( (>81 ) 



the line as on tlie other. Tlie result will be a certain distribution 

 of the whole nnnilier, entirely determined by chance. Let ns conceive 

 this operation to be very often repeated, say Q times, and let ns 

 calculate in ho\v many of these Q cases, a desired distribution of 

 the points over the p parts will occur. Dividing by Q we shall have 

 the probability of the distribution. 



The probability that there will be a, />,... ))i points on the 1'^'', 

 2"^, .... y>t'' part of Ihe line ('/ + ''> + ... + n^ = (/), is given by 



p J a! h ! . . . m I 



q 

 In the case of a vevx large value ^^, this probabilitv becomes 



P 

 extremely small, as soon as one of the numbers a, />,... m is far 



below—. Neglecting these small probabilities, we shall confine our- 



P 

 selves to those cases, in which each of the nuinV)ers a, h, . . . . m is 



very large. Then, by the well known formula of Stirling, 



a I = l/2ajr I — 1 ? etc. 



and, if we i)ut 



we shall find 



a ^ h , m , 



q ' q '' " q 



log P =: — 1 (p — 1) io(] {2jrq) — q log p — 

 — [(«V + \) ^og a 4- . . . + i^nq + h) % '«'] .... (32) 

 It is to be ]-emarked that the numbers «,/>,... ?>i can only increase 

 or diminish by whole units. The numbers a', h' . . . m' can change 



by steps equal to — ; this may be made so small that they may be 



q 



considered as continuously variable. 



§ 14. We shall hi the first place determine the values of (/', //, . . . m' 

 for which the probability P becomes a maximum. We have 



dlog P =1 — \ Q -\ , -\- Q^oq a' ]da-\- ... -J- ( V 'T 7; — , + qlogm' \din' 



[\ 2a ' J \ 2m ' y 



with the condition 



Ja' + . . . + (hn = 0, 



which is a consecpience of 



.,'-[-.... 4- m' = 1 (33) 



The maximum will therefore be reached if 



