F. M. Turner 499 
Too r+QO 
I I zxdxdy 
and X = '-j^j^ (III.). 
I zdxdy 
J hJ -'^ 
Since the centroid of every y array lies on the regression line, the centroid 
of the figure must also lie upon this line. Consequently 
y = x .r (IV-)- 
All the integrals in (II.) and (III.) ean be reduced to the probability integral or 
directly integrated. 
Putting e ' = V 27rp and f e ~ ^'"^ rfa; = V 27rq, 
J h' 
we have I I zdxdy = -j^^ — I e <rx' dx 
J hJ -00 \/27r ax-Jh 
= Nq, 
and I xzdxdy=-r:— — e '^^ixdx 
J hJ - oo v27r a-xJ h 
= Np.a-x, 
:.x = P^a, (V.). 
From (IV.) y = r.^^a, (VI.). 
Substituting the values from (V.) and (VI.) we obtain the following values for 
the three integrals in (II.) : 
hpq -\- ([' —p^ 
I I (x — xf zdxdy = Nax" 
J h J - 00 
(V - yy^d^ dy = Na,f —^^^——^ ^ , 
J h J —00 q 
h pq + q^ — p' 
•y ~ 
rx, r+oo 
I I {x — x){y — y) zdxdy — NraxCTy 
J h J — 00 
Substituting these values in (II.) we obtain : 
_ r-{hpq + q^-p") 
.(VII.). 
{hpq — p'') + q- 
or r- = (Vlin 
f/ + (l - r''){hpq - p') ^ 
From (V.) we see that the distance between the centroids of the curtailed and 
the whole figures, resolved along the axis of x, is —ctx- The distance of the 
limit of the curtailed figure from the latter centroid resolved in the same way is 
h' = ]i(Tx', so that ^ — }i is the expression for the distance of the curtailed centroid 
from the limiting straight line. It follows that - — /; is always positive. 
