180 aEODESIC IJfVESTiaATIONS. 



Until the science of G-eoclesy is tlius fully applied, not the least reliance can 

 be placed on the lengths of arcs of meridian determined from triangulations. 



1t is easy to prove that the principal theorems arrived at apply to 

 short arcs on any surface whatever, as well as to the spheroidal sur- 

 face of the earth, even when such surface is so irregular as to be inexpressible 

 by means of an equation. 



"We can assume any straight line cutting the normals at the stations S\ S" 

 as polar axis of reference, and any point P in it as a pole ; and then assuming 

 any point C in this polar axis as centre of reference, we can take the plane 

 through it perpendicular to the polar axis as the equatorial plane of reference. 

 Thus the figure can be constructed as already indicated in the case in which 

 the surface is a spheroid. 



And in this general case, it is evident that — 



sin A' cos ^ 2' = cos 9 sin cp^ ; 

 sin A" cos i 2" = cos 9 sin (p^^ ; 



sin A' 'B„ cos I" cos i 2' 



sin A" S, cos I' cos i- 2" 



and that all the formulae, not implicating peculiar properties of the spheroid, 

 hold good for the general surface, when the stations ;S', »S"' are so near to 

 each other as to p)ermit us to regard the normals as making angles with 

 the chord whose sines are equals, and the traces of the normal-choral planes 

 as equals in length and circular measure. 



If there be three stations S', S", S"', to be simultaneously considered, the 

 assumable position for the polar axis of reference is generally restricted ; as 

 such axis must cut the three normals to the surface drawn at the stations. 



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

 be assumed as the polar axis. If two of the normals cut each other, and 

 that neither of them is cut by the third normal, 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 intersection, then the 

 polar axis must lie in a ruled surface of the second degree, &c. 



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

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

 drawn to cut them, and .". but two assumable positions for the polar axis of 

 reference. 



However, with respect to all surfaces of revolution (whose normals must 

 all cut the axis of revolution), we can, by giving a more extended signification 

 to some of the involved entities, arrive at general theorems applying to any 

 stations whatever on the surface, of which the theorems already evolved may 

 be regarded as particular cases for short arcs, so stated as to be easier of 

 application when actual results have to be computed. 



For instance, we can easily demonstrate the following : — 

 Theoeem. If S', S", be any two stations on a surface of revolution of any 

 kind, and A'. A", the angles which the true " geodesic" joining the stations 

 makes with the traces of the meridian planes through the stations, and that 

 i?/, -Zi,„ are the normals terminating in the axis, and r^, r^,, the central radii 

 to the stations, then will — 



sin A' R^i . cos I" r,, . cos X " Cp„- 



sin A!' R, . cos V r, . cos \' Cp, 

 (See Fig. 2.) Conceive the "Geodesic" S'S", to be divided into infini- 

 tesimally small parts or elements, S'S,, S^So, S2S3, Sn S". 



Let A„ A^yA^, A , represent the azimuths of the stations S', S^So, 



