362 
ON THE CENTRO-SURFACE OF AN ELLIPSOID. 
[520 
The equations d$W = 0, dfW = 0 contain only (?, and are in fact identically the same as 
the equations d$V = 0, d^V — 0 ; the elimination of £ from the equations d$V= 0, dfW= 0 
would theiefore lead to the equation of the centro-surface: and the centro-surface is 
connected with the surfaces W = 0, d$. W = 0, df W = 0 and the parameters (?, rj in the 
same way as it is with the surfaces F = 0, d^V—0, dfV = 0 and the parameters (?, m. 
That is, if from the equations W=0, d$ W = 0 we eliminate (? we have a surface il = 0, 
depending upon 77 and having a cuspidal curve; and the locus of the cuspidal curve 
(as 77 varies) is the centro-surface. But the equation W = 0 divides by f — 77, and 
throwing out this factor it becomes 
a. 2 # 2 b 2 y 2 c 2 z 2 
(a 2 + (?) (a 2 + 77) + (b 2 + (?) (b 2 + 77) + (c 2 + (?) (c 2 + 77) 
so that the surface fi = 0 is obtained by eliminating (? from this equation and the 
derived equation in regard to (?; or, what is the same thing, by equating to zero the 
discriminant in regard to f of the cubic function 
(o* + f) (i* + f) (<?+?)- 2 „if,, (i ! + f) (o a + ?)• 
II ~T T/ 
This surface is in fact the torse generated by the normals at the several points of the 
curve of curvature belonging to the parameter 77; the cuspidal curve is the edge of 
regression of this torse, that is, it is the sequential centro-curve of the curve of 
curvature ; and we thus fall back upon the original investigation for the centro-surface. 
85. In verification I remark that if X, Y, Z be the coordinates of a point on the 
curve of curvature in question, and (x, y, z) current coordinates, then the tangent plane 
of the torse, or plane through the normal and the tangent of the curve of curvature, 
has for its equation 
Xx Yy Zz _ q 
and if in this equation we consider the point (X, Y, Z) to be the point belonging to 
the parameters (77, £), viz - we have - fiy X 2 = a 2 (a 2 + £) (a 2 + 77), See., then this plane 
will be always touched by the before-mentioned ellipsoid, 
a 2 x 2 
+ 
b 2 y 2 
+ 
cz 
(a 2 + (?) (a 2 + 77) ' (b 2 + (?) (b 2 + 77) 1 (c 2 + |) (c 2 + 77) 
The condition for the contact in fact is 
= 1. 
v JP (a 2 + f)(a 2 + 77) 
~ (a 2 + 77) 2 a 2 
viz. substituting for (X, Y, Z) their values, this is 
which is true. And this being so, the ellipsoid and the plane have each the same 
envelope, viz. this is the torse in question.