858 



L=^- — \b'R''-3hR+3){hR'+l)>''''^-'^'+{b'R'+ShR+3)(hR'-\)e-'>'f-'^']{i5) 



For the rest no alterations have to be made to section 9 and the 

 calculation of tj would proceed in the same manner as with an 

 internal oscillating sphere. 



21. Another case which is of practical importance and has found 

 experimental application '), is that of a hollow sphere completely 

 filled with liquid which is made to swing. It may be expected that 

 this case can be derived as a special case from our general formulae 

 by putting R' = 0. In that case according to (53) : 



aR' 



and 



_ R' {br—l)e'"- + (br'rl)e~'"- 



" '^ ^ 7' {bR -l)e''K+{bR+l)e-'''f ^ '' 



Physically, however, this is only possible, if for ;• ^ 0, ii does 

 not become infinite and, as a matter of fact, it does not, for with 

 /• =^ 0, u becomes 



n = 4 ab' — (48) 



" ^ ^bR—\)ti^-\-{bR+\)e-6Ji ^ ' 



In the general case the liquid cannot be at rest at the centre: 



the wave-motion starting from the oscillating sphere passes through 



the centre and expands again beyond it; this may al.so be formulated 



by saying, that the waves are reflected at the centre, this time as 



upon a free boundary, i. e. without reversal of phase. Only when 



bR is so large, that the motion is damped out before reaching the 



centre, i.'^ = practically and further 



R' br-l 



u = a — e-K^->) (49) 



r' bR—\ ^ ' 



22. In the case of a sphere filled with a liquid we have further 

 (by putting R' — in (45) ) : 



{b'R^-ShR + d)ei^ - {b'R'^ UR+^e-bR 



{bR—\)ebR + {bR^-\)e-l> 



If the wave-motion is damped out when arriving at the centre, 



i. e. if I'—^R may be put very small, the value of )■ is given by 



b'R' — SbR + 3 



L-^.-rRhi — (51) 



^ ' bR — l ^ ' 



which is obtained from (25) by reversing the sign of .S; in the same 



1) H. V. Helmholtz und G. v. Piotrowski, Wien. Ber. 40 (2), 607, 1S6(J. H. 

 V. Helmholtz, Wissensch. Abh., 1, p. 172. 



G Zemplén, Ann. d. Phys., 19, 791, 1906 ; 29, 90i2, 1909. 

 Vid. also Lamb, Hydrodynamics, p. 578. 



^ = i-^^i '^^.^.,UM^(bR^.Ue-bR • ^■'*^) 



