362] 
471 
362. 
NOTE ON LOBATSCHEWSKY’S IMAGINARY GEOMETRY. 
[From the Philosophical Magazine, vol. xxix. (I860), pp. 231—233.] 
Writing down the equations 
1 
> = cos a = 
cos a 
cos A + cos B cos G 
sin B sin G 
1 _ . cos B + cos C cos A 
cos 1) C0S sin G sin A 
1 cos G + cos A cos B 
> = cos c = .—.—.—„ 
cos c sin A sin B 
where A, B, C are real positive angles each <£7r: first, if A + B + G > tt, then a, b, c 
are real positive angles each less than r (this is in fact the case of a real acute- 
angled spherical triangle), but a, b', c are pure imaginarles of the form pi, qi, ri 
(where p', q\ r' are real positive quantities; and secondly, if A + B + G < tt, then a, b, c 
are pure imaginaries of the form pi, qi, ri (where p, q, r are real positive quantities), 
but a', b', c' are real positive angles each less than ^7r. Hence assuming A + B + G <ir 
and writing ai, bi, ci in place of a, b, c, the system is 
1 . cos A + cos B cos C 
. = cos ai = .—„ .—~ 
cos a sin B sin C 
1 
cos V 
cos bi = 
cos B + cos G cos A 
sin C sin A 
1 
cos c' 
= COS Cl = 
cos C + cos A cos B 
sin A sin B