396 
On the Partial Correlation Ratio 
are used, including of course the equations given by the "least squares" method 
or maximum correlation method. Even without introducing high moments, when 
there are three variable characters new equations may be obtained by the method 
of moments, by combinations of characters which do not arise in the other methods. 
The method of moments is most convenient for our purpose, but we shall only 
employ those equations which can also be justified by (say) the method of least 
squares. 
For convenience let the origin be taken at the mean of the three characters so 
that x = y = z — 0, let q^syt-u denote 
SSS [n^y.x'fz' '} ^ ps,t,u .27^ 
Ar_ s _ « _ « _ ,s- _ « _ M )• 
With this notation r^y and qx^yiz" are identical; when z does not appear in the 
product, it is sufficient to write qj^syt. 
The most reasonable next approximation to make when a linear function 
(}){x,y) does not adequately I'epresent the statistics is ^{x,'y) = s, quadratic 
function of x, y. 
T , z T ax by cxy ex^ fxfi 
Let — = d+ — + + — ^ + — + — „ (28). 
Multiply (28) hy n.^yz, sum for all values of y, z and divide by N 
0 = + cvxy + e +/ (29). 
Multiply (28) in turn by times — , — , and sum as before, 
(Jx o-y a^a-y o-x' o-y" 
and we obtain 
xy 
= « + hr^y + cq^iy + eg^:i +fqxy'' (30), 
= h + ar^y + cq^yz + eq^'y +fqy^ (31), 
qxyz = dr^y + aq:^iy + hq^y. + cq^iy + eq^^y +fqxy^ (32), 
qx^z =d+ aq^i + bqof-y + cq^.iy + eq^^ +fqxhf- (33), 
qy2^ =d + aq^y'- + hqy, + cq^y-^ + eq^-^y +fqyi (34). 
Actual numerical fitting shows that in many cases e and / are small compared 
witli c*. This is the case when the regression of z on x for a constant y and of 
2 on y for a constant x is linear. We shall therefore confine ourselves in the 
present preliminary paper to the case where we may write 
z , ax by cxy 
- = d+ — + ^ + — ^ (35). 
Here for constant x or y the regression o{ z on y or of z on x is linear. 
* Cf. Census of Scotland, 1911, Vol. iii. p. xlvii. where Mr G. Kae obtains by moments the 
regression of fertihty on age of husband and wife. Let Jr=age of wife, fl = age of husband, C = number 
of children in completed marriages. He finds 
C,^j^ = 20-149193 - 0-555812IF- 0-173804ff- 0-0028161^2 _ 0-003494H2 + 0 -012675 IFif. 
See also the paper by E. M. Elderton in the current part of this Journal, pp. 291 — 295. 
