[74] PROFESSOR STOKES, ON THE EFFECT OF THE INTERNAL FRICTION 



the factor k, of which the meaning has been already explained. The value of this factor, as 

 Bessel remarked, will depend upon the form of the body ; but he does not seem, at least in his 

 first memoir, to have contemplated the possibility of its depending on the time of oscillation, and 

 consequently he supposed it to have the same value for the long as for the short pendulum. 

 When the factor k is introduced, the equation obtained from the known difference of length 

 of the two simple pendulums contains two unknown quantities, namely k, and the length of 

 the seconds' pendulum. To obtain a second equation, Bessel made another set of experiments, 

 in which the brass sphere was replaced by an ivory sphere, having as nearly as possible the 

 same diameter. The results obtained with the ivory sphere furnished a second equation, in 

 which k appeared with a much larger coefficient, on account of the lightness of ivory com- 

 pared with brass. The two equations determined the two unknown quantities. 



Let \ be the length of the seconds' pendulum, t u t 2 the times of oscillation of the brass 

 sphere when swung with the short wire and long wire respectively, /„ l 2 the lengths 

 of the corresponding simple pendulums, corrected for everything except the inertia of the 

 air, m the mass of the sphere, m,i the mass of the fluid displaced ; then 



X*, 2 (l + t ^-k)- l = l l ; 

 m 



or, since m, is so small that we may neglect mf, 



X*, 2 (l -—&) = *,. 

 m 



The long pendulum furnishes a similar equation, and the result obtained from the brass 

 sphere is 



X (,t? - t?) (1 - — k) = k ~ h, (156) 



since L — l t is the quantity which is regarded as accurately known. The ivory sphere in 

 like manner furnishes the equation 



. x (<v - <?) (i - ^ *) - r, - r» (157) 



m 



where the accented letters refer to that sphere. The equation for the determination of k 

 results from the elimination of X between the equations (156) and (157). 



Now, according to the theory of this paper, the factor k has really different values for 

 the long and short pendulums. Let k t refer to the short, and k 2 to the long pendulum 

 with the brass sphere, k^ to the short, and k 2 ' to the long pendulum with the ivory 

 sphere. Then 



x#» (i - 2» &,) * /, \t» (i _ ^ k 2 ) = h, 



m m 



and therefore 



h - h = \t 2 * (1 - — k 2 ) - \t? (1 ! kj (158) 



m m 



In the equation resulting from the elimination of X between (156) and (157), let the 



