520] 
ON THE CENTRO-SURFACE OF AN ELLIPSOID. 
329 
Nodes of the Evolute. 
23. The Evolute is a curve with four nodes, all of them imaginary; viz. for the 
evolute in the plane of xy, the equation of which is 
(a 2 x 2 + try 1 — y 2 ) 3 + 27 fd 2 b 2 x 2 y 2 = 0, 
these are 
Coordinates of Nodes of evolute in plane of xy (imaginary), 
in fact these values satisfy as well the equation of the evolute, as the two derived 
equations 
6a 2 « [(a 2 « 2 + b 2 y 2 — y 2 ) 2 + 9y 2 6 2 y 2 ] = 0, 
Qb‘ 2 y [(a 2 « 2 + b 2 y 2 — y 2 ) 2 + 9y 2 a 2 « 2 ] = 0, 
or the points in question are nodes of the evolute. 
The evolute has the four cusps on the axes and two cusps at infinity, in all 
6 cusps as just mentioned; it has 4 nodes: and the order being 6, the class is 
30-2.4-3.6, =4. 
Section by the plane infinity. 
24. The surface itself is finite, and the section by the plane infinity is therefore 
imaginary; but by what precedes the nodal curve must have real points at infinity, 
viz. there must be real acnodal points on this imaginary section. The section by the 
plane infinity resembles in fact the principal sections; viz. writing successively £ = oo, 
and Tj = 20 , we have 
— /3ya 2 « 2 : — yab 2 y 2 : — a/3c 2 z 2 = a 2 + y : b 2 + y : c 2 + y 
= (a 2 + £) 3 : (6 2 +£) 3 : (c 2 + £) 3 , 
or 
giving respectively 
a 2 « 2 + b 2 y 2 + c 2 z 2 = 0, and (aax) 2 + (bj3yf 4- (cyzf = 0, 
where the first equation represents an imaginary conic which counts three times; and 
the second equation, the rationalised form of which is 
(a 2 a 2 « 2 + b 2 /3 2 y 2 + c 2 fz 2 f — 27a 2 6 2 c 2 a 2 /3 2 y 2 « 2 y 2 £ 2 = 0, 
an imaginary evolute. The conic and evolute have four contacts and four simple inter 
sections (in all 4.2 + 4 = 12 intersections) which are all of them imaginary. But the 
evolute has four real nodes (acnodes) d 2 a 2 x 2 = 5 2 /3 2 y 2 = c 2 y 2 z 2 ; or, what is the same thing, 
there are four real lines a 2 a 2 x 2 = b 2 /3 2 y 2 = c 2 <y 2 z 2 , which are respectively asymptotes of the 
nodal curve: viz. inasmuch as the equation of the surface contains only the squares 
c. yin. 42