66 M. C. Cellerier on the Distribution of 



we get 



F'(m) = ^« 2 -* 2 ) {*tfdy e~W, \=e~^ Cu du F"(«), 

 Jo Jo 



putting 



V'ii^ifdye-W. 

 Jo 

 The indefinite partial integral is 



\ef>* = \ u*W(u) -i j vfie-^du ; 



and between the limits there results 



On substituting in the above value of ^'(x) -we have 



Finally, partial indefinite integration gives 

 2/3 + 2/3, 



^^ 2 +Aj^-^^ 



or, between the limits, 



, X a x 2 3 



■whence results identically 



^ r '(x) = 0, and consequently ^r(«)=^ / (a?). 

 Formula (19) therefore well represents the law of the velo- 

 cities; and a, /3 will be found on substituting it in equations 

 (15) and (16), which gives 



J 00 r*w 



j?e-P"da:=l, *\ ^V^ 2 = A 2 ; 

 Jo 



it will, however, be simpler to employ the number A, putting 

 A 2 = |/i /2 , or &'=7*v/§=0-82A, 

 The known integral 



gives, on replacing x by x\/ j3 and differentiating twice with 

 respect to /3, 



fV*y*=-^L fV^^ito-^, C*0-i*dMmb/?. 



Jo 2*//3 Jo 4/3f Jo *£§' 



the above relations thus become 



U\/tT 1 3« \/*7T "7/91 /D -^ a ' 



~ *> 00, /7s =3*» whence /3= ^- 2 , « = p, 



W/3 8/3V/3 



