388 Lord Rayleigh on the Work done hy Forces 



In the case o£ incompressibility, h = 0. If we restore the 

 time-factor e'^^* and throw away the imaginary part of the 

 solution, we get 



3 3 ~ 



^^ s\i\{pt — hr)— ^..coipt \. . . . (18) 

 kr -^ k-r'^ ^ J ^ 



the value of ^ differing from that of u merely by the sub- 

 stitution of y for X. The value of A is given by (17), and 

 Z, cos pt is the whole force operative at the origin at time t. 

 At a great distance from the origin (18), (19) reduce to 



„_ Zi .vz cos (pf-kr) ^ .... (20) 



7= -::: - 



+ 



7= + 



Upon this (' Theory of Sound/ § 378) I commented : — 

 '"''TT. Konig (T^^'ied. ^?r?2. xxxvii. p. 651^ 1889) has remarked 

 upon the non-agreement of (18), (19), first given in a different 

 form by Stokes, with the results of a somewhat similar in- 

 vestigation by Hertz (Wied. Ann. xxxvi. p. 1_, 1889), in 

 which the terms involving cos pt, sin pt do not occur, and he 

 seems disposed to regard Stokes's results as affected by error. 

 But the fact is that the problems treated are essentially dif- 

 ferent, that of Hertz haying no relation to elastic solids. 

 The source of the discrepancy is in the first terms of (3) &c., 

 which are omitted by Hertz in his theory of the ?ether. But 

 assuredly in a theory of elastic solids these terms must be 

 retained. Even when the material is supposed to be incom- 

 pressible, so that B vanishes, the retention is still necessary, 

 because, as was fully explained by Stokes in the memoir 

 referred to, the factor {a^~b^) is infinite at the same time.'''' 



Although the substance of the above comment appears to 

 be justified, I went too far in saying that Hertzes solution 

 has no relation to elastic solids. It is indeed not permissible 

 to omit the first terms of (3) &c. merely because the solid is 

 incompressible ; but if, though the solid is compressible, it 

 be in fact not compressed, these terms disappear. Now 



