167 

 The weight of ,r is thus defined bj' 



_ _ 1 



a' 



It may calso be found by the following calculation 



krl 



— = [:EkiaiA] =: 2A [/•/«/] = A \kia;] + D [k;hi] + C [//c/] + . . 

 pi _| 



= A~A [k;a;]' - B [kj'^/y - c [k/y/y _ . . . 



= A — [k/:SajA]' z=A=-, 



9? 

 so that gx is also determined bv 



1 



«-■(• 



By considering the quantity U as (A^ -|- 1)^'' coordinate perpendi- 

 cular to the iV-dimensional space {d', y, z, . . .), the equation 



[/vT,'^]~2[^,u7]' = 2tr 



represents the quadratic space ¥^. The origin of the coordinates 

 ^',i/,5',..£" now lies at the point S, the projection of which on 

 Ü' ^= — f7^(f/=:0) is the required point. Now this point S is not 

 the summit of W. 



The linear space of conditions R of N — v dimensions is now 

 joined to (he point /''^ocbyan (iV — r -f- l)-dimensional space/?,, 

 which passes through S and intersects the quadratic space 'F in a 

 quadratic space ^V\ having the same character as W, in that it 

 also has its centre in U' = go, but is of fewer dimensions, viz. 

 iV^+(A^— r+l) — (iV+l) = iV — r. The quadratic space W^ has 

 its summit in S. 



We now have to determine the points Q in ^\, at which the 



((r-l-l)-dimensional) spaces of normals are parallel to the ^'-axis. In 



such a point Q ^V ^ is also enveloped by a parabolic cylindric space, 



the generator-spaces of which are parallel to the .r-axis, and which 



therefore has an equation of the form 



9x ê" = 2U'. 



. 1 

 lis parameter is — . 



93: 

 ^ 1 



In other words: — is the parameter of the parabolic cylindric 



9x 



space, which has its generator-spaces parallel to the .r-axis and 

 envelops the quadratic space W^ . 



V. We conclude this paper with a short summary of the results 

 -for the case of t/i'o xariables .c and >/. 



12* 



