374 PROBLEMS ON CAUSES. [CHAP. XX. 



In the second member of the above equation, performing the dif- 

 ferentiations and making x = 1 (since 9 = 0), we get 



D m (I + e e Y = fi (A O m ) 2M- 1 + **Yo ( A2 w ) 2/i " 2 + &c - 



J. t 



The last term of the second member of this equation will be 



M(M " 1) 'i ( 2:r 1)Amo " 2 "-"^^- i )--^- m+i)2> " m; 



since A m OT = 1 . 2 . . m. When ^ is a large quantity this term 

 exceeds all the others in value, and as fi approaches to infinity 

 tends to become infinitely great in comparison with them. And 

 as moreover it assumes the form p m 2* i ~ m , we have, on passing to 

 the limit, 



u\ m 



2/ "" 



Hence if (Z>) represent any function of the symbol D, which 

 is capable of being expanded in a series of ascending powers of D, 

 we have 



if = and JJL = oo. Strictly speaking, this implies that the ratio of 

 the two members of the above equation approaches a state of 

 equality, as p increases towards infinity, being equal to 0. 



By means of this theorem, the last member of (3) reduces to 

 the form 



Hence (2) gives 



as the expression for the probability that from an urn containing 

 an infinite number of black and white balls, all constitutions of 

 the system being equally probable, r white balls will issue in p 

 drawings. 



Hence, making p = m,r = m, the probability that in m drawings 



/\\m 



all the balls will be white is f - j , and the probability that this 



