of Attractive Forces. 329 



surface is less. To the approximate value of the velocity a term 

 must therefore be added, which vanishes if AV be constant, and 



TT 



also at the points where ^ = and &= -k- These conditions are 



all satisfied by the following expression for the velocity, 



dW 

 W sin 6— q . —rr sin 6 cos 6, 



^ at 



q being a small constant factor, always positive. 



Before proceeding further, this value will be shown to be con- 

 sistent with the usual hydrodyuamical equations. For this pur- 

 pose it will suffice to limit the analysis to small terms of the first 

 order. Taking then the equations 



"'^ pda^^dt-^' ""^ pdy^ dt-^' ""pdz^dt "^ 

 dp du dv dw _ _ 

 pdt dx dy dz ~ ' 



and putting, for the sake of brevity, h^ for k^o^, and P for Naj). 

 log p, the following partial diiFerential is readily obtained : — 



dt^~ 'Kda:'' "*" dy' '^ dz^ )' 

 Let us suppose that 'P=f{t) . cf>{x, y, z). Then 



The supposition is therefore allowable if we have also 



f"{t) + uj{t) = 0, 



u being a certain constant. But the integral of this equation is 

 f{t)=% .m sin {a.t-\-c), which is precisely the form of the func- 

 tion of t which applies to the problem before us. For, accord- 

 ing to this form, the condensation at every given point of space 

 is expressed by a periodic function of the time, or by the sum of 

 several periodic functions ; and it is the dynamical action of 

 vibrations admitting of such expression which is the object of 

 the present investigation. Thus it will be required to integrate 

 the equation 



d^ d^ d^ «2 



Let P be any point of space the coordinates of which are referred 

 to the centre C of the sphere, and let PC = r, and <PCA = ^. 



Then putting k^ for j^, and transforming to polar coordinates, 



