72 
ON THE CYCLIDE. 
72 ON THE CYCLIDE. [569 
569] 
In fact, representing this for a moment by 
/3 2 — r + V(0) + vw = 0, 
we have 
09*" 7 2 ) 2 + © - <f> = - 2 (/3 2 - 7 2 ) V(©), 
One ol 
a line; the 
or, substituting and dividing by /3- — 7-, we have 
the case in 
this purposi 
/3 2 — y 2 4- S 2 — 4 (a? 2 4- i/ 2 4- z 2 ) 4- 2 a/{(27« 4- f38) 2 — 4 (/3 2 — 7 2 ) y 2 } = 0, 
or, similarly 
/3 2 — y 2 — 8 2 4- 4 (af + y 2 + z 2 ) + 2 >^{(2/3# 4- yS) 2 + 4 (/3 2 — y 2 ) 2 2 } = 0, 
where /4-5 
«4-g, a 4-h, 
either of which leads at once to the rational form. 
the equatioi 
The irrational equation 
(.y 2 + < 
/3 2 — y 2 4- V{( 2 7^ + /3S) 2 - 4 (/3 2 - y 2 ) y 2 } 4- \/{(2/3« + 7S) 2 4- 4 (/3 2 - y 2 ) 2 2 } = 0 
is of the form 
p + V(pO + \/(st) = 0, 
or, what is 
0/ + Z 2 ' 
which belongs to a quartic surface having the nodal conic p = 0, qr —st = 0 (in the 
present case the circle at infinity), and also the four nodes (q = 0, r = 0, p 2 — st = 0) 
and (s = 0, t = 0, p 2 - qr = 0), viz. these are 
N ow assum 
only to the 
® = y = °> ± - 7 2 )(7 2 - S2 )}» 
and 
* = y = ±i^l(7 , -j8 , )08 , -8*)). *-0, 
or, what is 
and we hence again verify that the nodes form a system of skew anti-points, viz. the 
condition for this is 
that is, 
B 2 (/3 2 - y 2 ) + /3'-’ (y 2 - S 2 ) - y 2 (/3 2 - 8 2 ) = 0, 
where by a] 
It is s 
the axis of 
original con 
which is satisfied identically. 
viz. this is 
The cyclide has on the nodal conic or circle at infinity four pinch-points, viz 
these are the intersections of the circle at infinity with the planes /3 2 y 2 4- 7 2 z 2 = 0. 
the section 
and in par 
plane y = 0 
If /3 = 0, the equation becomes 
£7 + V(tf 2 + y 2 ) + V(iS 2 - * 2 ) = 0» 
if, to fix t 
in fig. 3; 
Bx in P, 
viz. the cyclide has in this case become a torus; there are here two nodes on the 
axis {x = 0, y = 0), and two other nodes on the circle at infinity, viz. these are the 
circular points at infinity of the sections perpendicular to the axes, and the pinch- 
points coincide in pairs with the last-mentioned two nodes; viz. each of the circular 
points at infinity = node + two pinch-points. 
similarly fo 
surface is : 
these plane 
2 = any smi 
axis ; z = 0 
C. IX