281 



taii^,-^- tan 2 - 



+ — si - i. 



if H3 



Expand the tangents into infinite series according to the law 



Z 3 2Z 5 17 Z ••7" exponent of Z. 

 tan Z = Z -i 1 1 h 



3 15 315 



and Ave find 



x x' 



V 





{- + — . . . . • f- + — + . 



[r 3 r 3 r 3 r 3 



fa a 3 "| 2 |b b 3 ) 2 



{-+ .... I- <- + +-..- [ 



= 1. 



[r 3 r 3 j [t 3 r 3 



Dividing r 2 from each fraction, and passing to the limit r = * and we 

 have the equation of an ellipse in the plane. 



x 2 v 2 



— + — = 1. 



a 2 b 2 

 Any equation in the "rectangular spheric" coordinates -will reduce, in the 

 limit when the sphere is made to increase infinitely, to the equation of a 

 corresponding locus in the plane. 



