520]
ON THE CENTRO-SURFACE OF AN ELLIPSOID.
361
the variable parameter be £, then this is a curve on the 12-thic surface 12=0 obtained
by the elimination of f from the equations V—0, S|F=0; viz. we have i2 = S 3 — T 2 = 0,
where S = (a, b, c)(m, l) 2 , T=(a,', b', c', d') (m, l) s , and the curve in question is the
curve 8= 0, T = 0, which is the cuspidal curve on the surface 12 = 0; the elimination
of m from the two equations 8 = 0, T = 0 gives as above the equation of the centro-
surface.
81. The surface il=8 3 — T 2 =0 obtained as above by the elimination of £ from the
equations V = 0, 3$ V = 0, (or, what is the same thing, by equating to zero the discriminant
of F in regard to f,) may be termed the sociate-surface: we have then the quartie
and sextic surfaces 8 = 0, T = 0 intersecting in the before-mentioned curve, which may
be called the sociate-edge ; and the locus of these sociate-edges is the centro-surface.
82. We may if we please, changing the parameter in one of the functions, consider
the two series of surfaces S = 0, T = 0 depending on the parameters m, m' respectively ;
a surface of the first series will correspond to one of the second series when the
parameters are equal, m = in', and we have then a sociate-edge. Taking a point
anywhere in space, through this point there pass two surfaces S = 0, and three surfaces
T= 0 ; but there is no pair of corresponding surfaces, or sociate-edge. If however the
point be taken anywhere on the centro-surface, then there is a pair of corresponding-
surfaces 8 = 0, T = 0; that is, through each point of the centro-surface there passes a
single sociate-edge; and if the point be taken anywhere on the nodal curve of the
centro-surface, then there are two pairs of corresponding surfaces ; that is, through each
point of the nodal curve there are two sociate-edges: this explains the method above
made use of for finding the equations of the nodal curve, by giving to the equations
S= 0, T = 0, considered as equations in m, two equal roots.
83. The d posteriori verification that the surfaces V = 0, V' = 0, V" = 0 intersect in
a point of the centro-surface, is not without interest; the parameters g 1} rj 1 of the point
of intersection are found to be | x = £, r] 1 = m — a 2 — b 2 — c 2 — 3£; whence in the equation
V = 0, writing — /3y<Fc 2 = (a 2 4- (a 2 + 7] x ) and m=a 2 + b‘ 2 + c 2 + S^ 1 + tj 1 , the resulting
equation considered as an equation in £ should have three roots f = £: the fourth
root is at once seen to be £ = ?7i, and we ought therefore to have identically
a (a 2 + gi) 3 (a 2 + Vi)
o 2 + £
— &C. + 0-8y (f — 3£ x — 7]y — ci- — If- — c-)
= (i; ~ Zi) 3 (Z - Vi) .
(a 2 + £) (F + g) (c* + £) 5
and by decomposing the right-hand side into its component fractions this is at once
seen to be true.
Third generation of the Centro-surface. Art. Nos. 84 and 85.
84. Instead of the foregoing equation V = 0, consider the equation
C. VIII.
46
