characteristic Quantities occurring in Central Motions. 21 

 Herein, for convenience we will put, instead of the fraction -, 



dt-u/l^ , «*> . . (41) 



V a ft+1 ft-fl 



This equation must be integrated in order to obtain the vibra- 

 tion-period. Therein, as limits, two values of x are to be taken 

 for which the expression under the root-sign, and accordingly 



dr 

 the radial velocity -=-, becomes =0. These values may be de- 

 noted by x and x v The integral from one limit to the other 

 gives the time needed by the movable point in order to arrive at 

 the highest from the lowest value; but as we understand by the 

 vibration-period the time employed by the point in attaining 

 from the lowest to the highest value and then again to the low- 

 est, the doable of the above-mentioned must be taken. We thus 

 obtain 



^ (42) 



/ 1— n r*x—x\ 



v k Jx=Xq . / __ 



* 71+1 71+1 



Comparing this expression for i Y with that given in (30), we 

 obtain for the function J x the equation 



*-) ___ e _ (43) 





3 2 



— x* 2 -x n + 3 



n + 1 72 + 1 



10. In order to effect this integration, we will first write the 

 equation thus : — 



v \ n+1 Ti + l y 



Into this we will introduce a new variable, z } determined by the 

 following equation : — 



71+1 7i+l V ; 



If we then imagine the quantity # 2 developed in a series ac- 

 cording to ascending powers of z, putting 



x' 2 = a + a 1 z + « 2 ^ 2 + ^ 3 +&c, . . : . (46 



and so determine the coefficients of this series that they satisfy 

 the preceding equation, we obtain the following values, in which, 



