AND EXTENSION IN VELOCITY. 67 



originally defined. Since A^ in this expression represents 

 the determinant of which the general element is 



the square of the preceding expression represents the determi- 

 nant of which the general element is 



Now we may regard the differentials of velocity dq t , d^ as 

 themselves infinitesimal velocities. Then the last expression 

 represents the mutual energy of these velocities, and 



d*e 



represents twice the energy due to the velocity dq { . 



The case which we have considered is an extension-in-veloc- 

 ity of the simplest form. All extensions-in-velocity do not 

 have this form, but all may be regarded as composed of 

 elementary extensions of this form, in the same manner as 

 all volumes may be regarded as composed of elementary 

 parallelepipeds. 



Having thus a measure of extension-in- velocity founded, it 

 will be observed, on the dynamical notion of kinetic energy, 

 and not involving an explicit mention of coordinates, we may 

 derive from it a measure of extension-in-configuration by the 

 principle connecting these quantities which has been given in 

 a preceding paragraph of this chapter. 



The measure of extension-in-phase may be obtained from 

 that of extension-in-configuration and of extension-in- velocity. 

 For to every configuration in an extension-in-phase there will 

 belong a certain extension-in-velocity, and the integral of the 

 elements of extension-in-configuration within any extension- 

 in-phase multiplied each by its extension-in-velocity is the 

 measure of the extension-in-phase. 



