168 Expectation of Moments of Frequency Distributions 
Noticing that (cf. Introduction, II, 8) 
' GJ =0 when m>2h, 
Ji = 0 
''T\-l)"6y'yS,_;,,, =l.«.5...(2/.'-l), 
11 = 0 
"T\- !)'<_„ /3,._;,.= I 
fj = 0 
where i?^,^ has the vahie given in the Introduction (12), (13) and (14), we see that 
the coefficient of l/N' in the development of Ei^i) d/j, is zero, if t < i. The coefficient 
of equals 
(- 1)' ri,,,-, 1 . 3 . 5 ... (2{ - 1) (- 1)* 1.3.5 ... (2/>- - 1) 
X 2 + ,...(19), 
where by .1/.^'"''^ ^ '■'^ denoted the sum of all the diiferent products of type 
("//j • • • M/'i-i- possible, subject to the condition that A,, //o, ...///_fc appear as 
sums of pairs of the numbei'S vy^, ?v, , ... '''f^i-^i-- 
When ?■] = ?•.,= ...= 7-^/, the aggregate of terms denoted by M ,^ reduces to 
(2/ - 2k - 1 ) {2i -2k-S)...r). :i . 1 , 
and 
1.3.:j...(2/,-l) % 
n 
(2t) 
-1" i> .1 o 
= 1.3.5... (2/.: - 1) C^~-^ /i'J.'' (2?; - 2k - l){2i - 2k - 3) . . . 5 . 3 . 1 
= 1.3.5...(2i-l)(75~Vf /^r* 
i—k 
The coefficient of l/i\'' consequently becomes (cf. (14) above): 
1. 3. 5... (2^•-l)(M,. -/./)'■. 
Similarly, we find that the coefficient of \IN' in the development of -£'(o/^.i, dyi, 
reduces to 
(-l)«n„,+„ 's'(-l)"CV:;S,;+,_;,,^ 
// = (I 
+ 's (-1)*^'' S ^^^^^ A'/''A'%-'>^-A+!) 
Kilt. n , ■ , 
+ (-l)^-S 2 - A-/'>r '•/-••■ W '--(SO). 
/.■ = 0 , = 1 y, (/_/,+,) 
X 2 (- 1 )'' C\!+,_t+k-l ^.2!+,-t-,k-)., k 
li = (l 
E lit. (//!')- 1 ^-2/.--l 17 
+ 
(-1)'- S 
(2("+l) 
