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NOTICES OF COMMUNICATIONS TO THE 
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7. On the Representation of a Curve in Space by means of a Cone and Monoid 
Surface. Report, 1862, p. 3. 
The author gave a short account of his researches recently published in the 
Comptes Rendus. The difficulty as to the representation of a curve in space is, that 
such a curve is not in general the complete intersection of two surfaces; any two 
surfaces passing through the curve intersect not only in the curve itself, but in a 
certain companion curve, which cannot be got rid of; this companion curve is in the 
proposed mode of representation reduced to the simplest form, viz. that of a system 
of lines passing through one and the same point. The two surfaces employed for the 
representation of a curve of the nth order are, a cone of the wth order having for 
its vertex an arbitrary point (say the point x = 0, y = 0, z = 0), and a monoid surface 
with the same vertex, viz. a surface the equation whereof is of the form Qw — P = 0, 
P and Q being homogeneous functions of (x, y, z) of the degrees p and p — 1 
respectively (where p is at most —n— 1). The monoid surface contains upon it 
p(p — 1) lines given by the equations (P — 0, Q = 0); and, the cone passing through 
n(p— 1) of these lines (if, as above supposed, p^>n— 1, this implies that some of 
these lines are multiple lines of the cone), the monoid surface will besides intersect 
the cone in a curve of the nth order. 
8. On a Formula of M. Chasles relating to the Contact of Conics. Report, 1864, p. 1. 
The author gave an account of the recent investigations of M. Chasles in relation 
to the theory of conics, viz., M. Chasles has found that the properties of a system 
of conics, containing one arbitrary parameter, depend upon two quantities called by 
him the characteristics of the system; these are, p, the number of conics of the 
system which pass through a given point, and, v, the number of conics of the system 
which touch a given line ; or, say, p is the parametric order, v the parametric class, 
of the system. And he exhibited a transformation obtained by him of a formula of 
M. Chasles for the number of conics which touch five given curves, viz., if (M, m) 
(N, n) (P, p) (Q, q) (R, r) be the orders and classes of the five given curves respec 
tively, then the number of curves is 
= (1, 2, 4, 4, 2, 1 )(M, m)(N, n) (P, p) (Q, q) (R, r), 
where the notation stands for 1 . MNPQR + 2XmNPQR + 4XmnPQR + &c. The trans 
formed formula in question was communicated by the author to M. Chasles, and had 
appeared in the Comptes Rendus; but it is, in fact, included in a very beautiful and 
general theorem given in the same Number by M. Chasles himself.