ON THE QUARTIC SURFACES (*$Z7, V, IF) ! = 0. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xi. (1871), 
pp. 15—25.] 
Among the surfaces of the form in question are included the reciprocals of several 
interesting surfaces of the orders 6, 8, 9, 10, and 12, viz. 
Order 6, parabolic ring. 
„ 8, elliptic ring. 
„ 9, centro-surface of a paraboloid. 
„ 10, parallel surface of a paraboloid. 
„ „ envelope of planes through the points of an ellipsoid at right angles to 
the radius vectors from the centre. 
„ 12, centro-surface of an ellipsoid. 
„ „ parallel surface of an ellipsoid. 
I propose to consider these surfaces, not at present in any detail, but merely for 
the purpose of presenting them in connexion with each other and with the present 
theory. It will be convenient to use homogeneous equations, but for thfe metrical 
interpretation the coordinate W or may be considered as equal to unity: I have 
not thought it necessary so to adjust the constants that the equations shall be homo 
geneous in regard to the constants; this can of course be done without difficulty, and 
in many cases it would be analytically advantageous to make the change. 
I take throughout (X, Y, Z, W) for the coordinates of a point on the quartic 
surface (so that (U, V, W) in the equation V, F) 2 = 0 are to be considered as 
quadric functions of (X, F, Z, IF)), reserving {x, y, z, w) for the coordinates of a 
point on the reciprocal surface of the order 6, 8, 9, 10, or 12. The reciprocation is 
performed in regard to the imaginary sphere x 2 + y 2 + z 2 + w 2 = 0 : the relation between