38 On Practical Geodesy. 



that the tangent lines to am^ infinitesimally small arc of 

 the first order, which forms part of a geodesic, have their 

 least distance apart an infinitesimally small of the third 

 order; and that the ratio of the lengths of these tangents, 

 from the points of contact to their points of least distance 

 from each other, is that of equality. We know also that 

 the plane of every two consecutive elements of any 

 "geodesic" contains the normal at their point of junction; 



and .*. that sin Aoj = sin A23 ; sin A32 = sin Ag^; 



; moreover, we know that the ratio of the cosines 



of all infinitesimally small arcs is unity. Hence we have — 



sin A12 



_ E,2 cos ^2 



sin Asi 



~ Ri cos li 



sin A23 



_ R3 cos ^3 



sin A32 



E.2 cos 1.2 





= 





= 



And from these we at once obtain the desired proof, by 

 equating the product of the first sides of the equations to 

 the product of their second sides. 



However, it may be proper to observe that this method of 

 proof holds good only when none of the normals R^, Rj, . . . 

 R„, is either = or = oo ; and that we shall suppose this to 

 be the case for all geodesies referred to in the present paper. 

 We may evidently write the above relation in the form — 



sin Aj 2 perpendicular from (jO to polar axis 



sin A^ „ _ 1 perpendicular from Ci) to polar axis 

 Or we may express it in words as follows : — 



THEOREM. 

 On any surface of revolution, the sines of the angles G^, 

 G^^, which the geodesic connecting two stations S^, S^^, makes 

 with the meridian traces through these stations are to each 

 other inversely as the perpendiculars from the stations to 

 the polar axis. 



For a spheroid, such as the earth's reputed surface, we 

 can prove, in like manner, that for any two stations what- 

 ever on its surface — 



sin^ A, tan^ I, + |, tan^' ^^ + 1-0068314987 



b' 



sin^ A„ tan^ 4 + -^ tan'- I,, + 1-0068314987 



