1032 PROFESSOR TA1T ON THE 



Heuce the number of encounters for which the relative speed is from u to u + du 

 proportional to 



uHuf^i . . (l) 



The limits of v x are v±u, or u±v, according as v>u, and those of v are to <x> , so that 

 the integral is 



The first term of this integral may be written as 



y»oo 



and the second as 







2 



Together, these amount to 







/ 2 7 -2fta?2 / _2Aar2 



/ xdxB +uf axs 



The first term vanishes, and the second is 



u 



2V 2h 



U I IT 



Thus the value of (l) is 



u 3 du -hupp I rr ro . 



HT 6 \/W (2) - 



But, on the same scale, the whole number of encounters in the same time is 



Thus the fraction of the whole encounters, which takes place with relative speed 

 u to u + du, is 



whose integral, from to oo , is 1 as it ought to be. 



§ 59. Now these relative motions are before encounter distributed equally in all 

 directions. Let us deal therefore only with those which are parallel to a given line. 

 The final result will be of the same character relative to all such lines; and therefore 

 the encounters will not disturb the even distribution of directions of motion. 



