characteristic Quantities occurring in Central Motions. 25 

 The factor preceding the series in the expression for J,, viz. 



. , which for n= — 3 is infinitely large, and for values of 



v»-f3 



n below —3 becomes imaginary, shows very clearly that, for 

 attractive forces proportional to a negative power of the distance, 

 the minus third power constitutes the limit to which stationary 

 motions are in general possible. 



Further, it is characteristic of both expressions that all the 

 terms which contain s have the common factor /it. From this it 

 follows that, when /^ = 0, both expressions become independent 

 of the quantity s, equations (53) and (57) changing into 



J,= . and J=l. 



\/w + 3 



Now, since according to (47) 



(n + 2)(n-l) 



/i ~ ^+3™"' 



this simple behaviour occurs in the two cases in which n has the 

 values — 2 and 1. 



These two values of n are the only ones with which the mo- 

 tions universally (that is, for all values of s) take place in closed 

 paths, That is to say, this can only be the case when the ratio 

 between the vibration-period and the rotation-period is invari- 



i .J 



able, and hence 4, which is identical with ~ } must be indepen- 

 % J 



dent of s. Thence it follows further that in the series under (55) 



the coefficients of all the powers of s, from the first onward, must 



be =0, so that the constant part alone remains — which only 



happens when yit = and consequently n is either = — 2 or =1. 



i 1 



For fi=0, the fraction 4 is represented by the formula . , 

 1 vm + 3 



which, according as n= — 2 or n = 1 } takes the value 1 or J. 

 By this is expressed that, in the former case, during a rotation 

 one radial vibration takes place, so that the radius vector has 

 one maximum and one minimum — while in the latter case there 

 are two radial vibrations,, so that the radius vector has two 

 maxima and two minima. 



All the preceding considerations relate to the motions of a 

 material point about a fixed centre. Considerations altogether 

 similar might be instituted in -relation to the motions of two 

 material points about each other, and would lead to correspond- 

 ing results. In my previous memoirs I have carried out this 

 extension for the case in which the motions take place in closed 

 paths ; and with motions in paths not closed the extension can 

 be made in substantially the same manner. 



