160] 



so that 



60) 



61) 



ELLIPSOIDS OF REVOLUTION. 



X = 



s 8 (6 2 -a 2 ) 2 



'/ 



4a6 8 y / s2 ^ s 



= (._ a ,v ' |7 



No 



w 



s*ds 



so that 

 62) Y= 



for if 



then 



-,/&_ a* -, /6 2 -a 



Jbor sm~~ ] I/ 2 we may write tan~ ] I/ 2 



423 



These formulae all serve for an oblate spheroid, where a < 6. 

 For a prolate spheroid, "b > a, they introduce imaginaries, from which 

 they may be cleared as follows. 



Call 



sin 1 (iu} = #, 



then 



iu = sin &, 1/1 -}- u 2 = cos -9 1 , 



therefore 



Put 



cos # i sin # = ]l + u 2 -f w, 



sin 1 (iu) = & = i log { 



u. 



