L80 Prof. Challis on the Hydrodynamical Theory 



the equation (#) in art. 7 the value of a to terms of the 



second order with respect to m, the following results may be 



obtained : — 



./ . „ 2fc 5 m 2 a A 

 n> = / ( m sm qg— — « — cos 2</£ j ; 



Now the principle already adopted in the foregoing theory 

 of musical sounds requires that the plane-waves of the present 

 problem should be supposed to be made up of sums of the 

 spontaneous vibrations defined by the above three equations. 

 I have been compelled, by a logical necessity, to abandon the 

 hypothesis of uncompounded plane-waves such as are usually 

 admitted in hydrodynamics, having obtained absurd results 

 when the reasoning on that hypothesis was made to include 

 terms of the second order. It will therefore be supposed 

 that the plane-waves of our problem are composed of an un- 

 limited number of spontaneous vibrations, all in the same 

 phase of vibration and having their axes all parallel, it being 

 presumable that under these conditions the transverse vibra- 

 tions will be wholly neutralized. 



12. The diameter of the sphere being assumed to be ex- 

 cessively small compared with the wave-length X, the condensa- 

 tion of the incident waves at any point of the sphere's surface 

 will be very nearly the same for the composite waves above 

 described as if they were uncompounded plane-waves. In fact, 

 as far as regards the hemispherical surface on which the waves 

 are immediately incident, the tw T o cases may be treated as iden- 

 tical ; but the amount of condensation at points of the other 

 hemispherical surface will, in the case of the composite waves, 

 depend in some measure on limited lateral divergence due to 

 the composition, as will be presently indicated. This effect is 

 smaller as the size of the sphere is less ; so that for a first ap- 

 proximation we may assume that the condensation on both 

 halves of the surface of a sphere of the size of an atom is the 

 same as if the incident plane-waves were not composite. 

 Since the motions in these waves consist of harmonic vibra- 

 tions which produce no residual motion of translation of any 

 particle of the aether, for given increments (dz) of space the 

 increments of condensation in the condensed portion of a 

 wave must at each instant be just equal to the decrements ; 

 and this must be the case also in the rarefied portion, although 



