NOTICES AND ABSTRACTS 



OP 



MISCELLANEOUS COMMUNICATIONS TO THE SECTIONS. 



MATHEMATICS AND PHYSICS. 



Mathematics. 



On a Formula of M. Chasles relating to the Contact of Conies. 

 By Professor Caxlet, F.B.S. 

 The author gave an account of the recent investigations of M. Chasles in relation 

 to the theory of conies, viz., M. Chasles has found that the properties of a system 

 of conies, containing one arbitrary parameter, depend upon two c[uantiti_es called 

 by him the characteristics of the system ; these are, /x, the number of conies of the 

 system which pass through a given point, and, v, the nimiber of conies of the 

 system which touch a given liue ; or, say, /j. is the parametric order, ^ v the para~ 

 metric class, of the system. And he exhibited a transformation obtained by him 

 of a formula of M. Chasles for the number of conies which touch five given 

 curves, viz., if (M, m) (N, n) (P, p) (Q, q) (E, r) be the orders and classes of the 

 five given curves respectively, 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+22mNPQIl+42»wPQR+&c. The 

 transformed formula in question was commimicated by the author to M. Chasles, 

 and had appeared in the ' Comptes Rendus j' but it is, in fact, included in a very 

 beautiful and general theorem given in the same Number by M. Chasles himself. 



On the Prohlem of the In~and-drcumscribed Triangle. 

 By Professor Catlet, F.B.S. 

 The general problem of the in-and-circumscribed triangle may be thus stated, 

 viz., to find a triangle the angles whereof severally lie in, and the sides severally 

 touch, a given curve or cm-ves ; and we may, in the first instance, inquire as to 

 the number of such triangles. The first and easiest case is when the curves are 

 all distinct ; here, if the angles lie in curves of the orders m, n, p, respectively, 

 and the sides touch curves of the classes Q, R, S, respectively, then the number 

 of tiiangles is = 2wwpQRS. The number may be obtained for some other cases ; 

 but the author has not yet considered the final and most diflicult case, viz. that in 

 which the angles severally lie in, and the sides severally touch, one and the same 

 given curve. 



1864. 1 



