vibrating in a resisting Medium. 4-65 



fixed or a moving centre may be deduced. I will not stop 

 to make the deduction, which presents no difficulty, but at 

 once employ the equations given in the Treatises on Hydro- 

 dynamics for motion propagated from a. fixed centre. (Pro- 

 pagation towards the centre is excluded by the nature of the 

 question.) These equations are (putting 1 + s for p), 



f'(r-at] f(r-at). c 

 J v -' J - - 



u = 



, . . 

 nic 



as they contain arbitrary functions, apply immediately to the 

 arbitrary disturbance given to the fluid. In the problem be- 

 fore us they apply, therefore, to the motion given to the fluid 

 by the vibrating sphere at its surface. For as the sphere is 

 supposed to be perfectly smooth and consequently to impress 

 motion only in a direction normal to its surface, the motion 

 at the surface is plainly directed to or from a moving centre. 

 The arbitrary condition of the motion is that at a given di- 

 stance (r), equal to the radius of the sphere from the centre 

 regarded as fixed, and at a given point of the surface of the 

 sphere, the velocity impressed follows either exactly or very 

 approximately the law of a vibrating pendulum. Let the ve- 

 locity of the centre of the sphere at any time t be V sin b t. 

 Then for any point the radius to which makes an angle 6 

 with the direction of the motion, we shall have the normal 

 velocity v equal to V cos 6 sin b t. Hence, putting for brevity 

 u = f(rat], and substituting in the equation (2.), it will be 



found that -7- -\ -- u + V a r cos 6 sin b t = 0, 

 dt r 



an equation in which u and t are the only variables, and 

 which is true whatever be t. The integral of this equation is 



_ at 

 u = Ce 7~ V r 2 cos 6 cos <J> sin (b <$), 



tan $ being put for - . The term involving C will be in- 

 sensible for all but very small values of , on account of the 



factor e -"T> and may therefore be omitted. Hence, by dif- 

 ferentiating and putting a tan $ for b r, 



12 = V a r cos d sin 4> cos (b t- <f>). 

 dt 

 Now the pressure at the point of the sphere we are consi- 



f(rat) 

 dering is equal to o 2 s, or by equation (3.) a . - - - , 



_ .--. . Hence this pressure is V a cos & sin $ cos (Z>/ 



/* U t 



Phil. Mag. S. 3. Vol. 17. No. 112. Dec. 1810. '2 H 



or 



