873] 
WHICH CUT EACH OTHER AT GIVEN ANGLES. 
561 
viz. this is 
F 2 cos 2 ¡3 + Z 2 cos 2 7 + 2 YZ cos /3 cos 7 cos A 
— 2Fcos /3 (cos B + cos Gcos A) — 2Z cos 7 (cos G + cos A cos B) 
+ cos 2 B + cos 2 C + 2 cos A cos B cos G = Y 2 + Z 2 + 2 YZ cos A'. 
Reducing by the relation A + B + G = nr, this becomes 
— 2 F cos (3 sin A sin G — 2Z cos 7 sin A sin B + 1 — cos 2 A 
= F 2 sin 2 ¡3 + Z 2 sin 2 7 + 2YZ (cos A' — cos ¡3 cos 7 cos A). 
Here A' = A + ¡3 + 7, and thence 
cos A' = cos A (cos /3 cos 7 — sin ¡3 sin 7) — sin A (sin 7 cos ¡3 + sin ¡3 cos 7); 
hence the right-hand is 
= F 2 sin 2 (3 + Z 2 sin 2 7—2 YZ (cos A sin ¡3 sin 7 + sin A sin 7 cos /3 + sin A sin /3 cos 7), 
or, reducing by 
F sin /3 = sin B, Z sin 7 = sin G, 
this is 
= sin 2 B + sin 2 G — 2 sin B sin G cos A — 2 F cos (3 sin A sin G —2Z cos 7 sin A sin B, 
and the terms in F, Z are equal to the like terms on the left-hand; the whole 
equation thus becomes 
— 1 + cos 2 A + sin 2 B + sin 2 G — 2 cos A sin B sin G = 0, 
where the last term is 
= 2 cos A {cos (B + G) — cos B cos G], 
= — 2 cos 2 A — 2 cos A cos B cos G, 
= — 2 cos 2 A + (cos 2 A + cos 2 B + cos 2 G — 1), 
= — cos 2 A + cos 2 B + cos 2 G — 1; 
the equation is thus 
— 1 + cos 2 A + sin 2 B + sin 2 G — cos 2 A + cos 2 B + cos 2 (7—1 =0, 
or, finally, it is —1 + 1+1 —1 = 0, which is an identity. The formulae for the inter 
section of the third and first circles, and for that of the first and second circles, are 
of course precisely similar to the above formula for the intersection of the second 
and third circles; and the verifications are thus completed. 
Cambridge, April 7, 1887. 
C. XII. 
71