520] 
ON THE CENTRO-SURFACE OF AN ELLIPSOID. 
325 
a*x 
by 
_ + = 1 is a cuspidal curve on the surface, and the section 
/3 2 a 2 r 
15. The ellipse 
by the plane z = 0 is consequently made up of this ellipse counting three times, and 
of the evolute; it is therefore of the twelfth order; and the order of the surface is 
in fact =12. 
X 2 F 2 
It is clear that the section of the centro-surface arises from the section — + -rr = l> 
a 2 b 2 
viz. the normal at any point of this ellipse lies in the plane Z=0, and its inter 
section by a normal at the consecutive point of the ellipse gives a point of the 
evolute; the evolute being thus the sequential centro-curve of this section: the inter 
section by the normal at the consecutive point on the other curve of curvature gives 
afx 1 
a point on the ellipse 4- 
by 
1, which ellipse is therefore the concomitant centro- 
X 2 Y 2 
curve. Observe that this other curve of curvature cuts the ellipse —- + -rr- = 1 at 
1 a 2 b 2 
right angles, and that the normals at the consecutive points above and below the 
point on the ellipse will meet each other and also the normal at the point of the 
same ellipse at the point on the ellipse 
mentioned ellipse is a cuspidal curve on the 
a 2 x 2 by 
/3 2 ^ a 2 
centro-surface. 
this shows that the last- 
16. The three principal sections of the centro-surface are consequently 
x = 0, = 1, and (byf + (czf = a*; 
y = 0, + a °Y = 1> an d (czf + (ax)* = ; 
CL <y 
z = 0, + h ~y = i, and (ax) 1 + (byf = y 3 ; 
viz. each section is made up of an ellipse counting three times and of an evolute 
(of an ellipse). I have for shortness represented the three evolutes by their irrational 
equations. It will presently appear that the section (imaginary) by the plane infinity 
is of the like character. 
17. Considering only the positive directions of the axes, we have on each axis 
two points, viz. 
of X, 
X — 
7 
a 5 
X = 
of y, 
y= 
a 
6’ 
y = 
of z, 
z = 
c * 
z =