134 M L. De Forest — XJnsyrnmttriccO Law of Error 



coefficient whose coordinates reckoned from ?/'=:^,^^ ,(. are adx^ hdy and 

 cdz will be 



^o + adM + hd,w + cd.io -\ — d^ ic^ — d^G -f —dho 



+ ahdJlyV: + acd^d^w + bcd/l^ic^ (50) 



and all the coefficients in the block will be successively represented 

 by assigning to a, h and c all the integral values between —m and m. 

 Suppose that all the values of I in (46) have been thus represented. 

 Collect the coefficients of w, d^w, d,jio, etc., remembering that ^L=l. 

 Let a ^^ a'g, a^ denote the sums of the products of each L into its 

 first, second and third sub-indices respectively. Let (9,, §2-> ^3 ^^^ ^'^^ 

 sums of the products of each L into the squares of its first, second 

 and third sub-indices respectively. Let y ^, Y21 7 3 ^^ ^'1*^' sums of the 

 products of each L into the product of its first and second, first and 

 third, and second and third sub-indices respectively. Let (J,, (J,, 6^ 

 be the sums of the products of each L into the cubes of its first, 

 second and third sub-indices respectively. Let tf^, 7/2, //g, 7/4, >^^, ij^ 

 be the sums of the products of each L into the product of the second 

 sub-index into the square of the first, the product of the third into 

 the square of the first, that of the first into the square of the second, 

 that of the third into the square of the second, that of the first into 

 the square of the third, and that of the second into the square of the 

 third, respectively. Let be the sum of the products of each L into 

 the continued product of its three sub-indices. We can now bring 

 (46) into the following form. 

 w—a^djo—a/IyW—cK^djc + \fi^d^'w -f \fi^d^w -f \fi^d'^o^ 1 



+ }\dxdyto + y„d/l,w + y/l„d^w=y. 



— a^w -\- fi^d.w + y/7„u' + y/Lw — ^S^d/w—^7/^d,j'w—h/^d,'iP 



— // d^djr — ?)d,d.w — 6d„d.w = V. 



— c)/„ic -t- y^dxio -f ft„d„v} -h y./l-ji' — hf^d^^io—^d./lJ'w—hj/Vtr \- (51) 



— iijIjIm — QdjdM — iid<d.xoz=. — '-- V. 



^. 



These equations may be simplified by a suitable choice of the 

 coordinate axes. Any coefficient L^ j_, represents the probability that 

 an error which occurs will fall at the point .r=(iJ.'\ y=zhJij, z^cJz. 

 If these coefficients are also regarded as the masses of material points, 

 and their center of gravity is taken as an origin, we shall have n ,=0, 



