OF FLUIDS ON THE MOTION OF PENDULUMS. [13] 



a sphere, and presently found that the corresponding differential equation admitted of integra- 

 tion in finite terms, so that the solution of the problem could be completely effected. The 

 result, I found, agreed very well with Baily's experiments, when the numerical value of a 

 certain constant was properly assumed ; but the subject was laid aside for some time. Having 

 afterwards attacked a definite integral to which Mr Airy had been led in considering the 

 theory of the illumination in the neighbourhood of a caustic, I found that the method which 

 I had employed in the case of this integral would apply to the problem of the resistance to a 

 cylinder, and it enabled me to get over the difficulty with which I had before beeen baffled. 

 I immediately completed the numerical calculation, so far as was requisite to compare the 

 formulae with Baily's experiments on cylindrical rods, and found a remarkably close agreement 

 between theory and observation. These results were mentioned at the meeting of the British 

 Association at Swansea in 1848, and are briefly described in the volume of reports for that 

 year. 



The present paper is chiefly devoted to the solution of the problem in the two cases of 

 a sphere and of a long cylinder, and to a comparison of the results with the experiments of 

 Baily and others. Expressions are deduced for the effect of a fluid both on the time and on 

 the arc of vibration of a pendulum consisting either of a sphere, or of a cylindrical rod, or of a 

 combination of a sphere and a rod. These expressions contain only one disposable constant, 

 which has a very simple physical meaning, and which I propose to call the index of friction 

 of the fluid. This constant we may conceive determined by one observation, giving the effect 

 of the fluid either on the time or on the arc of vibration of any one pendulum of one of the 

 above forms, and then the theory ought to predict the effect both on the time and on the 

 arc of vibration of all such pendulums. The agreement of theory with the experiments of 

 Baily on the time of vibration is remarkably close. Even the rate of decrease of the arc of 

 vibration, which it formed no part of Baily's object to observe, except so far as was necessary 

 for making the small correction for reduction to indefinitely small vibrations, agrees with the 

 result calculated from theory as nearly as could reasonably be expected under the circum- 

 stances. 



It follows from theory that with a given sphere or cylindrical rod the factor n increases 

 with the time of vibration. This accounts in a good measure for the circumstance that Bessel 

 obtained so large a value of k for air, as is shewn at length in the present paper ; though it 

 unquestionably arose in a great degree from the increase of resistance due to the close prox- 

 imity of a rigid plane to the swinging ball. 



I have deduced the value of the index of friction of water from some experiments of Cou- 

 lomb's on the decrement of the arc of oscillation of disks, oscillating in water in their own 

 plane by the torsion of a wire. When the numerical value thus obtained is substituted in 

 the expression for the time of vibration of a sphere, the result agrees almost exactly with 

 Bessel's experiments with a sphere swung in water. 



The present paper contains one or two applications of the theory of internal friction to 

 problems which are of some interest, but which do not relate to pendulums. The resistance 

 to a sphere moving uniformly in a fluid may be obtained as a limiting case of the resistance to 

 a ball pendulum, provided the circumstances be such that the square of the velocity may be 



