104 PKOCEEDINGS OF THE AMERICAN ACADEMY. 



(r(Ao.j 4- A(^J, the difference in the radiation reaching the surface 2,, when 

 there is and when there is not a different material in 2o from the me- 

 dium between 2^ and 2i, is an infinitesimal of the second order. This 

 supposition seems more reasonable than it really is. 



For let us suppose, if it is possible, that the body >S^ is composed 

 of a material which emits and absorbs only a partial spectrum, i.e. 

 there are wave lengths for which S emits and absorbs no energy. With 

 no body 2^ and ether for the medium throughout, the radiation e (A) 

 reaching any element of surface within the cavity will not contain 

 energy of these wave lengths which the body >S' does not emit. But if 

 we insert a body 2^, however small, which has a complete spectrum, the 

 energy of the forbidden wave lengths will build up until it reaches 

 such intensity that the amount absorbed by the small body 2,-, will be 

 equal to the amount emitted by it. Since both of these quantities will 

 therefore be infinitesimals of the first order, 8i(A.) and SjC-^) will not ap- 

 proach zero at all as 2,, decreases in size. 



We cannot therefore regard the invariance of the function ^(A) as 

 itself a statement of the non-existence of bodies of partial spectra.^ 

 For that invariance depends upon the assumption that A,^^ and A.^^ 

 approach zero as 2^ becomes smaller and smaller, and hence gives no 

 contradiction when S is a body of partial spectrum. 



However, bodies of partial spectra, if they exist, must satisfy Kirch- 

 hoff's Law. For the analysis of pages 101 and 102 applies. And a body 

 that has a partial emission spectrum must have an absorption spectrum 

 extending over precisely the same region of wave lengths ; and vice 

 versa. 



The second assumption of importance is the supposition that there 

 can be constructed a dense distribution of absorption coefficients be- 

 tween some two distinct functions of A.. This assumption does not by 

 any means demand the existence of an arbitrary absorption coefficient. 

 Indeed the requirements of the proof can be met by a denumerable 

 system^. And yet we cannot be certain of the possibility of physi- 

 cally constructing even this denumerable set. 



° See, however, in this connection W. Wien, Annalen d. Phvsik, 52, 163, 

 1894. 



^ That is, a system of the same order of infinity as the natural numbers, 

 or the rational fractions. 



If A'(A.) and A" \) (see p. 8) have the same limit at oo, a denumerable .set 

 sufficient for the demands of the proof can be set up by requiriuK that f (A) be 

 continuous with continuously turninp; tangent (except at a finite number of 

 point'^), and by replarinp, in the definition of density, the two curves f(A)± 5 (a) 

 by two curves distant 5 from the curve f(A). 



