16 ON THE NUMBER OF CONSTANTS IN THE EQUATION OF A SURFACE. [709 
If n = 4, there is in each of the four cases one system of values of p, q\ viz. the 
cases are 
P> 3 = 
2 1 No. = a 2 + + a. 2 + a 3 — a x — a x — 1 = 9 + 3 + 9 + 19 — 3 — 3 — 1, = 33, 
3 1 „ a 3 + iq + otj + a 3 — a 2 — a 0 — 2 = 19 + 3 + 3 + 19 — 9 — 0 — 2, = 33, 
1 1 „ ttj + Uj + iig + cig — eta — cio — 2 = 3 + 3+19 + 19 — 0 — 9 — 2, = 33, 
2 2 ,, tt2 + i£ 2 + tt 2 + ®2 — — 3= 9 + 9+ 9+ 9 — 0 — 0 — 3, = 33, 
and the number of constants is in each case = 33. This is easily verified: in the first 
case we have a quartic surface containing a conic, the plane of the conic is therefore 
a quadruple tangent plane; and the existence of such a plane is 1 condition. In the 
second case the surface contains a plane cubic; the plane of this cubic is a triple 
tangent plane, having the points of contact in a line; and this is 1 condition. In 
the third case the surface contains a line, which is 1 condition: hence in each of 
these cases the number of constants is 34 — 1, =33. In the fourth case, where the 
surface contains a quadriquadric curve, we repeat in some measure the general reasoning: 
the quadriquadric curve contains 16 constants, and we have thus 16 as the number 
of constants really contained in the equations P = 0, Q = 0 of the quadriquadric curve: 
the equation PS — QR = 0, contains in addition 9+10, =19 constants, but writing it 
in the form P (S + kQ) — Q(R + kP) = 0, we have a diminution = 1, or the number 
apparently is 16 + 19 — 1, =34. But the quadriquadric curve is one of a singly infinite 
series P + IR= 0, Q + IS = 0 of such curves, and we have on this account a diminution 
= 1; the number of constants is thus 34—1, =33 as above: the reasoning is, in fact, the 
same as for the case of a plane passing through a line; the line contains 4 constants, 
hence the plane, qua arbitrary plane through the line, would contain 1 + 4, =5 constants; 
but the line being one of a doubly infinite system of lines on the plane the number is 
really 5 — 2, = 3, as it should be. 
Cambridge, 2nd Sept., 1880.