characteristic Quantities occurring in Central Motions. 11 

 This being put in (10) gives 



or, differently arranged, 



SF(r) — J6>F'(r) = — mc 2 8^ — m-^cBc + m (—) Slogs',, 

 for which we may write 



S[P>)-i^^)] = -mcS^)+m(^ySlogz r . (12) 



5. We will now turn to the motion of rotation. 



In such a motion, in which the movable point need not have 

 the same distance from the centre at the end as at the commence- 

 ment of a rotation, the period of a rotation of the radius vector 

 also need not be exactly the same for several successive rotations ; 

 yet at all events for a greater number of rotations we shall obtain 

 a definite mean value of the period of a rotation. To this we 

 will refer the symbol i. Now, pursuant to equation (7), we can 

 write 



d6 = c^dt ; 



and when we integrate this equation for a whole number n of 

 rotations, consequently between the limits = and = n.27r, 

 there comes 



C ni l T 



n.27r = c\ -dt = c- ) ni, 

 Jo r 2 - r l 

 and consequently 



'l-T <>»> 



Putting this value of c-^ in equation (12), we obtain 



1 /dr\ 2 



S[F(r)-irF(r) = -27nwcS^+m(^J Slogi,. . (14) 



To this equation we can give a more symmetrical form ; for the 

 first term on the right-hand side can, in consideration of equa- 

 tion (13), be transformed thus : — 



— 2irmch - = —mc 1 -^ ih - =mc% 8 los: i. 

 i r i r 



Thereby the preceding equation is changed into 



8[F(r)-irF(r)]=mc 2 ^Slog2 + m^Jsiogf 1 . . (15) 



