and on the Theory of Tartini's Beats. 25 



that there is a constant flow of fluid from the rarefied parts of 

 the waves to the condensed parts, causing the condensation and 

 rarefaction to decrease more slowly with the distance than accord- 

 ing to the law of the inverse square. On calculating the amount 

 of fluid that thus flows, it will readily be found to be just the 

 quantity required to change the law of decrement from the in- 

 verse square to the simple inverse of the distance. Thus the 

 difficuky before stated is explained on the hydrodynamical prin- 

 ciples which I have proposed. 



The same comnumication contains the indication of another 

 method of solving the same problem, by means of which the fol- 

 lowing more general results might be obtained : — 



1 r 27r _ 1 



V = ^ . v= -%.< mcos — {r + fcat + c) > 



««S = /cftS.s=-S. -l ± mcos— {r + Kat + c) >. 



The circumstances of any given small disturbance may be satisfied 

 by these equations. 



There is also another point relating to the central motion of 

 an clastic fluid which requires elucidation. In the Number of 

 the Philosophical Magazine for November 1853, I have shown 

 that the integral of an equation which is admitted to be truCj 



h'^^K^l)=»' 



when applied to motion propagated uniformly from a centre, 

 proves that the condensation varies inversely as the square of the 

 distance. Tn reliance upon this reasoning, I have assumed the 

 law of the inverse square in the solution of several hydrodyna- 

 mical problems. But I now perceive that the assumption was 

 not allowable, because the above equation was obtained by geo- 

 metrical considerations applied to the motion and density of the 

 fluid, prior to any consideration of its pressure. The law of de- 

 crement of condensation which it gives depends therefore solely 

 on constancy of mass and continuity of the motion, and admits 

 of being modified by the action of forces. Thus this result is 

 compatible with the law which, as we have seen, is obtained 

 when the dynamical equation is taken into account. 



II. I pass now to the theory of Tartini's beats, in order to meet 

 an objection which might be raised against the fourth of the 

 general hydrodynamical resulta before mcutioued. It might be 



