of the Mechanical Theory of Heat from the First. 29 

 If Be is the amount of energy sought, then 



or also 



Be=?<m(x f Bx f + . . . -x"Bx- . . . ) ; 



multiplying by dt, and taking into consideration the significa- 

 tion of x f . . . , x" . . . , we find 



Be . dt=-*Zm(dxBx f + . . . —dx'Bx— . . . ). 

 By simple transformations we get 



Be . dt = B%m(x'dx + . . . ) — dl t m(x / Bx + . . . ). 



Integrating this equation between the limits corresponding to 

 the initial and final states, we have 



(\ . dt = B ( t 2m(x'dx+ . . . )-%(x\8x 1 + . . .) 

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+ 2<m(x / 8x + . ..). 



Transforming the integral on the right-hand side, taking into 

 account equations (4) and (5), gives 



\ i 8e.dt=28\ t Tdt-2T 1 8t + 2T BL 



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As T x and T are constant, the equation can also be written 

 thus — 



\'8e.dt=28\*(T--T 1 + To)dt. .... (6) 

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In a definite actually effected passage out of (f ??o) m *o (fi^i) 

 the quantity of energy is a certain function of the time. The 

 passage, however, from the same initial to the same final state, 

 by the same path, can, with respect to time, be executed quite 

 arbitrarily ; consequently Be is a different function of the time, 

 according to. the velocity of the importation of energy. By 

 suitably selecting this velocity, any indefinitely small value 

 whatever may be assigned to 



r 



Be . dt. 



Let the velocity of the importation of energy be such that 

 the period of the transition shall satisfy the following equation 

 of condition — 



i Be.dt = t.BQ, 

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where SQ signifies the amount to which more energy would 

 be required by the transition on the second path than by the 

 transition on the first path. 



