6\ ='1^ and 6; = —'-" {/) 



8(52 



at tlie same time and tlie periodic time of the damped vibrations 

 is in botli oases the same muhiple of tlie periodic time of the 

 undamped motion, or diflerently expressed: the reduced periodic 

 time is the same. 



Then it appears that for the similarity of the two motions it is 

 necessary and sufficient tiiat tiie expressions: 



n 



have the same value in both cases; for given R,ii and ii these 

 equalities determine for one of the systems /v and T^ and therefore 

 also M. since : 



K= C\nR' and 7'„ = - ' — , therefore .1/ = 4.-T —-„ — C, ^—, {II) 

 C\ li J,- ft 



where 63 and 6', again stand for equal values in both cases. Each 

 system of values of C\ and C\ defines a state of motion, of which 

 there is thus a doubly infinite series. 



If the oscillating system consisted of nothing but the sphere, we 

 should have K ^ -^^ jtn' R\ fi' being the density of the sphere, and 

 in that case it follows from the required proportionality of K to 

 fi R\ that ((' must be proportional to ji, i.e. the density of the sphere 

 would have to he the same multiple of the density of the liquid in 

 both cases. Seeing that the oscillating system can be more compli- 

 cated, the latter condition does not need to be fulfilled, if only K 

 has the required value. 



4. We may conclude therefore that it is possible to obtain a 

 conform representation of the oscillating movement of a sphere in a 

 liquid, by taking a ditferent sphere in a second liquid ; the radius 

 of the second sphere may even be chosen arbitrarily, but the moment 

 of inertia of the vibrating system and the rotational moment are 

 then completely determined. That there is no similarity in general, 

 is due to the fact, that the motion depends upon five quantities: 

 11 (I, /^/v' and .17, which can all i)e changed independently, whereas 

 by making a suitable choice of the fundamental units for each case 

 only thi-ee of these quantities can be made to assume equal values. 

 If in each case a system of units is chosen such that r^ = 1, fi ^ 1 

 and R =z 1 it does not follow that K und AI have the same value 

 in both systems; the equality of the values of /land J/ expressed 

 in the special units in the two systems is thus the condition to 

 which similarity is tied down. 



