520] 
ON THE CENTRO-SURFACE OF AN ELLIPSOID 
353 
C. VIII. 
45 
62. From the foregoing equations we have: 
\/aacc+ V/9 by + Vy cz = V — a/97, 
a* *Jax + (3% \tby + y^ \!cz = 0, 
the second of which is best written in the rationalised form 
(1, 1, 1, — 1, — 1, — 1) (a V■ a a#, /9 V/9 by, 7 Vy c.z) 2 = 0, 
and combining herewith the equation 
Va a# + V/9 by + \/<ycz = */ — a/97, 
then for any given signs of Va, V/9, V7 and V - a/9y the first of these equations 
represents a quadric surface, the second a plane, or the two equations together represent 
a conic. 
By changing the signs of the radicals (observing that when all the signs are 
changed simultaneously the curve is unaltered) we obtain in all 8 conics, but only 
four quadric surfaces; viz. the two conics 
Va ax + V/9 by + Vy cz = + V — a/9y 
lie on the same quadric surface. 
63. The conics form two sets of four, corresponding to the two sets of four lines 
on the ellipsoid. The analysis seems to establish a correspondence of each conic of the 
one set to a single conic of the other set; viz. the conics have been obtained in pairs 
as the intersections of the same quadric surface by a pair of planes: there is a like 
correspondence of each line of the one set to a single line of the other set, viz. the 
lines meet in pairs on the umbilici at infinity, but this correspondence is included in a 
more general property: in fact each line of the one set meets each line of the other 
set in an umbilicus; and the corresponding conics (not only meet but) touch at the 
corresponding umbilicar centre; and qua touching conics they have two points of 
intersection, and consequently lie on the same quadric surface. It is to be added that 
the two conics touch also, at the umbilicar centre, the cuspidal conic of the principal 
section. 
64. The 8 conics form two tetrads, and the principal conics (reckoning as one of 
them the conic at infinity) another tetrad: the complete cuspidal curve consists therefore 
of three tetrads of conics: with these we may form (one conic out of each tetrad) 16 
triads; viz. each conic of one tetrad is combined with each conic of either of the other 
tetrads, and with a determinate conic of the third tetrad, to form a triad. And the 
conics of each triad, not only meet but touch at an umbilicar centre, the common tangent 
being also by what precedes, the tangent of the evolute at that point, which point is 
also a node of the nodal curve.