66 EXTENSION IN CONFIGURATION 



kinetic energy due to V l and V 2 n combined is the sum of the 

 kinetic energies due to these velocities taken separately. And 

 the velocity V B may be regarded as compounded of three, 



*Y F 3"> *Y" of which v * is of the same nature as F i ' V * 

 of the same nature as V 2 ", while V B " f satisfies the relations 



that if combined either with Fi or V the kinetic energy of 

 the combined velocities is the sum of the kinetic energies of 

 the velocities taken separately. When all the velocities 

 Fg , . . . V n have been thus decomposed, the square root of the 

 product of the doubled kinetic energies of the several velocities 

 PI> JY' JY" ete *' ^H be the value of the extension-in- 

 velocity which is sought. 



This method of evaluation of the extension-in- velocity which 

 we are considering is perhaps the most simple and natural, but 

 the result may be expressed in a more symmetrical form. Let 

 us write e 12 for the kinetic energy of the velocities F x and V% 

 combined, diminished by the sum of the kinetic energies due 

 to the same velocities taken separately. This may be called 

 the mutual energy of the velocities V\ and F 2 . Let the 

 mutual energy of every pair of the velocities Fj , . . . V n be 

 expressed in the same way. Analogy would make e n represent 

 the energy of twice V 1 diminished by twice the energy of Fi , 

 i. e.y e n would represent twice the energy of Fi , although the 

 term mutual energy is hardly appropriate to this case. At all 

 events, let e n have this signification, and e 22 represent twice 

 the energy of F^, etc. The square root of the determinant 



n 12 ... i 



represents the value of the extension-in-velocity determined as 

 above described by the velocities V\ , . . . FJ,. 



The statements of the preceding paragraph may be readily 

 proved from the expression (157) on page 60, viz., 



A 



by which the notion of an element of extension-in-velocity was 



