a(l-e'cos^^)'/^d^ 



a cos Od\ 



a cos 6dX 



and since a = a (property of geodesies on surfaces of revolution, i. e. r sin a„ = t sin a„, 

 r = a cos ^),ds/aDSd= a(l - e^ cos^^)'''^ d0/ad0 = (l - e'' cos^^)'/=', which may be written 



S = a(d + Sd) = a I'd + /'^'[( 1 - e' cos'61)'/'' - 1] DSdl . (83) 



If (83) also represents the equator, then 5d = 0, when d = Oo = 0. Hence we add to the 

 integrand 1 - (1 - e^ cos ^doY'^ to get 



S = a(d + 5d) = afd + f^'\(l - e' cos' dY^' - (1 - e' cos'^o) '/'] DSdl , (84) 



and we note that when d = d^ = 0, 8d = 0; when = ^o , s = d = 5d= 0; when d^ = 7t/2, d, = 6i, 

 d2 = ^2» DSd = dd, d = 02- (9i then (84) represents the meridian. 

 Expanding (84) to 6th order terms in e, find 



d - (eV2) (1 + eV2 + 3eV8) / ' (sin'^o " sin'^)DSd 

 d, 



+ (eV8) (1 + 3e72) f '^Msin'^o " sin*0) DSd 

 - (eVl6) / ^^ (sin"6lo - sin '6*) DSd 



Now from (77), sin 6 = sin 6^ cos d, 



sin'0 = sin'^o cos'd = (1 + cos 2d). 



(85) 



(86) 



The value of sin'^ from (86) placed in (85) and the resulting integrations performed with 

 respect to d, leads to expressions in powers of the right hand quantities in (79) so that (85) 

 may be written finally as 



71 



