Miscellanea 
183 
o, a, (Ij! a, Oi! 02 ((12-1) a,-i 
(01-02)!! (01-02 + ^)1! ^!! 
Oi! O2 (02 -1) (02-2) (o, -3) a,,-i , 
+ (07-02^4)1! 4-!-! + 
J'ecriB entiu les moments jusqu'au sixieme ordre, cii employant la notation 
lia,a,...a,,= M [.I'l"' .Co"- ... .C,,"']. 
ftl = »-12- 
i34=3: 
/331 = /5*i3 = 3/-12, ft. = ^'-'12 + i; 
^6=15; 
fti = 15ri2. ^2 = 12r\2 + 3, ft^ = (ir^. + Orj.; 
ftn = 12ri2ri3 + Sr-ja, ftoi = *>''\2'"i3 + (j'"i2''23 + ''»\.)' ^22 = 8''i2'"i3'"23 + + '^'''i^ + ^/'-.j + 1 ; 
ftnt = (''■l2''l3'"l4 + '^'■l2''34 + ^'■l3''24 + ^'•l4''23> ^211 = ^'T^l^^'i + ^^\2>\J'-n + 4?i2>-14'-23 + "^''l^'u 
+ 2r23'-2i + r„^; 
^21111 = 2/-]2''i3''45 + 2ri2/-i4?-35 + ^.j + 2ryj^^r.^r^ + ■2l\■J■^^)\^^ + -Ir^^i^-^r^.^ + r.^.^r^-^ + r..^r.,^ 
ftiuii = '■i2'34'56 + riiU^rr,^ + ... (15 termes). 
Les moments jusqu'a Tordrc 4 a deux variables ont ete calcules par M. Pearson et puis par 
M. Soper qui a donne une formule generale pour les moments a deux variables; Biontelrika, t. ix, 
1913, p. 101. (On doit remarquer cependant que sa formule (xxxii) est atteinte d'uno erreur 
typographique. ) 
Le cas de trois variables a ete traite par M. Wicksell dans "The general chaiacteristies of the 
frequency function of stellar movements, etc." Lund 1915, p. 11. 
Enfin M. Isserlis a deduit notre formule (28) pour le cas 2*' = 4 ou de quatre vai iables dans 
Biomelrika, t. xi, 1910, p. 189. 
III. Formulae for determining the Mean Values of Products of Devia- 
tions of mixed Moment Coefficients in two to eight Variables in 
Samples taken from a limited Population. 
By L. isserlis, D.Sc* 
A. Let /)j2, /'34 denote the product moment coefficients, referred to -a, fixed origin in a sample 
of size )i extracted out of a population of size N, there being /o«r independent variables x^, x^, 
.T3, x^. The mean value of pj^^ many samples is P^.,, the corresponding product moment 
coefficient for the sampled population. Let dpi^ = Pn - Pri, denote the deviation of the moment 
coefficient of the sample from its mean value, tiien 
Mean value of djj^.^ dp.^^ in many samples 
= ^ (^^1234 - PviPn), 
N - V 
where 
and P1234 is the product moment coefficient with respect to the four variables. 
[* Dr Isserlis sent me a paper containing the results of the present note with others accomijanicd 
by proofs in the course of 1916. It has been impossiblo to publish that paper so far, but it is only 
fair to him, having regard to the fact that other investigators are now entering this field, to publish 
his formulae in association with the memoirs printed in this part of Biometrika. Editok.] 
