OF ARTS AND SCIENCES. 379 



of the given conic. This conic is called the supplementary conic of 

 the given one. It can be combined with the given conic so as to sim- 

 plify the solution of many questions in spherical conies. 



9. Conjugate Diameters. These are related to each other as in 

 plane conies ; that is, the diameter conjugate to the one through x'y' 

 contains the conic pole of the one through x'y', and vice versa. Its 



equation is therefore -^ -j- "-7^ = 0. Its extremity x y is found, as 

 in plane conies, to be such that 



x" y' y'' x' 



~a^^^ 6"' y ^= ^ oT- 



To find the lengths of a' and b 

 any two conjugate semi-diameters. 





COS a' = cos c cos 5 



tan2 a' = x'^ + y'' = b''-\-- 



a'i 62 



and tan^ b' = x"^ -4- y"^ =z a^ — x'"^ ; 



. •, tan^ a' -\- tan^ b' =^ a^ -\- b"^ =. constant. 



This might also be inferred from the corresponding properties of plane 

 conies, by the pi-inciples laid down in Art. 1. 



10. To find the perpendicular distance from the centre on a tan- 

 gent. ^ 



The trigonometric tangent of the perpendicular from the centre is 

 the cotangent of the distance of the pole from the centre. Calling 

 the perpendicular from the centre p, we have 



««t^ = VV* + Y*= ab = ab ' 



" ' tan V 



ah 



11. There are some curious properties of conies with reference to 

 the cyclic arcs. («) We have from the quaternion equation of the 

 cone 



cos d COS d' = k, 



