DE. J. CASEY ON CYCLIDES AND SPHERO-QTTABTICS. 
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through a common circle, the consecutive pair of inverse points on this circle will be 
common to all the surfaces ; such a pair of inverse points will be the two conic nodes of 
all binodal cyclides of the system kW . 
For let the generating spheres at the given pair of inverse points be a, [3, and Z»/3), 
and the equations of the cyclides may be written ab-\-w 2 , (3^ 'w' 2 , and 
where w 2 , w' 2 , w\ denote homogeneous functions of the second degree in a,j3,y; and it is 
evident that aW -j-dW' — W" is a binodal cy elide having the given pair of points as conic 
nodes. 
Cor. J will be of the sixteenth degree in the coefficients of W, W', W". For if in 
J we substitute for each coefficient a of W, a + Jca^ where a x is the corresponding coeffi- 
cient of a fourth cyclide W n it is evident that the degree of the result in It is the same 
as the number of cyclides of the system which can be drawn to have double 
contact with the curve of intersections of the cyclides W ; and W", and the degree is 
therefore sixteen (see art. 354). 
