Equilibrium of Vis Viva, 165 



m + ))i' m f ,__ m + m 1 



■m 



u i = —r—i — P -,u, i'i = v— q . v, 





, m + m! m / _ x 



10/ = -. — x -,w. . (7) 



m m 



The total increase of 2logF + 2log/ by all impacts of the 

 type specified is, therefore, rf«(logF / + log/ f 1 — logF — log/ x ). 

 From this we shall obtain the total increase previously 

 designated by S 12 (2logF + 21og/) if we integrate over all 

 values of the variables whose differentials are contained in dn*. 



This integration may be performed by means of a special 

 device. In conjunction with the above terms which are 

 furnished by the " specified " impacts, let us take the terms 

 furnished by the " opposite " impacts, and thus divide the 

 whole of the collisions into pairs. 



We shall consider an impact to be " opposite " to any 

 u specified " one when the condition of the colliding atoms at 

 the end of either is exactly the same as their condition at the 

 commencement of the other one. The centres of the colliding 

 atoms must, however, be interchanged of course, since they 

 approach each other before impact. The remaining variables 

 x . . . ic' will be included between the same limits for both 

 impacts. In the accompanying figures the largest circle repre- 

 sents a shell, the smallest a nucleus, and the intermediate one 

 a single atom; the arrows drawn through their centres represent 



* If we assume that the second of the impinging- atoms is not a single 

 atom, but a shell, we arrive by exactly analogous reasoning at a similar 

 equation 



28 x 2 log F=jdrc (log F'+log F/— log F -log F x ), 

 in which 



Fi =F(#i, y v Si, <', V, w x ", u v v 1} w x ). 



F/sF^i, y 1} z xi u x ", v x ", <', w/, y/, wj). 



u v v 1} W{, and u x \ v x , w{ are the velocity-components of the second shell 

 before arid after the impact, and must be expressed by equations of 

 similar form to (6) and (7). a 1} y v z v «/', v t ■ ', io x " are the remaining 

 quantities by which the position of the second doublet at the moment of 

 the impact is determined. Finally, 



dn=~D' ' 2 FF \\ edx . . . dw" du dv dw dx\ . . . dw" dp dq dr d\ bt. 



By similar processes of reasoning we should find 



2S 2 2log/=j^(log/'+log/ 1 '-log/- log/,) 



and 



dn=D r2 ff{V€ die dv dw dp dq dr dX dt. 



The meanings of the quautities in these equations will be clear without 

 further explanation. 



