characteristic Quantities occurring in Central Motions. 15 



must likewise be such, whence it follows that jt? is a function of 



z z z 



— alone, or vice versa — is a function of p alone. Regarding — 



z » z i ... ?1 



as a function of p, we can from this function derive others; and 



we will introduce one, denoted by J, which shall be determined 



by the following equation : — 



Then is 



(1—^)8 log 4- = 8 log J; 

 h 



and therefore equation (25) changes into 



^p81ogp = 81ogi-81ogJ. . . . (27) 



By transposition and integration of this equation we obtain 



1— n 



l°g * = — o — ^°S f> + l°g J + const. 



It is immaterial which value we attribute to the integration - 

 constant, since, in accordance with (26), we can suppose any 

 additive constant we please to be likewise contained in log J. 



The value most suitable for what follows is log 27r\/ rri . If we 



V fc 



put this quantity in the place of the constant, and then combine 

 the three logarithms of the right-hand side, we get 



logf=log(27r,y/fp 2 j); 



and from this we obtain, for the rotation-period z, the following 

 simple expression : — 



z = 2tt^/%Vj m (28) 



We can deduce a corresponding expression for the oscillation- 

 period ij. Equation (25), namely, can also be written in the 

 following form : — 



\-n 



5 log p = S log i, +pS log - • 



Introducing into this the function Jj, which is determined by 



