156 Expectation of Moments of Frequency Distributions 
Putting h — 1, we obtain 
Hence : 
and iVV,.,(.v) = Nil, + C,' "^2 - j)' (,v_;) (27). 
If we give r in turns the values 2, 3, 4, we find the reh\tions (26). 
IV 
(1) From the relations (15) and (17) we find : 
^•■<^:^'> = 1.3.5...(2r-l)]l+''24.- S (-l)^/3,-H.,^^^- ,^'--;r:'' 
As iV^ increases the ratio ^'"'—consequently tends to the limit 1.3.5 ... (2?' — 1), if 
.Ci iV' ^ ^ 1.3.5... (2r - 1) /./ 
tends to zero. 
But if this last expression tends to a limit different from zero as N increases, 
then the limit to which ^-''''^^^ tends cannot be equal to 1.3.5... (2r - 1). 
The quantity ^ 3~5"~~^2r — ^1) become infinitely great for 
any value of r and is independent both of the value of N and of the law of dis- 
tribution of values of X. In order that '^^^^ should tend to 1 . 3 . 5 . . . (2r - 1) 
it appears then to be a sufficient condition that ^ T..,.^ i—i+ti should tend 
to zero for i = 1, 2, 3, — 1. A sufficient condition for this, in its turn, is that 
expressions of type 
should tend to zero, when the quantities ji,^.^, ■■.]/, are connected by the relation: 
j\ +^2 + ••• +jf= U i^iid the quantities /?], lin, ... hy satisfy the relation : 
/'I ji + h,j,+ ...+ hfjj =2{i-h+ I), 
and I can take all integral values between 1 and 2 (i — h). Finally this condition 
is satisfied, if expressions of type 
