316 ON THE VIBRATION OF PENDULUMS IN FLUID MEDIA. 



correct time of a pendulum's vibration, it will not be sufficient 

 merely to allow for the loss of weight caused by the fluid 

 medium, but that it will likewise be requisite to conceive the 

 density of the body augmented by a quantity proportional to 

 the density of this fluid. The value of the quantity last named 

 when the body of the pendulum is an oblate spheroid vibrating 

 in its equatorial plane, has been completely determined, and, 

 when the spheroid becomes a sphere, is precisely equal to half 

 the density of the surrounding fluid. Hence in this last case 

 we shall have the true time of the pendulum's vibration, if we 

 suppose it to move in vacua, and then simply conceive its mass 

 augmented by half that of an equal volume of the fluid, whilst 

 the moving force with which it is actuated is diminished by the 

 whole weight of the same volume of fluid. 



We will now proceed to consider a particular case of the 

 motion of a non-elastic fluid over a fixed obstacle of ellipsoidal 

 figure, and thence endeavour to find the correction necessary to 

 reduce the observed length of a pendulum vibrating through 

 exceedingly small arcs in any indefinitely extended medium 

 to its true length in vacuo, when the body of the pendulum is 

 a solid ellipsoid. For this purpose we may remark, that the 

 equations of the motion of a homogeneous non-elastic fluid are 



. 



Vide Mfa CeL Liv. in. Ch. 8, No. 33, where < is such a func- 

 tion of the co-ordinates x, y, z of any particle of the fluid mass, 

 and of the time t that the velocities of this particle in the direc- 

 tions of and tending to increase the co-ordinates x } y, and z 



shall always be represented by -p , -p , and - respectively. 



Moreover, p represents the fluid's density, p its pressure, and 

 V a function dependent upon the various forces which act upon 

 the fluid mass. 



