493 



8 



SA — : , 



whence 



The vector ^, \ying in Ry , may be resolved in the directions 

 ii,l\c,... Denoting its components in these directions by X,Y,Z,... 

 we find 



é = Xa + rb + Zc + ..., 

 or 



öi=Xai+ Y ft i-\-Zy, -{-... 



Now, ^ being perpendicular to ^, <5,..., whence [/:?/(T/]=0,[y/a/ J=0,.., 

 we have 



\ = [öi^] = K[aiOi] 

 or 



1 



From the equations 



[«/ <^/ ] = ^ , [fti 0/ ] = 0, [y, <7, ] = 0, . . 

 wliich may also be written 



[«/'] X + [ctifti] Y + [aiYi] Z + ... = Y , 



or 



K] X^ -f [«,/?,] Xr + [rr, y;] XZ + . - 1 z= 0, 



[/y/«;] X^ + [(ie] XY + [,/:?rA] ^Z -f .. + == 0, 



[y/«/] A- + [yift,] XY + [y.^] XZ + .. + = 0, 



the first unknown quantity A' takes tlie value 



[ctiöiY e' (JA 

 The reciprocal \alue of the weight of A is therefore found to be 

 the lirst unknown of the "modified normal equations". 

 Putting further 



X = él/[rA«.-1, Y^ii\/[gihrY Z = g[/[gicr],... 

 the modified normal equations pass into 



{(/ic^ r + [giciibi] |ii + [giaia] §? -f ... - 1 = 0, 

 [gibiüi] è' + [gibn ^li + [gibiCi] ^i; + ... + = , 

 [giciüi] r + [gi^-ibi] in + [r//^/1 èS + -. + = 0, 



