358 Captain A. R. Clarke on the Course of 



curves. Thus, the difference of latitude of Q' north of Q is 



e 2 4smio-smi(r, . . . . . 



- — = — ~ — - cos wism w — sin u.) ; 



2 cos i cr' v ' 



(9) 



and if a$u be the distance of P north of Q, 



cm = 15- cos 



f cr si 

 L sir 



sin cr. 



Cr' 



cos u, cos a, — 



cr, sin a . . 



— -. j— cos tt' COS ex.' 



sin cr 



4 sin \ cr sin -jr c^ 

 cos-| a 3 



sin 



M /p 



(10) 



This expression completely determines the course of the geo- 

 desic ; and it is to be remarked that a has not been supposed 

 small. We may alter the form of (10) by eliminating a! through 

 the equation 



— cos v! cos a' = sin u { sin cr' — cos u t cos cr' cos ot r 

 Thus we get 



e 



Bu= -^-cosw{H cosu j co$u l + KsinW/}, 



w cr sin cr, — <r l sin cr cos cr' 



■ (ii) 



K = cr / sincr — 4 



smtcrsmier 



V.l . 



cos i a' 



If we desire to trace the geodesic line, not as passing through 

 two given points, but as starting from a point Ain a given azimuth, 

 we may refer its different points to the corresponding points of 

 the curve of intersection of the vertical plane at A which touches 

 the geodesic at that point. In order to do this, we must put in 

 (11) cr^ cr' — a, and in the result put cr' = 0, making the point 

 B move up to A along the geodesic. The result of this opera- 

 tion is 



«2 



hu=. — cosw{H cos w ; cosa y + Ksinw,}; 



H: 



CTCOS cr — Sin cr, 



t 



K = — a sin cr + 4 sin 2 i cr. 



(12) 



We may apply this equation to verify an expression given by 

 Bessel in the Astronomische Nachrichten, No. 3, for the difference 

 between the astronomical azimuth of a point B at A and the 

 azimuth of the geodesic AB at A. The difference is evidently 



du sin « . 



sin or 



