334 On a Mathematical Theory of Attractive Forces. 



the action of the sphere on the fluid to be the same as that of an 

 attractive force tending to the centre of the sphere, and equal at 



the surface to — . As also, by the law of continuity of mass, 



the fluid is compelled to fill space, the attractive force at any di- 



stance r from the centre is -^ . According to this view, the 



usual hydrodynamical equations will be 



dx r^ '^ dt ' dy r^ "^ It ' 



dz T^ dt ~ ' 



Now, whatever function of 6, that is of -, we assume V to be, 



the same partial diff'erential equation involving P, x, ?/, r and t, 

 will be found, namely that which has been used in the foregoing 

 theory. Consequently that equation does not embrace velocities 

 by which the fluid fills space apart from any change of density. 

 Por this reason I found that \rdO + Ut?/- is an exact diff'erential 

 when that part of the velocity is excluded. And this result is 

 in perfect accordance with an apj-iori demonstration I have given 

 in the Philosophical Magazine for May 1849,that udx + vdy + tvdz 

 is an exact differential for small vibrations of the fluid, the proof 

 applying only to vibrations accompanied by condensations. 



I am now able distinctly to point out in what respect the fore- 

 going investigation of the mutual action between a small sphere 

 and an elastic fluid in which it vibrates, differs from that which, 

 after Poisson, Professor Stokes has adopted in a memoir "On 

 some cases of Fluid Motion,'^ contained in vol. viii. part 1 of the 

 Cambridge Philosophical Transactions. That solution rests upon 

 the gratuitous supposition that udx + vdy + wdz is an exact dif- 

 ferential for the total velocitj^ The solution I have given shows 

 that this quantity is not an exact differential when the whole of 

 the velocity is taken into account; and upon this point I am 

 able to pronounce definitively, because by making P the principal 

 variable, I have conducted the reasoning independently of any 

 supposition as to the integrability of that differential function. 



Having now shown that waves of large breadth may attract a 

 small spherical body towards their origin, and having previously 

 shown that waves of small breadth may repel such a body in the 

 contrary direction, the main difficulty in forming a theory of 

 attractive and repulsive forces seems to be overcome. I hope to 

 be able to apply these results to the known physical forces. 



Cambridge Observatory, 

 October 15, 1859. 



