Intelligence and Miscellaneous Articles. 159 



If L be the mean vis viva of the atom M, this gives for the mean 

 vis viva L— |(MA 2 ) of its oscillatiDg motion the value 3M/4A. 

 The mean vis viva, however, of an atom m is equal to 3/4&. In 

 order to find the dependence of L on A, the author seeks the pro- 

 bability ^cTU 1 dV x dWj that, if the atom M before the impact had 

 the velocity- components U, V, "W, the resulting impact is exactly 

 such that the velocitv-components after it lie between the limits 

 Uj and U T + dU v Y\ and V, + dY v W 1 and ¥, _+ dW v Consider- 

 ing first an impact occurring upon any surface-element of the 

 sphere, aud then summing all the collisions resulting upon all the 

 surface-elements so that the condition mentioned is fulfilled, he finds 



odu, d\\ dw= (^) ww y^v 



,- ** rM±!ii o + u(TJ-n)+V(Y 1 -v)+-w(w 1 -Ay) -i 2 



L 2m ^ Q J 



«U, dV, tfW, 



in which R is the radius of the atom M, 



Q= + >/(U-Y?+(Y-Yf + (\V-YxY. 



Now, as the atom M in the course of the time-unit collides FN 

 times with m so that before the impact the velocity-components of 

 the former lie between the limits U and \J+dU, Y and Y+dY, 

 "W and "W + c?W without any further conditiou, 



~E9tedU 1 dV l dW 1 



is the number of impacts which M during the unit of time suffers 

 in such wise that the velocitv-components before the impact lying 

 between the limits U and ~U+dU. V and Y+dY, Y\ T and W+dW, 

 after the impact lie between the limits TTj and JJ 1 + d'U v Y 1 and 

 Y x + dY v "W^and V^^. If we integrate over all the differ- 

 entials contained in F, we obtain the number of impacts after 

 which the velocitv-components lie between the limits Uj and 

 JJ 1 + dJJ v Y 1 and Y, + dY v W, and "V^ + rfW,^ without any other 

 condition : and since this must be equal to the number of impacts 

 at which the velocity-components before the impact lie between 

 the same limits, without any condition for those after the impact, 

 we get 



F^^tnJ, dY 1 d\Y } ( * f I ""F^*. 



Jo Jo Jo 



If in this equation, which holds for all values of U lS V l3 W 13 we 

 assign to each of these quantities the value 0, we get, after carry- 



