IV. AXES OF CENTRAL QUADRIC. 567 



components to accord with the number of dimensions of the hyperspace. 

 The geometric product of the vector with an infinitesimal displacement 

 defines the generalized coordinates of the vector, so that 



Y r dlj r + Z r 



r=n = 



and we find that the reduced component of the vector is what is defined 

 by the formula 42) of page 116) 



In our application to mechanics the differential form in question is 

 2Tdt 2 , where T is the kinetic energy of the system It is immaterial 

 whether we speak of vectors in a hyperspace, as we have here done, or. 

 as Hertz does, speak of vectors with respect to our mechanical system. 

 The meaning in either case is plain. On dividing the above formulae 

 by dt 2 , we find that the generalized velocities and momenta have the 

 relation to each other of coordinates and reduced components of the same 

 vector in the hyperspace. The two reciprocal forms 14) and 23) have 

 the relation of the Lagrangian and Hamiltonian forms of the kinetic 

 energy. The equations of motion of the system say that, no matter 

 how the forces are applied, or how parts of them are equilibrated by 

 the constraints, the reduced components of the applied and the effective 

 forces are equal for every coordinate. 



NOTE IV, 



AXES OF CENTRAL QUADRIC. 

 The principal axes of a central quadric surface, 

 1) F(x, y, e) ~ Ax 2 + By* + Cz 2 -f %Vyz + 2Egx + 2Fxy = 1, 



are defined as the radii vectores in the directions for which the radius 

 vector is a maximum or minimum. If we put 



x = ra, y = r|3, z = ry, 

 we have 



2) ^ = F(, P, y), 



and the maxima and minima of r occurring for the same directions as 

 the minima and maxima of 1/r 2 , are obtained by finding the maxima and 

 minima of F(cc, ft, y) subject to the condition 



3) g>( a , 



