﻿Prof. Cayley on the Geodesic Lines on an Ellipsoid. 535 



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 f 

 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 repre- 

 sentation of any kind, but a mere diagram for the purpose in 

 question. We have A, A, B, C, C the extremities of the axes; 

 U 15 U 2 , U 3> U 4 the umbilici: P the point on the surface; 1 P2 

 and 1 P 4 the curves of curvature through P, viz. these are ovals 

 containing the umbilici U^ U 2 and JJ lt U 4 respectively. Then 



T, I J i 



UjP U 3 and U 2 P U 4 are the limiting geodesies passing through th 

 umbilici; the line TPT' represents a geodesic line of the one 

 kind, viz. this at T touches an oval (curve of curvature) U J U 4 , 

 and at T' the conjugate oval U 2 TJ 3 . Similarly S P S' is a geo- 

 desic line of the other kind, viz. this at S touches an oval (curve 

 of curvature) V l TJ 2 , and at S' the conjugate oval U 3 U 4 ; the 

 dotted figure-of-eight curves are the loci of the points of con- 

 tact T, T', S, S\ 



2N2 



