of two Points on the Earth's Surface. 33 



and by substituting this value of sin 9 A in the second term of 

 the right side of the last equation, we shall get, 



If in this expression we make e 2 = 0, b 2 = /3 2 , we shall have, 



ds „. „» /"* dX COS A. „ : 



— r- = 2A= /3 2 / — : — 3 frf A cos A v' /3 3 — sin 2 A. 



In order to perform the integrations, put /3 sin a = sin A, 

 /3 sin a' =a sin A' ; then 



2A = (3 2 fd a - S^fd a cos 9 a ; 



and, by integrating between the limits a! and a, 



2 A = — |l(a_fl') — i-/3 3 sin (a— a') cos (a + <*')• 



Now if the great circle of which <r is a part intersect the 

 equator, a and «' are the arcs between the intersection and 

 the extremities of <r: wherefore a— a! = <r, and 



2A = - il o- - -1 /3 3 sin <r cos (« + «')• ( s ) 



In like manner if we make e 2 = 0, and b 2 = /3 9 , in the dif- 

 ferential equation (1), we shall get, 



b db f* dXcosX - 2 . /? d X cos X _ 



" ede 1/ (0> -'sin* x)| "" ^ ~~ ? U v'^-TsTnTT = ° ' 



which is transformed, by the same substitutions as before, into 

 this which follows, 



-.LLL.±.fJl- - { l-p 2 )fda = 0; 



ede &%/ cos 5 a v r ' J 



from which we readily deduce, 



ML = _ p* ( i _ /3 3 )-^— cos a cos a'. (4) 



ede ' K ' ' sin <y 



In order to determine the other coefficient B, I shall write M 

 for t hat pa rt of equation (2) which is multiplied by the radical 



a/j-P^j-; and I shall expand the radical in a series, then 

 -*i-= M -4(l-& 2 )M-&c. 



ede 2 v ' 



By differentiating this equation and making e 2 = after the 

 operation, 

 ^/ ds \ 



ede ede 6rf6 ede v ~ ' 



Now in this expression, the value of — — is known by the 

 N. S. Vol. 8. No. 43. July 1830. F formula 



