288 



ROYAL SOCIETY OF CANADA 



Let OP = d, sin 6 tan F = tan /M/ = 

 cos tan ^. That is, tan d tan F = tan 

 (p. But cos /Man rf = tan ^. The 

 ehmination of F gives tan'^ d = tan' 

 6 -\- tan^ (f), an equation of the second 

 degree in tan 6 and tan and readily 

 seen from the hypothesis to be the equa- 

 tion of a small circle with the origin as 

 centre and angular radius of d. Mak- 

 ing the supposition that B becomes in- 

 finite, we arrive at the well-known form, 



' + y' = r\ 



VIII. The more general case will now be 

 attempted. Let P be (6 ^) and P', {6', (f)') ; 

 cos d = sin PT sin P'T' + cos FT cos jw 

 P' r cos {d - 6') = COS P r cos P' r 



|tanP7'tanP'7" + cos (6-6'). 



But 



cos d ian (f) = tan PT ; 

 and 



cos d' isincf)' = tan P' T\ 



/ 



N 



O 



T T 



whence cos d = (cos PT cos P' T cos d cos d') (1 + tan ^ tan ^' 



-}- tan </> tan ^'). 



That is 



But 



cos 



cos" 



1 + tan'^ d 1 -f tan'^ PT ' \ ^ tan'^ P' 7" 

 (1 -f tan ^ tan ^' + tan ^ tan cji')- 



(1 + tan' PT) ^ 1 -h tan' <9 + tan' 

 cos* <? i 



It therefore follows that 



(1 + tan' d) = ( 1 -f- tan' d -f tan' <i6) (1 -f tan' ^^ + tan^ 00 

 (1 + tan d tan ^' + tan <p tan </)')', 



which is an equation of the second degree in tan Q, and tan 0, and 

 as the hypothesis indicates the equation of a small circle with centre 

 at P' and with d as angular radius. 



