Al. a. Tchouproff 
187 
(3) If the law of distribution of the values of A'' satisfies the condition : 
/i,2,+i = 0, 1 = 0, 1, 2, ... Gc, 
then, as we saw above, fJ-2i.+i,(N) = 0, i = 0, 1, 2, ...oo and, as appears from (5), 
V2i+T,(]SD = 0, i = 0, 1, 2, ... 00 . When the law of distribution of the values of A"" is 
Gaussian and /u.^,- = 1 . 3 . 5 . .. (2t — i=l, 2, ...go, then, as we have seen 
A<-2/,(iV) = 1 . 3 . 5 . . . (2i — 1) /Lio' ^ and, consequently, 
/iy;-i\- 
V N 
N-l 
Vii, (N) = [ ^j' ) _ S G.-^- fl2i-ih fJ-ih, (jV-1) 
V N 
1 . 3 . 5 . . . (2i - 2h - 1) 1.3.5...(2A-1) . 
1.3.5 ...(2^•-l)/L^,'■ S C/' 
(1) Let us put 
/,ro ^ (N-iy 
- {N-iy 
II 
(iV) 
(8). 
We have : 
1 
.(9). 
' -im = J_ w''"> 
m 
.(10). 
h = 0 
h ^ r'' w^'""''^ J' i 
]^m-h ^ m " r, {N) ,;U\) ] 
.(11). 
(2) When m = 2 we find : 
= E S [X,-A(^.)]- + ^ i S [A,-Z(^)]'-[A-AV)]'- 
/=i /=i j4=t 
= Nv,,, ^^) + N{N-l)E [A\ - Z(^)]'- [A, - Zt^)]*- 
* The quantity can also be written in the form 
{N-l){N-2) (N-1){N-S) ■'^^^(^'^'-1^, 2 
K. Henderson ("Frequency Curves and Moments,"' J. I. A., 1907), while giving correctly the values 
M2,{iV). M3,(A-), M4,(.V), "■'AX) and i-ii.i^v), erroneously gives (p. 435) -. 
ij^l) {N - 2) {N -1)_{N - 3) 
''4,(.V) = 
M4 - 
(M4 - S^.j-)- 
The true value of «'4,(A0 exceeds the value obtained by Henderson by — — " 
13—2 
