159 



V; , the latter obtain the values v,- , repi-e.seMtiiig- the distances of the 

 point P to the spaces 1'/= 0. 



The distance from Vi=0 to P is to be considei-ed as a vector 

 ^i with tensor v; and direction cosines «/,/?/, y/, .. . 



We now imagine a force 5/ acting upon P (in ^Y-dimensional 

 space) in the direction of the normal tv (from P to F/=:0)and the 

 magnitude of which is proportional to the distance vi and a factor 

 Pi characteristic of the space V;. (The space Vi^=0, for instance, 

 may be considered as the position of equilibrium of a space ]^=i?j 

 passing through P by elastic flexion.) 



So the space Vi acts upon P with the force 



All the spaces Vi{i = \, . . . 7i) combined consequently exert on P 

 a resultant foi'ce, amounting to 



S=:[g,]=- [p.ul 



This resultant force depends on the position of the point P. 

 Hence we have in ^V-dimensional space a vector-field A. determined 

 by the above equation. 



Now the question to be answeicd, is: at which point Pare these 

 forces Til in equilibrium? For thi« point P we have 



or 



The "components" of this vector-equation in the directions of the 

 axes are 



[pi vi cii ] = 0, [pi vi^i J = . ipi vi Yi] = 0, . . . . 

 Substituting for vi the expression Vi:= ^ c(ix-\- (.li, we obtain 

 [picti'] X -f [p/«,|?,] y + [piaiYi] z -\- . . . -[- [p,«/|t/] = 0, 

 [pi^iai] X -f- [/?/^^■'] y + [pi^iTi^ s + . • . + [pi^iiii^ = 0, 

 [piYicti^ X H- [p/y//?/] y 4- [pr/i''] z -\- . . . + [piymi^ = 0, 



or by 



(li bi Ci mi 



l/^a," y/^ar l/^a/ l/-^ai' 



[giai''] X + [.9'/^''^'] y + [!fi"ic.] - + ••• + [gicv^u] = , 

 [gibiCii] X -|- [iJihr] y -f \<j;hici^ .~ -f . . . -f [;//^?n,] = , 

 [g.c^cii I X 4- [giCihi ] // -|- [inc,"] i ^ , . . -\- [<]ic, mi] -— , 



In this way the "normal equations" are found. 



