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

94 Spherical Excess.
Analytical Proof.-The triangles ABa, A Ca having the same
spherical excess, we have, by Cagnoli's theorem, ~ 80,. c.Ba. b. Ca.
sin si - sin B sin sin - sm C
2 2 2 2
cos Aa cos i Aa
Hence Ba c
sin- cos -
2 2
Ca- b
sin cos
and from this and two similar equations we get
sin 2Ba sin - CP n A7 = sin A = a sin RaC sin 2 -B.
(1)
Also the triangles ABa, 7yBC having equal areas,
tan c tan i Ba = tan yB tan a. (Art. 81, Cor. 2.)
Hence,
tan 1 Ba tan I C,/3an n -yA = tan 2 aC tan i/3A tan -yB.
(2)
From (1) and (2) we have
cos2 Ba cos CP cos Ay= cos -aC cos BA cos -yB. (3)
From (1) and (3) we get
sin Ba sin Cf3 sin Ay = sin a C sin,3A sin yB. (4)
Hence the arcs Aa, Bl, Cy are concurrent. (NEUBERG.)
Cor.-The triangular co-ordinates of the point of intersection
of the arcs Aa, B/3, Cy are
cos a - cos b cos c, cos b - cos c cos a,
cos àC - cos a cos b. (Ibid.)
These values are obtained from the equation
sin Ba cos - c
sin ( a - j-Ba) cos b'
which gives cot 2Ba, and thus sinBa.