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XX. Intelligence and Miscellaneous Articles. 



ON THE EXPRESSION OE THE WORK RELATIVE TO AN ELEMEN- 

 TARY TRANSFORMATION. BY J. MOUTIER. 



MCLA USIUS has recently given a demonstration of Carnot's 

 ♦ theorem, founded on the expression of the work relative to 

 an elementary transformation on the hypothesis now generally ad- 

 mitted, in which heat is considered to be a mode of motion. M. 

 Ledieu arrived at the same result by a different path. These solu- 

 tions leave the nature of the motion undetermined, and by that very 

 fact present the greatest generality. My intention is to treat the 

 same question by admitting that heat consists of a vibratory motion. 

 The analogy existing between heat and light permits us to suppose 

 that it is so ; and as the vibration theory suffices for the explana- 

 tion of all optical phenomena, there is reason to investigate whether 

 it can likewise account for the phenomena of heat. This particular 

 hypothesis is not new in science. It is true that it restricts the 

 generality of the solution ; but, on the other hand, it permits us to 

 state precisely the nature of certain phenomena. 



The vibratory motion with which each point is animated can be 

 decomposed according to three rectangular directions ; each com- 

 ponent motion is an oscillatory rectilinear motion of the same 

 period, produced by a force proportional to the distance from the 

 material point to a fixed centre. 



If we represent by m the mass of the material point, by <p the 

 acceleration at the unit of distance, by a the amplitude of the oscil- 

 lation, the mean value of the force is f=%m<pa. 



The duration i of an oscillation is i= -7=. 



V $ 



The maximum velocity of the material point is U = — • — 



The maximum semi-vis viva is |mU 2 =fa. 



The mean semi-vis viva \mu 2 is the half of the maximum semi- 

 vis viva ; ^mu 2 =-£fa. This mean is considered to be proportional 

 to the absolute temperature T. 



The elementary work which corresponds to a rise of temperature 

 dT is composed of two parts : one is equal to half the increment of 

 the mean vis viva ; the other arises from modifications brought into 

 the vibratory motion by supposing that the mean vis viva preserves 

 the same value, or else that the temperature remains constant. 



The temperature remaining constant, the amplitude of the oscil- 

 lation may change, provided that the duration of an oscillation varies 

 in the same ratio. If the amplitude of the oscillation is increased 

 by the quantity da, there results a work which is expressed by the 

 product of the mean value of the force into the increment of the 

 amplitude, or f da. 



Now, as the ratio - is to remain constant, 

 a 



da= — di; 

 1 



