104 Prof. Thomson on the Restoration of Mechanical Energy 



negative quantity of heat given to the part of the engine at the ^ 

 temperature T in this process. Now if such a process be com^ 

 pleted for every part of the body without either taking or giving 

 heat by communication with other bodies, the whole quantity of 

 heat given to the refrigerators at T by the engines worked from 

 all the parts of the body for which ^ > T, must be equal to the 

 whole quantity given by the sources at T to the engines working 

 to raise the temperature of those parts of the body for which 

 < < T ; or, algebraically, the sum of the quantities of heat given 

 by all the engines to the parts of them at T must be 0, that is, 

 according to (e), 



jyjdx dy dz [f^odt . eT/y -f\dt . frf^'^ 



— — — — y y. 



From this we deduce 



jyjdx dy dzj'^cdt . i'y'^^'''=:jyyicdydzf\dt.e^fy\2), 



by which the value of T may be found. When T is determined, 

 the whole work obtained in the process may be calculated by 

 adding all the terms given by the expression {d) ; and thus, if W 

 denote its value, we find 



'^ = ^jyjdxdydzj'\dt.{\-frfy') . (3). 



The first member of the former of these equations may be put 

 into a simpler form if we take to denote the thermal capacity 

 of the whole body at the temperature 6, a quantity which must 

 generally be considered as a function of 6, Thus, if cq dx dy dz 

 denote the thermal capacity of the portion dx dy dz when at the 

 temperature 6, we have 



Iff-' 



c^dxdydz^^ (4); 



and the equation becomes 



J"^®de . ^^^'^"'^jyjdx dy dzj^'cdt . fr/y^ (5) . 



In order that there may be data enough for solving the pro- 

 blem, the nature of the given body must be specified so that the 

 value of c for each point xyz is known, not only for the given 

 initial temperature t of that point, but for all other tempe- 

 ratures through which we have to suppose it to vary in this 

 investigation ; and is therefore to be regarded as a known 



function of Q. Hence the value of the integral / %dQ e^J '* 



