and the Laws regarding the Nature of Heat. 9 



this, let the bag during the expansion be brought into contact 

 with a body A of the temperature t, from which it shall receive 

 heat sufficient to preserve it constant at the same temperature. 

 While this expansion by constant temperature proceeds, the 

 pressure decreases according to the law of M., and may be repre- 

 sented by the ordinate of a curve ah, which is a portion of an 

 equilateral hyperbola. When the gas has increased in volume 

 from oe to of, let the body A be taken away, and the expansion 

 allowed to proceed still further without the addition of heat ; 

 the temperature will now sink, and the pressure consequently 

 grow less as before. Let the law according to which this pro- 

 ceeds be represented by the curve be. When the volume of the 

 gas has increased from of to og, and its temperature is lowered 

 from t to T, let a pressure be commenced to bring it back to its 

 original condition. Were the gas left to itself, its temperature 

 would now rise ; this, however, must be avoided by bringing it 

 into contact with the body B at the temperature r, to which any 

 excess of heat will be immediately imparted, the gas being thus 

 preserved constantly at t. Let the compression continue till 

 the volume has receded to h, it being so arranged that the de- 

 crease of volume indicated by the remaining portion he shall be 

 just sufficient to raise the gas from r to t, if during this decrease 

 it gives out no heat. By the first, compression the pressure in- 

 creases according to the law of M., and may be represented by a 

 portion cd of another equilateral hyperbola. At the end the in- 

 crease is quicker, and may be represented by the curve da. This 

 curve must terminate exactly in a ; for as the volume and tem- 

 perature at the end of the operation have again attained their 

 original values, this must also be the case with the pressure, 

 which is a function of both. The gas will therefore be found in 

 precisely the same condition as at the commencement. 



In seeking to determine the amount of work performed by 

 these alterations, it will be necessary, for the reasons before 

 assigned, to direct our attention to the exterior work alone. 

 During the expansion, the gas produces a work expressed by the 

 integral of the product of the differential of the volume into the 

 corresponding pressure, which product is represented geometri- 

 cally by the quadrilaterals ea, bfandfbcg. During the com- 

 pression, however, work will be expended, which is represented 

 by the quadrilaterals gcdh and hdae. The excess of the former 

 work above the latter is to be regarded as the entire work pro- 

 duced by the alterations, and this is represented by the quadri- 

 lateral abed. 



If the foregoing process be reversed, we obtain at the conclu- 

 sion the same quantity abed as the excess of the work expended 

 over that produced. 



