520] 
ON THE CENTHO-SUBFACE OF AN ELLIPSOID. 
335 
36. The nodal curve passes through (I) the umbilicar centres, (II) the outcrops, 
(III) the nodes of the evolute. The geometrical construction led to the conclusion that 
the real umbilicar centre was a node on the nodal curve, and that the real outcrop 
was a cusp (the tangent lying in the principal plane). It will presently appear 
generally, as regards the several points real or imaginary, that the umbilicar centre is 
a node on the nodal curve, and the outcrop a cusp—the tangent at the outcrop 
being in the principal plane: as regards the node on the evolute this is a simple 
point on the nodal curve, and by reason of the symmetry in regard to the principal 
plane, the nodal curve will at this (imaginary) point cut the principal planes at right 
angles. Hence considering the intersections of the nodal curve by a principal plane, 
the umbilicar centre, outcrop and node of the evolute count respectively as 2 points, 
3 points and 1 point, and as for each kind the number is 4, the whole number of 
intersections is 4(2+ 3 + 1), =24. It may be shown that these are the only inter 
sections of the nodal curve with the principal plane ; and this being so, it follows 
that the order of the nodal curve is = 24; which agrees with the result of a 
subsequent analytical investigation.