L. ISSERLIS 
397 
Equations (29) to (32) become, when the regression surface is given by (35), 
0 = d + crxy (36), 
rxy = a + br^y + cqxh, (37), 
= b + avxy + cq^y 
.(38), 
fjxyz = dfxy + aqx^i + hq^yi + cq^iyi (39), 
= - cr'''xy + aq^iy + hq^y^^ + cq^-^y^ by (36). 
Solving these equations we obtain 
a h 
xy 
1 ^xy'' "^yz 
qxy'' ^x-y- ''"it;/ ^xyz 
^xy'^ ^'xy '^'yz 
qx'y- ~~ ^"xy 1x-y Ixyz 
1 r„ 
^'xy 
xy 
liz 
xy 
1 
q.x-y 
qxy- 
qx'^y 9xy- '-Ix'y- '' xy 
.(40), 
r, 
Txy 1 r, 
qx^y 9.xy- 9xyz 
we have already denoted the partial regression coefficients by 73 and 73' so that 
1. = ^^^ and 73' = '-^^ 
^ ' xy ^ ' xy 
In addition let 
and 
1 _^.2 
xy 
^xyqx^y ^xy' 
(9) and (10). 
(41), 
1 - r' 
xy 
After some reductions the determinants in (40) yield 
u = js + cO 
^ = 7:/ + 
„ _ Orxz + ^Ty^ + q^yz 
also 
Note that 
and 
0qx-iy + ^qxy2 + qx-2y -r'xy 
d = — Cjxy 
ri<ix?y + y^qxy = - {6nx + <i>r,y) 
1z'i'zx + ^%rzy = xy'R^ (cf. eqn. 11). 
.(42). 
.(43), 
.(44), 
.(45), 
.(36). 
.(46), 
§5. By definition (1 - ,,Zr/) = ^i^^^^^i^^ 
•. using (35) 
<Jx O-y 
axCTy) 
ax h y cxy \ n. 
' X" y 
"xyz 
N 
Biometrika x 
+ + + hi- + c-qx^yi + 2dcrxy + 2abrxy + 2acqx%y + 2bcqxyi, 
51 
