64 



Mr. J. D. Hamilton Dickson. 



[Jan. 21, 



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 % meet them on the line viz., on OP. 



3. Sections of the " surface of frequency " parallel to the plane of 

 xg, 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 zOP. 



Sections of the same surface parallel to the plane of yz are got 

 from the exponential factor (1) by making x constant. The result is 

 simplified by taking the origin on the line OM. Thus putting x—x^, 

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



Vi c fol- 2 ?/l -Q 



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

 the exponential takes the form 



1 (1-22)2 9(1*50)2 J* 7 T l(i-22) 3 9(l-50) 2 J - ' 

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



e 2 (1-22)2^9(1-50)2 ■ 



or [on referring to (3)] 6=1*50*/ ^ 



V 17-6J 



(5) 



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



nearly -i_ times that of those parallel to the plane of xz, and the 

 v J!i 



locus of their vertices lies in the plane 2OM. 



It is important to notice that all sections parallel to the same co- 

 ordinate plane have the same probable error. 



4. The ellipses (2) when referred to their principal axes become, 

 after some arithmetical simplification, 



20-68 5-92 



4-^— -= constant, (6) 



the major axis being inclined to the axis of x at an angle whose 

 tangent is 0'5014. [In the approximate case the ellipses are 



^_ + ^L-= const., and the manor axis is inclined to the axis of x at an 

 7 2 J 



angle tan -1 \.~] 



5. The question may be solved in general terms by putting 

 YON = #, XOM=0, and replacing the probable errors 1*22 and 1*50 

 by a and b respectively : then the ellipses (2) are 



