354 Captain A. R. Clarke on the Course of 



have passed. The grand chain of triangles upon the great 

 European arc of parallel between Valentia and Orsk runs along 

 the parallel, and does not approach a great circle or geodesic 

 course. 



1. 



It would be difficult to make the subject we are going to con- 

 sider intelligible without going to the root of the matter. There- 

 fore, at the .risk of being tedious, we propose to investigate, in 

 the briefest possible manner, the leading characteristics of the 

 geodesic curve. If we suppose the position of a point on an 

 ellipsoid of revolution to be determined by its distance f, mea- 

 sured from one of the poles along the curve of a meridian, and 

 by its longitude on, then, r being the distance of the point from 

 the axis of revolution, the length of a curve traced on the sur- 

 face is 



If this length is to be a minimum between the given extremities, 

 we shall most quickly arrive at the distinctive character of the 

 curve by giving a variation Seo, a function of f, to &>. Thus 



and consequently for the minimum, 



r 2 dco _p 

 ds 



To fix our ideas, let longitude be measured positively from 

 west to east, and azimuths from north through east, back to 

 north. Let a be the azimuth of the element ds of the curve, 

 then 



ds cosa = — d%, 



ds sin a = rdco; 



and the latter of these, substituted in the characteristic of mini- 

 mum, gives 



r sina = C. 



Now, if u be what is sometimes called in treatises on the conic 

 sections the eccentric angle, or the reduced latitude, r=a cos u, 

 a being the major semiaxes of the spheroid; and if u l u i be the 

 values of u a. at the initial point A, 



cos u t sin « i = cos u sin a. 

 The relation here expressed is that which exists between two sides, 



