100 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Let us now put our apparatus in operation. At the start we put 

 everything that has a temperature (ether, of course, has not) at one 

 temperature T, and throughout succeeding time maintain S at temp- 

 erature T. The body 2^ will thereafter, according to the Second Law 

 of Thermodynamics, remain at temperature T. According to the First 

 Law of Thermodynamics the amounts of energy absorbed and emitted 

 by 2o are equal. We have then the equation 



aj E,, (A) d\ + ^21 + ^^^ E,, (X) dX + ^12 = (10) 



or 



J\, (A) dX + f\, (X) dX = 



J\ (X) e,^ (X) dX + J^^ (X) e,^ (X) dX. (11) 



We can at this point deduce that 



J^ Ea, (X) dX + J^ E,, (X) dX = 



Ax (X) e^ (X) d\ + J A.^ (X) e^ (X) dX. (12) 



For if this equation did not hold, but if, on the other hand, the left 

 hand member were equal to the right hand member plus a (quantity e, 

 we could by taking 2o small enough secure a contradiction from equa- 

 tion (11), and the assumption that A,^^ and A,,.^ approach as 2^ de- 

 creases indefinitely. 



If we replace S by any other solid body fulfilling the conditions we 

 get an equation 



Ea,iX)dX-\- I E<r,iX)dX = 



A, (X) eY (X) dX + J^ A^ (X) e,' (X) dX. (13) 



The question now arises as to what is the relation among the ^'s, — 

 'ei(X), e2(X), ei (X), e^ (X). If the functions ^1 (X) and ^l2(X) were 

 perfectly arbitrary it could easily be shown that all the functions (^ (X) 

 are identical ; but we have no right to make such an assumption.* 



* See page 104. This is in fact the assumption in the proof of E. Pringsheim. 



