116 ON THE MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. V. 



The solution in this case may however be obtained ab initio 

 by a simpler analysis, as follows*. 



Let OX be the direction of motion of the centre of the 

 sphere at any instant, and V its velocity. Let P be any point of 

 the fluid, and let OP = r } and the angle POX = 6. It is evident 

 that &amp;lt;f) will be a function of r, 6 only; and we know from the 

 theory of Spherical Harmonics -that any such function which 

 satisfies (1) and has its derivatives zero at infinity can be expanded 

 in a series of the form 



&amp;lt;/&amp;gt; = const. + * + j + j + &c, 

 r r r 



where Q n is the zonal harmonic of order n, multiplied by an 

 arbitrary constant. The .condition which &amp;lt;/&amp;gt; has to satisfy at the 

 surface of the sphere is 



^ = Fcos0, 

 dr 



so that, if a be the radius, 



Hence - Hr = Fcos 0, 



and Q = Q 2 = Q 3 = c. = 0. 



We have then finally 



$ = const. J ^- cos 6 (12). 



It is easy to verify the fact that this value of &amp;lt; really satisfies 

 all the conditions of the problem. 



If we impress on the whole system the moving sphere and 

 the fluid a velocity - F, in the direction OX, we have the case 

 of a uniform stream of velocity F flowing past a fixed spherical 

 obstacle. The velocity-potential is then got by adding the term 

 Vr cos 6 to (12), so that we now have 



= const. - V(T + | ^] cos 0. 



* This solution, generally attributed by continental writers to Dirichlet (Monats- 

 berichte der Berl. Akad. 1852), was given by Stokes, Camb. Trans. Vol. vin. (1843). 



