traced upon the Surface of the Sphere. 273 





Cor. 1. Let P = s that ' ls > ^ e circle be a great circle; then its equa- 



tion becomes 



cot</>= tan\ cos (0 /c), ................................. (2.) 



or cos (6 K) = cot A. cot<, ................................. (3.) 



which are the equations of a great circle whose centre has the position \, K ; 

 according as it is resolved for one or other of the variables. 



Cor. 2. Let X= 0; that is, let the centre of the circle (1.) be at the 

 pole of the equator, then the circle is 



cos p = cos <, .................................................. (4.) 



which is the equation of a parallel of latitude, or declination, according as 

 the terrestrial or celestial sphere is used. 



Cor. 3. At the same time, let X = 0, and P = ^ ; then the equation be- 



comes 



co s = 0, or < = 1, .......................................... (5.) 



which is the equation of the equator itself. 



Cor. 4. Let 7v. = ^ ; then equation (1.) becomes 



m 

 cos > =r sin (f) cos (6 K),..., ................................ (6.) 



the equation of a less circle, whose pole is in the equator. 





Cor. 5. Let also p = s 5 then 



sm$cos(0 K) = ........................................ (7.) 



This is fulfilled both by sin (j> = 0, and cos (6 K) = 0. The interpreta- 

 tion is, that whilst sin <p = 0, we have cos (0 K) indeterminate, and whilst 

 cos (0 K) = 0, then sin < is indeterminate. The former of these shews, 

 that whatever value be given to 6, the circle will pass through sin < = 0, or 

 P ; and the latter, that whatever value be given to sin cf>, the equation of 

 longitude is always the same. The latter is properly the equation of any 

 one definite meridian, the longitude of whose centre is K ; the former shews, 



