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



amplitudes of oscillation decreased in geometric progression; secondly, that with different 

 disks the moment of the resisting force was proportional to the fourth power of the radius. 

 From these laws Coulomb concluded that each small element of any one of the disks expe- 

 rienced a resistance varying as the area of the element multiplied by its linear velocity. It 

 should be observed that Coulomb was only authorized by his experiments to assert this law to 

 be true in the case of oscillations of given period, inasmuch as the time of oscillation was 

 nearly the same in all the experiments. 



Let a be the radius of the disk in the fluid, r the time of oscillation, the angular dis- 

 placement of the disk, measured from its mean position, i" the moment of inertia of the whole 

 system ; and let 1 : 1 — m be the ratio in which the arc of oscillation is diminished in one 

 oscillation. According to the formula (15) we have 



e -»/3< 



for the factor which expresses the ratio of the arc of oscillation at the end of the time t to the 

 initial arc. At the end of one oscillation t = t, and the value of the above factor is 1 — m, 

 which is given by observation. Putting for /3 its value, in which My° = I, and nr = ir, 

 we get 



lo&a-^-^V^ (162) 



Let T be the time of oscillation, and I the moment of inertia, when the under disk is 

 removed : then /= I t* T~ 8 . Also if M be the mass and R the radius of the large graduated 

 disk, we have i~ = ^ MR' 2 , neglecting, as Coulomb did, the rotatory inertia of the copper cylin- 

 der. Substituting in (162), we get 



log e (l -m)- 1 = 2-lTip f i'lT-lT 2 a i R-' ! M- 1 (163) 



Let W be the weight of the disk in grammes. Then the mass of the disk is equal to that of 

 W cubic centimetres or 1000 W cubic millimetres of water. Hence M => 1000 p W, a milli- 

 metre being the unit of length. Substituting in (163), and solving with respect to \Ztx', 



we get 



vV = 1000 x 2* log, 10 . Tr-i WR? T- 2 a- 4 T* log 10 (l - to)" 1 , . . ( 164) 



and the same value of -y/V ought to result from different experiments. 



The weight of the disk is stated to have been 1003 grammes, and its diameter 271 milli- 

 metres, and it made 4 oscillations in 91 seconds. Hence 1^=1003, R =» 135-5, T=2275. 

 The last three factors in (164) vary from one experiment to another. After making experi- 

 ments with three disks of different radii attached to the copper cylinder, Coulomb made 

 another set with nothing attached, for the purpose of eliminating the effect of the imperfect 

 elasticity of the wire. The following table contains the data furnished by experiment, together 

 with the value of \Z/i deduced from the several experiments. The latter is reduced to the 

 decimal of an English inch, by including 2*5952 (the logarithm of the ratio of a millimetre to 

 an inch) in the logarithm of the constant part of the 2nd member of equation (l64). 



