G. F. Becker — Maximum Dissipativity. 115 



Aet. X. — A Theorem of Maximum Dissipativity ; by GrEORG-E 



F. Becker. 



The proposition which I desire to prove in the following 

 pages is, in general terms, that in all moving systems there is a 

 constant tendency to motions of shorter period, and that, if 

 there is a sufficient difference between the periods compared, 

 this tendency is always a maximum, so that all natural phe- 

 nomena occur in such a way as to convert the greatest possible 

 quantity of the energy of sensible motion into heat, or the 

 greatest possible quantity of heat into light, etc., in a given 

 time, provided that the interval of time considered exceeds a 

 certain fraction of the period of the most rapidly moving par- 

 ticles of the system. 



The simplest case of motion which can be selected for exami- 

 nation is that of a particle describing a path which returns upon 

 itself. In any actual system, such a movement must be a 

 "stable "* one. In a very important class of such movements, 

 there is a position of closest approach to a fixed center and a 

 position of greatest departure from it corresponding to the 

 perihelion and aphelion of planetary motion, and in such cases 

 the symmetry with reference to an axis shows that the " action" 

 between these points must be the same in either direction. The 

 nature of- these cases indicates the investigation of half the 

 action for an entire recurrent path. 



If an actual, recurrent, stable path is compared with that 

 which would result from an infinitesimal conservative disturb- 

 ance of the same movement, it appears that the distance between 

 the points of intersection is infinite compared with the perpen- 

 dicular distance which separates the paths at any intermediate 

 position, and at such a position the paths are infinitely nearly 

 parallel. It is easy to show that lines perpendicular to the two 

 paths intercept arcs on which the action is equal. If p is the 

 radius of curvature at any point on one of the curves, old- the 

 angle which tangents taken at an interval of time dt make with 

 one another, ds the arc described in this time, and if the sub- 

 script numeral , indicates the corresponding quantities for the 

 other path, 



p*d$=p*d$ + p^SSp; 



for the curves being parallel have the same center of curvature. 



* The motion of a system is stable if after any infinitely small disturbance un- 

 accompanied by a change of total energy, it returns to some configuration belong- 

 ing to the undisturbed path after a finite time and without more than an infinitesi- 

 mal digressioo. 



