FOR THE EARTH'S SURFACE. 



299 



If N be the length of the normal PN terminated by the axis of 

 rotation, 



sin & = sin 6 ; 



hence 



1 



N _ r sin & __ 

 a a sin 6 J\ e 2 cos 2 ' 



N 1 



and 

 also 

 and 





^ = (l-e 2 )cos0; 



r x ' 



N 

 sin (6 1 0) . e" sin 6 cos 6. 



By these formulae the values of cos 6', sin 0', cos (9' 6), sin (#' #) for 

 values of 6 differing by 1 from to 90 have been computed, employing 

 Bessel's dimensions of the Earth, as given in Encke's paper and tables in 

 the Berliner Jahrbuch, 1852. 



Encke gives log w e = 8 '9 122052075, 



also, n being = 



Now 

 also 



Hence 



also 



therefore 



and 



a b 

 ~a + l' 



log IO Jl - e 2 = 9-9985458202, 

 Iog 10 n = 7-2238033861, 



Iog 10 (l+?r) = O'OOOOO 12 173. 



e 2 



& = - 



and 



e 2 2ft 



2-e 2 ~ 1+n 2 



Iog 10 2 = 0-3010299957; 



log w (2 - e 2 ) = 0-2995782505, 



Iog 10 e 2 (2-e 2 ) = 8-1239886655. 



These agree with the results obtained by Bessel, whose tables extend 

 only to the seventh place of decimals. 



382 



