BY SMALL OBSTACLES OF CYLINDRICAL AND SPHERICAL FORM. 269 



Note, April 12th, 1910. 



Prof. LARMOR has kindly pointed out to me that it is not legitimate to apply the 

 formula (4) to fogs without further consideration. Although the inertia of the water 

 particles is so much greater than that of the surrounding air, yet in consequence of 

 the viscosity of the air it does not follow that we may regard the water particles as 

 approximately fixed. I have investigated the problem of a free spherical obstacle. 

 The analysis is very similar to that in the problem of the fixed obstacle. The 

 secondary waves diverging from the obstacle are affected only in the terms containing 

 spherical harmonics of the first order. If \Je lvt is the velocity of the obstacle along 

 the axis of x we obtain 



' (ka) = -Jnfa(ha) + U ...... (1), 



a il la (ha) ..... (2), 



together with 



dS .......... (3) 



where p rx is the component of the stress across the surface of the obstacle in the 

 direction of the axis of x and the integration is taken over the surface of the obstacle. 

 The last equation reduces to 



}] ....... (4) 



where p l is the density of the obstacle. 



Eliminating A, and Bj between the equations (1), (2), and (4) we obtain approxi- 

 mately when ka is small 



Pi J Pi 



Hence, if L be the ratio of the amplitude of the motion of the obstacle to that of 

 the waves of sound, we have 



V /2 ........ (5). 



Again, eliminating Bj and U from (l), (2), and (4) we obtain approximately 



Al = - * ^W/ 2 (*) ["/ (*a)-f 2 



Pi 



Hence we obtain without difficulty when ka is small 



- 4 - 4 }- 1 ....... (6), 



where [A^, [AJo denote the real part of the value of A in the case of the fixed and 

 free obstacle. 



