56 Mr. G. G. Stokes on &ome Points in the 



must take the complete integral of (3.), and determine the two 

 arbitrary functions which it contains. We are at liberty, for 

 example, to suppose the condensation and velocity when / = 

 given by the equations 



r r r^ 



from r=b io r=c, and to suppose them equal to zero for all 

 other values of r; but we are not therefore at liberty to sup- 

 press the second arbitrary function in the integral of (3.) The 

 problem is only a particular case of that considered by Poisson, 

 and the arbitrary functions are determined by his equations (6.) 

 and (8.), where, however,it must be observed, that thearbitrary 

 functions which Poisson denotes by^, F must not be confounded 

 with the given function here denoted by^, which latter will ap- 

 pear at the right-hand side of equations (8.). The solution pre- 

 sents no difficulty in principle, but it is tedious from the great 

 number of cases to be considered, since the form of one of the 

 functions which enter into the result changes whenever the 

 value of r + a^ or of r — at passes through either b or c, or 

 when that of r — at passes through zero. It would be found 

 that unless /(6) = 0, a backward wave sets out from the inner 

 surface of the spherical shell containing the disturbed portion 

 of the fluid; and unless /(c) = 0, a similar wave starts from 

 the outer surface. Hence, whenever the distui'bance can be 

 propagated in the positive direction only, we must have A, or 

 f{c)—f{h), equal to zero. When a backward wave is formed, 

 it first approaches the centre, which in due time it reaches, 

 and then begins to diverge outwards, so that after the time 



- there is nothing left but an outward travellinfj wave, of 

 a ° n » 



breadth 2c, in which the fluid is partly rarefied and partly 

 condensed, in such a manner ihaX Trsdr taken throughout the 

 wave, or A, is equal to zero. 



It appears, then, that for any outward travelling wave, or 

 for any portion of such a wave which can be isolated, the 

 quantity A is necessarily equal to zero. Consequently the 

 conclusion arrived at, that the mean condensation in such a 

 wave or portion of a wave varies ultimately inversely as the 

 distance from the centre, proves not to be true. It is true, as 

 commonly stated, that the condensation at corresponding points 

 in such a wave in its successive positions varies ultimately in- 

 versely as the distance from the centre ; it is likewise true, as 

 -Professor Challis has argued, that the mean condensation in 

 ^'iiny portion of the wave which may be isolated varies ultimately 



