706 On the Probability Distribution of a Particles. 



The particular case in which ?i = has been known for 

 some time (Whitworth's * Choice and Chance,' 4th ed. 

 Prop. 51). 



If we use the above analogy with radioactive trans- 

 formation, the theorem simply tells us that the amount of 

 primary substance remaining after an interval of time t is 

 e~ xt if a unit quantity was present at the commencement. 



The probable number of a particles striking the screen in 

 the given interval is 



The most probable number is obtained by finding the 

 maximum value of Wn. 



W ' x 



Since w n = • , this ratio will be greater than 1 so long 



as n < a?. Hence if n > #, 



W B >W n _i; 



if n = #, W B =W„_i. The most probable value of n is 

 therefore the integer next greater than x ; if, however, so is 

 an integer, the numbers x— 1 and x are equally probable, 

 and more probable than all the others. 



The value of X which is calculated by counting the total 

 number of a particles which strike the screen in a large 

 interval of time T, will not generally be the true value of X. 

 The mean deviation from the true value of X is calculated 

 by finding the mean deviation of the total number N of 

 a particles observed in time T from the true average number 

 XT. This mean deviation D (mittlerer Fehler) is, according 

 to the definition of Bessel and Gauss, the square root of the 

 probable value of the square of the difference N— XT, and so 

 is given by the series 



D* = i (N-XT) 2 ^p e-^ 



71=0 -^ ' 



_ ,.« v r m* . (*t)» _ 9 (\T)»+i ^ o^2iH = XT 



n=oL(N-2)! + (N-1)! '(N-l)! + (N) ! J 

 Hence D == V ^XT, and the mean deviation from the value 



