150 Expectation of Moments of Frequency Distributions 
Hence we find : 
,.[-(71+;)] 
ill 
where the summation with regard to i extends to all integral values from 1 to the 
smaller of the two numbers Ji and r — Ji, and the second sunnnation extends to all 
positive integer values not less than 2 of h^, lu, ... //,-, satisfying the condition: 
or 
Ih + /'2 +... + /(.; = /;+ i, 
•(5), 
where the summation for i extends to all integer values from 1 to the smaller of 
the numbers Ji and — and the second summation extends to all positive integer 
values of y'], ji.,, ...j/, satisfying the equation : 
and to all positive integer values of A,, lio,...hf, satisfying the conditions: 
2 ^ //!<//.,<... <///, 
//] ji + liij., + . . . + h fjf = li -\- i. 
We find hence : " 
Ry, ;■ = in^' 
R,-^r-\ = C,? m/~" in 2 
=1.3 0/ m/-* m.;^ + C,? m,'-^ m^, 
Rr,,-3 = 1.3.5 C'/ m,'-'' mi + 10 G,!- + 6'/ "'•i''"' m, j 
E,., ,_4 =1.3.-5.7 6'/ 7;ii'-8 m.^ + 105 C,7 m,'-' m^-vh, j 
and on the other hand 
R,.^] = m,. 
1 \ (7), 
v...(6), 
and so on. 
(2) The calculation of the coefficients Ri;,--h '""I'T-y 'i-'so be effected by another 
method. 
Noting that 
E 
r+i ( 
■ A' 
S X; 
i = l 
1 
