GEODESIC INVESTlGATIOlfS. 181 



S^^ii^ if observed from S,. S., S3 S^-, and let A' represent 



the azimuth of S, as if observed from S' ; and let A" represent the azimuth 

 of S^ as if observed from ;S". 



Put E', R„ Rn, ^n, -E"for the normals (terminating in the axis) at 



S', S,, S.,, S^^, S" ; and ;•', r„ r,„ r^, r" for the central radii to 



the stations taken in like order. 



'Now, from the elements of analytic geometry, we know that the tangent 

 luies to any infinitesimally small arc of the first order, which forms part of a 

 " geodesic," have their least distance an infinitesimally small of the third 

 order ; and that the lengths of these tangents, from the points of contact to 

 their points of least distance from each other, are equals. We know also 

 that the plane of every two consecutive elements of any " geodesic " contains 

 the normal at their point of junction ; and therefoi'e that sin A^, sin A.2, sin A3, 

 . . . . sin ^jj, are respectively equal to the sines of the azimuths of the stations 



S-2, 83, S^' 8" as if observed from S^, 8.., ...... 8^, which are their 



supplements. Hence — (as the cosines of infinitesimally small arcs are 

 equals) ^^ 



sin A' R^ cos I, sin A^ R cos lo sin A^ B^ ^^^ h 



sin A^ R' cos V ' sin A^ R^ cos I sin A^ R2 cos I2 



sin An.i Rn cos ^n siu An R" COS I" 



sin Aa Rn.i. COS la.l ' siu A" Rn COS la 



sin A' R" COS I" 



•■• evidently- ^^TaJ'^^R' cos I' " 



And in a similar manner it may be shown that — 



sin A' r" cos A." 



sin A^' r' cos k' 



sin A' perpendicular from S" on the axis 



sin A" perpendicular from 8' on the axis 



For a spheroid such as the earth's surface, we can prove, in like manner, 

 that for any two stations whatever — no matter how distant from each other, — 



sin^^' tanM'+l^ 



sin2 A" 



tan- t "T To 



in which A' , A", are the angles made by the geodesic joining the stations, 

 with the traces of the meridian planes through the stations. And if the 

 earth's surface were that of a circular cylinder or cone, or of any other known 

 surface of revolution expressible by means of an equation, we could easily 

 find the particular equivalents to substitute for the perpendiculars from the 

 stations to the axis in the above general expression for sin A' -f- sin A". 



The following theorems (and others of a kindi'ed character) are evident 

 from what has been shown in the few last notes : — 



1°. If )3/ be the point in which a geodesic on any surface of revolution is 



cut perpendicularly by the trace t, of a meridian plane, and that yS^^ is the 



point in which it is cut obliquely by the trace t^^ of another meridian plane : 



then, putting jp^, ^^,, to represent the lengths of the perpendiculars from jS^, j8^, 



N 



