495 



So the vector ^ is perpendicular to tiie space R, "of condition'* 

 determined by the coordinate-axes r„_|_j and therefore cannot generally 

 be an}- longer assumed to be perpendicular to the space Z?^v('^,'-S,^v) 

 of the variables. ^ lies in the ?i-dimensional space R'„ ,t„^j =: M„-\-j 

 (J=:l,.r), which is parallel to the space Rn "of observation" determined 

 by the axes Xh (/i=l, . . n). 



The parallel-space R'n cuts the space Ry of the variables in a 

 linear space of xV -|- ^^ — {?i-\-v) = N — v dimensions, which we shall 

 denote by q'x—,. This latter is parallel to the space gy-v of inter- 

 section of the space Rn of observation with the space Ry. 



We now project the extremity of ^1 lying in /^'„ m this space 

 on the space Q'y-, of intersection. The projecting vector will now 

 be the "correction-vector" ^. 



Translating iv to the origin into the -vector OP, OP will be per- 

 pendicular to the space Qy-j common to Ry and /?„. 



Next we construct the normal space of Qy-, which passes throngh 

 the origin 0. This space has 7i-\-v — {JST — v) = 7i-\-2v — JSI dimens- 

 ions. It contains the space R, of condition (as normal space of R,,), 

 further the line OP, and also the normal space of 7i -\- v — JSf 

 dimensions which can be drawn from P perpendicular (o Ry. 

 This latter space therefore lies together with R., in a space of 

 n-\-2v — ^dimensions and thus cuts R, in a space of (?i-|-r — A^) + 

 -{- V — {7i-\-2v — iV) = dimensions, consequently in a point. As for 

 this point Q, it thus lies both in R, and in the normal space drawn 

 from P perpendicular to Ry, from which among other things 

 follows, that PQ makes right angles with each line of Ry, more 

 particularly with the vectors 21, 03, ^, . . . So, projecting OP and OQ 

 on 21, these projections are equal. The same holds for the projections 

 on ^, (è, . . . 



Representing OQ by the vector ^\ {[i',üi', lu'), we have, as ^' 

 lies in R.>, 



Kh' = and y,h = 0. (/t=l,...n) 

 From 



01,31) = (J?',2I), (je,5g) = (.^V^B), i^,^) = (^',e), . . 

 follows 



As }{„_^j = for J =: 1, . . . r, the sum [x/«;] is only to be ex- 



tended from 1 to ii; hence [x/«/] = ^"" x/j«/, = [>c/j«/,]'; and since 



1 



jj/,' = for /i = l,..?i, the sum [x/V.v] is to be extended from ?i-[-l 



