2B6 



ROYAL SOCIETY OF CANADA 



as < I a I < r, with a similar relation for /?. In the particular 

 case where a = ^ = o the equation is readily obtained. 



cos d tan (f) = tan PM = sin 6 tan a or 

 tan — tan tan d = o. This becomes 

 tan ^ = m tan ^ where w = tan a. This 

 has included the case where a = ^ = n. 

 Suppose, further, that a = and 

 < I /? I < TT. This is readily seen to 

 have as equation tan 6 = o. When 



a ± 0, and /? =— ;j-the equation is 



tan d = c, which for different values of 

 c may become any great circle passing 

 through the poles of OX. We conclude, 

 therefore, that in all cases the equation 



A tan d -\- B tSLU (f) -{- C = o 



represents a great circle; furthermore, all great circles are represented 

 by this equation. 



IV. It is interesting to note that \{ a = a R, h = ^ R, x = 6 R, 

 and y = <}> R the equation ' 



tan 6 I tan _ 

 tan a tan /3 ' 



as R becomes infinite, becomes the familiar equation in plane 

 geometry, 



-+4=1. 



Similar remarks apply to the two type forms, 



tan <f) — m tan d = o 

 tan d — c = o. 



V. The fundamental equation of all great circles could be 

 taken as 



tan cf) tan 1 

 tan (f)' tan 0' 1 

 tan (f)" tan 0" 1 



