96 
Proceedings of the Royal Society of Edinburgh. [Sess. 
closed curves on the ellipsoid (being in fact the intersection of the ellipsoid 
with a confocal quadric) : the region between these two portions of the 
line of curvature is a belt extending round the ellipsoid : and all the 
geodesics which belong to this line of curvature are comprised within this 
belt,* and touch the two portions of the line of curvature alternately. The 
matter is represented schematically in the diagram, where ABCDEF and 
PQRSTU are the two portions of the line of curvature, and AJRKELPMCT 
is an arc of one of the geodesics belonging to it, touching one of the 
portions of the line of curvature at A, C, E, and touching the other 
portion at R, P, T. 
In order that the geodesic may be closed, it is necessary (as in all poristic 
problems) that a certain parameter (depending in this case on the value of 
the constant pci of the line of curvature) should be a rational number : the 
geodesic is unclosed if this parameter is an irrational number. If it is 
closed, then there are oo 1 other geodesics which belong to the same line of 
curvature and which are also closed ; but if it is not closed, then no other 
geodesic belonging to this particular line of curvature can be a closed 
geodesic.f 
* Ignoring the exceptional case of those geodesics which pass through an umbilicus. 
f This is obvious in the case when the ellipsoid is of revolution : for then the two 
portions of the line of curvature are parallel circles on the surface, and the oo 1 geodesics 
which belong to this line of curvature are obtained from each other by mere rotation about 
the axis of symmetry. 
