traced upon the Surface of the Sphere. 285 



tan a, cos {$, tan X, cos K, 



sin K = _4_ / (7) 



v tan 8 a, 2 tan a, tan \ cos fi.K, + tan 2 X, 



tan a. sin 8. tan X, sin K, 



COSK=-, (8) 



* tan 2 a, 2 tan a, tan X, cos /3, K, + tan 2 X, 



Again, cot X = tan X, (cos K cos K, + sin K sin K,) 



tan X^cos K, (tan a, sin /3, tan X, sin K,) sin K, (tan a, cos/3, tan X / cos K,)} 

 ^ tan* a, 2 tan a, tan X, cos /3, K, + tan 2 X, 

 tan X, tan a, sin /3 K, 



V tan 2 a, 2 tan a, tan X, cos /3,- K t -f tan* X, 

 The equation of the circle is, therefore, 



/ COt X. COt Ct. . n , n -\ n , o 



cotcp =:-i_- J { cos (tan a, sin p, - tan X, sin /c ; ) - sin 6 (tan a ; cos p, - tan X, cos K)}.. (10) 



sin/3, ^ 



Cor. 1. If a / = x/ and /3 / = K /S or which is the same thing, if the given 

 point coincides with the centre of circle (1), we shall have 



, . ..I ', .- . V 



f ' 



cot <= ^ (1J) 



indicating that the circle is indeterminate,* as from other considerations we 

 know it should be. 



Cor. 2. The same result would have been obtained by considering that 

 the great circle, which is perpendicular to another, passes through its centre, 

 and therefore that the great circle sought passes through a, (3 t and X, K, . 

 Hence by (IV.) we should have a result similar to this one ; and, indeed, 

 dividing both numerator and denominator of (9, IX.) by tan X, tan a /f we 

 shall have the same form as well as the same value of cot X, as is furnished 



* It is indeterrainate, because the equation (11) is fulfilled by the co-efficients, which 

 should determine the circle, not from a particular value assumed by the variable (, 

 which would merely indicate an extraneous factor, that might be eliminated by dif- 

 ferentiation. 



VOL. XII. PART I. 00 



