Full text: The collected mathematical papers of Arthur Cayley, Sc.D., F.R.S., sadlerian professor of pure mathematics in the University of Cambridge (Vol. 7)

140 
A MEMOIR ON QTJARTIC SURFACES. 
[445 
For k =14, the cones are 3 1} 1, 1, 1, or 2, 2, 1, 1: it is easy to see that we have 
only the three cases 
Cones 3J, 1, 1, 1 : 2, 2, 1, 1 Singular tangent planes 
gives (14. 3 + 0.2) 4- 6, = 7 
No. may be 14 , 0 
„ 8,6 
2 , 12 
(8.3+ 6.2) 4- 6, =6 
( 2.3 + 12.2) 4- 6, =5 
and we may in the like manner limit the number of possible cases, for other values 
of k. But I do not at present further pursue the inquiry. 
As to the Number of Constants contained in a Surface. 
14. We say that a surface P — 0 contains or depends upon a certain number of 
constants; viz., this is the number of constants contained in the equation P = 0 of the 
surface, taking the coefficient of any one term to be equal to unity; thus the general 
quadric surface contains 9 constants; the surface can in fact be determined so as to 
satisfy 9 conditions; or, as we might express it, the Postulation of the surface is = 9. 
[I have elsewhere said Postulandum and Capacity: I prefer this last expression.] 
And if, in the general equation so containing 9 constants, k of these are given, or, 
what is the same thing, if the quadric surface be made to satisfy any k conditions, 
then the number of constants, or postulation of the surface, is =9 —k. 
15. But a different form of expression is sometimes convenient; the conditions to 
be satisfied are frequently such that, being satisfied by the surfaces P = 0, Q = 0,..., 
they will be satisfied by the surface aP + /3Q+ ... = 0, where a, /3, ... are any constant 
multipliers whatever. When this is so, there will be a certain number of solutions 
P = 0, Q = 0,... not connected by any such relation, or say of asyzygetic solutions, such 
that the general surface satisfying the conditions in question is aP + ¡3Q + ... = 0; and 
hence, taking one of these coefficients as unity, the number of constants, or postulation 
of the surface, is equal to the number of the remaining coefficients, or, what is the 
same thing, it is less by unity than the number of the asyzygetic solutions P = 0, 
Q = 0.... Instead of considering the number of constants, or postulation, we may consider 
the number of solutions (that is, asyzygetic solutions) or surfaces P — 0, Q = 0, ... which 
satisfy the conditions in question. 
16. Thus, for the quadric not subjected to any conditions, there are 10 surfaces 
(for example, these may be taken to be the surfaces x 2 = 0, y 2 = 0, z 2 =0, w 2 = 0, yz = 0, 
zx = 0, xy = 0, xw = 0, yw = 0, zw — 0); and the general quadric surface is by means of 
these expressed linearly in the form (a, ...$#, y, z, w) 2 = 0. So for the quadric surfaces 
through k given points, the number of these is = 10 — A;; thus for the surfaces 
through 4 given points, say the points (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1), 
the 6 given surfaces may be taken to be yz =0, zx = 0, xy = 0, xw = 0, yw = 0, zw = 0, 
and every other quadric surface through the 4 points is by means of these expressed 
linearly in the form (a, ...\yz, zx, xy, xw, yw, zw) — 0; for the quadric surfaces through 
8 points there are two surfaces P = 0, Q = 0; and every quadric surface through the
	        
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