8.— A GEODESIC ON A SPHEROID AND AN ASSOCIATED 



ELLIPSE. 



By Lawrence Crawford. M.A., D.Sc, F.R.S.E. 



1. In this paper the length of the arc of a geodesic drawn 

 from a given point on a spheroid in a given direction is found as 

 the length of an arc of an ellipse, and the difference of the longitude 

 of any point on the geodesic and the given point is expressed as an 

 elliptic function of an angle connected with the corresponding 

 points on the same ellipse. An expression is then found for the 

 change in longitude on return along the geodesic to the same 

 latitude. 



2. The equation of the spheriod is' — ni^+"'^=i 



a'" c- 



and a geodesic is given bv /'- sin-^' / = constant, where ^ 



(Is 



is colatitude, measured from OZ and f is longitude. 



Also </5- = </(T- + ;'^ sin"9(/^-, where da- is element of arc of 

 meridian ellipse, and the co-ordinates of the point in the plane of 

 the meridian may be written {n cos //, c sin //) where a cos /.' = /'sin H, 

 ( sin //=:;- cos B. 



By substitution for ila-- in terms of liti-, and for H in terms of //, 

 we find equation of geodesic becomes 



a- cos- // ( -„ cos- // — i) d<t>^=a- ii—c'^ cos^ //) dir 



where in is the constant of the geodesic and r- = i — %. 



a- 



Also by the same substitutions we find 



ds^= cos^ // (i—c- cos'^ //) dii^K ^ ^ cos- // — I ). 

 Ill'' I \ III- / 



3. If \p be the angle at which the geodesic crosses the 

 meridian, /- sin 6 d(f)=ds sin \p, 



and r- sin^ 6 -^=^m becomes r sin 9 sin \L^iii, 

 as 



i.e., a cos // sin ■dy=iii, 



or a cos //q sin « = /;/, 



where Hq is eccentric angle of starting point and a the angle 

 geodesic makes with meridian at //,> 



H. ,9 rt- cos- // (i — f- cos- //) , .7 

 ence we get ds- = ^ ^ — ^ .— s— <^" • 



cos- 7;— cos-^ Uo sm- a 



