AND EXTENSION IN VELOCITY. 65 



But the systems which initially had velocities satisfying the 

 equation (172) will after the interval Bt have configurations 

 satisfying equation (177). Therefore the extension-in-con- 

 figuration represented by the last integral is that which 

 belongs to the systems which originally had the extension-in- 

 velocity represented by the integral (171). 



Since the quantities which we have called extensions-in- 

 phase, extensions-in-configuration, and extensions-in-velocity 

 are independent of the nature of the system of coordinates 

 used in their definitions, it is natural to seek definitions which 

 shall be independent of the use of any coordinates. It will be 

 sufficient to give the following definitions without formal proof 

 of their equivalence with those given above, since they are 

 less convenient for use than those founded on systems of co- 

 ordinates, and since we shall in fact have no occasion to use 

 them. 



We commence with the definition of extension-in- velocity. 

 We may imagine n independent velocities, V l , . . . V n of which a 

 system in a given configuration is capable. We may conceive 

 of the system as having a certain velocity F~ combined with a 

 part of each of these velocities V l . . . V n . By a part of V\ is 

 meant a velocity of the same nature as V\ but in amount being 

 anything between zero and V r Now all the velocities which 

 may be thus described may be regarded as forming or lying in 

 a certain extension of which we desire a measure. The case 

 is greatly simplified if we suppose that certain relations exist 

 between the velocities V\ , . . . V w viz : that the kinetic energy 

 due to any two of these velocities combined is the sum of the 

 kinetic energies due to the velocities separately. In this case 

 the extension-in-motion is the square root of the product of 

 the doubled kinetic energies due to the n velocities Fi , . . . V n 

 taken separately. 



The more general case may be reduced to this simpler case 

 as follows. The velocity F 2 may always be regarded as 

 composed of two velocities Vj and V 2 ", of which VJ is of 

 the same nature as V l , (it may be more or less in amount, or 

 opposite in sign,) while V 2 " satisfies the relation that the 



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