394 ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 



As we may have occasion to perform similar integrations, we may state 

 here, to save trouble, that if U and r are functions of x not vanishing with x, 

 whose variations are very small between the limits x = + r and x = r, 



< 27 >- 



When m is an odd number, the upper sign only is to be considered ; 

 when m is even or zero, the upper sign is to be taken with positive values 

 of x, and the lower with negative values. Applying this to the case before us, 



f +* MNv 

 - ~2T 



We have now to integrate 



r-i 





n being taken from to <x> . We thus find for the quantity of matter trans- 

 ferred across unit of area by the motion of agitation in unit of time, 



where p = MN is the density, v the mean velocity of agitation, and I the mean 

 length of path. 



PROP. XV. The quantity transferred, in consequence of a mean motion of 

 translation V, would obviously be 



Q=Vp ......................... ........... (29). 



PROP. XVI. To find the resultant dynamical effect of all the collisions 

 which take place in a given stratum. 



Suppose the density and velocity of the particles to be functions of x, 

 then more particles will be thrown into the given stratum from that side 

 on which the density is greatest ; and those particles which have greatest 

 velocity will have the greatest effect, so that the stratum will not be generally 



