A. Ritchie-Scott 99 
•■• - (X22 - X12 - X21 + .Yii) = (^22 - -42,) dm.^. - (.4i2 - All) d'lh- 
+ {B22 - ^12) dm. 2 — (-B21 - Bii) dm. I - dru^ (21). 
Reference to the diagram shows that ^22 ~ -^21 ^I'^d the other coefficients are 
the proportions of the areas of the trapezettes bounding the briquette of volume 
^22- These area smay be systematically named for the whole table thus, 
I 
that is, the areas of the planes meeting in the line of which the point s, t is the 
projection, in the direction shown, are named from the point so that 
Bs, t ~ -Ss-i, t = ^st- 
Hence we may write ^22 ^21 = ^22 > 
^12 ^11 — C-12' 
^22 ~ Bi2 = ^22^ 
Bzi - Bii = ^21- 
If now we notice that since m^. = N — n^. etc., and m^. = tiy., 
.'. dni^. = — dn^., 
dm. 2 = — dn.^, 
dm-i. = dn-i., 
dm. I = dn.i, 
we then have 
- (Xu - Xvz - Xn + X22) dr = - {a22dn^. + a.y.dni. + ^22 ^^^^ 3 + ^ndn.-, + fZ/t.a) 
(25). 
Expandiiig this in terms of frequency volumes this becomes 
(Xn - X12 - X21 + X22) d>- = (ai2 + ^2i) d'ln + ^2id'>hi + (ago + ^21) '^"31 
+ ai2dni2 + dn.2 + a22f?%2 + (^12 + ^22) ^'hs + ^22''^'^23 + ("22 + ^22) d)h^ 
(26). 
It has already been shown by Pearson {Biometrika, vol. ix, p. 1) that when 
random samples are taken from a population so large that its composition is not 
appreciably affected by removing the samples we have the following relations: 
-n,,= ns,(l-'^) (27), 
Mean {dn,,'dntt') = -^^^^ {28a), 
Mean {dn, . dut .) = - (28&), 
Mean {dn, . dn.,) = n,, (28c), 
Mean {dn,,'dn . t ) = - (28fZ), 
