182 V. OSCILLATIONS AND CYCLIC MOTIONS. 



changed from their real values (unless $13 $23 = 0), while 



c 2 

 there appears the term - ~ independent of the velocities, depend- 



r Vss 



ing on the coordinates q lf q 2 . This is, since it gives rise to a 

 conservative positional reaction, undistinguishable in its effect from 

 potential energy. In reality, the reaction to which it gives rise is 

 motional, instead of positional, as it appears to be. If we could 

 explain all potential energy in this manner, namely as due to concealed 

 cyclic motions, we should have solved the chief mystery of dynamics. 

 In his remarkable work on dynamics, Hertz treats all energy from 

 this kinetic point of view. In order to have a successful model for 

 this representation of potential energy, which needs in order to be 

 perfect no linear terms, we must have Q 13 = Q 23 = 0. 



We can now see why the simple example of 48 showed no 

 linear terms, since by putting all the Q's with one suffix 2 equal to 

 zero we pass to the case of a system with two degrees of freedom. 

 If at the same time the coordinates are orthogonal, 13 = 0, so that 

 the single linear term disappears. This was the case above. 



Let us now pass to the general case. We have for the momenta 

 the equations 53) 37 and, for the first r, 137) which are written out, 



Pi = 



' 2 m 



157) 



Pr =Qr 



Pm Qml ( 



Let us now form the kinetic energy from the definition, 36, 38), 

 158) 



Multiplying the above equations, the s th line by q a ', and adding, we 

 obtain from the first r lines on the right, 



The terms coming from the last m r lines, and the first r columns, 

 as marked off by the dotted lines, are found to be, on collecting 

 according to columns, 



