traced upon the Surface of the Sphere. 405 



3. To find the point of intersection of two given lines, referred to po- 

 lar co-ordinates. 

 Let them be 



r-a, seed ft 1 ni x 



J- \ il -) 



r= a, sec 6 ft, ) 



From which by equating we get, 



a, cos 6 ft = cos ft, , 

 or, by expansion, etc. 



_ 



0,, cos ft / cos ft, 

 Again, from (11) we have 



cos ft = 



a, 



cos ^ ft, = ^ 

 or by taking cos" 1 of each equation and transposing 



a. 



ft ft, = cos- 1 cos- ' - , or 



cost 



Transposing, squaring, etc. gives 



a, 2 2 a, a w cos ft ft, + a,J = r* sin *ft ft, 



or = _ 



va, 2 a, a,, cos ft ft, + a,, 2 



which may be compared with (VI. 6) of my former paper *. 



It may here be remarked, that the varied form of the equation referred to was acci- 

 dentally omitted in printing that article. Divide all the terms by cot A, cot * and we 

 get for (VI. 7) the following : 



_ tan X, tan X,, sin , 



tan \ 2 tan A, tan A,, cos * , + tan 2 A,, 



3 F 2 



