On Practical Geodesy. 9 



arc on the earth's spheroidal surface whose circular measure 

 is a.s great as 1°^^ 30', and the latitudes of whose extremities 

 differ by as much as 1°, we may, with due respect to the 

 utmost attainable precision in geodetic surveying in Vic- 

 toria, assume — 



cos a, T / \ 



'- = 1 (so) 



COS a,, 



For by means of (27) it can be easily shown that even in 

 this extreme case a^^ — a^ is less than a sixth part of a 

 second, and that the logarithms of cos a^ and cos a^^ will be 

 the same to 8 places of decimals, and differ in the ninth 

 place by less than 4. Hence also, in the actual practice of 

 trigonometrical surveying, we may, for some purposes, 

 assume — 



a,, _ tan a,, _ sin a^, _ E,^ / \ 



a^ tan a^ sin a, R^^ 



their logs, being the same to at least 8 places of decimals. 

 Formulae 27 and 82 will be found very useful in the com- 

 putation of the angles of depression of the chord of the 

 geodesic arc; but, when worked by means of logarithms, 

 the best way is to find, in the first instance, an angle x such 

 that — 



R 



^.. 



and then equations (27) and (32) can be written in the 

 forms — 



tan J (a,, — a,) = tan {x — 45°) • tan J S (34) 



a,, — a, = tan {x — 45°) • %" (3 5) 



And since the angle x — 45° can never be more than a few 

 seconds in magnitude we have, in lieu of 35 — 



a,, — a, = )§" • {x — 45°) sin 1" (ae) 



Moreover, it is evident, that in actual practice, we infer — 

 from (31) and (15)— that— 



approximately (37) 



tana; = ^' (33) 



Zj Oi, 2// OL., 



and .'. z, a. sin a. R 



z.. a,. sin a.. R. 



(as) 



shewing that the auxiliary angle x of (33) has its tangent 

 equal to the ratio of the angles of depression of the chord, 

 and also equal to the ratio of the arcs z^^ and z^. 



