608 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
and we see that each of the points x=y=z= 0, and x—y—W— 0, are nodes; this 
surface is therefore quadrinodal. 
Let us now suppose that each of the spheres (x, y, z) is real, so that the coefficient n 
must be imaginary. It is simplest to take for system of reference the three principal 
spheres ( x , y, z) and their two points of intersection ( w , v), so that the equation of the 
absolute is 
ar+y 3 +z 3 = 4 wv. 
The equation H(</>-|-&i//) = 0 becomes now 
(ct-\-k)(b-\-k)(c-\-k){de — {2k —n) 3 } = 0, 
which has equal roots if either d or e=0, so that the equation may be reduced to the 
form 
ax 2 + 6y 3 + cz 2 -f dvr -f- 2nwv = 0 ; 
or by means of the absolute to the form 
ax~ + by 2 -\- cz 3 + dvr = 0. 
The surface represented by this equation has only three principal spheres and it has 
one node, viz., w— 0. 
Similarly, if the discriminating quintic have two pairs of equal roots, and the sphere 
x correspond to the unequal root, we can show that x being real, the equation can be 
reduced to the form 
ax°-\-by 2 -k-bz 2 -\-div 2 = 0 ; 
which represents a surface having three nodes, viz., the point ( w ) and the points 
common to the spheres ( x , u, u') ; u, u being such that they form with (x, y, z) an 
orthogonal system. 
255. Let us now suppose that three of the roots of the quintic H(<£+Z;i//) = 0 are 
equal. Taking for spheres of reference ( x , y) corresponding to the remaining roots, and 
(z, w, v) any other spheres forming with ( x , y) an orthogonal system, then the equation 
of the surface must take the form 
ax 2 -\- hy 9 --\-cz 2 - l rdiv 2j r ev 2 -\-2fwr-\-2<jzv-\-'2hziv== 0 , 
and the equation of the absolute the form 
x 2j r if-\-z 2 -\-w 2j rV 2 =0 ; 
so that the discriminating quintic becomes 
(&+«)(&-)-&) 
k-\-c, h, g 
h, k-\-d, f 
g, f 
=o. 
