246 
Goodness of Fit " in Statistics and Physics 
I now rearrange this into terms not involving the factor 1 — and those 
involving it. The former are: 
3h 
N 
P 2 
since S {riqpXg) = 0. Here p/x^ is the sth moment coefficient of the ^th array about 
2 
the mean. I now take the terms involving the factor 1 — They give 
1 /^_2 
N 
1 "'iP "g p I '-"'gp "9 3) \ ~ 2 
1 9,Q \ "9P^'9' g"9"p / 1 _ ' ^"gp 1 7. 2;s ^ 
J ^ ^ 1 ^ _^ j -'g 
q'' Q Q ''-Q 
3/2 
N 
iV2 
where the equivalence is most easily verified by expanding the latter expression 
and ticking off the corresponding terms in the previous one. 
Since S [ngptCg) = 0, we are left with the single term 
Thus we deduce combining our two sets of terms : 
V NJ ' 
pB-i 
S (8mj,)2| 2 _ p/i^a (^^2 +3(1 
A ?! 2 I }) 
where ^,62 i''^ tbe second ^-coefficient of the distribution of nij, for the j^th array, and 
is the second ^-coefficient for the pth. array itself in the sampled population. 
Now to the degree of approximation needful 
A n 
Accordingly 
N 
(^^)H-I('-I) 
