520] 
ON THE CENTRO-SURFACE OF AN ELLIPSOID. 
323 
or squaring, and substituting for X 2 , &c., their values as given by 
- fiyX 2 = a 2 (a 2 + £) (a 2 + 77), &c., 
the equations become 
— /3y a 2 x 2 = (a 2 + £) 3 (a 2 + 77), 
— 7a by = (6 2 + £) 3 (ò 2 + 77), 
— a/3 c 2 ^ 2 = (c 2 + £) 3 (c 2 + *7), 
viz. these equations give (x, y, z) the coordinates of a point on the centro-surface, the 
intersection of the normal at the point (X, Y, Z) of the ellipsoid, (determined by the 
parameters 77), by the normal at the consecutive point along the curve of curvature 
X 2 F 2 Z 2 
FT“ "b tz~. 1—TT~ 
a 2 + 77 b 2 + rj c 2 -f 77 
or say 77 is the sequential parameter ( x ). 
Of course by interchanging f and 77 we should obtain the coordinates of the point 
of intersection of the normal at the same point (X, F, Z) by the normal at the con 
secutive point along the other curve of curvature : £ being in this case the sequential 
parameter. 
12. I stop for a moment to consider the foregoing two equations 
\ = l d\ = -\d^ 
which at first sight appear inconsistent. But observe that in the foregoing solution 
X is the parameter of the point (x, y, z) of the centro-surface considered as a point 
on the normal at (X, F, Z) ; X + d\ is the parameter of the same point considered as 
a point on the normal at the consecutive point (X+dX, Y+dY, Z + dZ): the value 
X 4. d\ = i; + dl; would belong to a different point, viz. the consecutive point of the 
centro-surface considered as a point on the consecutive normal—wherefore the dX of 
the solution ought not to be =df. In further explanation, observe that the equations 
where X = £, 
if we pass from (x, y, z) to the consecutive point on the centro-surface, give 
but since by what precedes, 
this is 
1 The expressions are given in effect, but not explicitly, Salmon, p. 14B. 
41—2