On Practical Geodesy. 



37 



If the three normals intersect in one point, any line through 

 this point can be assumed as polar axis. If two of the 

 normals cut each other, and that neither of them is cut by 

 the third, then the polar axis must pass through the point 

 of intersection and lie in the plane of this point and the 

 third normal. If the three normals have no point of inter- 

 section, then the polar axis must lie in the surface of a ruled 

 quadric, &c. 



And when there are four stations, then should no two of 

 the four normals lie in one plane, there can be but two 

 transversals drawn to cut them, and therefore but two posi- 

 tions for the polar axis. However, with respect to all sur- 

 faces of revolution (whose normals must all cut the axis) we 

 can arrive at general theorems applying to any stations 

 wliatever on the surface. 



For instance, we can easily demonstrate the following 



THEOREM. 

 If (T), r«"), be any two stations on a surface of revolution 

 of any kind, and A^ 2, A^ „_i, the angles which the true 

 " geodesic " joining the stations makes with the traces of the 

 meridian planes through the stations, and that Ri, R„, are 

 the normals terminating in the axis, then will 



sin Ai 2 _ E,„ cos 4 . 



sin A„ „ _ 1 Ri cos li 



Conceive the " geodesic " to be divided into infinitesimaUy 



small parts or elements, 1, 2; 2, 3; 3, 4; 



n — 2, n — 1 ; n — 1, n. 



^H-i,n represent the azimuths of 

 the stations 



0, 0. CO- 

 CO. 0. 0' 



A21, 

 the stations 





 



Let Rj, I\/2' • 

 mals at stations 



0. • • 







as if taken at the stations 

 n - 1 respectively. 

 A«, „ _ 1 represent the azimuths of 



„ _ 1 as if taken at the stations 

 n?) respectively. 







be the lengths of the nor- 



respectively. 



Then from the elements of analytic geometry, we know 



