860 



Physics. — "The viscosity of liquefied gases II. On the similarity 

 of the oscillations of spheres in viscous liquids." By Prof. 

 J. E. Verschaffkt.t. Coinin. N°. 148f from the Physioal 

 Laboratory at Leiden. (Communicated by Prof. H. Kamerlingh 



Onnes). 



(Commuoicated in the meeting of October 30, 1915). 



1. When two different spheres are swinging- in two different 

 liquids, the question may be raised, whether the one movement 

 niiglit he a conforui representation of tiie second, that is to say, 

 whether it is possible in each of the two cases to choose the units 

 of length, mass and time such that, quantitatively, the two systems 

 become identical. It is easily seen, that in general this is not possible. 



Indeed it is clear, that the numerical values of quantities of 

 dimension 0, such as: logarithmic decrements per time of swing of 

 the oscillations of the spheres, are not changed by a change of the 

 units. For two states of motion to be "similar", the logarithmic 

 decrements have thus to be equal, which would naturally not be 

 the case in general. Similarly in order that there may be corre- 

 spondence in the two states of motion, the damping of the waves 

 over corresponding distances must be the same in both systems; as 

 the radii of the spheres ars corresponding lengths, the quantity 6'/?, 

 according to the previous communication, would have to be numeric- 

 ally equal in the two systems; this again would not necessarily be 

 the case. In general therefore the two states of motion would not 

 be similar. 



On the other hand, when a detinite state of motion is given, it 

 is possible to produce a similar motion in a different liquid, and 

 we shall now inquire, to what conditions this similarity is subjected. 



2. In the lirst place there must be similarity in the motions of 

 the spheres. These motions are represented by equation (2) of the 

 previous paper: we may also write this formula as follows 



a ^ ae-^' cos 2ytT = ae'-^+'^^''' (real part), 



I 

 where t= ^ i.e. the time measured in the time of swing as unit; 



if we take the time of swing in both cases as the unit of time, the 

 expression no longer contains anything specific, if at a given 

 moment a has the same value in both cases and ö is also equal in 

 the two cases. We can of course arrange the experiments in such 

 a manner, that the first condition is satisfied; we shall see imme- 

 diately, how the second condition may be fulfilled. . 



