Andrew W. Young and Karl Pearson 
221 
Mean 
s s' V AfAs' / s \ / s s' \ AjAg' / 
where *S denotes the same as S, i.e. summation for all values of s, and S S denotes 
s s s' 
summation for all values of s' except s = s' followed by summation for all values 
of s*. 
Substituting from the equations (a), (b), (c), (d), (e), (/), we have 
^^*^^?1A7^^ l^^^ +^')\ 
, ?i, (iV - H,) (iV - 2?),)) .r, of".,' (iV - 2n3) 
+ ^'S -,7^ \-^^'^.\y^. XiXi 
A^2 A1A2 ^2 j (A,A,' 
^ (iV - n,Y \ ^ {n,v, {N - n,) {N - «,) 
f { ^2 [X4". + (3X3 + 4X2 - Xi - I) + (4 - (6X3 + 1%2 - ^) 
+ (3X3 + 8x2 - Xi) ^i) 
(X3 - Xl + + (Xl - X3 - 2X2) ^ ^ 
+ (- ^ + (3X3 + 8x2 - Xl) -^-2) ^^sV 
+ 55 
' A,-A„ 
after expansion and rearrangement. 
Now it is evident that in numerical work the double summation would involve 
much extra labour, but we can get rid of it by using the identities 
(!©r^?(S)-?f(a)' 
?(ir^©=?(S)M?(S>f©-*f?(S'^--~')' 
2 1 
(?(¥)) -(a-??(^")' 
and so reducmg all to single summations. 
* As this notation may be somewhat unusual, it may be better to make it clear by taking a case 
with three variates only, for example : 
