281 
a) (y) 
tan? — \ tan? = \ 
r J | 
a| ioe b| ae 
tan24— > tan?4— } 
Expand the tangents into infinite series according to the law 
Z3 Die 17 Z “7” exponent of Z, 
tan Z = Z+— + —— + — + 
3 15 BN) 
and we find 
= xe | 2 (y y3 * 
(Se al ‘Ease ese. t 
Lr or? } lr 3) ie 
p= ile 
a ai | 2 (b bs : 
sae te Pe se aU 
r oue J r eres 
Dividing r? from each fraction, and passing to the limit r — ~%, and we 
have the equation of an ellipse in the plane, 
Any equation in the “rectangular spheric” codrdinates will reduce, in the 
limit when the sphere is made to increase infinitely, to the equation of a 
corresponding locus in the plane. 
