I 3S E. L. De Forent — Unnipmnetrical Lav) of T^'ror, eto. 



This final form is the product of three functions like V in (1), «nie 

 in X, one in y, and one in z. Differentiation gives 



ffe Va^&. + z 7 j 



These become zero when we take 



<— — ^ — ^ — ' 



~~ a,' " ~ «2' <^3' 



and at this point W is a maximum. They are also zero when W = 0, 

 and this occurs when x:=—a^b^, or when y=^—cioJ>2-> ^^' when 

 z=— ttgig, and also when either £c, y or z are equal to itoo , the + or 

 — sign being taken according as a^, «g or a^ are respectively + or 

 — . If we suppose parallel planes to be drawn at the distances 



— a^h^, — agig? ~~^s^ 



3 5 



from the YZ, XZ and XY planes respectively, the values of W will 

 all be included within only one of tb'e eight solid angles formed by 

 the planes so drawn. In other words, the law of probability repre- 

 sented by (66) is such that any error which occurs must fall at some 

 point within this portion of infinite space. But in the special cas^s 

 when either »^, cio or a^ is infinite, the limit of possible error is 

 extended to infinity in the a*, y or z direction respectively, and the 

 function W becomes symmetrical in that direction, depending on 

 x'^, y^ or z^^ as (39) shows. If a^, a^i a^ are all infinite, the errors 

 may fall in any portion of infinite space, and (66) is reduced to the 

 entirely symmetrical form 



K=A-- ('0) 



Compare Analyst, vol. ix, p. 68. The expression (69) was there 

 obtained as the limiting form of the system of coefficients in the 

 expansion of a polynomial of three variables, when only first ditter- 

 ences were taken into account. The same result would have been 

 obtained here, if we had neglected the second differences of )r in (50). 



