TRANSACTIONS OF THE SECTIONS. 5 



But - expresses the time of oscillation of any one vibrating particle ; 



this time would therefore be nearly constant, if the particles attracted 

 each other according to the law of the inverse square of the distance ; 

 and consequently this law is inadmissible, as being incompatible with 

 the law of dispersion. It had appeared to Sir William Hamilton im- 

 portant to reproduce these results, though he remarked that they 

 agree substantially with those of Cauchy, because the law of the in- 

 verse square was one which naturally offered itself to the mind, and 

 had, in fact, been proposed by at least one mathematician of high 

 talent. There was, laowever, another law which had great claims on 

 the attention of mathematicians, as having been proposed by Cauchy 

 to represent the phenomena of the propagation of the light in vacuo, 

 namely, the law of a repiilsive action, proportional inversely to the 

 fourth power, or to the square of the square of the distance. M. 

 Cauchy had, indeed, supposed that this law might hold good only for 

 small distances, but in examining into its admissibility, it appeared 

 fair to treat it as extending to all the neighbouring particles which act 

 on any one. But against this law also. Sir William Hamilton brought 

 forward objections, which were founded partly on algebraical, and 

 partly on numerical calculations, and which appeared to him decisive. 

 The spirit of these objections consisted in showing that the law in 

 question would give too great a preponderance to the effect of the 

 immediately adjacent particles, and would thereby produce irregu- 

 larities which are not observed to exist. In particular, if it be supposed 

 that 



S. r' A ar 2 = S. r '■ A y« = S. r '■ A z% 

 S.r' Ax*= S. r' At/* = S.7-iAz*, 

 S.r'Ax'^Ai/' = S.r'Ai/^Az" = S.r' Az^ Ax\ 



and also, in (5.), that c = o, a =^ b, and that X is much greater than e, 

 it is found that the two values v'^ and v/^ of the square of the velocity 

 V, corresponding to vertical and to horizontal but transversal vibra- 

 tions, are connected by the relation 



t;« = — - , 

 31;% 



being expressed as follows : 



r«=^S(^5r Ax*-r \ 



v« = — S(r -5r Ax*); 



In conclusion, he offered reasons for believing that the law of 

 action of the particles of the ether on each other resembles more the 

 law which Poisson has in one of his memoirs proposed as likely to 



