MOTIVE POWER OF HEAT. 151 



Now, <r, k, and p, being quantities which depend 

 upon the temperature, may be considered as func- 

 tions of t; and it will be convenient to modify the 

 integral so as to make t the independent variable. 

 The limits will be from t T to t 8, and, if we 

 denote by M the value of the integral, we have the 

 expression 



dp 



dt. . . . (1) 



for the total amount of mechanical effect gained 

 by the operations described above. 



21. If the interval of temperatures be extremely 



dp 



small, so small that (1 cr) -r- will not sensibly vary 



for values of t between I 7 and 8, the preceding 

 expression becomes simply 



dp 



T). . . (2) 



This might, of course, have been obtained at once 

 by supposing the breadth of the quadrilateral 

 figure AA^A^A to be extremely small compared 

 with its length, and then taking for its area, as an 

 approximate value, the product of the breadth into 



