94 



while all possible values must be given to the number n^ . . . n^ in 

 the other elements. If now we tirst take the ?i^ . . . n^. constant and 

 determine the average, we tind 



. „Ax =zpu ni -{-... pky njc — ?iy. P . 

 If then we proceed to determine the average according to ?ij .. .n^ 

 and keep in mind that ??, = ..,:= 7i^ = v, we find 



„Ax = v{piy. ^ . . . pk.) — n,, P=(v — 71,) P . 



In order to find „A/ we proceed in quite an analogous way, we 

 bring (7) into the square. Then we find 



A/= A,,^ 4- ...Afo'' + A.'' 

 4- 2 Aix As, + . . . 

 - 2 A. (Ai. + . . . Aav) . 

 If now we apply (5) and (6) and determine the average with 

 given /I, ; . . Ilk and n,, we find 



„a;* = (/^i^ - m) pu' -^ puni + ... -f P' (n,' - n,) + ,v, P 

 + 2 «1 ri'ipu P2/ + . • . 

 — 2nP {pu ni -\- . pjcy. nk) • 

 Here the average must be determined keeping constant 7i^ with respect 

 to n-^ etc. And we must bear in mind that ??,j' = ii^^ = . . . 7iJ=z r" -|- v '), 

 that further n^ = v and n^7i^ = r'. Consequently we find 



,M.'= {V' +V){pu' + .. .pic.'-) 



4- 2 r' {pup2A -f- • • • ) 



~r{pu' + . . ) 



— 2 n r P' f P' (n' — ii) 4- n P . 



The tiiree first terms together yield P* r*. The result becomes thus 



^\, = \ (n-v)' P^-«' P' I 4 (n 4 r) P. 

 from which by determining the average according to 7i the relation 



A ^ = 2 r P 

 arises. 



2. The extension of the given formulae may be obtained to the 

 case that the deviation of density in the various elements of volume 

 are not independent, where however concerning the emission of the 

 particles we must still presuppose independence of the events. 



In order to introduce the correlation of the densities I make use 

 of the function g, which was defined by Dr. Zernike and myself. *) 



1) We have n^ = v + è , Wi» = v^ + 2 v r+ i^ = ,3 + v. 



^yi^= (v4^i)(v + ;2) = v3 -I- V (17 + ^) + hh = ^'■ 



2) Chance deviations in density in the critical point of a simple matter. These 

 Proc. XVII, 1914. p. 582. 



