PROPERTIES OF PHOSPHOR-BRONZE WIRES 225 



The most striking point to be discussed in connection with the 

 data of these suspensions of different lengths, is that the num- 

 ber of vibrations required for the vibrating systems to fall be- 

 tween given amplitudes, increases ;is the wire is shortened. "We 

 can readily see that when the wire is shortesl the displacement 

 for a certain amplitude per unil length is smallesl in magnitude. 

 Since the periods were kept practically constanl in these tests 

 the angular velocity must have been greater in the longer sus- 

 pensions. 



Let us say that the average velocity varies as the angle of dis- 

 placement and inversely as the period. Since the angle of dis- 

 placement is arbitrarily taken proportional to the length, we 

 have, 



v=kL/T 

 where v represents the mean velocity of the vibrating system. 

 L the length of the wire, T the period and k a constant of the 

 alloy. 



Now assume the friction, both internal and external, to vary 

 as the mean velocity of all the moving parts. 7 Then 



cv = KL/T = f 



So, if we have variable lengths and periods we may say 

 that if the friction is to be the same in all cases, the ratio of 

 the lengths to the corresponding periods should be constant. 

 (This would hold true no matter what assumptions are made 

 in regard to the power of the velocity with which the friction 

 varies.) Or we have 



L/T=LVT' 



Now let A T ' be the number of vibrations between any two am- 

 plitudes per unit length and let N" be the number for another 

 length of the same wire between the same given amplitudes. 



If N is inversely proportional to the friction, 



N'=k/f 

 and X"=k/f" 



or N7N"=f"/F' 



but f=KL/T 



thus N'L'/T'=N"L"/T" 



but since the periods are kept constant T'=T", and 



.V /,' N"L" 

 or N'/N"=L"/L' 



7 The velocity of course Is a continually varying quantity but the integrated 

 value of the velocity over a whole period varies from v by only a constant, 

 which is included in the constant fc. 



15 



