L. ISSERLIS 407 
Now {r^-y Hi — a.,; j?/) — ^ = 2c^/'\.y if we neglect second order terms. If 
we use the value of c given by (73) in (70), and adopt the notation JJ^ for 
x^lz - rzi, we have 
- (^\r„?.n/'2 - (lxi,J) Ji (x U, - y-i-xUy)] (74^ 
Let us write xUz - j.-!'xUy and for yll, - j-i-yU,^., then (74) becomes 
(^Xz - uXz)' ^'V'/ = 2 {'■x.q^iy - ry,q.j,yi) [7/ {r,,yq,,,y - q^.y^) yXz - Js (rxyqxy^ - qx^y) xXz\ 
(75). 
(75) is a relation between second order terms and it is sufficient to use 
equation (55) for q^iy with yS^ replaced by zero and /S,, by 3, so that 
q,y, = \'^yU^, 
•'■ ixXz - yXzf = 4 {rxz ^Wy - ry, ) [j,' yXz ('■xy 'Jjj'y - ^^x) 
- ^,xXz C'Vv - "JjTy)] (76). 
This identity between ^77^, yiq,, j^Vy <'''"*1 yVx is symmetrical in x, y. Two more 
such identities may be obtained by interchanging the letters x, y, z in cyclic order*. 
There are therefore three identities between the six correlation ratios : 
i/^i x'^yy yVz> zVy> xVz> zVx- 
I have not so far succeeded in reducing them to simpler forms, although possibly 
such exist. In special cases simplifications result. These are illustrated in the 
following section. 
§ 9. We defined 73, 73' the regression coefficients of z on x, y by the equations 
73 = - ^V^Vv (9)^ ^^^ ry,-^Y^i' (10). 
-■■ ^"xy ^ ^"xy 
Let us now introduce the remaining regression coefficients 
(77). 
"^^yx '^'zx'^'yz i ^'zx ^'i/x'^'yz \ 
7i = ~i 3 , 7i 
^ I yz 1 '^"yz 
'^'yz '^'xz'^'xy / ^'xy ^'yz'l'zx 
^ ' XZ 1 — 1 ^Ji- 
lt will simplify the algebra if we use X\ /x^, v'-, A,'-, fi'-, v'- for yU^, zUy, xUz, zUx-, 
^Uy, yU, respectively and P, Q, R, P', Q', R' for yXx, zXy xXz> zXx, xXy yXz 
respectively so that 
P = \" — ji'^v'", Q = fjr — 7o'-A,'-, R = v' — 7/V''|^ 
P' = X'" — 7iV". Q' = f^'' — JiV', R = v'-- 'y.^X" j ^ ^ 
* We are supposing here that the regression surfaces of .c 011 y, z and of ij on z, x are also hyper- 
boloids of type similar to (35). 
