132 E. L. De Forest — Unsymmetrical Law of Error 



3log(l +j) + 3log(l +-\ — .43429(;r + ;y)— logc=:0. (44) 

 Assigning to Z' the values .01, .02, .03, .04, .05 in succession, with 



Kj=K2=:1.0211, 



the values of c are by (37) 



.26205, .52410, .78615, 1.0482, 1.3102, 



and substituting these in (44), we have the equations of five curves 

 of contour surrounding the point x=z — \, y=: — 1, at which the vertex 

 of the surface is located. The curves can be constructed by approxi- 

 mation, and appear as in Fig. 3. The surface cannot extend beyond 

 the dotted lines 



.x——a^b^ = — i, y^—a^b^ — — ^, 



and is tangent to the XY plane along them. The curves lie wholly 

 within the angle formed by these lines. 



In the foregoing examples we have supposed b^z=b„ and a^^za^. 

 But if these were not equal, or if a^ or 0^2 were negative, it is evident 

 that considerable variety would be occasioned in the form and position 

 of the curves of contour. Moi-eover, the value «,=:±1 or ^2=^1 i^ 

 rather an extreme assumption, and implies a degree of c. m. inequality 

 beyond anything that would be likely to occur in practical applica- 

 tions. The peculiarities in the form of the curves are thus exagger- 

 ated, merely to make them more readily visible. 



To find the unsymmetrical law of error in the position of a point in 

 space of three dimensions, the function which expresses the limiting- 

 form of the series of coefiicients in a polynomial of three variables is 

 to be obtained in a manner strictly analogous to the foregoing. 

 Indeed, the processes for one or two dimensions are special cases of 

 that for three dimensions, and might be demonstrated as such. The 

 coefiicients L, regarded as the masses of material points, are supposed 

 to be arranged equidistantly in the directions of three rectangular 

 axes, the common intervals between them being /Ix, Jy, Az, and the 

 polynomial and its expansion to the nX\\ power may be written 



Then, as shown in Analyst, ix, p. 36, the relation between the 

 whole (2m + 1)3 coeflScients L in the first power, and any similar 

 block of an equal number of coefficients / in the expansion, will bo 



