40] 
ON THE THEORY OF INVOLUTION IN GEOMETRY. 
263 
of points in the curve of intersection of two surfaces of the orders m, n,—or through 
[r, #] — [r : m, n, p : 0} of the mnp points of intersection of three surfaces of the orders 
m > p,—passes through the curve of intersection, or through the mnp points of inter 
section.” 
Thus a surface of the second order which passes through eight points of the curve 
of intersection of two surfaces of the second order passes through this curve ; and any 
surface of the second order which passes through seven of the points of intersection 
of three surfaces of the second order passes through the eighth point. (The first 
theorem obviously fails if the eight points have the relation in question, i.e. if they 
are the eight points of intersection of three surfaces of the second order.) 
Again—“ The curve of the r th order which passes through [r, 0] — [r : m, n : 6} of 
the points of intersection of two curves of the orders mn, passes through the remaining 
points of intersection.” e.g. “Any curve of the third order which passes through eight 
of the points of intersection of two curves of the third order, passes also through 
the ninth point.” 
Consider next the following question, which [as regards particular cases] has been 
treated of by Jacobi in the memoir already quoted (Creile, tom. xv.). “To find the 
number of relations which must exist between K (0 +1) variables, forming K systems, 
each of which satisfies simultaneously equations of the orders m, n, p... respectively; 
the number <£ of these equations being anything less than 0 ; or cf) being equal to 0, 
provided at the same time K=mnp....” 
Suppose m n, n <£p... and write 
[m, 0] — {m : m, n, p ... : 0} = A, 
\n , 6] — [n : n, p : 0} = A', 
&c. 
Imagine the equations of the orders n, »... given. Any function of the m th order 
which is satisfied by A r of the systems of values which satisfy the given equations, 
and any particular equation of the m th order, is satisfied by the remaining K — A 
systems of values. Hence assuming A systems, satisfying the equations of the orders 
n, p ... but otherwise arbitrary, the remaining systems must satisfy these equations, 
and a completely determinate equation of the m th order; i.e. there must be </> rela 
tions between the variables of each system, and consequently <£ (K — A) relations in 
all. Similarly, if the equations of the orders p ... were given, A' systems of variables 
might be assumed satisfying these equations, but otherwise arbitrary ; the remaining 
A — N' systems satisfy (</> — 1) determinate equations, or the number of relations 
between the variables is (0 — 1) (A —A')... ; continuing in the same manner the total 
number of relations between the variables is 
(f> (K — A) + (<£ — 1) (A — A') + (</>- 2) (A r/ - A") + ... 
in which however any term (</> —1) (A — A') or (<£ — 2) (A—A') ... &c., which becomes 
negative, must be omitted. It is obvious that we may write more simply 
A = [m, 0] — 1 — {m ; n, p.. .0} , 
A' = [n, 0] — 1 — [n \ p ...0} , &c.