222 NATURAL INHERITANCE. 



by differentiating (2) and putting the coefficient of dy equal to zero, 

 to meet the ellipses on the line, 



(1-22)2 9(1-50) 2 



6 



y 9(1-50) 2 6 



that is = 



x 14 17-6 

 (1-22)2 9(1.50)2 



. . . (3) 



or, approximately, on the line y = - s 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 = ?/, 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 fre- 

 quency 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 a; constant. The result is 

 simplified by taking the origin on the line OM. Thus putting x = x l 

 and y = 2/i + y '> where by (3) 



(1-22)* 9(1-50)2 



the exponential takes the form 



i -I +- 4 l,/'+ f yi + i-i 

 \ (1-22) 2 9(1-50) 2 ) J \ (1-22) 2 9(l-50j 2 



whence, if e be the probable error of this section, 

 1 14 



* (1-22)2 9(1 . 50)2 



__ ) . . . . . 



or [on referring to (3)] e = 1-50 / 



that is, the probable error of sections parallel to the plane of yz is 

 nearly ~j=- times that of those parallel to the plane of xz, and the 



locus of their vertices lies in the plane zOM. 



