294 Expectation of lloments of Frequency Distributions 
When r = 2, 3, 4 we find similarly* 
1 ^ r ii) 
1 ^ (i) 
N-2 1 
1 
''4, (AT) — 
^ 2 [wj" - /H^i,^]]^ + -^/xp. iv] _^ 2^ - ivi]' 
-2 1 ^' 
6 
1 ^ 
(2). 
Noting that 
2N-S 2 
II 
2 [////^ - m[i,iv]? 
AT - 
2 
2 (//'i )- - --^ 
/ = 1 
2 
N 
(0 
9 riv 
i Hi, 
+ 
2, 711, 
2 + 
iV 
2 (m^ T 
AT 
2 (m|'>+ 2 2 (m*''^)^?/^;''^)^ 
-^ 22 2 (m;'-''y'(mi'^')+ 2 2 2 (m;^'')Hm<;^V%^') 
we find, after some fairly tedious but quite straightforward transformations f, 
E 
N 
1 (A7-X,iV))^ 
.'■=1 
iV^^-2i\^+3 
2 CAr 
N 
2 /A., 
('■) 
A^ 
2 [^r']^ 
+ 4 (i^ - 1 ) |1 2^ - ;u^3, iV] 'Hti, 
2i^ - 1 / ^ (0\ / ^ . (0 -.A 
+ 
AT 
+ 4 2 jii.^'*[j?ij''-m[i,.v]p, 
* I7(?e Tschuproff, " Zur Theorie der Stabilitat statistischer Reihen " {Skandinavisk A ktuarietidskrift, 
1919, p. 84, equations (1) and (2)) and Bionietrika, Vol. xii, p. 186, equation (7). 
t Cf. Biometrika, Vol. xii, p. 192, equation (19). 
