of Edinburgh . Session 1874-75. 
593 
Taking Q, P, P x , P 2 , P 3 to be these points, and cr~, p, p x , p 2 , p 3 
to be their vectors — 
by (5) 
<T~ = cos E 
(A--+ A-), 
\ sin p l sin p 2 sin p. 2 / 
by (1) p = sin r 
therefore 
\ sin 
p 
+ — 4- — — — 
p x sin p 2 sin ■ 
Sa'a ^ 
Syy_' 
sin 2 ^ 
si»A 
sin 2 p 3 , 
' sin Pi + 
sin P2 + 
sin p z 
,sin 2 pi 
sin 2 _p g 
sin 2 ^ 
(noticing that Sa/3 = 0, &c.), 
therefore 
cos QP = cos E sin r ( — — + — ^ — + — — ^ . 
V sin p x sm p 2 sin p 2 ) 
Again, by (2) 
therefore 
cos P x ( 
and 
cos ' 
and 
cos P„Q = - Scrp 3 = cos E sin r, ( + -r— ^ ^ 
\sm p x sin p % smjp 3 / 
Adopting the usual notation, sin p x sin a = &c., = zn , we have (see 
Todhunter’s Spherical Trigonometry) 
cos E sin - 
Pi = sin r x 
( ~ — r— — + 
A_ + 
— )> 
V sin pi 
sm p 2 
sin p % J 
■■ - Scrpi = 
cos E sin r x ( 
1 
+A— + A- 
1 \ 
^ sin pi 
sin p, 2 sm p 3 
\ = Scrp = 
cos E sin r 9 1 
f 1 „ 
A— + A— Y 
J \ 
vsm Pi 
sm Pi sinjp 3 y’ 
cos PQ = - 
2 n 
(sin a + sin b + sin c) 
cos P X Q = cos ^ s * n Ti ( - sin a + sin b + sin c) . 
&c. &c. 
Ex. (2.) To find the arcual distances between the orthocentre 
and the poles of the inscribed, escribed, and circumscribed circles. 
