83 
dy? yo ay dy i 
BATT sop aden a 
The integration is in this case more simply effected by resolving 
the equation into its factors, which are manifestly 
Ce 
dau ie 
dy L 
gee pied 
These equations being integrated give 
y = ax 
ye + a? = 9? 
a and r* being arbitrary constants.. The first, for each value of a, 
is, the equation of a right line through the origin. The second, for 
each value of r?, is the equation of a circle of which the origin is 
the centre. As these constants are not restricted by any condition, 
it follows that all right lines whatever in the plane xy passing 
through the origin, and all circles described upon the same plane, 
with the origin as the centre, are projections of lines of curvature; 
provided that there are points of the surface corresponding to the 
points on the plane zy, through which these right lines and circles 
pass. The condition a’ = a” renders the surface, one of revolution 
round the axis of z. If therefore it be an hyberboloid or parabo- 
loid all circles and right lines whatever thus drawn are actual pro- 
jections of lines of curvature, since these surfaces have unlimited ex- 
tent. But if the surface be an ellipsoid, the greatest circle which 
can be a projection of a line of curvature is the section of the sur- 
VOL. XIV. P 
