14 Prof. R. Clausius on the Relations between the 



and consequently, employing the quantity p, 



from which it follows that 



^-L%^3) E J (23) 



After substituting, with the aid of this quantity p, for the mean 

 value of a power, represented by r n+l , a simple power p n+1 , we 

 can forthwith carry out the variation of it, and write 



8r"+*=(n+l)p n 8p. 



Accordingly, by the introduction of p, equation (20) changes 

 into 



^^kp n 8p=kp n + l 8logi-m(^8\ogl-. . . (23) 



Into this equation we will introduce a second simplifying 

 quantity, p ; for as the mean vis viva of the rotatory motion and 

 the mean vis viva of the radial vibratory motion together make 

 up the total mean vis viva, we can represent the two former as 

 fractions of the latter, which fractions shall be denoted by p and 

 I- p. Thus 



m 1 k 

 2 C ?^2 P 



_L_ y • • BB (24) 



m /dr\ 2 ,, x k m 



-Ait)^ l -php 



The value of p may vary between and 1 ; and on it depends 

 which out of all the forms possible with a given value of n the 

 path takes. When jo = 0, the point moves in a right line to and 

 from the centre ; and whenjo = l, it moves in a circle round the 

 centre. Between these two limits lie all the other possible forms. 



By introducing p into equation (23) we obtain first 



i^p kp n 8p = kp n+1 8 log i- (1 -p)kp n +>8 log I ; 



and this equation, divided by kp n+1 , gives 



1 ~~~ n 2 



-^-8hgp=8logi-(l-p)8\og^. . . . (25) 



As two of the terms of this equation are complete variations, 

 the third term, 



(1— p)81og — 3 



»+i 



