520] 
ON THE CENTRO-SURFACE OF AN ELLIPSOID. 
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opening out the two component parts thereof: the two upper cusps of the curve (b) 
are situate on the ^¿-ellipse of the centro-surface, and the two lower cusps upon two 
detached portions respectively of the ^-ellipse of the centro-surface. And as the curve 
of curvature gradually broadens out and ultimately coincides with the XF-section of 
the ellipsoid, the four-cusped curve continues to open itself out, and ultimately coincides 
as shown figure (c) with the icy-evolute of the centro-surface, viz. this evolute is the 
sequential centro-curve belonging to the horizontal curve of curvature or XF-section 
of the ellipsoid. The successive sequential curves are also shown (so far as regards 
an octant of the surface) in the figure (A). 
27. We consider next the generation of the portion (B) by means of the major- 
mean concomitant centro-curves. Starting as before with the vertical curve of curvature, 
the concomitant centro-curve is a finite portion (terminated each way at an umbilicar 
centre) of the ¿¡^-ellipse of the centro-surface. As the curve of curvature opens itself 
out into an oval, the concomitant centro-curve in like manner opens itself out into 
an oval, the two further vertices thereof situate on two detached portions of the 
¿r^-evolute of the centro-surface, and the two nearer vertices on the y^-evolute of the 
central surface. And as the curve of curvature continues to open itself out, and 
ultimately coincides with the horizontal curve or XF-section of the ellipsoid, so the 
concomitant centro-curve continues to open itself out and ultimately coincides with the 
¿ry-ellipse of the centro-surface. The successive forms (so far as relates to an octant 
of the surface) are shown in the figure (B). We have in each case attended only to 
the curves of curvature below the plane of xy, and the corresponding centro-curves 
above the plane of xy, but of course everything is symmetrical as regards the two 
sides of the plane. 
28. There is a precisely similar generation of the portion (A) by the minor-mean 
concomitant centro-curves, and of the portion (B) by means of the minor-mean sequential 
centro-curves. 
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