262 
ON THE THEORY OF INVOLUTION IN GEOMETRY. 
[40 
or the new curve contains \m + n — r — l] 2 more arbitrary constants than it would 
do if the mn points, through which it was made to pass, had been perfectly arbitrary; 
a result given before in the Journal, [5]. 
In the case of surfaces, if r^>m-\-n+p — 4. Then assuming r>m + n — 4, m+p — 4, 
or n+p — 4, we have 
(r : 7)i, n, p : 3} = [r — m, 3] + [r — n, 3] + \r —p, 3] 
— [r — m — n, 3] — [r ~7)i —p, 3] — [r — 7i —p, 3] 
= [r, 3] — mnp — [r — m — n—p, 3] ; 
or the surface contains ^ [to + n + p — r — l] 3 more arbitrary constants than it would do if 
the 7)\7ip points, through which it was made to pass, had been perfectly arbitrary. 
Similarly, in the case where r is not greater than one or more of the quantities 
m + n — 4, m+p — 4<, n+p — 4. Thus in particular, if r be not greater than the least of 
these quantities 
{?’ : 7n, n, p : 3] = [r, 3] — )nnp + [r — n —p, 3] + [r — 7tn—p, 3] 
+ [r — m — 7i, 3] — \r — m — n—p, 3]; 
or the surface contains 
£ [to + n +p — r — l] 3 — £ [n +p — r — l] 3 — ^ [m +p — r — l] 3 — ^ \m + 7i — r — l] s 
more arbitrary constants than it would otherwise have done. Again, for a surface of the 
r th order, subjected to pass through the curve of intersection of two surfaces of the 
orders m, n, 
[r : 7n, n, 3} = [r — 77i, 3] + [r — n, 3] — [r — m — n, 3]; 
in which the last term, or \[m + n — r — l] 3 , is to be omitted when r^>7)i + 7i — 4. 
The function of the r th order, which is satisfied by the systems of values which 
satisfy the equations of the orders m, 7i... contains, we have seen, [r, m, n, p ... 6] 
arbitrary constants; hence it may be determined so as to pass through this number, 
diminished by unity, of arbitrary points. But the equation being determined in general 
by the condition of being satisfied by [r, 0] — 1 systems of variables, it will be com 
pletely determined if, in addition to the above number of arbitrary systems, we suppose 
it to be satisfied by a number N = [r, 0] — {r\ ni, n, p... : 0) of systems satisfying the 
equations above. Hence the theorem 
“ The equation of the r th order which is satisfied by a number 
N = [r, 0~\ — [r\ m, n, p... : 0} 
of systems satisfying the equations of the orders m, n, p... is satisfied by any systems 
whatever which satisfy these equations.” 
In particular—“ The surface of the r th order which passes through a number 
[r, 0] — [r : 7)i, n : 0}