12 Prof. It. Clausius on the Relations between the 



The half of the factor rnc 2 ^ is the mean vis viva of the motion 

 of rotation ; for the component of velocity originated by the ro- 



JQ 



tation is r-rr, and the part of the vis viva corresponding to 



this component is -~ r2 \~Jf) > ^ or which,, according to (7), we 



m 1 /dv\ 



can even put — c 2 — T Just so the half of the factor m \-ri) is 



the mean vis viva of the motion of vibration. Accordingly the 

 equation is symmetric in relation to the motions of rotation and 

 vibration*. Since, according to (9), the sum of the two factors 



1 /dr\ 2 . 



mc 2 -^ and m(j,) is equal to rY'(r), we can even give to the 



preceding equation the following forms, more convenient for 

 many applications : — 



B{f{7)-ir¥{r)]=rF^)B\ogi^m(^ . (16) 



S[l^-irl^]=^F^)Slog« 1 +mc 2 - 2 81ogi . (17) 



r t l 



This equation, given in various forms under (14), (15), (16), 

 and (17), is the expression of a new relation, universally valid 

 for motions about a fixed centre of attraction. 



When the central motion is of such a kind that a constant 



ratio subsists between the period of a rotation and the period of 



t i 



a vibration, the fraction — is invariable, and therefore 6Mo£-r- = 0. 



Thereby the two preceding equations become accordant and are 



* Pursuant to equation (13), we can also form the following equation : — 



W*2*l ffl & / 2 1\ , 2 T * l 



Now, as the expression mc 2 -^j represents a force of attraction equal to the 



m 1 

 centrifugal force, we may regard the quantity — -~ c 2 — 2 , arising from the 



integration of that expression, as the ergal relative to the motion of rota- 



m 1 

 tion. Remembering further that -^ c 2 -^ represents the vis viva of the 



motion of rotation, we see that the preceding equation has precisely the 

 same meaning for the motion of rotation as (10) has for the motion of vi- 

 bration. The sum of the two equations, together with the consideration of 

 (9), gives the preceding equation (15). 



