characteristic Quantities occurring in Central Motions. 23 



their values derived from (48), taking into account only the 

 upper of the two signs which stand before the roots, as the lower 

 sign would give a negative period. Thereby we obtain 



+***^£F^*-*r+*l- . (5i) 



Hereby the quantity Jj as a function of q is so far determined 

 that, for values of q not deviating too much from unity, it can 

 be calculated with tolerable approximateness. 



From this series, with the aid of (38), the corresponding series 

 for J can be derived. 



11. This derivation and other calculations become somewhat 

 easier when the series is developed according to ascending powers 

 of 1 — q. We will represent the latter difference by a special 

 letter, putting 



s=l-q (52) 



Then 



l-q 2 =-{l-s) 2 = 2s-s*; 



and the insertion of this value changes the above series into 



J ~\AT3L i+ 2 2 .3 5+ 2«.3* S 



2 4 .3 3 .5+2 2 .3 2 .31/ i +13V, i „ 1 ,_ 

 -fi 28.3S.5 -**+&(!.]. (53) 



In order from this to calculate J, we can write equation (38) 

 in the following form : — 



d /J,\ 1 dJ 



( J f)=i^S <"> 



Putting herein for Jj the preceding expression, we obtain first 

 the differential coefficient of -j, and then, by integration, -~ 



itself. The added integration-constants can be readily deter- 

 mined, because for 5 = 0, consequently for the circular motion, 

 the rotation-period i, and accordingly the value of the function 

 J, can be directly determined. It follows, namely, that for this 



J, 1 



case J is to be put =1, whence -^ = , Taking 



J v^ + 3 



into ac 



