60 Mr Green on the Vibration of Pendulums 
Hence we readily get for the total pressure on the body tending to in- 
crease, & 
zy =f a8(p! —R’) )=4 red ate abe x fasten Pr = 
v representing the volume of the body, 2” the pressure on that side where 
w is positive, p/ the pressure on the opposite side, and ds an element of the 
principal section of the ellipsoid perpendicular to the axis of w. 
If now we substitute for yz its value given from (8.), the last expression 
will become 
(or) 
alic p of. ie 
Y a@oc dx 
oneal. ge FoR 7 Fe (10.) 
—aic af 
@be 
Having thus the total pressure exerted upon the moving body by the sur- 
rounding medium, it will be easy thence to determine the law of its vibra- 
tions when acted upon by an exterior force proportional to the distance of 
its centre from the point of repose. In fact, let p, be the density of the 
body, and, consequently, p,v its mass, g"X’ the exterior force tending to de- 
crease X’. Then, by the principles of dynamics, 
aX’ , 
aarp Fi + gX'— 
If, now, in the formula (10.) we substitute for \ its value drawn from 
(9.), the last equation will become 
oo) 
avef of. i 
abe »tx x 
ic 
zo vef 5he 
which is evidently the same as would be obtained by supposing the vibra- 
0= ak 
tions to take place in vacuo, under the influence of the given exterior force, 
provided the density of the vibrating body were increased from 
