36] REPRESENTATION BY HYPERSPACE. 113 



ds* = 



in space of three dimensions , 



ds* = E rt dq r dq, 



r=l 



so by analogy , in space of m dimensions, 

 37) ds' 



That is to say a quadratic form in m differentials may be interpreted 

 as the square of an arc in m dimensional space. Thus we may 

 assimilate our system depending upon m coordinates to a single point 

 moving in space of m dimensions, characterized by the expression 

 for the element of arc, 



To each possible position of this point corresponds a possible con- 

 figuration of our system. No matter what be taken as the mass of 



the point, M, its kinetic energy, --M\-j*-\, is equal to the kinetic 



energy of our system, the coefficients in the quadratic form for ds 2 

 and T being proportional. 1 ) The advantage of this mode of speaking 

 (for it is no more) may easily be seen from the many analogies 

 that arise, connecting the dynamical theory of least action with the 

 purely geometrical theory of geodesic lines. This method is adopted 

 by Hertz in his Prinzipien der Mechanik and is worked out in a 

 most interesting manner by Darboux in his Theorie des Surfaces, 

 Tom. II. The ideas involved were first set forth by Beltrami. 2 ) 



1) Since by the nature of the above transformation, we have 



1 



ds -=w 



if as in 32 we consider each mass m r to be the sum of m r unit mass-points, and 



then ds is the quadratic mean, or square root of the mean square of the dis- 

 placements of all the particles. 



2) Beltrami, Sulla teorica generate dei parametri differenziali (Memorie 

 della Accademia delle Scienze dell' Istituto di Bologna, Serie 2, t. VIII, 

 p. 549; 1869. 



WEBSTER, Dynamics. 8 



