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



long narrow cylinder, placed with its axis horizontal and its middle point in the prolongation 

 of the axis of the vertical copper cylinder. In these experiments, the arcs did not decrease 

 in geometric progression, as would have been the case if the resistance had varied as the 

 velocity; but it was found that the results of observation could be satisfied by supposing the 

 resistance to vary partly as the first power, and partly as the square of the velocity. In 

 Coulomb's notation, 1 : 1 - m denotes the ratio in which the arc of oscillation would be 

 altered after one oscillation, if the part of the resistance varying as the square of the velocity 

 were destroyed. The several experiments performed with the same cylinder were found to be 

 sufficiently satisfied by the formula deduced from the above-mentioned hypothesis respecting 

 the resistance, when suitable numerical values were assigned to two disposable constants m 

 and p, of which p related to the part of the resistance varying as the square of the velocity. 



Conceive the cylinder divided into elementary slices by planes perpendicular to its axis. 

 Let r be the distance of any slice from the middle point, 9 the angle between the actual and 

 the mean positions of the axis, dF that part of the resistance experienced by the slice which 

 varies as the first power of the velocity. Then calculating the resistance as if the element 

 in question belonged to an infinite cylinder moving with the same linear velocity, we have by 

 the formulae of Art. 31 



dF = k'M'n-± , where M' = irptfdr, -i = r — . 

 dt r dt dt 



If G be the moment of the resistance, I the whole length of the cylinder, we have, putting 



n = 7TT -1 , 



G = 



^Jc'paH 3 d9 



12 T dt* 

 whence 



log e (l-m)-'= J 7 (165) 



/ being the moment of inertia. 



Expressing / in terms of the same quantities as in the case of the disk, we get from (147) 

 and (165) 



log, (l-m)-' = log 10 e. 7 ^^.^.^.tn% . . . (166) 



and gp is the weight of a cubic millimetre of water, or the 1000th part of a gramme. The 

 numerical values of p.', T, R, Whave been already given, but p! must be reduced from square 

 inches to square millimetres. The cylinders, of which three were tried in succession, had all the 

 same length, namely, 249 millimetres. Their circumferences, calculated from their weights 

 and expressed in millimetres, were 21.1, 11.2, and 0.87, and the time of four oscillations was 

 92 s , 9l"» 91 s - The values of m calculated from these data by means of the formula (147) are 

 0.4332, 0.2312, and 0.01796. For the first and second of these values, Wk' may be obtained 

 by interpolation from the table given in Part I. ; for the third it will be sufficient to employ 

 the second of the formulae (115). 



