58 Mr. G. G. Stokes on some Points in the 



In order to simplify as much as possible the analysis, 

 instead of an expanding envelope, suppose that we have a 

 sphere, of a constant radius b, at the surface of which fluid is 

 supplied in such a manner as to produce a constant velocity V 

 from the centre outwards, the supply lasting from the time 

 to the time t, and then ceasing. This problem is evidently 

 just as good as the former for the purpose intended, and it 

 has the advantage of leading to a result which may be more 

 easily worked out. On account of the length to which the 

 present article has already run, I am unwilling to go into the 

 detail of the solution ; I will merely indicate the process, and 

 state the nature of the result. 



Since we have no reason to suspect the existence of a 

 function of the form F (r + at) in the value of f which belongs 

 to the present case, we need not burden our equations with 

 this function, but we may assume as the expression for <p 



f{'>'-at) ,^ . 



^ = -^. — (70 



For we can always, if need be, fall back on the complete in- 

 tegral of (3) ; and if we find that the particular integral (7.) 

 enables us to satisfy all the conditions of the problem, we are 

 certain that we should have arrived at the same result had we 

 used the complete integral all along. These conditions are 



(p = when ^ = 0, from r=6 to r= 00 ; . . (8.) 

 for p must be equal to a constant, since there is neither con- 

 densation nor velocity, and that constant we are at liberty to 

 suppose equal to zero; 



-^ = V when r=6, from /=0 to ; = t; . . (9.) 



-r = when r=:b, from t — r to t= oo . . . (10.) 

 dr ^ 



(8.) determinesy(2) from z = Z> to 2" = oo ; (9.) determines /(z) 

 from s = 6 to is=^b—ar; and (10.) determines y(r) from 

 z = b — ar to z= — 00, and thus the motion is completely 

 determined. 



Jt appears from the result that if we consider any particular 

 value of r there is no condensation till at =.r—b, when it 

 suddenly commences. The condensation lasts during the time 

 r, when it is suddenly exchanged for rarefaction, which de- 

 creases indefinitely, tending to as its limit as t tends to x . 

 The sudden commencement of the condensation, and its 

 sudden change into rarefaction, depend of course on the 

 sudden commencement and cessation of the supply of fluid at 

 the surface of the sphere, and have nothing to do with the 



