Mr Green on the Vibration of Pendulums. 55 



the motion of a pendulum by the action of the surrounding me- 

 dium, we have insisted more particularly on the case where the 

 ellipsoid moves in a right line parallel to one of its axes, and 

 have thence proved, that, in order to obtain the correct time 

 of a pendulum's vibration, it will not be sufficient merely to al- 

 low 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 vacuo, 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 endea- 

 vour to find the correction necessary to reduce the observed length of a pen- 

 dulum vibrating through exceedingly small arcs in any indefinitely extend- 

 ed 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 



*-i=m : {m*m+mt f 



da? 2 dy l dz* ' '' 



Vide Mec. Cel. Liv. iii. Ch. 8. No. 33, where d> is such a function of the 



