SECT. 3.] PLACES IN ORBIT. 140 



but since this derivation of the formula might perhaps seem open to some doubts, 

 we will give another not depending upon the ellipse. 

 Putting, for the sake of brevity, 



tan i v = 6, tan if = & , we have r = i p (1 -f 66), r = i p (1 -f- & & ), 



1 60 . l O tf . 20 20 



= 1 - R ^ J CO S , :. fTW ,, S1 n^ T -^,Sm* : =rfFF . 



Hence follow 



r cost/ r cos# = jt?(d & &\ r smi/ rsmv=p(& 6), 

 and thus 



Now it is readily seen that 6 6 = c^ i^^ iv ^ s a positive quantity : putting, 

 therefore, 



\/(l-fJ(0 -f) 2 ) = 77, we have 4,^^(5 6)r). 

 Moreover, 



r + / = * j (2 + ^ + 6 6 ) = 

 wherefore, we have 



From the former equation is readily deduced, 



as TJ and d ^ are positive quantities; but since i (& 6} is smaller or greater 

 than r], according as 



= - cos/ 



COS | f COS 



is positive or negative, we must, evidently, conclude from the latter equation that 



in which the upper or lower sign is to be adopted, according as the angle de 

 scribed about the sun is less than 180, or more than 180. 



