466 On a small Sphere vibrating in a resisting Medium. 



And by integrating in the usual way to obtain the pressure on 



4*7rVar 9 

 the whole sphere, it will be found to be . sin <}> 



\J 



cos (bt $). This is reckoned positive in the direction con- 

 trary to that of the motion of the sphere. Hence if cr = the 

 ratio of the specific gravity of the fluid to that of the sphere, 

 the accelerative force of the resistance in the positive direction 



of the motion is . sin <t> cos (b t <). If A = the di- 



r 



stance to which motion is propagated in the fluid in the time 



of one vibration of the sphere, b = , and consequently, 



A 



tan $ = - This is an exceedingly small quantity. Hence 



A 



T . 2 TT r br 



very approximately sin $ = = , and the accelera- 

 tive force of resistance = V bv cos b t. Again, if x the 

 distance of the centre of the sphere at the time / from the 



A 



mean place about which it is oscillating, = V sin b t, 



(I 



and y-V = Vbcosbt. Hence the accelerative force of the 

 at* 



resistance = <r . -TTJ-* The length of the pendulum being 

 / and the force of gravity g, the accelerative force of gravity, 



taking account of the buoyancy of the fluid, is =- (1 a-). 



I 



Hence, 



and consequently 



d? x gx /I <r\ 



TP '' T ' \T+irf 



This is the result I obtained by my two former methods. 

 As it does not contain a, it is applicable to any resisting me- 

 dium, supposing the vibrations to be slow. Putting the 

 factor in brackets, under the form 1 n <r, we shall have 



2 

 n . For a brass ball of specific gravity 8, vibrating 



in air, n = 2 very nearly; and for the same vibrating in water, 



