8 Prof. R. Clausius on the Relations between the 



approaches and recessions is accomplished. If, further, during 

 one approach and recession several revolutions take place, or if, 

 generally, the period of an approach and recession be merely 

 commensurable with the period of a revolution, the point will, 

 certainly not after each revolution, but yet after a certain num- 

 ber of revolutions, come again to the same place and then repeat 

 in like manner the motion just completed, so that a closed path 

 will result from several revolutions. If, on the contrary (which 

 is the most general case), the period of approach and recession 

 be incommensurable with the period of revolution of the radius 

 vector, each successive revolution will take place in another path, 

 and we shall then have no longer to do with a closed path. 



Now, in order to apply my first equation to the central motion 

 of a pointy we can first give the virial a simpler form. If the 

 force acting on the point is an attraction or repulsion proceed- 

 ing from the origin of the coordinates, the intensity of which is 

 represented by the function F'(r), a positive value of the function 

 denoting attraction, and a negative value repulsion, then is 



-|(Xi+Ty + Zjr).= grP(r), 



and hence equation (2a) changes into 



m 



2 2 



,2 — 



rV'(r). (6) 



In this equation the distinction whether the path is closed or 

 not is of no consequence. For its applicability it is merely ne- 

 cessary that the motion be stationary, so that the quantities v 2 

 andrF'(r) shall not continually change in the same direction, 

 but only oscillate within certain limits and hence have definite 

 mean values. 



It is not so with my second equation ; in this occurs the period 

 i of the motion under consideration. If, then, the motion takes 

 place in a closed path and therefore regularly repeats itself in 

 exactly the same manner, we can confine our attention to a single 

 run through this path, and take the time required for it as the 

 quantity i. I have treated of this case in my previous memoirs. 

 But when the motion takes place in a path which is not closed, 

 and therefore does not present any periods repeating themselves 

 in the same manner, the contemplation cannot thus be limited 

 simply to one definite period ; and a further investigation is re- 

 quired in order to ascertain whether and in what way my second 

 equation can be applied to such a case. 



4. For this purpose we will carry a little further the already 

 above intimated division of the total motion into two — the an- 



