lOG The Correlatkm Coefficient of a Polychoric Table 
of this latitude might in some cases be obtained by using another cell But as the 
r from this cell would probably difler from that previously found, and as neither of 
them would be identical with that of the normal surface from which they are sup- 
posed to be sampled, we must find some means of approximating to the " best " r. 
The most general method of doing this, following on the above, would seem to be to 
weight each of the frequency blocks and determine the weights so that the re- 
sulting probable error of the weighted r is a minimum. In doing this we must 
have regard to the fact that the variates we are dealing with are not independent 
but correlated. We must consider this method. 
Let the polychoric table have p rows and q columns so that the indices of the 
last row and column are Iq, 2q, ... 'pq and pi, p2, ... pq respectively, and let each of 
the frequency volumes be weighted by an arbitrary weight ^.^n, iv-^^o, ... indicated by 
the same suffix as their respective cells. 
Then ^''ii'^ii + '''^ 12**12 + ■•• 
= «'n''"ii + ^"12 ("'12 - '"n) + ■"'21 ('«2i - "Hi) + ■'<'22 ('"-22 - "h2 — »*''2i + *%i) + •■• 
+ w,t {nist - "Vi.i - WH/-1 + ''"^-l,i-l) + ••• 
= (Wii - IVj^.^ - «'2i + ?«u -h ... + {n'st - '^''s-Hl, t - 'X's, t + 1 + <-n) + ••• 
= coumn + 60i2Wh2 + •■• + ^^t^n.,t (60). 
Then ("'ii'«ii + ■?'^]2'h2 + ••■) = ]^ ("^ii'^ii + -^12 "H2 + ■••) 
= ^11 (i^o i^n -1- ^11) + <^i2 (1^0 2^0 + ^'12) + (61), 
which is an equation to find r, using as an abbreviation for 
sr^.tO^r -\- ,r,.,e.^ f" + .r.^-iB.^f'' + etc. 
(Compare the usage in equations (2) to (5).) 
m^q, JHg,,, ... Jf pi, Wp.^ ... are complete segments of the normal solid and are inde- 
pendent of r, i.e. -~ = ^t^^jOq, and these terms disappear from both sides of the 
equation. The ?f''s of the cells in the last row and last column of the table may therefore 
be dropped and we have (pt —\){q—l) iv's to determine. The probable error may 
be written down in the manner already shown and as there are {p — I) (q — 1) 
independent frequencies there is snfiicient data to determine the w's so that the 
probable error is a minimum. 
There is no essential difficulty in carrying this out except that the coefficients 
rapidly become very cumbersome as tables increase beyond 3 x 3 for then the 
simplification dm2. = — dn.^ is no longer available. It will be found however that 
the same result may be derived more simply from the method discussed below. 
§ G. Polychoric Method. 
The frequency surface divided into p columns and q rows is divided at each 
point 11, 12 ... into four c^uadrants, and for each of these divisions a value for r, 
viz., rii, ■ ■, may be found by the tetrachoric method. These may be regarded 
as approximations to the true value of r, and their weighted mean found, the weights 
