62 M. C. Cellerier on the Distribution of 



equal quantity is suppressed and created during the time t; 

 this condition, which characterizes the final state, amounts to 

 ^}r(,v) = yjr / (x). The function c})(x) is to be determined so as 

 to satisfy the preceding equation for every value of x. 



The unit of length, at first chosen arbitrarily, must now be 

 reduced to one metre. The alteration thence resulting in the 

 formulas is the same as if we had supposed the enclosure a 

 cubic metre; also, whatever this enclosure maybe, it amounts 

 to taking for N the nnmber of molecules contained in a cubic 

 metre. 



Accessory Conditions. 



The number of molecules of a group being N</>(V)cfo, on 

 adding it for all the groups, N will be found, whence it follows 

 that 



I 



<t>(x)dx=l (15) 



Another datum is furnished by the value of the pressure; 

 this for a portion of side S of 1 square metre has for its value 

 p=Sf, the summation extending to all the molecules of the 

 medium, / being for each its normal action upon S. But, 

 these actions being intermittent, it is preferable to replace it 

 by its mean value during a time t, this being short enough 

 for the passage of a molecule during it to be inconsiderable in 

 comparison with the dimensions of the enclosure. We thus 

 get 



*-H fd> - 



The quantity j"/ dt does not differ from for a mole- 

 cule unless it strikes the side during the time t; and in that 

 case it measures the impulse given by the side to the mole- 

 cule — that is, 2mv, v being the normal component of its velo- 

 city. Consequently 



2m „ 



P=- r 2v, 



the sum 2 extending to all the impacts against the side S 

 during the time t. 



Case I. Let us suppose that all the molecules have velocities 

 h, equal and parallel, carrying them towards the side, and 

 making with the normal to this an acute angle 6. In this 

 case, for a molecule to produce a collision it must at the com- 

 mencement of the time t be within a prism having S for its base, 



