Egon S. Pearson 
43 
and if be the correlation of the successive residuals and cr/ and a.! the corre- 
sponding standard deviations in Groups I and 2, we have finally 
p; <r/ a: = p, a, a, - ^ - 1 ) - [(n + l ) y, + (v - 1 ) - 2nd] . . . (xxiii). 
Similarly we have 
= 2dl (y,) - 2nd^ + 2b |(i - "^^^j (y, - - [t - '^^j + 1 Fr 
-= % IV-i- 1), 
whence it follows that / 
o-/' = cTi - £2 1) (xxiv). 
And again, 
+ 2 F^^i - 7« ( "+^— Am - nh' - 2nbd -2{b + d) (>/n+^ - y,- nb) 
t=\ \ n / 
= na.:- + 2bnpu + nb" - b- - (/i - 1) i + ^''^ ~ ^ 
(n + 1 11 — 1 A 
= «<^2'' + ^ n (n- -l}+b {(n + 1 ) + {n - 1) y„+, - 2nd] 
= ^^2' - f2 - 1) - {(^ + 1) ^1 + ('^ - 1) 2/n+i - 2«(^1 (xxv). 
If the values of p/ have been calculated by this means for each of the m series, 
we shall have for the combined series, 
S (p/ 0-/0-2') 
1 = — , ^ . (xxvi), 
/2K^)S(<x/^) 
V m m 
a modified form of equation (xii). 
As we are subtracting the ordinates of a different straight line from each 
series, a modification of the first-diiference equations may be necessary. The 
