in the Position of a Point in Space. \^>\ 



If we have both a^ = oo and aj^oo, (34) becomes the ordinary 

 symmetrical probability surface, and the curves of contour reduce to 

 ellipses 



i':+i!.=_l^i-^. (41) 



To show the modifications of form which these curves undergo 

 from the introduction of the unsymmetrical elements represented by 

 the constants a^ and a^, let ns first consider the ellipses (41) as circles, 

 with Ji=&2=-*- l*«t,ting x^ -{-y^znr^, the squared radius of such a 

 cii'cle is 



loge 



Giving to Z' the values .01, .02, .03 in succession, remembering that 

 K^ = l and K3=:l, (37) gives for c the values 



.25133, ,50266, .75399, 



and the corresponding radii r are by (42) 



3.324, 2.346, 1.503, 



with which the three concentric circles in Fig. 1 are described. 



If we now suppose that there is a c. ra. inequality in the x direc- 

 tion, so that aj has a finite value, for instance «i = ], while b-^ and h^ 

 remain as before, then by (40) the equation of.any curve of contour is 



5.262 log ( 1 +-j — 8,<- — 18.421 log c. (43) 



Giving to Z' the values .01, .02, .03, .04 in succession, with 



K, = 1.0211, K2 = l, 



the values of c are by (37) 



.25663, .51326, .76991, 1.0265, 



which being substituted in (43), give us four equations by which the 

 four curves of contour can be constructed as in Fig. 2. They sur- 

 round the point a5= — 1, .V=0, for which Z is a maximum according 

 to (36). The surface cannot extend beyond the dotted line drawn at 

 the distance x=i—a^h^z:z—4: from the Y axis, so that all the curves 

 of contour lie wholly to the right of this line. 



Again, suppose that while b^, b^ and a^ remain as before, ci^ has 

 also the finite value a.^^l due to a c. m. inequality in the y direction. 

 Then by (38) the equation of any curve of contour is 



y 



