16;^ 



coincides witli tliQ set of' points of the space r = at infinity, thus 



tlie tangent cylindric space, the generator-spaces of which are parallel 



to the ,r-axis, we find for this cylindric space tlie equation 



g^x'-' ^- 2 U'. 



1 

 Its parameter is — , or the reciprocal value of the weight of the 



direction x. 



III. We now snppose, that the variables x, y, z, . . . must at the 

 same time satisfy the following v ricforous equations of condition 

 ^j{x,y,z,. ..) = {j=\,...v) 

 Then the point P is constrained to the common {N — r)-dimensional 

 space fP of intersection of the r (A^ — l)-dimensional spaces 0; . 



Now the point P, subjected to the elastic forces %i, is in e(pii- 

 librium, when the resultant % =: [5i] is perpendicular to *P. 

 Let the normal at P to 0^ have the direction cosines 

 b^j ö<2>y h^j 



d.v -. 0?/ d; 



' iy = 7-^-7^-^2 ' rj' = ^r.-r-rT, , ^^■ 



^K^') ^^{^) ^^{j:: 



The normals at P to the spaces 0/ form a linear r-dimensional 

 space. In this space 5 must lie, which means: '^ can be resohed 

 in the directions of these normals, the unit-vectors of whicii will be 

 denoted by \Vj . 



So we have 



Ö — [qj wj]' 

 where [ ]' signifies the summation over j from 1 to v. 



The components of this vector-equation are 

 [j),v,ai] + [qj a,j']' = 0, [p^vi^i] + [qj ^y]'=:0, [p.ivy/] + [qjYj']' — {), etc. 

 or 



[piCt,'] X 4- [piai'^,] y + [p,«n'/] ^ + . . . + [piiiilH] + ['Ij «./]' = ^' 



[Pi^iui] X + [picir] y 4- [/>//ir//] ^ -1- . . . + [piiiiiii] + Vqj ^j] = 0, 

 [piYiai] X + [pr/i^i] y + VpiYi'] c + . . . + [piyi^ii] 4- [qj y/J' = 0, 



Putting 

 we may write the above e(juations in the foi'm 



00/ 



qj' = qjV/2[^], (./ = l,..r) 



d0; 



[$'/«<•'] 'V -f [gid^hi^ y 4 l.^Afl/c, J c -t . . . + L'Aff/m/] -j- [qj' 1' = 0, 



Ox 



