hence 
Sin a A p B = | cos A sin 1B- ¡3 +cos B sin A- a —sin A sin B sin aft • a[31 (2). 
Equation (1) expresses what is held to be the fundamental theorem of 
spherical trigonometry; but the complementary theorem expressed by 
(2) is never considered. So far as magnitude is concerned, it may be de 
rived from (1) by the relation cos 2 0 + sin 2 0 = 1; but it is not so as regards 
the axis. Equation (1) is the generalization of the theorem of plane trig 
onometry 
cos (A + B) = cos A cos B — sin A sin B; 
while equation (2) is the true generalization of the complementary theorem 
sin (J. + B) = cos A sin B + cos B sin A. 
The one theorem may perhaps be derived logically from the other, when 
restricted to the plane, but it is not so in space. The two equations form 
together what is called the addition theorem in plane trigonometry. Why 
do we have addition on the one side of the equation, while we have mul 
tiplication on the other? Because A-\- B is the sum of two indices of an 
axis which is not expressed, the complete expression being 
cos A cos B — sin A sin B 
Sin a AJrB = (cos A sin B + cos B sin A) • a ‘ 
Prosthaphaeresis in spherical trigonometry. 
The formula for a A /3 B is obtained from that for a A ft B 
by putting a 
minus before the sin B factor. Hence 
cos a A fr B = cos A cos B -f- sin A sin B cos ct/3, and 
Sina A 13 B =— cosAsinB• /3“ -f-cos BsinA' aC + sin A sin B sin a(3- ap*’ 
Hence the generalizations for space of 
cos (H—B) + cos (A-\-B) =2 cos A cos B, 
cos (H—B) — cos (A+B) = 2 sin A sin B, 
sin {A-\-B) + sin (A—B) — 2 cos B sin A, 
sin (A-\~B) — sin —B) = 2 cos A sin B, 
are respectively 
cos a A ft B -f- cos a A [l B — 2 cos A cos B, 
cos a l- 
Sin a A p B — Sin a A [1 B = 2 { cos A sin B • ¡3 — sin A sin B sin «/3 ' a p |' 
Let 
v i W