14* On Practical Geodesy. 



Multiplying both sides of these equations by the chord h, 

 and remembering that the projection k^ of the chord on the 

 plane of the equator is equal to k. sin 0, we have — 



k • sin A, cos a, = k^' sin <f)^ 

 k • sin A,, cos a,, = k^' sin <f>^, 



But from the plane triangle p.C^p,^ we know that 



R, cos I, sin o> R,, cos l^, sin o> 

 ° ~ sin 0, ~ sin </>,^ 



.*. we have — 



(«o) 



k ' sin A, cos a, = R,^ cos I,, sin w 

 ^ . sin A,, cos a,^ = R, cos l^ sin 'w 



And, since ^ = 2s • sin |^ 5 -f- ^ • sin V, we have — 



2s • sin A, sin -^ ]§ • cos a, . 



^ . g^ j„ = R,, cos I, sin w (e i) 



2s • sin A,, sin J ^ • cos a,^ 



;^-; — i — Yi, = R^ cos l^ sin <o 



And since for any pair of mutually visible stations cos a, = 

 cos a,, = cos J ^, 



s • sin A, • sin 3 _, , . 



:§ • sin 1" = ^'' ^°' ^'' "'^ " (e .) 



s • sin A,, sin 5 ^ , . 



— ^ -. — t77 — = K, cos C, sin o) 



2 • sm 1" / / 



When the geodesic arc s is such that its circular measure 5 

 is not more than 1°, we immediately deduce the relations — 



s • sin A, 



0) = 



R,, • cos I,, • sin 1" / X 



s • sin A., 



0) = —, 



R, • cos I, ' sin 1" 



l^° In Chambers' "Practical Mathematics," and in the 

 article on " Geodesy" in Spon's Dictionary of Engineering, 

 the formulae (63) are given in an erroneous form which must 

 inevitably lead to incompatible results when applied in 

 trigonometrical surveying. The erroneous formulae given 

 there and elsewhere are — 



s • sin A, s • sin A., 



(0 = i = 11 



R, • cos I,, ' sin 1" R,, • cos I, * sin 1" 

 (See note 6 to problem 10 given in the sequel.) 



22. From 50 or 60 we have- 

 cos a, _ R,, cos l,^ sin A,, 



cos a^, R, cos I, sin A, 



(") 



