NOTE ON THE GEODESIC LINES ON AN ELLIPSOID. 
[From the Philosophical Magazine, voi. xli. (1871), pp. 534, 535.] 
The general configuration of the geodesic lines on an ellipsoid is established by 
means of the known theorem (an immediate consequence of Jacobi’s fundamental formulae, 
but which was first given by Mr Michael Roberts, Gomptes Rendus, vol. xxi. p. 1470, 
Dec. 1845) that every geodesic line touches a curve of curvature; that is, attending 
to the two opposite ovals which constitute the curve of curvature, the geodesic line is in 
general an infinite curve undulating between these opposite ovals, and so touching each 
of them an infinite number of times (but possibly in particular cases it is a reentrant 
curve touching each oval a finite number of times). The geodesic lines thus divide 
themselves into two kinds, accordingly as they touch a curve of curvature of the one 
or the other kind; and there is besides a third limiting kind, the lines which pass 
through an umbilicus : any such geodesic line passes through the opposite umbilicus, 
and is in general an infinite curve passing an infinite number of times alternately 
through the two umbilici; but possibly it is in particular cases a reentrant curve 
passing a finite number of times through the two umbilici. I annex a figure giving 
a general idea of the configuration of the geodesic lines drawn in different directions 
from a given point P on the surface of the ellipsoid: this is drawn (as it were) on 
the plane of the greatest and least axes; but it is not a perspective or geometrical 
representation of any kind, but a mere diagram for the purpose in question. We have 
A, A, B, G, G the extremities of the axes; U 1} U 2 , P 3 , P 4 the umbilici; P the point 
on the surface; 1P2 and 1P4 the curves of curvature through P, viz. these are ovals