528] 
ON THE NON-EUCLIDIAN GEOMETRY. 
413 
We may, from the formulas 
a? = sin 2 q + sin 2 r — 2 sin q sin r cos a, a sin a = sin q sin r sin a, &c., 
but, more simply, geometrically as presently shown, deduce 
sin a cos B + sin c = - sin B sin q (sin q — sin r cos a), 
a 
sin a cos C + sin b = - sin C sin r (sin r — sin q cos a), 
and thence 
.. „ . w . „ . 1 . „ . ~ . . ism q sin r (1 + cos 2 a) 
(sin a cos B + sin c) (sin a cos G + sm b) = — sm B sin G sin q sm r < , . „ . , x 
7 a 2 cos a ( sin O' + sm 2 r) 
= 2 sin B sin G sin q sin r (sin q sin r sin 2 a — a- cos a) 
CL 
= sin B sin G (sin 2 a — sin q sin r cos a), 
which is the equation in question. For the subsidiary equations used in the demon 
stration, observe that the four points 0, X, A', B lie in a circle, and consequently that 
GO. GX = GA'. CB; or multiplying each side by sin G, then GO . GX. sin G = A'K. GB, 
that is, 
sin r (sin r — sin q cos a) sin G = a (sin a cos G + sin b), 
and the other of the equations in question is proved in the same manner. 
From the formula for cosh a we find 
sinh a = 
where - 
A, 
sin B sin G 
A 2 = — (1 — cos 2 A — cos 2 B — cos 2 (7—2 cos A cos B cos (7), 
whence also 
sinh a : sinh b : sinh c = sin A : sin B : sin G; 
and we can also obtain 
—t cosh a — cosh b cosh c . 
cos A = ; ; &c. 
sinh b sinh c 
So that the formulae are in fact similar to those of spherical trigonometry with only 
cosh a, sinh d &c. instead of cos a, sin a &c. The before-mentioned formula for cos a in 
terms of d, q, r is obviously a particular case of the last-mentioned formula for cos A. 
Cambridge, 11 May, 1872.