On Frccctical Geodesy. 5 



in which F is the same function of the latitudes in the 

 equation (4) and (5). 



SoP, — SooP. = (R. sin I, — U, sin IJ . (1— e^) (e) 



C,Z, — C,Z,, = (R, sin I, — R,^ sin ^J . e' (,) 



SoP.-S,,/).. : Z,Z,, :: (1-6^): e^ (s) 



3. From the expressions for the magnitudes of Q„ Q,„ we 

 have 



R; + Q; = 2-Ii; (1 _ e2 sin%) + F = 2a2 + F; 

 R^/ + Q^/ = 2-R,;(l — e^sin^/,) + F = Sa^ + F. 

 And therefore it is obvious that we have the relation — 



R/ + q; = R^/ + Q^; (,) 



Hence it follows that if N be the middle point of the 

 segment Z^Z^^ of the polar axis intercepted by the normals, 

 we have — 



And from this it is obvious that the stations So, S^^, are in 

 the surface of a sphere whose centre is N, and that we have 



^. -y Q. (-) 



(See formulae 81 'A and 81'B in the sequel.) 



4. If in each of the triangles Z^Z^^^S^, Z„Z^ ^S^o, we 

 express the base Z^Z..^ in terms of the other two sides and 

 the included angle, it is evident from (9) that — 



R^ • Q, • cos S^ = R,, • Q,^ • cos 8,^ (12) 



. cos S, _ R,, • Q ,, 

 cos "8,, ~ R, • Q^ 

 .•• ^.-Q. 7 R, -Q, (13) 



absolutely; but in all ordinary cases they are equals to at 

 least 10 places of decimals in their logarithms. 



5. It is evident that the plane through the middle point 

 N, of the segment Z^Z^^, perpendicular to the geodesic chord 

 S^S^o, must bisect this chord or pass .through its middle 

 point M. And therefore, since the portions NZ^, NZ„„, of 

 Z^Z^o, which Lie on opposite sides of this plane are equals, it 

 follows that the planes through Z^, Z^^, perpendicular to the 

 geodesic chord S^S^o, cut it in points T^, T^^^ equidistant 

 from its middle point M. Hence — 



sin a, = cos T,S,Z, = S,T, 



