DR. J. CASEY OX CYCLIDES AND SPHERO-QUARTICS. 
709 
we substitute for each coefficient a, a-\-kci we get the condition that Aa-f 
shall be a generator of the cyclide W+UW'. The condition will be of the form 
<r+*r + jfcV+£V=0 (204) 
In terms of the functions a, r, r', d can be expressed the condition that the sphere 
Xa-J-^$ + yy+§^ shall have any permanent relation with the cyclides W, W' ; as, for 
instance, that it should intersect them in spliero-quartics w, w' connected by such perma- 
nent relations as can be expressed by relations between the coefficients of the discriminant 
of iv-\-kw'. Thus if we form the discriminant with respect to k of equation (204), we 
get the condition that the sphero-quartics in which Xa+g-0-j-j'y-b^ intersects W and W 
shall have double contact ; in other words, the discriminant is the condition that the 
sphero-quartics shall have a common generating circle which touches both quartics at 
the same points. Again, r=0 is the condition that the sphero-quartics w and w' are so 
related that the harmonic circles (see ‘ Bicircular Quartics,’ art. 184) of one are generators 
of the other. 
329. The coefficients <r, r, r , a of equation (204), and the discriminant of the same 
equation, have another geometrical interpretation. Thus a and d are the equations in 
tetrahedral coordinates of the focal quadric of W and W', A , gJ, v, o being the current 
coordinates (see art. 27), and r, d are quadrics covariant with <r, d. Thus r “is the 
locus of a point whence cones circumscribing a and d are so related that three edges 
of one can be found forming a self-conjugate system with regard to the second, and 
three tangent planes of the second which form a self-conjugate system with regard to 
the first” (see Salmon’s ‘ Geometry of Three Dimensions,’ page 159). The discriminant 
of (204) is the developable circumscribed to a and d ; in other words, the locus of the 
centre of Xa+g/0 + v is the developable. Hence we infer : — The locus of the centre 
of a variable sphere which cuts two cyclides in sphero-quartics having double contact is 
the developable circumscribed about the focal quadrics of the cyclides which correspond 
to their common inversion sphere. 
330. If we suppose the cyclide W' of the last article to become U 3 , we have the 
following theorem : — The locus of the centre of a sphere S intersecting the cyclide W in 
a spliero-quartic WS which has double contact with the spliero-quartic WU is the deve- 
lopable % formed by tangent planes to U along WU (see Chapter VIII., art. 142). 
331. If W=aa 2 +50 2 +cy 2 ffi($$ 2 =O, then v=Ax 2 -|-Bg, 2 -j-Cf' 2 +D^ 2 , where A =bcd, 
B — cda, See. ; and changing a into a-\-ka', Sec., we get 
(V d d'\ (c J d' a' \ id' a! b'\ „ , (cl V c' \ 9 /ontx 
■-h + 7+7) A ^+0+^ + «)V+( S +^+j)C/+( S Hr I +-)D ? *; . (205) 
and the cyclide which has (205) for a focal quadric will be got by reciprocating (205) 
and substituting a, 0, y, b for the variables. Hence the required cyclide will be 
S 2 
b'cd-^bc'd + bcd'^ c'db + cd'b + cdb 1 ^ d'ab + da'b + dab 1 1 a'bc + ab'c-^abc' ^ ^ 
This will be the locus of the nodes of binodals circumscribed to W W', the same points 
5 d 2 
