a Medium to small Oscillations. 115 



successive amplitudes of vibration are sensibly in geometric 

 progression. 



2nd. That the pendulum is deprived (so to speak) of a cer- 

 tain portion of its weight by the resistance of the medium, in 

 addition to what it loses by the buoyancy of the medium*. 



From these facts I shall now endeavour to show, that the 

 resistance of the medium must be a linear function of the velo- 

 city, involving not only the velocity itself, but also the differ- 

 ential coefficients of the velocity with respect to the time. In 

 other words, that the resistance is expressed by the formula 



where v is the velocity of the ball, and Cj, Cg, &c. certain 

 constants. Poisson states, as an experimental fact, that the re- 

 sistance to small motions is proportional to the velocity, but 

 this law is manifestly insufficient to account for the loss of 

 weight of the pendulum caused by the resistance (see Poisson, 

 Mecatiique, torn. i. p. 363), and therefore it cannot be the true 

 law. I am not aware whether the law 1 have just stated has 

 been proposed before or notf, but it appears to me to be in 

 complete accordance with the results of experiment, and to be 

 capable of proof, as the following reasoning is intended to 

 show. 



27. Let 9 be the angle which the string of the ball-pendu- 

 lum makes with the vertical at any time /, c the length of the 

 pendulum, c R the resistance tending to check the motion of 

 the ball, —a the value of d at the commencement of any oscil- 

 lation, and a' the value of 9 at the end of that oscillation, g the 

 force of gravity deducting the proper allowance for the buoy- 

 ancy of the medium, and n^ = — . 



Then we find for a small oscillation, 



J = —''-^ ('•) 



which gives 



which, taken between the limits of the oscillation, gives 





(2.) 



[* On this subject some remarks, by Sir John Herschel, will be found in 

 our last Number, p. 53-54. — Edit,] 



t I may observe that, as far as I can judge from a hasty perusal, the me- 

 moirs written upon the oscillations of the ball pendulum seem to confirm 

 the truth of this law. 



