L. ISSERLIS 
189 
where t.^y^ is the mean vahie of t for given values of x, y and z, so that as is well 
known 
where 
1, 
' xy > 
1, 
' yt 
.(31), 
.(32), 
rzt 
1 
Thus 
+ A,,(?,,,.)/A,, (33), 
' xz> ' yz > 
'^xt> '''yt' '^zt> 
and Apq is the minor corresponding to r^,,. 
^xyzt — ~ {^xt1x-vz + ^ytlx 
SO that ^ can be evaluated approximately in the case of linear regression 
without employing any mixed moments beyond the simple product moment 
occurring in a correlation coefficient. 
For normal distributions we may use (27), 28) and (29) giving 
xy ^xz 
) + A,,(2r,,r,, + r,,J]/A,, ...(34). 
By well-known properties of first minors of a determinant we have from (32) 
A,,, + r.^yA.t + r.^,A,t + r^,;A.( = 0 (35), 
r,yA,t H- ^ut + TyAzt + ryt^ti = 0 (36), 
Txz^xt + ry,Ayt + A,i + r.^A,, = 0 (37). 
Multiply these equations by r,j., r^y respectively and add, 
•"• k^uz + ^^''xz'^xy) l^xt + h'xz + ^/yj^ 
xy) ^yt 
+ {'Txy + 2r,,r^.J A,^ + {ry,r.^t + r,j-,t + r^.r.i) Au = 0 (38). 
Combining this result with (33) we see that for normal distributions 
dxyzt ^xy'zt ^yz^xt ~l~ '^zx'^y 
.(39)*, 
an interesting result likely to prove useful in other applications and probably 
capable of generalisation Particular cases of (39) are obtained by putting t = x 
so that q^2y^ = Vy, + 2r^j,r^2 which is (27) and t = x, z = y giving q^.y^. = 1 + 2r\.j, 
which is well known. 
If we now substitute these values in equation (21) we find 
= ^^y.r^t + ^r.^r^t - 2f^.,r„,r,( - 2f^jr„(r,i 
" '^'f xz'^' xt'^ xy ' ^fyz'>'vt'^xy + ^xy^zt (''\-2 + ''^xJ + '^^yz + ''^i)- 
The right-hand member can be put in the form 
~ *'a;a^z() (^yz ~ '>'xy'^xz) + i'^xi ~~' '''xy'^yt) {'''yz ~ ^!/<*'z«) 
+ i'^'xz ~ f'xy'^'yz) ('"?/( ~ "'yz'^'zi) + h'xz ~ '^'xtTyz) i'^'yt ' ^xy'^'xt)} > 
* This result, which is accurate for normal distributions, is given as approximately true for such 
distributions by H. E. Soper, Biomctrika, Vol. ix. p. 100- 
