491 



operates with vectors, restricts liimself to two variables and one 

 rigorous equation of condition. 



It is our purpose not only to extend their method to the case of 

 an arbitrary number (iV) of variables and an equally arbitrary 

 number (r) of conditions, but also to derive the ?y6^((//</ of the unknown 

 quantities in the same way. 



I. There are given JSf quantities x, y. z, . . . which are to be deter- 

 mined from n (approximate) equations of condition (equatio]is of 

 observation) : 



ttix -i- biy -\- Ciz -\- . . . -}- mi =0 i=l, . . .71. 



These equations have the weights gi resp., and so are equivalent 

 to the equations 



ai [/gi . X + hi \/gi .y + Ci \/gi . z -\- . . . + nii j/r/,- =: i = l, .. .n, 



each of which has the weight unity. 



We now introduce 



ai \/gi hi \/gi q \/gi mi [/gi 



a;: 





v\9i<^n \/[9ihi'\ vhi^n u'bim'] 



A r= w \/[gia,% B = y Vlgih^l C = z \/\gi<^-'\. - ■ • M= [/ \gimi'] 

 Ai = Aai = ai \/gi . .v, Bi = B(3i = hi \/gi .y, Ci— Cyi = ci \/gi . z, . . 



. . . Mi = Mm — mi \/gi , 

 [ ] denoting summation over i from 1 to n. 



So the equations of observation run in the form 



Ai^Bi-^ Ci+ ... J/,-=0 i = l,...n. 



We now consider J/, Bi, Q, . . . Mi as the components of the 

 vectors 31, 55, (^, . . '?^i, resolved parallel to the rectangular coordinate 

 axes of an ?z-dimensional space. Thus the tensors are A,B,C, . . . M, 

 «; 5 /?/, Ym . . . ft/ representing the direction cosines. 



The set of n equations of observation may now be condensed in 

 the single vector-equation 



31 ^ ÏB + (£-!_... 4- .?3^ = 0, 

 which expresses, that the vectors 21, 35, (i, , . .93i must form a closed 

 polygon. The coefiicients ai, bi,Ci, . . . and the weights ƒ// being given, 

 the unit vectors a, b, c, . . . of the vectors '^l, ^, 5, . . . are determi- 

 nate. So the vector-equation requires that ^'il may be resolved in the 

 jS^ directions n, b , c , . . . , in other words : that ?)i lies in the A'-dinien- 

 sional space Ry, determined by the vectors a,b,c,.,. and called 

 the space of the variables (or unknown quantities). 



In consequence of the errors of observation this condition is not ful- 

 filled. The most probable corrected value of ^ is the projection of 

 ^^i on the space Ry of the variables. 



