861 



In the second place the spherical shells, at corresponding distances, 

 must perform the same motion, reduced to the time of swing as 

 unit, that is in 



71 must have the same value for corresponding values of r in the 

 two systems. Seeing that the radii of the spheres are corresponding 

 values of r, we shall tind all corresponding values of ?• by taking 



eipial multiples of R. Calling q = — the reduced distance from the 



R 



centre, the function ii, reduced to the time of swing, must be the 



same for reduced distances. For an unbounded liquid we have 



according to (15) 



Q' /:* + ! 

 where ^=^hR; it appears, therefore, in order that this expression 

 may not contain any specific quantity, that the quantities b must 



be such in ihe two cases that O^R^ = h,R.^. As 6 = 1/ - k, it 

 follows that ( — (i-\-2ni) must have tlie same value in both cases, 



in other words must have the same numerical value. 



In order therefore that similarity may exist between the two states 

 of motion, R and T cannot be chosen at Avill -. the radii of the 

 spheres being given, at least one of the spheres must have a pre- 

 scribed time of swing and in order to obtain this value the moment 

 of inertia of the sphere and the rotational moment are at our disposal. 

 As we shall see, both these quantities are thereby completely determined. 



3. The motion of the sphere is determined by equation (26), which 



we may write in the form 



LT M 



K ^ K 



If kT =^ — ff -|- 2.T is to have the same value in both cases, 



LT M T' 



the quantities — - and — T'' or ~^^ have to be equal. Owing to the 



,. LT ^ R'liT 



equality ofbR, the equality of^ involves on account of (25) that of — — -; 



T' 

 according to (28) the condition of the equality of —^ is then satisfied 



