566 
NOTICES OF COMMUNICATIONS TO THE 
[555 
where a is the order, A the class of the curve; a is the number, three times the 
class + the number of cusps, or (what is the same thing) three times the order -t- the 
number of inflexions. 
2. On a Correspondence of Points and Lines in Sjmce. Report, 1870, p. 10. 
Nine points in a plane may be the intersection of two (and therefore of an 
infinite series of) cubic curves; say, that the nine points are an “ ennead ”: and 
similarly nine lines through a point may be the intersection of two (and therefore of 
an infinite series of) cubic cones; say, the nine lines are an ennead. Now, imagine 
(in space) any 8 given points; taking a variable point P, and joining this with the 
8 points, we have through P 8 lines, and there ig through P a ninth line completing 
the ennead; this is said to be the corresponding line of P. We have thus to any 
point P a single corresponding line through the point P; this is the correspondence 
referred to in the heading, and which I would suggest as an interesting subject of 
investigation to geometers. Observe that, considering the whole system of points in 
space, the corresponding lines are a triple system of lines, not the -whole system of 
lines in space. It is thus, not any line whatever, but only a line of the triple system, 
which has on it a corresponding point. But as to this some explanation is necessary ; 
for starting with an arbitrary line, and taking upon it a point P, it would seem 
that P might be so determined that the given line and the lines from P to the 
eight points should form an ennead,—that is, that the arbitrary line would have upon 
it a corresponding point or points. 
The question of the foregoing species of correspondence was suggested to me by 
the consideration of a system of 10 points, such that joining any one whatever of 
them with the remaining nine points, the nine lines thus obtained form an ennead; or, 
say, that each of the 10 points is the “ enneadic centre ” of the remaining nine. I 
have been led to such a system of 10 points by my researches upon Quartic Surfaces; 
but I do not as yet understand the theory. 
3. On the Number of Covariants of a Binary Quantic. Report, 1871, pp. 9, 10. 
The author remarked [see 462] that it had been shown by Prof. Gordan that the 
number of the covariants of a binary quantic of any order was finite, and, in particular, 
that the numbers for the quintic and the sextic were 23 and 26 respectively. But the 
demonstration is a very complicated one, and it can scarcely be doubted that a more 
simple demonstration will be found. The question in its most simple form is as 
follows: viz. instead of the covariants we substitute their leading coefficients, each of 
which is a “ seminvariant ” satisfying a certain partial differential equation; say, the 
quantic is (a, b, c,..., k\x, y) n , then the differential equation is (ad b + 2bd c ... + njdk) u = 0, 
which qua equation with n +1 variables admits of n independent solutions: for 
instance, if n= 3, the equation is (ad b + 2bd c + Scd d ) u = 0, and the solutions are a, ac — b 2 ,.