OF FLUIDS ON THE MOTION OF PENDULUMS. [67] 



spheres, Nos. 5, 6, and 7 ; and one 3-inch sphere, No. 66. Nos. 8 and 9 are the same spheres 

 as Nos. 5 and 7 respectively, swung by suspending the wire over a cylinder instead of attach- 

 ing it to a knife-edge apparatus. As this mode of suspension was not found very satisfactory, 

 and the results are marked by Baily as doubtful cases, I shall omit the pendulums Nos. 8 and 

 9, more especially as with reference to the present inquiry they are merely repetitions of Nos. 

 5 and 7. 



In the case of a sphere attached to a fine wire of which the effect is neglected, and swung 

 in an unconfined mass of fluid, we have by the formula? (52) 



■ *■*♦£</£. 



(148) 



2a v 2tt V ' 



2 a being in this case the diameter of the sphere. Before employing this formula in the com- 

 parison of theory and experiment, it will be requisite to consider two corrections, one for the 

 effect of the wire, the other for the effect of the confinement of the air by the sides of the 

 vacuum tube. 



I have already remarked at the end of Section IV., Part I., that the application of the 

 formula? of Section III. to the case of such fine wires as those used in pendulum experiments 

 is not quite safe. Be that as it may, these formulae will at any rate afford us a good estimate 

 of the probable magnitude of the correction. 



Let I be the length, a, the radius, V 1 the volume of the wire, V the volume of the sphere, 

 / the moment of inertia of the pendulum, / that of the air which we may conceive dragged 

 by it, H the sum of the elements of the mass of the pendulum multiplied by their respective 

 vertical distances below the axis of suspension, IT the same for the air displaced, <j the density 

 of the air. Then the length of the isochronous simple pendulum is IH~ l in vacuum, and 

 (/+ /') (H - H')~ l in air, and the time of vibration is increased by the air in the ratio of 

 liH~i to (I + I')i (H — H')~i, or, on account of the smallness of <r, in the ratio of 1 to 

 l + ^ (/'/"' + H'H~ l ) nearly. Now ^H'H~ l is the correction for buoyancy, and there- 

 fore 



n "l = ^7-f (149) 



We have also, if &, be the value of the function k of Section III., Part I., 



/' = kaV(l+ay+%k 1 <TV 1 l\ H' = aV(l + a) + ^aVJ, . . (150) 



and HI' 1 = (/+a) _1 very nearly. Substituting in (149), expanding the denominator, and 

 neglecting V* t we get 



tt-l-ft+A^*,/- ) -l-±k— — , 



•* V \l + a) ■ V l + a 



Now V l is very small compared with V, and it is only by being multiplied by the large factor 

 &! that it becomes important. We may then, without any material error, replace the last term 

 in the above equation by ^V^V~ l P {I + a)~ 2 , and if \ be the length of the isochronous simple 



33—2 



