more kinds of Moving Particles among one another. 23 



In integrating this expression, we must remember that N, v, 

 and / are functions of x, not vanishing with x, and of which the 

 variations are very small between the limits x = — nl and 

 x= -f nl. 



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, 



+ Vx m dx=-?-,4-(V rm+2 )' • • • (27) 

 ~ m + 2 ax v ' v ' 



When m is an odd number, the upper sign only is to be con- 

 sidered ; 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, 



t+nl MNvxdx . d ,Z*-*r or, 



I 



1 



i 



+ ~2r^ =: ~'^ (MNmZ) * 



We have now to integrate 



i 



■~^(MNviye-»dn, 



n being taken from to oo. We thus find for the quantity of 

 matter transferred across unit of area by the motion of agitation 

 in unit of time, 



«=-i £ (pf). (28) 



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

 and / the mean length of path. 



Prop. XV. The quantity transferred, in consequence of a mean 

 motion of translation V, would obviously be 



Q= V (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 func- 

 tions of x y then more particles will be thrown into the given 

 stratum from that side on whicli the density is greatest ; and 

 those particles which have greatest velocity will have the great- 

 est effect, so that the stratum will not be generally in equilibrium, 

 and the dynamical measure of the force exerted on the stratum 

 will be the resultant momentum of all the particles which lodge 

 in it during unit of time. We shall first take the case in which 



