from an unequally heated space. 105 



may be regarded as a known function of 6, and may be tabulated 

 for different values of this variable. The value of 6, for which 

 this function is equal to the second member of equation (5), is 

 the required quantity T. 



The solution of the problem may be put under a very simple 

 form, if the thermal capacity of each part of the body be inde- 

 pendent of the temperature, in the following manner. Let the 

 temperature of the body be measured according to an absolute 

 scale, founded on the values of Carnot's function, and expressed 

 by the following equation, 



<= ^ -« (6), 



where a is a constant which might have any value, but ought to 

 have for its value the reciprocal of the coefficient of compressi- 

 bility of air, in order that the system of measuring temperature 

 here adopted may agree approximately with that of the air-ther- 

 mometer. Then we have 



_ Z/o^ = , and 6 Vo^ = 



t + oc' " e + u' 



and since, according to the hypothesis that is now made, © 



is constant, the first member of equation (5) becomes simply 



T + a 

 ©a log ; and we have explicitly, for the value of T, the 



equation lffflo^'±^.e<lrdydz 



fff\osit+«-).cdxdydz Y . , (7)^ 



rn Jjjcdxdy^ 



or T = e ^'''^ —oi;~> 



and equation (3) takes the simpler form, 



W=Jj(]15^</«;<;y&(/-T-(T + «)log^) . . (8). 



If the given body be of infinite extent, and if the temperature 

 of all parts of it have a uniform value, T, with the exception of a 

 certain limited space of finite extent through which there is a 

 given varied distribution of temperature, any of the equations 

 (2), (5), or (7) leads to the result 



T=r, 



which might have been foreseen wiibout analysis. In this case, 

 then, equation (3) gives (in terms of a definite integral of which 

 the elements vanish for all points at which / has the value T) 

 the work that may be obtained by bringing the temperature of 

 all the matter to T; and the same result is expressed more sim- 

 ply by (8) when the specific heat of all the matter in the space 

 through which the initial distribution of temperature is non^ 

 uniform does not vary with the temperature. 



1 Acton Square, Salford, Manchester, 



December 28, 1852. 

 FUl Mag. S. 4. Vol. 5. No. 30. Feb. 1853. I- 



