10 On Practiced Geodesy. 



Lave, rigorouslj 



(-) 



11. Again, from the triangle S JS,,, we have, rigorously- 

 sin Q, cos a,, 



sin O,, cos a, 



Hence it follows that for any pair of mutually visible 

 stations, such as occur in trigonometrical surveying, we may 

 assume — 



sin O^ 

 sin Q, 



1; 



. j2 1 their logarithms being the 



' = 1 j \ same to at least 8 places - (40) 

 of decimals. 



tan O 



cos O, 



1; 



cos O, 

 (See formulae (30) and remarks as to its approximate accuracy.) 



12. From what has been already shewn or observed, it is 

 evident — 



O,/ — O, = € — A (41) 



and .'., we have from (23) — 



tan 1 (O, - OJ = «i^lK_ZL^) . tan i A (4.) 



cos 2" -^ 

 COS J S 



and, since a^^ — a^ is but a fraction of a second, even when 

 2 is as much as 1°^^ 80'; and that a can be but a few 

 seconds in all cases that occur ; it is easy to prove that, in 

 the actual practice of trigonometrical surveying, the angle 

 O,, — O, will never exceed the to part of a second. And 

 from this and equations (40) it follows that we can regard 



In the account of the trigonometrical survey of Great 

 Britain and Ireland, the magnitude of O,, — O, is shewn to 

 be always less than too 00 part of a second ; but it is not 

 shewn that the ratio of the sines or tangents of the angles 

 O,,, 0„ may be regarded as equal to unity for all pairs of 

 mutually visible stations : yet this is necessary, as, in some 

 instances, O,, and Q^ are extremely small arcs. 



13. And if we put H, and S, to represent the small 

 spherical angles S,,D^D,,, S,D,,D„ it is evident that, in like 

 manner, we have — 



sml(DA-I>.E.).^ 

 " ' cos 1 (D,E,, + D,E,) ^ ^ 



and it can be easily shewn that the difference of the angles 

 H,; and S, is as extremely small as the difference of the 



