330 ON THE CENTRO-SURFACE OF AN ELLIPSOID. [520 
x 2 , y 2 , z 2 , the lines in question will be not merely parallel to, but will be, the 
asymptotes of the nodal curve. 
The plane infinity may be reckoned as a principal plane, and we may say that in 
each of the four principal planes there are four umbilicar centres, four outcrops, and four 
evolute-nodes. 
The generation of the surface co?isidered geometrically. Art. Nos. 25 to 28. 
25. I have deferred until this point the discussion of the generation of the 
centro-surface by means of the centro-curves, for the reason that it can be carried on 
more precisely now that we know the forms of the principal sections and of the nodal 
eurve. The two figures exhibit (as regards one octant of the surface) the portions 
already distinguished as (A), and (B): they intersect each other in the nodal curve, 
shown in each of the figures. 
26. Consider first the generation of the portion (A) by means of the major-mean 
sequential centro-curves. The major-mean curves of curvature (attending to those below 
the plane of xy) commence with a portion (extending from umbilicus to umbilicus) of 
X 2 Z 2 
the ellipse — + — = 1, this may be termed the vertical curve, and they end with the 
X 2 Y 2 . 
whole ellipse —- + -yr- = 1, which may be termed the horizontal curve. The normals at 
1 a 2 b 2 
the several points of the vertical curve successively intersect along a portion (terminated 
each way at an umbilicar centre) of the evolute in the plane of xz or umbilicar 
plane; viz. this portion of the evolute, shown fig. (a), is the sequential centro-curve 
belonging to the vertical curve of curvature. The curve of curvature is at first a 
narrow oval surrounding the vertical curve; the corresponding form of the sequential 
centro-curve is at once seen to be a four-cusped curve as in fig. (h), and which we 
may imagine as derived from the curve (a) by first doubling this curve and then