G. F Becker — Maximum Dissipativity. 117 



turbed and undisturbed paths at the end of the circuit at right 

 angles, and the action from the last focus to this line on each 

 path is consequently the same. 



Whatever the recurrent path of a particle may be, the action 

 upon it will be the same as it would be on a circle of appropri- 

 ate diameter upon which the particle should move at its mean 

 velocity, completing the circumference in the same time which 

 it occupies on the real path. This condition of the equality of 

 time implies that, if dd- is the elementary angle which the 

 radius of the circle makes with a fixed radius, dd-=dt. If the 

 mean velocit}^ of the particle is u, the elementary circular arc is 

 udt=rdd- and consequently u=r. Now the area of the circle is 



t\ t\ H 



*r' z d^ P u*dt 



/rdS f wdt r m , 



t\ to to 



T being the kinetic energy of the particle in its real path per 

 unit of mass ; the area of the circle is therefore half the action of 

 the particle per unit of mass for an entire circuit. The ratio of 

 the area to the circumference of a circle is of course the greatest 



possible. This ratio is -— - and if the length of the circular 



path is regarded as the independent variable and as a given 

 constant, say s, 



/*Tdt = 



su 



max. 

 2 



In comparing the mean velocity, the corresponding path, and 

 the product, or the action, one or other, must be considered as 

 given, and the conclusion that one-half the action for an entire 

 circuit is the greatest possible is therefore entirely general. 



In cases where the disturbed path returns to the same start- 

 ing point as the undisturbed path, the action for an entire cir- 

 cuit cannot be an absolute maximum, because the action is the 

 same on various paths between the same terminal configurations, 

 but a proof independent of that in the last paragraph can be 

 given that the action will differ infinitely little from an absolute 

 maximum. Any circuit must be reversible, or the action must 

 be the same in whichever direction the particle traverses the 

 path. As has been shown, the action for an entire circuit is 

 constant and the action for an infinitesimal movement is the 

 least possible. The action from the starting point over the real 

 path, back to a position infinitely near the starting point, is 

 therefore a constant minus a quantity which is the least possi- 

 ble, and is therefore the greatest possible. It also differs infinite- 

 ly little from the action for an entire circuit. 



