520] 
ON THE CENTRO-SURFACE OF AN ELLIPSOID. 
359 
Another generation of the Centro-surface. Art. Nos. 77 to 83. 
77. By what precedes, the equation of the centro-surface is obtained as the 
condition in order that the equation 
{'!a~x 1 (a 2 + f)~ 2 } — 1 = 0 
may have two equal roots. But taking m an arbitrary constant, this is the derived 
equation of 
[Xu 2 # 2 (a 2 + £) -1 } + | + m = 0, 
and as such it will have two equal roots if the last-mentioned equation has three 
equal roots; and conversely, we have thus the equation of the centro-surface by 
expressing that the last-mentioned equation, or, what is the same thing, the quartic 
equation 
(f + m) (£ + a 2 ) (f + 5 2 ) (f + c 2 ) - 2aV (£ + 5 2 ) (£ + c 2 ) = 0 
has three equal roots. The conditions for this are that the quadrinvariant and the 
cubinvariant shall each of them vanish; the two invariants are respectively a quadric 
and a cubic function of m; viz. the equations are 
(a, b, c)(m, l) 2 = 0, (a, b', c', l) 3 = 0 ; 
where the degrees in (x, y, z) of a, b, c are 0, 2, 4 and those of a', b', c, d' are 
0, 2, 4, 6 respectively: the equation of the centro-surface then is 
a, b, 
c = 0, 
a, b, c 
a, b, c 
a', b', c', d' 
a', b', c', d' 
which is of the right order 12 ; but it would be difficult to obtain thereby the 
developed equation. 
78. For the nodal curve the cubic equation must be satisfied by each root of 
the quadric equation, or, what is the same thing, the quadric function must completely 
divide the cubic function ; the conditions are 
a, 
b, c 
a, b, 
c 
a', b', 
c', d' 
where the degrees may be taken to be 
0, 0, 2, 4 
0, 2, 4, 0 
0, 2, 4, 6