284 Expectation of Moments of Frequency Distributions 
On the other hand, 
/^[2, A'] 
(m/'')^] = »«[2,ivi-^,|jm,<''p 
1 ^' 
r 
iV ; = i 7, = o 
r- a 
+ (_iy-i(,._l)^_|[„,^(0]. 
...(1). 
II 
Now, 
■ A' 
S X; 
r ! 
= 2 S S ... S S 
where the summation for j is for all integer values from 1 to either r or N which- 
ever is the smaller, the summation with regard to i^, i^, 4, ... ij extends to all 
integer and mutually unequal values of ij from 1 to N, and the last 
summation to all integer values of 1\, r^, ... Vj satisfying the equation 
Hence, 
ri + r., + ... +rj = r. 
r N 
(=1 
(h) ('2) ('V) 
or, when ?■ S N, 
1 r ' 
= 1 S S . . . S S — , — ,- — , m m - . . . ni.. 
.(2), 
S m.'V S 2 S 
m 771, 
N'-l^.^ '■ i.ri ;,=7;+i ,-,=1 n!(r-r,)! 
N-'2 N-l N r-2 c-r,-! 
+ S S S S 2 
r! 
('1) ('2) (4) 
+ 
2 
+ Z i ... > 2 2" ... " "2 
/,=1 /.,= i|+l /'r— 1 = '!— 2 + 1 (■i=l 1'3=1 r)-_2=l 
...m m 
r,—2 J— r, ->-.,-... -j-r-.. 
^ (3)^ 
+ 2 2 ... 2 r!m;'^n;'^..m;"-' 
= 1 i., = ii + \ i,. = /r- 1 f 1 
+ 1 . 3 C/ Wijj"*^^ Z^'^, JV] +'Cr' ^'^[i"'!^] /^L3, Nil + 
* Cf. "On tbe Mathematical Expectation of the Moments of Frequency Distributions" (Biometrika, 
Vol. XII, Ohap. 1, § II, equation (10), p. 151). 
