464 bepobt— 1878. 



transformations, by which it is shown that the in distinct periodic integrals of the 

 symmetric oscillation system, corresponding to m equal roots of the equation for 

 k, can be made to appear by a series of successive reduction, while the same 

 reduction applied to an unsymmetric system leads necessarily to terms in which t 

 occurs outside the sine or cosine. 



Lastly, the same general conclusions are established for the linear equations. 



dx. 



m — -s = a n x 1 + a 12 x 2 + I. , (i\ n x m 



m at 



dv 



-=^ = a^x^ + fl 22 r 2 + .... a 2n x n , 

 at 



&c, &c. 



where a n = a 22 = &c. = o a ii + «2i = °> & c * 



9. On Halphen's New Form of Chasles's Theorem on Systems of Conies 

 satisfying Four Conditions. By Dr. T. Archer Hirst, F.B.S. 



The theorem in question is expressed by the formula 



n = afx + /3i>, 



where p and v are the characteristics of the system of conies satisfying four of the 

 five conditions, and a and j3 are numbers which depend solely upon the fifth con- 

 dition. Amongst the n conies given by the formula degenerate ones are, of course, 

 included, provided they satisfy the five conditions. 



From Halphen's new form of the theorem, however, these degenerate conies are 

 excluded, and the result is the number of proper conies which satisfy the five 

 conditions in question. Halphen's paper will be published shortly in the ' Proceed- 

 ings of the London Mathematical Society.' 



10. On the Laiv of Force to any Point when the Orbit is a Conic. 

 By J. W. L. Glaisher, M.A., F.B.S. 



1. Sir W. R. Hamilton, in the 'Proceedings ' of the Royal Irish Academy for 

 1846 (vol. iii., p. 308), proved that if a body be attracted to a fixed point, with a 

 force varying directly as the distance from that point, and inversely as the cube of 

 the distance from a fixed plane, the body will describe a conic, of which the plane 

 intersects the fixed plane in a straight line, which is the polar of the fixed point 

 with respect to the conic. The author showed that if the distance of any point 

 from the fixed point be denoted by r, and if the perpendicular from any 

 point P upon the fixed plane be denoted by p, so the law of force to is, ac- 

 cording to Sir W. R. Hamilton's law, ^ 3 ,then the periodic time is i—p$, where 



p denotes the perpendicular from the centre of the orbit upon the fixed plane. 

 For example, if the plane of motion is parallel to the fixed plane, p is constant, 



ft, 2?r , _ 2tt 



=p 0i so that the force to = — s r = /xV, say, and the periodic line = j—Po 1 ~~ j — . 



It follows also that all the orbits having their centres in the same plane parallel to 

 the fixed plane are described in the same periodic time. 



2. Considering only the case of motion in the plane of xy, and combining the 

 above theorem with formulae given by MM. Darboux and Halphen (' Comptes 

 Rendus', t. 84, pp. 760-762, 936-941), it follows that if the orbit 



{ax + by + cy = Ax* + 2Hry + By 2 



