1 36 Mr. Ivory on the Shortest Distance, fyc. 



station being the angle between the meridian and the vertical 

 circle passing through the other station. Understanding the 

 term in this sense, I shall put m and m! for the azimuths re- 

 ciprocally observed at the stations of which A and A' are the 

 latitudes; and it is obvious that m and m' will coincide with 

 \l and jx' when e 2 = 0. In this Journal for September 1828, 

 I have found these two equations, viz. 



A = \/l — e"-sin 3 A, A'= vT — e 2 sin 2 x', Q 



sm X sin X 



sin -d/ . . cos X a . in 



+ cos \f/ sm A — cos A tan A' = ■ T • £ A Q, 



tan m cos X 



sin -J- i • i i cos X' o . ^v 



+ cos \J/ sin X' — cos \' tan A = — . e~ A Q 



tan m cos A. 



And if we suppose e 2 = 0, these equations will be changed 

 into those which follow : 



sin -d> , I „ 



h cos \J/ sin A — cos A tan A' = 0. 



tan ft 



r + cos \|/ sin A' — cos A' tan A = : 



tan fjt, 



we therefore have, 



1 1 COS x 



tan m tan ^ cos x' sin \£ 



1 1 cos x 



- x e 2 A' Q, 



- e 3 A Q. 



tan m tan (J cos X sin ^ 



Now cos A' sin \J/ = sin <r sin /*, and cos A sin 4/ = sin <r sin [*'; 

 and if we put, 



X + X' X — x' 



X = cos 



2 2 



X + x' X — X' 



i/ = sin — — cos — — 



R = 0-43429 &c. 

 we shall easily obtain the following formulas ; viz. 



cos A sin m x —. — ) -f- e 2 y sin A R, 



log sin (m—fjJ)== log ( cos A' sin in! x — — ) + e 2 y sin a' R, 



which serve to compute ju, and /x' when m and ml are given. 

 The same formulas will likewise serve to compute m and ml 

 when i*. and ft/ are given : namely, by first substituting sin ft, 

 and sin ft/ for sin m and sin m 1 . in order to find near values of 

 m and m' ; and these being used in a second operation will 

 bring out the required quantities with all the accuracy that 

 can be desired. 



Applying the formulas to M. Puissant's example, I have 

 found, m — 139° 17' 4"-07, 



m' = 40 ] 6 45 "95. 



Of 



