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LVIL The Second Laiv of Thermodynamics. 

 By S. H. Burbtjry, F.B.S* 



IN a statistical system of molecules let T be the mean 

 kinetic energy, and v 1 . . . v r certain " controllable " co- 

 ordinates, according to the notation used by Messrs. Larmor 

 and Bryan in their Report on Thermodynamics. Let ^Q be 

 the energy that must be supplied, or spent in external work, 

 when T becomes T + ^T, and i\ becomes ^i + cH'i? &c. The 

 object of this paper is to find the most general condition 



"csO 

 that will make -^~ a complete differential of a function of 



T, Vi . . . v r . It will be found to be the same condition which 

 is necessary to make the motion stationary with T, v t . « , v r 

 constant. 



1. Let the state and position of a molecule be defined by 

 the n generalized coordinates % . . . x n with corresponding- 

 velocities, and the r controllable coordinates v x . . . v r . It is 

 assumed that the v's vary so slowly that their velocities may 

 be neglected, but their components of momentum shall be 

 Y 1 . . . V r . Let U be the potential, r the kinetic energy of a 

 molecule, E = U + t. 



2. Let us assume provisionally that if the molecule be 

 started from any position with any velocities, v l . . . v r being 

 maintained constant, its motion will be periodic ; and there- 



dV 



fore on average of time -5- = for each v. 

 to dt 



3. Generally, if only the conservative forces act, v x . . . v r 

 will vary. We must then apply external constraining forces 

 P x . . . V r to prevent %\ . . . v r from increasing. 



By Lagrange's equations, 



dV dr dJJ T > r, , 



V =-r 1 "" tor each v; 



dt dv dv 



dV 

 and since —j- = on average of time, 



-r> dr dXJ P , . 



r= -. 7— on average or time. 



dv dv ° 



4. For any given constant values of v 1 . . . v r we may have 

 many different motions of the molecule, all of this character, 

 but with different values of t, the mean kinetic energy. So 

 with the same t we may have different values of v± . . . v r . If 

 w T e wish to make the system pass from an original motion, in 



* Communicated by the Author. 



