XXXIV. On the Vibrations of an Interrupted Medium. By the Rev. PHILIP 

 KELLAND, M.A., F.R.S.S.L. & E., Professor of Mathematics in the University 

 of Edinburgh. 



(Read January 15. 1844.) 



IN certain investigations which I have presented to the Societ} r , relative to 

 the modifications which light undergoes when it meets with a medium more dense 

 than that in which it is travelling, the law of force has been supposed to be that 

 of the inverse square of the distance. On re-examination of this subject, I find 

 that there is no necessity for restricting the computations by the hypothesis of 

 any particular law. The conclusions are perfectly independent of the law, pro- 

 vided one of the equations of reduction, the value of which it is not possible to 

 compute by any known methods of analysis, be admitted as an experimental 

 result. The object of the present Memoir is twofold : 1st, To present the analysis 

 of the general theorem of vibrations at the surface of an interrupted medium in 

 its most simple form ; and, 2d, To apply the results to the case of reflection unac- 

 companied by refraction. To accomplish the former object, I have, after deducing 

 the equations of motion, sought to determine the values of the different constants 

 which enter them, by means of the condition of symmetry. This investigation 

 has led me to the generalization of some of the remarkable results which M. Cauchy 

 has given relative to this subject, besides presenting me with some other conclu- 

 sions, one of which is remarkable for its simplicity and completeness. It is this : 

 A particle in a medium substantially symmetrical, will, when displaced from its 

 position of equilibrium, be acted on by no accelerating paces ; it will consequently 

 not be urged farther from its position of rest, at least until the other particles 

 shall have been displaced. This is an important theorem, for it removes one 

 objection to the possibility of attractive particles forming a system of stable equi- 

 librium. In reference to the latter part of my investigation, I have only to 

 remark that the equations cannot be solved in all then* generality, inasmuch as 

 they contain two more unknown quantities than the number of equations. But 

 by approximation (omitting all terms of high orders of small quantities), I find 

 that, in general, when there is no refraction, the reflection is equal to the inci- 

 dence, and the rays suffer no retardation. But when the medium is crystallized, 

 that is, when the forces within it have a different value from those in air, the 

 reflected light is less than the incident, and the vibrations in the plane of incidence 

 suffer retardation. 



VOL. XV. PART IV. 6 Y 



