16 Prof. R. Clausius on the Relations between the 



the following equation, 



logJ^-J*-^*, .... (29) 

 the result is similar to the preceding : — 



H=^\/- k P~h (30) 



Between the functions J and J„ occurring in these two ex- 

 pressions, there subsists, according to equations (26) and (29), 

 the following relation : — 



^logJ , -..dlogJ 



As, with the aid of this equation, one of the two functions can 

 be deduced from the other, we can say that, in the two expres- 

 sions of i and i v only one undetermined function of p occurs. 



7. In equations (28) and (30), the periods i and i x are ex- 

 pressed by the quantities p and p. According to equation (22), 

 p is a simple function of the energy E, which remains invariable 

 during the whole motion, and hence can be taken as known. It 

 is otherwise with the quantity p. It is true this has a simple 

 signification (the mean vis viva of the motion of rotation as a 

 fraction of the total mean vis viva); but its value cannot be 

 stated so simply, because for the calculation of the mean value 

 of a variable quantity the whole course of the motion must be 

 taken into consideration. Hence it is advisable to introduce 

 instead of p another quantity the value of which is obtained im- 

 mediately from the data usually employed for the determination 

 of the motion. 



These are the energy E and the already discussed quantity c f 

 the half of which represents the area described by the radius 

 vector in the unit of time, and which, as well as the energy, 

 remains constant during the whole motion. We will now bring 

 in a quantity q determined by the following equation, 



2 = m¥^ C p£^E] 2 ~% . . . (32) 



and which, consequently, can be calculated in a very simple 

 manner from E and c. 



In order to find the connexion of this new quantity q with p, 

 we can first, with the aid of equation (22), transform the expres- 

 sion of q into the following : — 



= a A? _£_ 

 V k „»+! 



(33) 



