56 M. C. Cellerier on the Distribution of 



Second Disposition. 



The molecules are of two hinds, of ichich the numbers are n, 

 n'. Those of one and the same kind have equal velocities, par- 

 allel and in the same direction, of which the value is u for the 

 first, and v for the second; the angle of the velocities is 9. 



Let us represent these velocities, in magnitude and direc- 

 tion, by OK = u, OB = v; let V, V be the diagonals AB, OG 

 of their parallelogram; we shall have 



V = y/ 1? + v 2 - 2uv cos 0, V'= s/u? + i? + 2uv cos 0. (4) 

 To these let us draw parallels DOE, AD, BE. OB may be 

 regarded as the resultant of two other velocities E, I ; in 

 like manner A is that of D, I. Thus the velocities of 



■> c 



O A 



all the molecules are exactly the same as if the entire mass 

 with its boundary had a velocity of translation represented in 

 magnitude and direction by 1 or ^V', while the relative 

 velocities of all the molecules at the commencement of the 

 time t amounted to D for the first species, to E for the 

 second. As in the first disposition, these velocities are par- 

 allel, in opposite directions, and all equal to ^V. Conse- 

 quently in the relative motion every thing will take place as 

 we have already seen. The total number of the collisions will 

 again be the value of ft given in equation (1); besides, desig- 

 nating by 6' the acute angle made by one of the new veloci- 

 ties with I, there will be, according to equation (3), a num- 

 ber /Jb cos « of collisions, for which 0' > u, a. being an acute 

 angle chosen arbitrarily. 



All these new absolute velocities will be found by com- 

 pounding the preceding ones with I or |V / , which makes the 

 angles 6' and tt — 6' with those resulting from one and the 

 same collision. These absolute velocities will therefore be 



x/iY' + iY ,2 ±±YV'cosd 

 It can be verified from formula (4) that the sum of their 

 squares is w 2 + v 2 , as it ought to be, the vis viva not having 

 changed. 



