On Practical Geodesy. 27 



These have been arrived at by other means in the "Account 

 of the Principal Triangulation of Great Britain and Ireland." 

 Moreover, from the spherical triangle S^PS^^, we have — 

 sin I., cos /, — sin I, cos I,, cos w 



cot A^ 

 cot A, 



cos I,, sin (0 

 sin l^ cos 1,1 — sin l,, cos Z, cos w 

 cos l^ sin w 

 E,, . , • 7 \ e^ ' cos I. 



' '* cot A, — cot A_ = (— ^sinZ, — sinZ,, ) , . 



^R,, ^ cos I., sin w/ V 



cot A,j — cot Aoo= (— ^' sin I,, — sin Ij • ^-^ 



\ix, ^ cos I, sin CO 



These also are given in the "Account of the Principal 



Triangulation of Great Britain and Ireland " (see page 231 



of that work). 



32. From (60) it is evident that for any pair of mutually 

 visible stations, we have — 



7 _ E,^ cos Z, sin w 



sin A,, cos 4 2 

 7 E^/ cos Ij, sin 0) , , 



sm A, cos J 2 



7 XV, rV,, sm O) / 27'OA 97'0*\ 



(R^, — R^ji sm A, sm A,, ^ 



the last of which is rigorously accurate for any two stations 

 on the earth's spheroidal surface, and a direct expression in 

 terms of the two latitudes and difference of longitude ; but 

 it assumes the form § when the latitudes I,, l^^, are equal. 



^^ ^ sin^ a, W,, 1 — e^ sin^ L . ,, 



33. From . ., = -^ = = o . . 7 > we have the 



sm- a,, R^, 1 — e'' sm'' I J 



rigorous formulae — 



2 sin^ a,, — sin^ a, , v 



sm'' I, sur a,, — sm^ l,^ Bmr a, ' 



6^ _ cos^ I, sin^ a,, — cos^ l^, sin^ a, . . 



2? sin^ Z, sin^ a,, — sin^ Z,, sin^ a, 

 applying to any two stations whatever on the earth's 

 spheroidal surface. 

 From (53) we have — 



2 _ sin^ A ,, sec^ Z, — sin^ A, sec^ I,, , . 



sin^ A,^ tan'^ I, — sin^ A, tan^ l„ 

 h^ _ sin^ A, — sin^ A,^ , . 



a^ sin^ A,^ tan^ Z, — sin^ A, tan^ I,, 

 (Holding true to at least 8 places of decimals in their logarithms.) 



