Al. a. Tchouproff 
285 
When ?• = 2 we find 
2 1 
?« 2, (iV) = _ + /t^ra, iv] 
Similarly we find* 
3 3 1 
i 62 1 ^2 
.(4). 
1 ^ 
(5). 
Comparing these relations with the corresponding relations established in (11) 
of Chapter I of the first Part of my paper on the " Mathematical Expectation of the 
Moments of Frequency Distributions^," we see that when each variate follows its 
own law of distribution we may obtain vi2^ (j^r) and (a^) by replacing TOj, fio and /jl-. 
in equations (11) by the corresponding arithmetical averages for all the N variates. 
But in the case of ot^ ^jvt) this method would only produce the first three terms; 
the term of order ^ is not equal to .vj — Syu,^^ but to the arithmetic mean of 
the differences //'^ - 3 [fi-'^] 
III 
If in (2) we put 7/ti<'' = 0 and replace the m's by the corresponding /i's, we find 
1 r! 
(>j) 
Vilr^l ... Tjl 
where the summation with regard to j extends to all integer values from 1 to r or 
N whichever is the smaller, the summation with regard to io, ... ij extends to 
all mutually unequal integer values of i^, i^, ... ij from 1 to N, and the last sum- 
mation to all integer values, not less than 2, of i\, r^, ... rj satisfying the relation 
ri^r.2, + ... + Vj = r. 
Hence, 
H-2r. (N) = 
N 
2 S 2 
N-\ N 2r-2 
[2r]! 
-2r 
[2r]! 
N-i N—l A* 2j'-4 2( — r,-2 
+ 22 2 2 2 
f, = l /,= ;, + U3 = /,, + l»-, = 2 r, = 2 T-^l r^\{2r - — ?'.,]! 
('■,) ('...) (/;,) 
+ ... + 1.3.5...(2r-l)r! 2 2 ... 2 .a 'Vi 
(/,■) 
^'2 
= 1.3.5...(2r-l)a;.;^^,,4-^ 
, , [-3] r-3 2 , [-2] r-2 / 1 ^ / 
+ /^[2, ^1 f^l,, A'l - /^[2. iV ] (]V 
+ 
(<3)j, 
* Cf. Tschuproff, "Zur Theorie der Stabilitiit statistischer Reihen " [Skandinaviak Aktuarietid- 
gkrift, 1919, p. 82). 
t Biometrika, Vol. xii, p. 151. + Cf. Biometrihi, Vol. xii, p. 154, equation (22). 
