of Edinburgh, Session 1874-75. 
589 
2. On the Circumscribed, Inscribed, and Escribed Circles of 
a Spherical Triangle. By C. G-. Colson, Esq. Communi- 
cated by Professor Tait. 
In the following paper I propose to investigate expressions for 
the vector of the following six points of a spherical triangle : — 
(1.) Pole of inscribed circle. 
(2.) (3.) (4.) Poles of escribed circles. 
(5.) Pole of circumscribed circle. 
(6.) The orthocentre or intersection of arcs drawn perpendicu- 
larly from angles upon the opposite sides. 
These vectors will all be found in terms of the vector of the 
corners of the triangle drawn from the centre of the sphere. 
Throughout the investigation a, (3, y will denote the vectors of 
A, B, C, the corners of triangle ABC, A'B'C' will represent the polar 
triangle of ABC (A! being pole of BC), &c. ; a (3' y will denote the 
vectors of its corners ; and following the notation usual in spherical 
trigonometry, a, b, c, A, B, C will denote sides and angles of the 
triangle ; p Li p 2 , p 3J the perpendicular arcs from A, B, C on BC, 
&c. ; R, r, r lt r 2 , r 3 the radii of the circumscribed, inscribed, 
and escribed circles. 
After finding these vectors we proceed to deduce certain well- 
known results, among others, to find the radius of the circle (analogous 
to that discovered by Feuerbach in the case of a plane triangle) 
which touches the inscribed circle and the three escribed circles. 
To find the vector of the pole of inscribed circle. Let p be the 
vector (from centre of sphere) of P, the pole of inscribed circle of 
the triangle ABC. Then we may express p as follows : — 
p = xa -f- yf3 + zy , 
where xyz are scalars to be determined. Operating by S .V/?y on 
the expression, we have 
SpV/?y = xSoNfiy. 
But 
Y/3y = a sin a ( a being the vector of A') , 
therefore 
Spa — xSaa ' , 
or 
cos PA' «=. sc cos A A' , 
