Al. a. Tchouproff 161 
(3) When r = 2, in the case when the law of distribution is Gaussian, we find 
from (8) : 
Hence noting that 0!,"^ = (^V - 1) we find : 
Lot us assume that for all values { ^ m, 
v^'\ ^. = .V (.V + 2) + 4) . . . (N + 2i - 2) 
In this case : 
.^;';;^>=i^/..;+^|i.3.5...(2vM + i) 
+ S C,„n.3.5...(2m-2/( + l)(iV-l)(i\^+l)...(i\r+2//,-3) , 
h = l I 
\!^,^N^i.r\l.-i.5...{2vt-l) 
+ S C' l.S.5...{2i,i-2k-\){N-l){N+l)...(N+2h-3)\, 
and, consequently, 
(N + 2m) = Nfil'^-'l 1.3.5... {2w, - 1 ) {2ra + 1 ) + (iV - 1 ) 1 . 3 . 5 . . .{2m - 1 ) 
/« - 1 
+ S C'' 1 .3.5 ...(2m-2/i - l)(2«i- 2/( + l)(i\^- \){N + 1) ... (N -\-2k - S) 
h=\ 
+ S 6'';^_jl.3.5...(2/u-2/t-l)(iV-l)(iV + l)...(i\^ + 2/i-l)[ 
= i\^/x;+'|l.3.5...(2m+l) 
m \ 
+ :i 6Vl.3.5...(2//i-2/t + l)(iV-l)(iV^+l)...(i\r+2/(-3)'r 
/, = i J 
^^('" + 1). 
2.(A^) 
Thus when the law of distribution of values of A' is Gaussian, 
^f^lm = i^m ^(^V + 2) (iV + 4) . . . (iV + 2m - 2)/.,"' for m = U, 1, 2, 3, ... 00 (13). 
Biometrika xii 11 
