

60 Mr Green on the Vibration of Pendulums 



Hence we readily get for the total pressure on the body tending to in- 

 crease, x 



00 00 



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

 x is positive, pj the pressure on the opposite side, and ds an element of the 

 principal section of the ellipsoid perpendicular to the axis of x. 



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

 will become 



00 



a'b'd p v fJ£- 



J a?bc d\ 



p = *-b — > -ai (io.) 



2 — a'b'd fAf 



J aPbc 



o 



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 t be the density of the 

 body, and, consequently, p t v its mass, g~K' the exterior force tending to de- 

 crease X'. Then, by the principles of dynamics, 



If, now, in the formula (10.) we substitute for \ its value drawn from 

 (9-), the last equation will become 



a'Vcf M- 



J a 5 bc \ d 2 X.' 



°=\p,+ 2 — w—p h^r+s- x 



J a°oc 



df 



which is evidently the same as would be obtained by supposing the vibra- 

 tions to take place in vacuo, under the influence of the given exterior force, 

 provided the density of the vibrating body were increased from 



