[bbatty] an outline OF ANALYTICAL SPHERICAL GEOMETRY 269 



And 



ry, p,rp^ (1 + tan'^ 6 + t aii^ cj>) (1 + tanW- +tan' ^ _ . 

 (1 + tan ^ tan a" + tan 4> tan /3") ('^1 + tan^ r") 



The equating of these two, since (1 + tan' d + tan' 0) is never zero, 

 gives the same equation as before. Either definition would, there- 

 fore, suffice for the radical axis. 



XXXI. We shall now direct our attention to the consideration 

 of a treatment of the r and 6 system of co-ordinates. As before, a 

 fixed point is taken as origin, and a fixed great circle, usually OX, 

 is taken as the line of reference. By r we mean the angular interval 

 separating any point on the sphere from the origin, measured along 

 the arc of the great circle. The angle between the great circle so 

 used and the great circle of reference goes by the name of d. Let 

 our first problem be to find the equation of the great circle cutting 

 OX at the point (a, 6) and making an angle of a. 



tan /9 sin (« + 3/) = tan a sin M = 



tan P V. 



That is 



sin a (cot ^f + cot a) = tan a cot d. 



But 



cos d tan r = tan {a + M). 



Or 



cot a cot M — \ 



sec a cot r = 



cot a + cot M 



This becomes on reduction by eliminating M 



1. sec 6 cot r + cot a cosec a tan 6 = cot a, 

 or further, tan r (cos 6 tan a — sin 6 sec a) = tan a tan a. 

 2. tan r (cos 6 — sin 6 sec a cot a) = tan a. 



XXXII. This can readily be seen to be correct, for if the radius 

 of the sphere becomes infinite the equation is 



r cos 6 = a + r sin 6 cot a. 



Equations 1 and 2 are identical ; the second seems to be a more com- 

 pact form, however. The above equation would scarcely indicate 

 that if a = o, the equation becomes d = o. The equation is applic- 

 able to any other case, and so is the general equation of a great circle 

 in polar co-ordinates. We get at once in this system that the equa- 

 tion of a small circle with centre at the origin is tan r = c. 



XXXIII. To extend this to the case of a small circle of radius 

 Q. about the point (r', 6') is our next problem. The equation here 



ia evidently 



cos «5. = cos r cos r' 4- sin r sin r' cos {d — d'). 



