XVIL.—NORMAL CORRELATION. | 
array of z, or of z, in its mean, and as the distribution of every 
array is symmetrical about its mean, RR must bisect every 
horizontal chord and CC every vertical chord, as illustrated 
by the two chords shown by dotted lines: it also follows that 
RR cuts all the ellipses in the points of contact of the horizontal 
tangents to the ellipses, and CC in the points of contact of 
the vertical tangents. The surface or solid itself, somewhat 
truncated, is shown in fig. 29, p. 166. 
6. Since, as we see from fig. 50, a normal surface for two 
correlated variables may be regarded merely as a certain surface 
for which » is zero turned round through some angle, and since 
for every angle through which it is turned the distributions of all 
x, arrays and x, arrays are normal, it follows that every section 
of a normal surface by a vertical plane is a normal curve, ze. the 
distributions of arrays taken at any angle across the surface are 
normal. It also follows that, since the total distributions of x 
and x, must be normal for every angle though which the surface 
is turned, the distributions of totals given by slices or arrays 
taken at any angle across a normal surface must be normal 
distributions. Rut these would give the distributions of functions 
like a.z,+b.x, and consequently (1) the distribution of any 
linear function of two normally distributed variables x; and z, 
must also be normal ; (2) the correlation between any two linear 
functions of two normally distributed variables must be normal 
correlation. 
To find the angle § through which the surface has been turned, 
from the position for which the correlation is zero to the position 
for which the coefficient has some assigned value r, we must use 
a little trigonometry. The major and minor axes of the ellipses 
are sometimes termed the principal axes. If &, & be the co- 
ordinates referred to the principal axes (the &-axis being the 
x, axis in its new position) we have for the relation between £5 
&y xy, x, the angle 6 being taken as positive for a rotation of 
the z-axis which will make it, if continued through 90° coincide 
in direction and sense with the z-axis, 
§ =x,. cos 0+x,. sin 0 8) 
§y=a,. cos 0 — z,. sin 6} ( 
But, since ¢; &, are uncorrelated, 2(£,¢6,)=0. Hence, multiplying 
together equations (8) and summing, 
0= (03 - 03) sin 26 + 25.00, cos 20 
27.000, 
tan 26 = He 
321 
(9) 
01