322 ON THE YIBEATION OF PENDULUMS IN FLUID MEDIA. 



which is evidently the .same as would be obtained by supposing 

 the vibrations to take place in vacuo, under the influence of the 

 given exterior force, provided the density of the vibrating body 

 were increased from 



-a'b'c 



J a* 



We thus perceive, that besides the retardation caused by the 

 loss of weight which the vibrating body sustains in a fluid, there 

 is a farther retardation due to the action of the fluid itself; and 

 this last is precisely the same as would be produced by aug- 

 menting the density of the body in the proportion just assigned, 

 the moving force remaining unaltered. 



When the body is spherical, we have a V c', and the 

 proportion immediately preceding becomes very simple, for it 

 will then only be requisite to increase p, the density of the body, 



by ^ , or half the density of the fluid, in order to have the 

 correction in question. 



The next case in point of simplicity is where a = c ; for 

 then 





If a > V, or the body is an oblate spheroid vibrating in its 

 equatorial plane, the last quantity properly depends on the 

 circular arcs, and has for value 



I - arc ( tan = V(a *- ft-))} " a- (a* - ft") - , 



If, on the contrary, a < &', or the spheroid is oblong, the 

 value of the same integral is 



V 



