Fundamental Theorem in the Mechanical Theory of Heat. 83 



corresponding to any given change in the condition of the body 

 is completely determined by the initial and final conditions of 

 the latter, and is independent of the path pursued in passing 

 from one to the other. Conceive a body to pass successively in 

 different ways from the one condition to the other, but always to 

 return in the same manner to its initial state. It is evident that 

 the quantities of internal work produced along the different paths 

 must all cancel with the common quantity produced during the 

 return, and consequently must be equal to each other. 



It is otherwise with the external work. With the same initial 

 and final conditions, this can vary just as much as the external 

 influences to which the body may be exposed can differ. 



Let us now consider at once the internal and external work 

 produced during any given change of condition. If opposite in 

 sign they may partially cancel each other, and what remains 

 must then be proportional to the simultaneous change which 

 has occurred in the quantity of heat. In calculation, however, 

 it amounts to the same thing if we assume an alteration in the 

 quantity of heat equivalent to each of the two kinds of work. 

 Let Q, therefore, be the quantity of heat which must be im- 

 parted to a body during its passage, in a given manner, from 

 one condition to another, any heat withdrawn from the body 

 being counted as an imparted negative quantity of heat. Tlien 

 Q may be divided into three parts, of which the first is employed 

 in increasing the heat actually existing in the body, the second 

 in producing the internal, and the third in producing the ex- 

 ternal work. What was before stated of the second part also 

 applies to the first — it is independent of the path pursued in the 

 passage of the body from one state to another : hence both parts 

 together may be represented by one function U, of which we at 

 present only know that it is completely determined by the initial 

 and final states of the body. The third part, however, the equi- 

 valent of external work, can, like this work itself, only be deter- 

 mined when the precise manner in which the changes of con- 

 dition took place is known. If W be the quantity of external 

 work, and A the equivalent of heat for the unit of work, the 

 value of the third part will be A . W, and the first fundamental 

 theorem will be expressed by the equation 



Q=U+A.W (I) 



When the series of changes are of such a nature that through 

 them the body returns to its original condition, or when, as we 

 shall in future express it, these changes form a circular process, 

 we have U = 0, 



and the foregoing equation becomes 



Q=A.W (1) 



G2 



