326 Prof. R. Clausius on the Application of a 



to the length of the path ; and since in our case the path is a 

 circle with the radius r, we obtain 

 \=27rr, 

 or 



X 



This, inserted in equation (12) for r, gives 



Kl)=™ < 13) 



Hereby the function /(X) is determined. By differentiation ac- 

 cording to X we obtain further : — 



For simplicity, we will now introduce the new sign p with the 

 signification 



p=b < 15 ) 



we then obtain 



/(X)=F(p), ...... (16) 



/'W = 2^'W, ..... (17) 



Xf'(X)= P ¥(p) (18) 



Applying this to equation (11), which has taken the place of 

 (6), and to equations (7), (8), and (9), we arrive, for the case 

 in which the effective force is an attraction proceeding from a 

 fixed centre and represented by a function of the distance, at the 

 following equations : — 



W) = ?(P), (19) 



|p=|pF'M, (20) 



; = 27ri/j^-, .... (21) 

 V F(p) v ' 



B=¥(p) +*/>*&»), . . . (22) 

 wherein all the functions are known. 



As a still more special case, we will assume that the attractive 

 force is proportional to any positive or negative power of the 

 distance j but we will except the minus first power, which, in the 

 integration, leads to the logarithm, and hence it will be better 

 to treat it specially. We therefore put 



F(r) = Ar» (23) 



