494 



Now, from A = x\/[giai^'] ensues 



hence 



1 a/ 1 ba' X' 



which is the well-iinown theorem on the weights of the variables. 

 Example: 3 equations of observation with 2 variables. 



The unit-vectors a and b determine a 

 plan R^. The extremitj^ Moï — ^=OJI 

 is projected on this plane in the point 

 M\ OM' is resolved parallel to a and b 

 Q into the components 0A = ^ and 0B=^. 

 In the plane H^i^, 53) the vector é is 

 erected perpendicular to ^. On this vector 

 "A OM = —^^i and 0A = ^ have the same 

 projection OAs = Ms. This segment 3fs 

 has the mean error 8 ; the variable A, i.e. the segment OA there- 



8 



fore has the mean error 8^ = -. 



cos AOAs 



III. We now suppose that besides the n approMinale equations 

 of condition (equations of observation) v rigorous equations of con- 

 dition are given, viz. : 



anJ^jX + hnJ^jy + Cn+jZ + - f ^^n+j = (_y"=:l , ... v). 



For the sake of regularity in the notation, ^ve will also provide 

 these equations with factors gn-\-j fw4iich afterwards disappear from 

 the calculation). Thus we really operate with 



anJ^j Vgn+j-X + hnJ^ygn^j.y + CnJ^j V9n+j.Z "h - + m^^j \/ CJ,,^j=0{j=\,. r). 



Agreeing, that [j] now means a summation over / from 1 to 

 n -\- V, we may, retaining the notation used above, consider 3i. ^5, 

 ^. . . .,^yt as vectors in a space of n -\- v dimensions. 



The vector-equation 



51 -f ':^ -f (1 -t- ... + ^yt ^ 



is again not fulfilled on account of the errors of obser\ation. The 

 last r component-equations {n -f 1) . . . {n + »') however hold exactly 

 this time. 



Putting again 



51 + 05 + ^' + . . + m = ^ 



the V projections Kn-\-i, . • Kn-^. of ^ must be zero, whence 



y.n-\-j = (;•— 1, ..r). 



