704 Mr. H. Bateman on the 



by starting 1/8 minute later and adding B + C, D -i- E, &c. 

 The results are given in the appended Tables. In the final 

 horizontal columns are given the sum of the occurrences in 

 Tables A and B and the corresponding theoretical values. 



In the cases for 1/4 minute intervals, the agreement between 

 theory and experiment is not so good as in the first experi- 

 ment with 1/8 minute interval. It is clear that the number 

 of intervals during which particles were counted was not 

 nearlv large enough to give the correct average even for 

 the maximum parts of the probability curve, and much less 

 for the initial and final parts of the curve, where the pro- 

 bability of an occurrence is small. However, taking the 

 results as a whole for the 1/8 minute and the 1/4 minute 

 intervals, there is a substantial agreement between theory 

 and experiment, and the errors are not greater than would 

 be anticipated, considering the comparatively small number 

 of intervals over which the a particles were counted. 

 We may consequently conclude that the distribution of 

 a particles in time is in agreement with the laws of pro- 

 bability and that the a particles are emitted at random. As 

 far as the experiments have gone, there is no evidence that 

 the variation in number of a particles from interval to 

 interval is greater than would be expected in a random 

 distribution. 



Apart from their bearing on radioactive problems, these 

 results are of interest as an example of a method of testing 

 the laws of probability by observing the variations in 

 quantities involved in a spontaneous material process. 



University of Manchester, 

 July 22nd, 1910. 



Note. 



On the Probability Distribution of a Particles, 

 By H. Bateman. 



Let \dt be the chance that an a particle hits the screen in a 

 small interval of time dt. If the intervals of time under 

 consideration are small compared with the time period of the 

 radioactive substance, we may assume that A, is independent 

 of t. Now let W n (T) denote the chance that n a particles hit 

 the screen in an interval of time t, then the chance that 

 (n + 1) particles strike the screen in an interval t -f dt is the 

 sum of two chances. In the first place, n + 1 a particles may 

 strike the screen in the interval t and none in the interval dt. 

 The chance that this may occur is (1— \dt)W n +i(t). 

 Secondly, n a. particles may strike the screen in the interval t 



