160 



The force ^i= — 7?/'-i has the potential 



Ui=^piVi' = hpiVi^ ; 



for 



{Fi)x = — ^-=: — piVi—- = — piviai etc. 



Ox Ox 



The whole potential therefore amounts to 



As the equation Vi^^aix -\- ^n =^0 has the weight p/, the 

 mean error of weight 1 is determined by 



' - n-N ' 

 hence 



8^ = 



n-N 



At the point P satisfying tlie normal equations the potential and 

 consequently also e^ is a minimum. The "weight" of the distance i\ was 

 /)/. This weight may be determined a posteriori, if we know the 

 influence of the space Vi alone acting upon any point. We then 

 have but to divide the amount Fi of the force ;5/ by v, . 



II. In order to find the weights of the unknown quantities, we 

 now remove the origin by translation to the point P, which satisfies 

 the normal equations. 



Calling the minimum potential U^, denoting the new coordinates 

 by x', y\ z', . . . and introducing 



Vi' = aix' + ^iy' + Yiz' + ... = 2aix', 

 we obtain 



[piVi'^ = 2{U- U,) = 2U'. 



So Ü' is the difference of potential existing between a point 

 [x', y', z', . . .) and the minimum point P. 



The equation Qj/ I^- "^] = 2 6'' represents a quadratic {N — 1)- 

 dimensional space 52, closed (ellipsoidal) and having P as centre. 

 This space is an equipotential space and at the same time the locus 

 of the points of equal s. We shall call these spaces ^i, briefly hyper- 

 ellipsoids. The hyperellipsoids JÖ are homothetic round P as centre 

 of similitude. 



Introducing the principal axes as axes of the coordinates X,y, .^, ..., 

 we obtain for ii an equation of the form 



AX' + BY' -\- CZ' + ... = 2U'. 



The components of ^ in the directions of the principal axes are 

 found to be 



