102 PROCEEDINGS OF THE AMERICAN ACADEMY. 



It will then be unequal to ei (A) throughout a small neighborhood 

 Xi < \ < X2, — let us say that ei (X) — ^1 (X) > c > throughout the 

 interval. 



Now if we subtract (12) from (13) we have 



j -i 1 (^) W(X) - er (X)] d\ + /^-^2(A) [e/ (A) - e^ (X)] dX = 0. (14) 



In particular we have 



f\'(^) K(A) - e,(X)]d\ + n-l2'(A) [^/(A) - e,(X)] dX = 0, 



whence 



[^,(x) -a;{x)] [./(X) - .,(X)] dX + 



nA,(X) - A/(X)] [e,XX) - e,iX)] dX = 0. (15) 



X 



CO 

 



On account of the freedom of choice for Ai(X) and A2(A) we can now 

 get a contradiction out of (15). For we can choose A^(X) in reference 

 to Ai'(X) and A2(X) in reference to Ag^A) in such a way that the only 

 significant part of the integrals in (15) will be 



r^AriX) - ^/(X)] [e/(X) - ^x(X)] dk, 



which can be made unequal to zero. But this contradicts (15). 

 Hence ei(X^) = ei(X„) and ei(X) = ei(X). 



And not only for every body S but also for every position of 2„ in 

 the cavity is ei(X) = ei(X). For different positions of 2,, can be re- 

 garded as the same position of 2,, with different S. Hence in particu- 

 lar ei{X) = ^2 (A). We have then the theorem that the character of the 

 radiation impinging on either face of any element of surface a- anywhere 

 in the cavity S is a. function merely of the temperature of the cavity 

 and the nature of the medium in which the element of surface is im- 

 mersed. 



If we denote the function ei(X) = e^iX) by e(X), we may call f (X) per- 

 fectly black radiation. It is approximately the radiation emitted 

 through a hole of unit area in the bounding surface of a large cavity 

 whose interior walls are kept at the uniform temperature T. 



We can now easily prove KirchhofTs Law. Suppose that in the 

 cavity S we have any body K, enclosed by a surface ^S", just outside the 



