386 Scientific Proceedings, Royal Dublin Society. 
are four points on the surface, tangent cones from which all 
envelop the surface whose equation in tangential coordinates is 
a =0 (the existence of an infinite number of double tangent planes 
all passing through a point outside the surface is excluded, as we 
saw that its reciprocal has not an infinite number of double points 
all lying on a plane), and the equation in Cartesian coordinates of 
the surface represented by a= 0 is 
xe y 3 1 
(a? +b") Y Bee @ (PP) C+ Ht 
as stated above. 
It now appears that the equation of the surface may be written 
in tangential coordinates in three other ways, showing the exis- 
tence of the twelve imaginary conical points. I shall notice one 
case only, namely, 
o + pip2psps= 9, 
where o stands for 
(0? +0) N+ (P+ a") p+ (a+ 8) Vv? - 2. 
This is obtained from the value of S of the reciprocal surface just 
as @ was obtained from the value of P of the reciprocal; similarly 
the values of Q and & for the reciprocal would furnish the values 
of the enveloped ellipsoid in the two cases which I have omitted. 
Also, p, stands for 
N/A PO ern, She tan PO 
This is the case where the four conical (imaginary) points are, at 
infinity, and the tangential equations of these points are 
pi=9, p2=9, ps=9, ps=9; 
hence the Cartesian coordinates of these points are subject to the 
relations! 
a y” g2 
P-@ ¢-a@ &-b” 
that is, to 
et+yt+2=0, and @a’+By?+c3*=0. 
It may, however, be remarked that a double point is not always a 
1 See Salmon’s Surfaces, lust paragraph on page 423. 
