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222   NATURAL INHERITANCE.

by differentiating (2) and putting the coefficient of dy equal to zero, to meet the ellipses on the line,

y _ 23x-2y_0~ (1.22)2   9(1.50)2

6   '   .   .   (3) that is   y=   9(1.50)2   = 6

X 1   4   17.6

(1.22)2 + 9(1'50)2

or, approximately, on the line y = x. Let this be the line OM. (See Fig. 11, p. 101.)

From the nature of conjugate diameters, and because P is the mean position of p, it is evident that tangents to these ellipses parallel to the axis of x meet them on the line x = ly, viz., on OP.

3. Sections of the " surface of frequency " parallel to the plane of xz, are, from the nature of the question, evidently curves of frequency with a probable error 1.50, and the locus of their vertices

lies in the plane z OP.

Sections of the same surface parallel to the plane of yz are got from the exponential factor (1) by making --constant. The result is simplified by taking the origin on the line OM. Thus putting x = xl and y = v1 + y', where by (3)

y1 = 23x1- 2y1 = 0

(1.22)2   9(1.50)2

the exponential takes the form

1 +   4   t Z,12+

(1.   ?/12 2 + (3x1- 2y1)2 t   (4)

22)2   9(1.50)2 )   1(1-22)   9(1.50)2

whence, if e be the probable error of this section, 1 __ 1   4

e2   (1.22)2 + 9(1.50)2

or [on referring to (3)] e = 1.50   9   (5)

17.6

that is, the probable error of sections parallel to the plane of yz is nearly 22 times that of those parallel to the plane of xz, and the locus of their vertices lies in-the plane zOM.