12 



ever, to be fundamentally different from that of M. Cauchy ; 

 and in fact he was led to this form of the integral by other 

 considerations before he was aware that he had been pre- 

 ceded in the deduction. 



The remainder of the present communication is taken up 

 with the discussion of the relation between the coefficients 

 u and ki which expresses the law of dispersion. Following 

 M. Cauchy,*" the author has transformed this relation by 

 converting the triple sums into triple integrals ; and he has 

 found that, by applying this transformation at an earlier 

 stage of the investigation, the resulting relation is deduced 

 with great simplicity. 



The relation between u and k, for the vibrations in the 

 plane of the wave, has already yielded to M. Cauchy the 

 probable result, that the molecules of the ether repel one 

 another according to the inverse fourth power of the distance. 

 When this law of force is substituted in the corresponding 

 relation for the normal vibration, the author finds that the 



resulting value of t, or of the velocity of propagation, is in- 

 finite ; so that the normal disturbance is propagated instan- 

 taneously ^ and gives rise to no wave. Thus the hypothesis 

 of transversal vibrations seems to be established on theore- 

 tical grounds. 



The author finally gives reasons for concluding that the 

 theory, in its present form, is insufficient to explain the phe- 

 nomena of light in bodies ; and that it becomes necessary in 

 this case to take into account the action of the material 

 molecules. This extension of the theory will be given in 

 a future communication. 



3. " On the Composition of Thebaine." By Robert J. 

 Kane, M. D., M.R.I. A., Professor of Natural Philosophy 

 in the Royal Dublin Society. 



♦ Nouveaux Excrdces de Mathematiques. Livraison 7™*^. 



