G. F. Becker — Maximum Dissipativity. 119 



that the action for a natural motion is a minimum for a given 

 period means, that the average rate at which the kinetic energy 

 has decreased is a maximum or that the average rate at which 

 the kinetic energy has increased is a minimum. The assertion 

 that the action is a maximum means that for the given period 

 the kinetic energy has decreased as slowly or increased as rap- 

 idly as possible. 



There is every reason to suppose that every natural system 

 is conservative and that there is no natural system in which 

 movements of various periods do not go on simultaneously. 

 To take a simple illustration, it is impossible to vibrate an open 

 string without exciting harmonics as well as the fundamental 

 tone. If there is but one sensible or molar movement in a 

 system, there is at least friction and therefore molecular motion. 

 If we consider an open string for a period equal to that which 

 elapses between the initial movement and that at which the 

 fundamental movement is about to reach its first kinetic focus, 

 it is clear that all the harmonic vibrations must have passed 

 their first kinetic foci. The action of the fundamental motion 

 up to this time will be a minimum and that of the other 

 motions for the same time will be a maximum or minimax. 

 Of the total energy of the system, then, the fundamental 

 motion has a minimum portion and the harmonic motion a 

 maximum or a minimax. If the fundamental motion is com- 

 pared with one of the higher harmonics it is clear that the order 

 of the latter will be a very moderate one when its action will 

 have reached its period of absolute maximum, while the action 

 of the fundamental vibration is still the least possible. A 

 fortiori must this be the case when molar motions are compared 

 with molecular motions or heat-giving vibrations with those 

 which yield light. 



It might be objected that since the total energy of the graver 

 vibrations in this instance, or of any corresponding movement 

 in any other instance, does not remain constant, the principles 

 of the action of a conservative system are inapplicable to it. 

 But in the first place if there is a great difference between the 

 periods of the motions compared, the energy of the motion of 

 longer period may be considered as constant for the time of 

 the shorter vibration, and this assumption is commonly made 

 in discussing simultaneous molar' and molecular motions. 

 Furthermore, if the periods are not very diverse, the movement 

 of longer period may be reduced to the consideration of a con- 

 servative motion by supposing the energy withdrawn from it 

 to be potentialized. For the problem in hand this treatment 

 appears to be something more than a method of approxima- 

 tion ; for it is well known that every complex vibratory 

 movement may be resolved into simple vibrations. At the 



