EVANS. — NOTE ON KIRCHHOFF's LAW. 99 



o- rii(X) e^^X) d\ + ^12 (5) 



and 



<rf^M^) ^'tX^) ^^ + ^21. (6) 



Ai (X) and A2 (X) are called absorption coefficients and denote func- 

 tions that depend on the nature and temperature of o-i and o-g, and 

 possibly on the temperature of the body iS. The existence of such 

 functions is assumed. In the physical case the functions IlJ, e, and 

 A seem to be continuous in A. And accordingly although it is not 

 strictly necessary for the development of the proof, we shall make this 

 assumption. We assume also that the functions £J, e, and A are pos- 

 itive or zero at every A, and that the functions A are ^ 1. 



Before seeking to obtain relations among the several kinds of ener- 

 gies, it is well to consider more in detail the nature of the functions 



/'OO 



e<ri (A) and e^r^ (A). Let us denote by o- / ei(A) d\ the amount of energy 



that would reach o-' — the section of Sq half way between o-x and 0-2 



Jr»oo 

 ^2 (A) ^A the amount of energy that 

 

 would reach o-' from the o-g side, if 2o were filled merely with the same 

 medium that occupies the space between 2o and 2i. 

 Let 



e.,{X)=eU\) + 8.,iX) (7) 



e.,(X)=e,(k) + 8^^(\). (8) 



Now as we change the slab 2^ in size, the functions 8^^ (A) and 8<r, (A) 

 will change. If the body 2^ is small, the energy that reaches it will 

 depend little on its own influence,^ and 8^^ (A) and 8^^ (A) will be small. 

 We make the assumption that as 2^ approaches zero, keeping the same 

 relative proportions, both o-i and 0-2 approach zero, and approach zero 

 in such a way that 



A.,=^°'|8<.,(A) \d\ 

 and (9) 



both approach zero. 



-i: 



8^, (A) I^A 



* See, however, p. 103. 



