93 



tliese elements. Afler a time t lias passed these luiinbers have become 

 changed. Let fii, then represent (he chaiu'o that a particle which 

 at the time ^ = is in the element 1, is Ibnnd at the time t in 

 the element x, and let />^i represent the probability of the reversed 

 transition. Then, if there is no predilection for any direction in the 

 movement of the particles, it goes wilhont saying that /;i, = />^,. 



Knriher S /)/, = P if the snm is taken according to all values / 



)—\ 



except X = ;., for the snm represents the probability that the [)article 

 has come after the time t in one of (he k — 1 other elements, i.e. 

 ontside the element x. 



If an element ?. contains ii, particles the nnmber of parti<'les 

 having passed from P. to >i iii a given case will be A,^. I shall now 

 calculate tirst the average values of L,y, L*,, and A,, L,j,. The 

 number of cases whei-e A;;, has the value .s' and thus?*, — .v particles 

 have i-emained in the element, amounts to: 



n\-—s Is ! 



as is easily seen; to determine the three average-values this expres- 

 sion must be multiplied by ,v resp. .v* and summed from zero to n,. 

 Then after quite an elementary calculation of these finite sums, we 

 find 



A^/' =P//' («/.*—'';,)+ P/)- "/, (5) 



and 



E,y. = iK,n,, (4) 



To determine the avei-age of a double product we need only replace 

 (3) A by n and s by t (where t represent the numbei' of emitted 

 particles in a definite case). 



If the result obtained in this way is multiplied by (3) and summed 

 with respect to r from to n,, and with respect to t from to ?/.,;,, 

 we find 



Ly, A, ,,:==. Ip;,. ][>:,., riyn., ....... (6) 



With the help of the relations (4), (5) and (6) Smoixchowski's 

 formulae can now immediately be deduced. The change „A^, i.e. 

 the total change of the number of particles in the element h may 

 be represented by 



„A. = Ai. H Ao. . . . + A/,. - (A„ + . . . A.^.) ... (7) 



Now we can write A;, for L,\ -\- . . . L,k, ie. the total numbei' of 

 particles that leaves the element in the time t. 



Then we must determine the average of (7) with constant n,. 



