or 



it)2 



1 ' ' 



1 y' z' 



Ifnai^] \- [piaibi] , + [fiiaiCi] ^ -f ... — 1 = 0, 



Ox !U-''' 9x-v 



\flibiai]~ + [g^br] -^ 4 [9ib^c!] -^ + . . . -f- =: 0, 



9x 9^^ 9x^ 



1 y' z' 



[giciai] j- [giCib,] , -f [91 ci-] + ... + = 0, 



9x 9x-i^ 9^^'^ 



So — is apparently found as the first iinl^nown quantity in the 



"modified" normal equations, modified in this way, that the constant 

 terms are replaced by — 1,0,0,... resp. 



Considering Ü (c.q. Ui) as an (iV -|- 0''' coordinate perpendicular 

 to the xV-dimensional space {x, y, z, . . .), the equation 



represents a quadratic space of N dimensions, built up of x> (xY— 1)- 

 dimensional linear generator-spaces, all parallel to ( Fi= 0, t7=0), 

 the intersections of which with the planes perpendicular to (Fi = 0, 

 U=^0) are congruent parabolae. The parameter of these congruent 



parabolae is — . 

 pi 



The quadratic space pi Vi^==1Ui will briefly be called a 2:>«rö!^o//c 

 cylindric space imt/i pai^ameter — . 



Pi 



The equation 



[piV,^] = 2U 



represents a quadratic space ¥^ of A^ dimensions, the centre of which 

 is at ^=Go, and the intersections of which with the xV-dimensional 

 spaces U =: const, are hjperellipsoids i2. Thus W is the extension 

 of the elliptic paraboloid. 



The point T of »r with minimum U {(Jo), and hence closest to 

 t7=:0, which is called the summit of W, is projected on ^^=0 

 in the point P, satisfying the normal equations. 



By displacing the system of coordinate axes {,c, y, z, . . . , U) (by 

 translation) from to T, W obtams the equation 

 [p,F,'^] = 2 U'z=z2{U-U,). 



By constructing the enveloping cylindric space, the vertex of which 



