418 Mr DAVIES on the Equations of Loci 



From (1, 2) we have 



crt d0==:.^[* =-J4r- .................. (3.) 



sin cos si ii 2 



Or integrating, we get 



+ c6 cot '"' ................................. (4.) 



This equation (4) divides itself into two parts, according as + or is 

 taken ; and hence we have 



tan(+0) = + c, e cot *< .................................... (5.) 



tan(-0) = + c,,e cot "' .................................... (6.) 



We shall consider these equations separately : and, first, to find the con- 

 stants c, and c,,. 



To find c, put -j- ; then tan = 1, and log tan 0=0, and therefore 



* 



cot a . 6 + const = , or 



if we take that meridian where = , for the origin of 6, we shall have 



c, = l, 

 and our equation (5) becomes 



tan0 = e cot *-' ............................................ (5') 



3 7T 



Again, take = -7 ; then tan = 1, and log tan ( 0) = log 1 = 0. 



And hence, as before, we have 



c a = 1> 

 which converts (6) into 



tan( 0) = 6 coU " ........... . .............................. (6') 



and the united equation (4) may be written 



6 cot " ......................................... (4') 



