404 Prof. Stokes on the Communication of Vibration 



geneous gas, which is at rest except in so far as it is set in 

 motion by the sphere ; and let it be required to determine the 

 motion of the gas in terms of that of the sphere supposed given. 

 We may evidently for the purposes of the present problem sup- 

 pose the gas not to be subject to the action of external forces. 



Let the gas be referred to the rectangular axes of x, y, z, and 

 let u, v, iv be the components of the velocity. Since the gas is 

 at rest except as to the disturbance communicated to it from the 

 sphere, u, v, w are by a well-known theorem the partial differen- 

 tial coefficients with respect to x, y, z of a function <£ of the co- 

 ordinates; and if a 2 be the constant expressing the ratio of the 

 small variations of pressure to the corresponding small variations 

 of density, we must have 



d*$_M d*<f> d*$\. () 



~W- a xax* *"df ^r ' ■"■ ' K) 



and if s be the small condensation, 



S ~ a 2 dt ' 



As we have to deal with a sphere, it will be convenient to refer 

 the gas to polar coordinates r, 6, w, the origin being in the 

 centre. In terms of these coordinates, equation (1) becomes 



d 2 cf>_ Jd*<f> 2d4 _J d_f • a#\ _1 dty\ , 9 x 



dt 2 ~ U { dr*+r dr + r 2 sin d0Y mU d6/ + r*sm*6d to J' ^ 



and if u\ v f , iv' be the components of the velocity along the 

 radius vector and in two directions perpendicular to the radius 

 vector, the first in and the second perpendicular to the plane in 

 which 6 is measured, 



U dr' % rd6' W ~rsm0dco' ' ' [6) 



Let c be the radius of the sphere, and V the velocity of any 

 point of its surface resolved in a direction normal to the surface, 

 V being a given function of t, } and co ; then we must have 



-^=V when r = c (4) 



Another condition, arising from what takes place at a great 

 distance from the sphere, will be considered presently. 



The sphere vibrating under the action of its elastic forces, its 

 motion will be periodic, expressed so far as the time is concerned 

 partly by the sine and partly by the cosine of an angle propor- 

 tional to the time, suppose mat. Actually the vibrations will 



