49^ 



move in the space S, common to the normal space of ^'^r—v (of « -|- 2r 

 — jS/ dimensions) and the space /^'«parallel to /i„. The space /S obviously 

 has (??-|-2r— i\^ ) + ^2- — in-{-v) z=7i-\-v — N dimensions. A component 

 of ^ in this space has no effect on the vectors 51, 03, <5, ... A com- 

 ponent of ^ will only have any effect on '^1, 0\ (5, ..., when it lies 

 in the normal space ^S'^V of S, which has n -\- v — {n-\-v — A') = JS/ 

 dimensions. By translating this normal space S'n to 0, it contains 

 both Rj and ^a_„ (intersection of Rn and /t„). 



The variation of ^yit will exclusively influence vl, when the com- 

 ponent of ^1 undergoing this variation is perpendicular to 05, (Ï, . . . 



These considerations lead to the result that we want that direction 

 t> , which lies in S'y and is perpendicular to 03, ^", . . . The vectors 

 ^^, <J, . . . determine together a space of iV — 1 dimensions. The vector 

 é must lie in the normal space (of n -\- v — iV -|- 1 dimensions) of 

 the space (33, (i, . . .). This normal space cuts S'n in a space of 

 {n -{- V — iV + 1) + iV — {?i -\- v) =z 1 dimension, hence in a straight 

 line. So there is always one and only one line i fulfilling the 

 imposed conditions. 



Since ^ lies in S' ]sj, i. e. in the space joining Rj with qjv—^, the 

 projection t of é on Rn will fall into qn—^- 



Now we have for the direction cosines t/ of the projection t of 

 c> on Rn ■ 



n = — f-^T^, (^* = ^ V ^0 ; rn+J = (j = 1 ,..,r). 



As t, being a line of Qy_,„ also lies in the space Ry and therefore 

 may be resolved in the directions 01, 03, (i, ..., we have 



Tk = Pak + Qfh + Rn + ... , (/* = 1 v-.^O 



T„+/ = Pitn+j + Q,-^n+j + Rrn-\-j + ... = 0. (J = l,- .l') 



Putting 



Pi/[<^in'=P' ' Q[/[<yh^'=Q' , RVle,']' =^ R' ... 

 we obtain : 



txhP' + /^i>Q' + YhR' + - = ^/' . (^* — 1^ "^0 



an+jP' + /^n+jQ' + Yn+jR' + .•• = , {j = l,...r) 

 and, é being perpendicular to ^, ^,..., 



[^,(7.] = 0, [YiOi] = ... [ar\ = l. 



In this way we have collected n -\- v -{- N' equations to determine 

 the n -{- V unknown quantities a^ and the N unknown quantities 

 P\ Q', R',... . 



S'n being perpendicular to ^, ^ is also perpendicular to S\. By 

 multiplying the equation 



