178 Prof. Challis on the Hydrodynamical Theory 



gether depends on the a priori determination of the laws of 

 those aerial vibrations which I have designated as spontaneous, 

 which, apart from any particular mode of disturbing the air, 

 are found to be expressible by definite mathematical formulae. 

 The reality of this class of vibrations had been established by 

 experience, inasmuch as it had been ascertained that indepen- 

 dently of particular producing circumstances sounds are recog- 

 nized by the sense of hearing as consisting of fundamental 

 notes and accompanying harmonics. Obviously, therefore, the 

 accounting for such vibrations by abstract argument founded 

 on the general equations of hydrodynamics was a problem 

 demanding solution. The appropriate solution I claim to 

 have given by the process the steps of which are indicated 

 in arts. 5-7 of the present communication. In this research 

 no use was made of Fourier's theorem, because that theorem 

 depends on principles that are analytical, and only as such 

 admits of being applied in physical questions. But after 

 the laws of the primary aerial vibrations have been deduced 

 independently from hydrodynamical principles, Fourier's 

 theorem is properly applied in proving that all sounds may 

 be analytically represented by sums of such vibrations, and 



d'ty 

 consequently in justifying the expression for -=- assumed in 



art. 8. For these reasons I maintain that the mathematical 

 investigation of spontaneous vibrations, which I propounded 

 originally in the Number of the Philosophical Magazine for 

 February 1849, is an indispensable part of the theory of 

 Acoustics, being required to fill up a lacuna whereby the logic 

 of that theory would otherwise be vitiated. I have dwelt the 

 more on the foregoing evidence of the reality of spontaneous 

 vibrations of the air, because the same class of vibrations in 

 the aether will have to be taken into account in the theory I 

 now proceed to give of attractive and repulsive forces due to 

 aetherial undulations. Other physical forces are more im- 

 mediately referable to steady motions than to vibrations of the 

 aether. 



10. The aether being by hypothesis a perfect continuous 

 fluid defined by the relation p = a 2 p between its pressure p and 

 density p, I propose, in the first instance, to attempt the solu- 

 tion of the following definite problem : — A series of plane 

 periodic waves of the aether being incident on a very small 

 fixed sphere of given radius, it is required to find the pressure 

 at any point of the surface of the sphere at any time. 



It is supposed that in plane periodic waves the motion is in 

 directions perpendicular to a fixed plane and wholly vibratory, 

 and that the velocity is at each instant a function of the dis- 



