846 



conducted are such, that ff is a small number, of the order of 

 magnitude 0,1; in that case the expressions (18) can be developed 

 into series progressing according to the ascending powers of ■/ := 



In 



which leads to; 



-{/% 



/'=[/|(i + ix + ix^ + 



èx + ix'^ +•■•)] 



) 



so that 



=l/i[<'+^'-<'-'l+<'+4+ 



(19) 



(20) 



7. As mentioned above in section 1, the real part of (8) may in 

 general be written in the form 



«, = et'(-A'r [X, COS {k"t—h"r) + F, sin {k't—b"r)\ 



_)_ gt'(+iv [A', (;os(F<4-è"r) + F, «m(^"^4-6'V)J, . . (21) 



where A'j, .Y^, }", and Y^ are again functions of r, but now real 

 quantities. This form shows, that the motion of the liquid is the 

 result of the propagation of two waves, the one moving away from 

 the oscillating sphere, the other moving towards the sphere ; writing 



li^'t ± b"r in the form 

 to be 



t ± 



the speed of propagation appears 



2jr 



(22) 



this velocity therefore depends not on the specitlc properties of the 

 liquid only, but in addition on the time of swing of the sphere. 



The wave-length is I 



h" 



For <i very small we have by (19'), 



V- 



^K^-^'-l/v 



(22') 



When the liquid extends to infinity (practically), we have only to 

 deal with the former of the two waves : but when the liquid is 

 bounded, the wave which is emitted by the oscillating sphere is 

 reflected on the fixed wall, in such a manner that the phase is 

 reversed, and tiiereby the amplitude u becomes zero at the wall. 



In addition the waves undergo a damping effect during propagation, 



