158 



"e(ir.ations of observation") also j-i(/orous equations of condition 

 are given. 



Moreover, in either of tliese cases also the weights of the unknown 

 quantities will be derived from mechanical considerations. 



The method here developed is founded on the statics of a point 

 acted upon by elastic forces and is in principle closelj^ related to 

 the procedure of the h\st-nientioned mathematicians. 



To obtain general results, we will operate with an arbitrary 

 number (xY) of unknown quantities or variables, which are consi- 

 dered as coordinates in A'^dimensional space. In order to render the 

 results more palpable, we shall, at the end, recapitulate them for 

 the case of two variables. 



I. To determine the N^ unknown quantities 



a?, y, z, . . . . {N) 



the n (approximate) equations of condition or equations of observation 



a; w -\- hi y -)- ci z ■-\- . . . -\- ?n< r= (i =: 1 , . , . n) , 



are given, with the weights g-, resp. 



In the sums, frequently occurring in the sequel, we will denote 

 by -2" a summation over the coordinates x,y,z,... or over the 

 corresponding quantities (for inst. their coefficients a; ,h; ,Ci , . . .) and 

 by [ ] a summation over the n equations of obser\'ation, thus over 

 i from i to n. 



Putting accordingly 



or + br + c,^ + ...:=^ai^ 

 and introducing 



a; hi Ci m: 



we may write the equations of observation in the following form 



Vi = m .r + ^,- ?/ + y^ 2 + . . . + fi/ = (/ = 1, . . . n) 



or 



Vi ^ ^ai X + \ii = (i == 1, . . . n). 



These equations have resp. the Aveights 



ƒ>,•=:ƒ//JS'aJ■^ 



The equations F«=:0 represent (lY — 'I)-dimensional linear spaces; 

 their normals have the direction cosines («/ , ;j/ , y? , • . •) resp. 



In consequence of the errors of observation, the approximate 

 equations F/ = are incompatible; in other words: the n linear 

 spaces F/=0 do not meet in the same point. By substituting the 

 coordinates x, y,z, . . . of an arbitrary point P in the expressions 



