222 



NATURAL INHERITANCE. 



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

 to meet the ellipses on the line, 



that is 



y 9 3a; - 2 # -o 



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



6 

 9(l-50) 2 _ 6 



1 4 17-6 



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



(3) 



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 = §y, 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 1 

 and y = y 1 + y', where by (3) 



Vi 



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



the exponential takes the form 

 4 

 (1 



I (1-22V 2 



ty» + r 



.'/r 



(3x l - 2y A 



f ' 9(1-50)- j * \ (1-22 f 9(1 -50; 2 

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



1} 



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



or [on referring to (3)1 e = 1-50 /■ 



° v /J V 17-6 



(4) 



(5) 



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 

 V - 



locus of their vertices lies in the plane 2OM. 



