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MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
and if this be satisfied the surface has the circle at infinity for a cuspidal edge ; and so 
may be called a nodal-Cartesian. 
271. If the coefficients in the equation 
are connected by the relation 
ax z + by 2 -j- cz z + dw z = 0, 
f.+r-.+A+-.=o, 
the surface represents a nodal circular cubic. It clearly passes through the centres of 
the three principal spheres, and the tangent planes at these points are parallel to the 
plane 
ax . by , cz , dw 
-+r+r+ =o 
and so the tangent planes at the centres of the principal spheres are the tangent planes 
drawn from the line at infinity on the surface. 
(C.) Cyclides having a Binode. —§§ 272-276. 
272. The general equation of a cyclide having two principal spheres and a binode 
can, by § 254, be expressed in the form 
ax 2 -f- by z +2 h zw =0 ; 
the system of reference being the two principal spheres ( x , y) and the sphere (z) 
passing through the node (w), and cutting (r, y) orthogonally in the points ( w, v), the 
equation of the absolute being 
x z -f- y z + z~ = 4 ivv, 
and the coordinates of the plane at infinity —, 
J r 2 r 3 e e 
273. The coordinates of any tangent sphere at the point ( xy'z'w'v '), must 
satisfy 
£ rj f —2nr —2(o 
(a + k)x' (b + ]c)y' hw' + kz' hz' — 2kv' —2kw r 
The equation of the tangent plane at the point (x'y'z'w'v) will be 
fax' by' hid hz' 
-f fi-j-— \(xx-\-y'y-\-z’z — 2ivv — 2vw) 
r i r 2 r 3 6 / 
= (- + -+- — 2- ' \(ax'x-\-by r y-\-hw'z-\-hz'w). 
(296) 
