Equilibrium of Vis Viva. 



1G7 



2S 12 (2 log F + 2 log/), because we count every impact twice ; 

 once as a " specified/' and the second time as an u opposite," 

 impact. As V, e, and dX are not changed by the impact, we 

 have 



dn'=Jf-Wft ( m + 3 m ') 3 Ve dx . . . div" did dv' dw' dp dq dr dX St. 



It can easily be shown (most simply by a geometrical 

 proof) that du' dv' dw' = du do dw, and hence 



S 12 (5 log F + 2 log/) = | j' (log F + log// - log F -log/,) 



x(F/x-F/,') 



(m -\-m'f 



YeWdu dv dw dp dq dr dz...dw" 8k. (8) 



Similarly it can be shown that 

 8 1 21ogF = 2a«j(logF' + logF/-logF-logF 1 )(FF I -F'F 1 ') 

 X Ye D l,2 dx . . . dio n dx l . . . dw" du dv dw dp dq dr d\ 



S 2 Slog/=28«j-(log/' + log/ 1 '-log/-log/ 1 )(// 1 -///) 



x Ve D' 2 du dv dw dp dq dr dX, . . . . (9) 



in which the variables are the same as in the footnote on 

 page 165. 



We shall confine our attention to the consideration of the 

 stationary condition, in which F and / are at all times the 

 same functions of the variables contained in them. In this 

 case no causes except those already considered can effect 

 changes in XlogF and Slog/; and the total change in E 

 during the time St is therefore 



SE = S l S (S log F + 2 log/) + Si2 log F + S 2 2 log/. 



Since everything (and therefore E) remains unaltered, SE 

 must be zero. But in the integrals of equations (8) and (9) 

 the two factors in brackets have necessarily opposite signs, 

 while the remaining quantities are essentially positive. The 

 expression to be integrated is therefore necessarily negative, 

 and the sum of the integrals of which 6E is composed can 

 only vanish if at each impact 



Fyi'=F/i, F'F/=FF„ /'/,'=//;. . . . (10) 



Let us now consider the most simple case, namely, that in 

 which the shells always pass each other without impinging, 

 and similarly with the single atoms ; but between a shell and 

 a single atom let there be always an impact if they approach 

 within distance I) of each other. We have then only the 

 first of the equations (10), but it holds good for every possible 



