270-271] VIBRATING SPHERE. 491 



the wave-length are only slightly affected in this way. To find the energy 

 expended per unit time in generating waves in the surrounding medium, we 

 must multiply the frictional term in (x), now regarded as an equation in real 

 quantities, by u, and take the mean value ; this is found to be 



In other words, if p l be the mean density of the sphere, the fraction of its 

 energy which is expended in one period is 



It has been tacitly assumed in the foregoing investigation that the ampli- 

 tude of vibration of the sphere is small compared with the radius. This 

 restriction may however be removed, if we suppose the symbols M, v, w to 

 represent the component velocities of the fluid, not at a fixed point of space, 

 but at a point whose coordinates relative to a system of axes originating at 

 the centre of the sphere, and moving with it, are #, y, z. The only change 

 which this will involve is that we must, in our fundamental equations, replace 



f* b -- 



The additional terms thus introduced are of the second order in the velocities, 

 and may consistently be neglected*. 



271. The theory of such questions as the large-scale oscilla- 

 tions of the earth's atmosphere, where the equilibrium-density 

 cannot be taken to be uniform, has received little attention at the 

 hands of mathematicians. 



Let us suppose that we have a gas in equilibrium under certain 

 constant forces having a potential XI, and let us denote by p and 

 p the values of p and p in this state, these quantities being in 

 general functions of the coordinates x, y, z. We have, then, 



dp = p dl (1). 



The equations of small motion, under the influence (it may be) of 

 disturbing forces having a potential XI', may therefore be written 

 du _ dp p dp, d& 



dv dp p dp dtl' 



*+$;-* 



dw _ _ dp p_dp _ dl' 

 " 



r 1+ T > i ~ Po -T 



dt dz p dz dz 



* Of. Stokes, Camb. Trans., t. ix., p. [50]. The assumption is that the maximum 

 velocity of the sphere is small compared with the velocity of sound. 



