718 
DE. J. CASEY ON CYCLIDES AND SPHEEO-QTJAETICS. 
given by the equation 
(m'- f) ( M '_ A) (**- (**-£) (**- A) («/- g) =o 
( 233 ) 
a sextic in Hence the proposition is proved. 
351. Given four cyclides W, W', W", W'", required to find the locus of a qxiir of 
inverse 'points such that their polar spheres , with respect to the four cy elides, may pass 
through the same pair of inverse points. 
The required locus is the Jacobian of the four cy elides 
d W 
d w 
d w 
d W 
da. 3 
43 ’ 
dy 3 
dl 3 
dW 
d\T 
dW 
dW' 
da 3 
dy ’ 
dZ 3 
dW" 
dW 
dW" 
dW> 
da 
W 7 
dy 7 
dZ ’ 
dW" 
dW" 
dW"' 
dW"' 
da 1 
dp 7 
dy 3 
do 3 
Cor. The envelope of a sphere whose pole points with respect to four cyclides are homo- 
spheric is the Jacobian of the four cyclides. 
352. The Jacobian is the locus of the nodes of all binodals which can be represented 
by FW+/JW , -F/CW''d-/J , 'W m . Thus, there being given six pairs of inverse points, 
the locus of the nodes of all binodes which can pass through them is an anallagmatic 
surface of the eighth degree. For if W, W', W", W" be any cyclides through them, 
every cyclide through them can be represented by W -f-FW" since 
this last form contains three independent constants, which are necessary to complete 
the solution. 
Cor. 1. If in any case IcW + h'W + k"W" + h"'W" can represent two spheres, the 
intersection of these spheres is a circle on the Jacobian. 
Cor. 2. If one of the cyclides W be a perfect square L 2 , the Jacobian consists of a 
sphere and an anallagmatic surface of the third order in a, /3, y, ci, that is, a surface whose 
deferente is a surface of the third class. 
Cor. 3. If the cyclides have in common four pairs of inverse points which are homo- 
spheric, the sphere through these points is a part of the Jacobian. 
Cor. 4. If the four cyclides have a spliero-quartic curve common to all, the sphere 
through the sphero-quartic counts doubly in the Jacobian, which therefore reduces to 
a cyclide and the square of the sphere. 
Cor. 5. The Jacobian of four Cartesian cyclides is a Cartesian cyclide. 
353. If F, F', F", F m be the focal quadrics of W, W', W", W"' in tangential coordinates, 
the deferente of the Jacobian of W . . . W" is the Jacobian in tangential coordinates of 
