On Practical Geodesy. 11 



angles O,, and O,, and that they too can be regarded as 

 equal to each other. Moreover, the points D,OI),, are on 

 one great circle. 



14. Now, since for all pairs of mutually visible stations 

 on the earth's spheroidal surface, we have — 



A, + A, = A, + A,, 



and that we can express the angle w in terms of the angles 

 Ag + Aqo and the sides V, I", of the triangle S^PS,, ; there- 

 fore by substituting, in such expression, A, + A,, for its 

 equivalent, we have — 



tan J a, = "?' f f/ 7 \\ ■ cot i (A, + A„) 

 Sin ^ [L, -f- L,i) 



This formulse is known as Dalhy's Theorem, for the history 

 of which see the "Account of the Principal Triangulation of 

 Great Britain and Ireland," page 236. 



15. By appljdng Delambre's analogies to the same spheri- 

 cal triangle S^PS^^, we find in like manner — 



sin 1 (A, + A,) = ^^ii^ • cos J {l" — (46) 



cos J V 



COS I (A, + A J = ?^^-f-^ • cos 1 (l" + V) (4,) 



and 



cos J 



tani(A, + A„)= -|j;:-;:j -cot|. 



cot i (A, + A,J - ^^^ 2 (I" + ^') . foT. 1 ,., 



(-) 



cos 1 {I" — I') 



From (48) it is evident that when the latitudes of 

 the stations are of constant magnitudes, then the greater the 

 difference of longitude w is, the less will the sum of the two 

 azimuths be. 



''CONVERGENCE OF MEEIDIANS." 



The stations being supposed on the same side of the 

 earth's equator, the sum of the azimuths A, -f- A,, is always 

 less than 180°; and it is customary to call the defect or 



180° — (A, + A,,) 

 the " convergence " of the meridians as respects the stations. 

 Putting C to denote this convergence, it is evident from 48 

 that we have — 



taniC = ^^f (^- + H-tan^o, 

 cos J {I, — ;„) ^ 



