7 
487] 
ON THE QUARTIC SURFACES (U, V, Wf= 0. 
The quartic surface is consequently the envelope of the quadric 
viz. this is 
V & b ) 
Hence the quartic surface is 
6 2 (— + Y ~) + 26 (Z 2 + Y 2 + Z 2 ) + aX 2 + bY 2 - 2ZW = 0. 
(? + ~t) (aZ2+bY2 ~ 2ZW ^ - ( Z2 + 72 + z ^ = °> 
or, what is the same thing, 
X 2 Y 2 (a - b) 2 - 2ZW (bX 2 + aY 2 ) - 2abZ 2 (X 2 + Y 2 ) - abZ 4 = 0. 
This has four conic nodes ; viz. considering the equations 
these give the point Z = 0, Y=0, Z = 0 four times, and four other points which are 
the nodes in question; the point (Z = 0, Y = 0, Z = 0) is a singular point of a higher 
order; the reduction caused by these singularities should be =8 + 19, so as to make 
the order of the surface of centres = 9 ; that is the reduction on account of the point 
(Z = 0, F=0, Z= 0) must be =19; but it is not by any means obvious how this is so. 
Parallel surface of the paraboloid. 
This is given, Salmon’s Solid Geometry, 2nd Edit., pp. 146 and 148, [Ed. 4, p. 180], 
for the paraboloid aX 2 + bY 2 + 2rZW = 0, as the envelope of the quadric surface 
+ 2 drzw — (6 2 r 2 + k 2 ) w 2 =0. 
da +1 6b+ 1 
The reciprocal quartic is thus the envelope of 
that is 
whence the equation is 
Z 2 Y 2 
viz. this is a quartic having the nodal line-pair Z = 0, — + = 0; and a further 
singularity at the point Z = 0, F = 0, Z = 0. It would require some consideration to 
show that the order of the parallel surface is thence =10, as it should be.