ON THE VIBRATION OP PENDULUMS IN FLUID MEDIA. 321 



solid. Then the pressure upon any point on its surface will be 

 had by making /= in the last expression, and is 



df 



Hence we readily get for the total pressure on the body 

 tending to increase x 



df C , dfju f df 



v representing the volume of the body, p " the pressure on that 

 side where x is positive, jt? ' the pressure on the opposite side, 

 and ds an element of the principal section of the ellipsoid per- 

 pendicular to the axis of x. 



If now we substitute for p its value given from (8), the last 

 expression will become 



ofWprTi:, 



Having thus the total pressure exerted upon the moving 

 body by the surrounding medium, it will be easy thence to 

 determine the law of its vibrations when acted upon by an 

 exterior force proportional to the distance of its centre from 

 the point of repose. In fact, let p f be the density of the body, 

 and, consequently, p f v its mass, gX' the exterior force tending 

 to decrease X 1 . Then by the principles of dynamics, 



If, now, in the formula (10) we substitute for X its value 

 drawn from (9), the last equation will become 



