543] 
525 
543. 
ON AN IDENTITY IN SPHERICAL TRIGONOMETRY. 
[From the Messenger of Mathematics, vol. i. (1872), p. 145.] 
In a spherical triangle, writing for shortness a, /3, 7 for the cosines and a', /3', 7' 
for the sines, of the sides: also 
A 2 = 1 — a 2 — /3 2 — 7 2 4- 2a/3y ; 
we have 
cos 
A - * ~ #7 ~; n A - A 
^ _ /3y '» sin ^ “ ¿sy ’ 
with the like expressions in regard to the other two angles B, G respectively. 
Hence 
cos (A + B + G) = cos A cos B cos G — cos A sin B sin G — &c. 
(a. - /3y) (/3 - 7a) (7 - a/3) — A 2 (a + /3 + 7 — ¡3y — 7a — a/3) 
(i-oa-^a-T 2 ) 
The numerator is identically 
= (1 — a) (1 — /?) (1 — 7) [A 2 — (1 + a) (1 + /3) (1 + 7)], 
viz. comparing the two expressions, we have 
(1 - a) (1 - /3) (1 - 7) A 2 - (1 - a 2 ) (1 -/3 2 ) (1 - 7 2 ) 
= (a — ¡3y) (/3 - 7a) (7 - a/3) + A 2 (— a — /3 — 7 + /3>y + 7a + a/3) ; 
or, what is the same thing, 
(1 - a/37) A 2 = (1 - a 2 ) (1 - /3 2 ) (1 - 7 2 ) + (a - /3 7 ) (/3 - 7a) (7 - a/3), 
which is the identity in question and can be immediately verified. We have thus 
_ A 2 — (1 + a) (1 + (3) (1 + 7) 
(l + a)(l+/3)(l+ 7 ) ’ 
and thence 
cos (A + B + G) 
1 + cos (A + B + G) = 
(1 + a) (1 + /3) (1 + 7) ’ 
1 ro-M 1 B l + a ) G+/3X 1 + 7) - A 2 
1-cos (A+B + C) (1 + a)(1+i8)( r+-^-» 
giving at once the values of cos 2 \ {A + J3 + G), sin 2 ^ (A + B + G), sin (A + B + C), and 
tan 2 ^(A + B + G): these are known expressions in regard to the spherical excess.