Probability Distribution of ol Particles. 705 



and one in the interval dt ; the chance that this may occur is' 

 \dtW n (t). Hence , : , 



W n+1 {t + dt) = (l-\dt)W n+1 (t) + \dtW n (t). 



Proceeding to the limit, we have 



Putting n = 0, 1, 2 . . . in succession we nave the system 

 of equations : 



dW Q 



= -xw ( 



dt ""•»' 

 dw, 



dt 



= KWi-W 2 ), 



dW 2 



dt 



which are of exactly the same form as those occurring in the 

 theory of radioactive transformations *, except that the time- 

 periods of the transformations would have to be assumed to 

 be all equal. 



The equations may be solved by multiplying each of them 1 

 by e xt and integrating. Since W (0) = 1, W»(0)'=r-0, we' 

 have in succession : 



w„=«- M , 





|(Wi«*)=\, .-. w 1 = 



.\te~ Kt 3 



j t (W 2 e")=\% .: W 2 = 



2! ' 



and so on. Finally, we get 



n ! 



' ';■ 



The average number of a particles which strike the screen 

 in the interval t is \t. Putting this equal to at, we see that 

 the chance that n a. particles strike the screen in this s (] 

 interval is 



n ! . . : ; = 



* Rutherford, ' Radioactivity,' 2nd edition, p. 330. The chance that 

 au atom suffers n disintegrations in an interval of time t is equal to the 

 ratio of the amount of the nth product present at the end of the interval 

 to the amount of the primary substance present at the commencement. 



Phil. Mag. S. 6. Vol.' 20. No. 118. Oct. 1910. 3 A 



