BM, CM divide BC, CA, AB in the spherical
ratios nc: n5, na: n,, nb: na. M is called the spherical mean
centre of the points with respect to the system of multiples
na nb, n,.
76. If M be the mean centre of the points A, B, C for the
multiples n,, nb, n,, and P be any other point, then
ncos AP + nbcos BP + nccos CP = n. cos MP. (347)
DEM.-We have by Stewart's Theorem (Euc. III. 17), from
the triangle BPC,
cos PB sin A'C + cos PCsin BA' = cos PA' sin a;
G