390 FOURTH REPORT — 1834. 



as applied to the wave-theory of light, in a memoir read to the 

 French Academy in the year 1830 *. 



Having assigned the general equations of motion of a system 

 of molecules, acting on one another by attracting or repelling 

 forces which vary according to any function of the distance, 

 M. Cauchy observes that it is not necessary to have recourse to 

 their general integrals in order to determine the laws of undula- 

 tory propagation. It is sufficient, in fact, to determine the law 

 of propagation of a plane tvave. For if we consider a great 

 number of plane waves inclined to one another at small angles, 

 and which are at first superposed in the neighbourhood of the 

 point which is considered as the origin of the disturbance, the 

 vibrations in the elementary waves, to which each of these gives 

 rise, may be supposed too small to affect the sense separately, 

 and these waves become efficacious only by superposition. Con- 

 sequently the general wave-surface will be the locus of all the 

 pomts in which the elementary plane waves are superposed ; and 

 will therefore be the surface touched by them all at any instantf . 

 Hence the problem is reduced to the determination of the law 

 of propagation of a plane wave. 



M. Cauchy then shows that a disturbance, confined originally 

 to a given plane, will in general give rise to three pairs, of plane 

 waves parallel to the original plane, and propagated with uni- 

 form velocities, — the two waves of each pair moving with equal 

 velocities in opposite directions. The velocities of propagation 

 of the separate pairs, he proves, may be represented by the re- 

 ciprocals of the axes of a certain ellipsoid, whose form depends 

 upon the position of the plane wave and upon the nature of the 

 system ; and the absolute displacements of the molecules will be 

 parallel to the directions of these axes. Accordingly, a system 

 of plane waves, superposed at first at the point of original dis- 

 turbance, will be subdivided into three corresponding systems ; 

 and these, by their superposition, will generate a curved surf ace 

 of three sheets, each sheet being touched by all the plane waves 

 of the same system. From these principles it follows that a 

 single ray of light will be, in general, subdivided into three po- 

 larized rays ; — a ray being said, in this theory, to be polarized 

 parallel to a certain line or plane, when the vibrations of the 

 ethereal molecules are parallel to that line or plane. M. Cauchy 



* " Memoire sur la Tli6oriede la Lumiere," Mem. Inst., torn. x. 



t M. Poisson docs not admit the legitimacy of this conception of the wave- 

 surface ; and he thinks that an assemblage of indefinite plane waves, having a 

 small part in common at the origin of the motion, cannot represent the initial 

 condition of a mediuni disturbed at that point. 



