268 Prof. Challis on Hydrodynamics, 



fluid on the sphere is 



87r 3 fca 3 Sm l 



\ 3 A m ' 



which is independent of the magnitude of the sphere. Now since it 

 has been shown that the function S contains, together with pe- 

 riodic terms, others which are always positive, it follows that the 

 sphere, if free to move, would not only vibrate, but also perma- 

 nently change position by the action of an accelerative force. If 

 the unknown factor m 1 be positive, the acceleration would be 

 towards the origin of the waves, and the movement would resemble 

 that produced by an attractive force. Supposing that the axes 

 of the component vibrations all pass through a central point, the 

 magnitude of the force due to the same composite waves would, 

 at different distances from the centre, be proportional to the 

 number of the axes included within a given transverse area, and 

 consequently would vary inversely as the square of the distance 

 from the centre. 



It remains to discuss the other part of the particular integral 

 of the equation (77), namely that obtained by supposing that 

 gr = <£ 2 sin 6 cos 6. Putting, as before, / for f[r — icat) , and F for 

 Y(r + /cat), the equation (1) for determining ^> 2 admits of the fol- 

 lowing exact integral, 



(See Peacock's f Examples/ pp. 469-473. In the course of ob- 

 taining this integral, it appeared that / could not be an arbitrary 

 function of icat — r.) Retaining both functions, the following 

 results may be obtained by processes analogous to those applied 

 to the first part of the value of qr, viz. 



//+F /' + F' /" + F"\ cos 2 6 

 ''"^"VF r^ + ^r~~)-2~ J 



Ka \ r 4 r 6 or z J 



£ = / 8(/i-Fi) _ 8 (/- g ) . !(/' -Z) /"-i*Wg 



ica \ r* t* 3r 2 3r ) 2 ' 



IT 



These equations show that U = andW = where 6=— } for all 



values of r at all times, and that in the same case <j is equal to 

 the condensation of the incident waves. In fact the excursions 

 of the particles of the fluid must now be regarded as small com- 

 pared to the dimensions of the sphere, and in this respect the 

 above equations are quite distinct from those given by the first 



