A. Ritchie-Scott 115 
The relations between two superimposed tetrachoric divisions involve the deter- 
mination of ten constants, four as, a^, ajg, cr-zii ^ti< ^^^^ six i2's, ^n-ia, -Rn-aiJ 
■^ii-22j -^i2'2i5 Ri2-22- -^2i-22- The ct's follow the example already given, the proper 
suffixes being attached. The value of S 11.21 has already been given. The remaining 
five (S's are as follows : 
'^11'12 = JpidP 11 dPl2'>'>'n + cPllcPlZ ("21 + ^'31) ~ 6^11(i-f*12 '%2 ~ aPucPl?. {^^22 + ''32) 
6^ 116-f*12 '^'iS + aPnaPl2 ("23 + %3)} (91)- 
'S'll-22 = {dP 11 dP 22 "^hl ~ c^*ll-if 22 "21 + cP 11 cP 22 ""^Zl ~^ bPlldP 22*^12 + a^*lld^*22 *^22 
~ aP llc-f* 22 ''*32 + 6-f 116-^*22 '"'13 ~ aP llbP 22 "^^2^ + aP 11 aP 22''^ ss\ 
'^12'21 = ^ {rf-Pl2(i-^21 '''■11 ~ cPl2dP 21^^21 + cP 12 cP 21**31 ~ dP 12bP 2l''^12 + C^126^*21 ''^22 
~ c-P 12a^ 21*^32 + bP 12bP 21 ~" aP 126^21 '*23 + a-Pl2o^21 33} (93). 
'5l2'22 = 2V^{(i^l2(i-P 22 (%1 + '^'■12) ~ c-Pl2(Z-f*22 ("'21 + '''22) + (;Pl2c-f*22 ("si + '*'32) 
6^ 126-^22 'hs ~ 126^22*^23 + aPl2aP22 (*'33)} (9i). 
1 
'S'21'22 jy^{d-f*21d^22 ('hi + '*2l) + cP 21 cP 22 '^^31 ~ bP 21^-^*22 ('^2 + "22) ~ 0^*21 C-P22 **32 
+ 6^216^*22 (*^13 + ''^as) + 210-^*22 ^'^33} (95). 
A more convenient form of the above for actual computation purposes will be 
found on page 120. 
We may now by means of the P's express the standard deviation r^, in a form 
consisting of sums of squares. 
- (SC'J dr = S (^^-^ SP J (96), 
{ \X.^t J) \Xst/ X>^tXs't' 
^11 / "-'II Q , '-^12 o , '-'13 
,„ *^11,11 + , ^ '^ll, 12 + 7^ 13 + ... 
Xll VXll Xl2 Xl3 
+ ^f^^S,,,,2+-'^S,,,,2+^'S,2.i,+ ...) + eto (97). 
Xl2 VXll Xl2 Xl3 / 
^"12 fC^U a I t*12 CY I ^1 
■^11, 12 *-'l2, 12 "T 
X12 VXll Xl2 Xl3 
Now consider the S's to be expanded in terms of the frequencies and pick out 
all the coefficients of the frequency n,f say. The coefficient of the taken from 
Sim, I'm' say will be : P;,„. : Pj',„. in which the quadrant suffixes will be determined 
by the relative position of jij^ to hn and I'm'. Let these undetermined P's be de- 
noted by ]). We shall then have as the complete coefficient of 
(~ Ihl -Ihl + 1*11 -1*12 + 2^11 -1*13 + • • • ) 
Xll \Xii X12 Xi3 / 
+ ~ Ihl -Pvz + ^ V12 ■ P12 + ~ P12 . Pr, + ...) + etc. 
X12 \Xii X12 Xl3 / 
Since we are dealing throughout with the cell n^t the quadrantal suffix (i.e. the 
8—2 
