282 Mr DAVIES on the Equations of Loci 



Also, from (5) we readily obtain the sine and cosine of e/ ; viz. 



cos e, = + sin A sin K /3, ........................................ (6.) 



, = :tl sin 2 Xsin 2 /c ft, ................................. (7.) 



VIII. 



To determine the angle contained by two given great circles. 



Let the circles be denoted by 



cot0 = tan X, cos (6 K,) ..................................... (1.) 



cot0 = tan A.,, cos (0 /c ;/ ) ................................... (2.) 



The angles made by each of these with the meridian through their points 

 of intersection, are found, by Art. VII., putting K for the longitude or ab- 

 scissa of intersection. 



For from (VII, 6) we have 



/ = cos" 1 sin X, sin (K K,) ........................................ (3.) 



e,, = cos -1 sin X /7 sin (K K,,) . ..................................... (4.) 



Hence 



cos" 1 sin X, sin (K /c,) cos" 1 sin \ sin (/c K a ) = e, e,; ............ (5.) 



or taking cosine function, and reducing the result, we should obtain a com- 

 plicated expression for cos(e, ), involving a radical. The better way, 

 then, if this method be adopted at all, is to compute f, and f tl separately 

 from (3 and 4). But there is a more direct method of effecting this object, 

 at least in an analytical point of view. It is as follows. 



The distance of the centres of two great circles on the sphere is the mea- 

 sure of the inclination of those circles. If, therefore, we denote by 8 that 

 distance, and recur to our equations (1 and 2), we shall have 



cos 8 = cos X, cos X,, + sin A, sin \, cos K, ............................... (6.) 



and 8 is the angle made by the two circles. 



Cor. 1. If X zi 5 , then one of these circles is a meridian, and we obtain 



cos d = sin X, cos K tl /c, (7.) 



