Now 



so that 



Complete Radiation at a given Temperature. 465 



^(.») = - I I cos u (v — x) (bOv) du dv 



■^JO J-oo 



= - I I Gos,uvGosuxe~''^"^dudv. . . C9) 

 '''''' COS uvdv=^e-'''^"''; .... (10) 



J. 



+ 0D 



e' 



Cv/ttJo ^ 



This equation exhibits the resolution of (8) into its harmonic 

 components ; but it is not at once obvious how much energy 

 we are to ascribe to each value of u, or rather to each small 

 range of values of u. As in the theory of transverse vibrations 

 of strings, we know that the energy corresponding to the 

 product of any two distinct harmonic elements must vanish ; 

 but the application of this, when the difference between two 

 values of u is infinitesimal, requires further examination. 

 The following is an adaptation of Stokes's investigation * of a 

 problem in diffraction. 



By Fourier^s theorem (9) we have 



IT . (f){ip)=\ fi(ii) COS ux du + \ f 2(11) sm uxdu, . . (16) 

 Jo Jo 



/j(w) = j cosuv<j)(y) dv, .... (17) 



•/ — 00 



smuv^{v)dv (IS) 



— 00 



In order to shorten the expressions, we will suppose that, 

 We have 



/^ CO /^oo 



•TT^. \(t>(x) P= I /i(w)/i(mO cos ux cos u'x dudu'. 



This equation is now to be integrated with respect to x from 

 — CO to +00; but, in order to avoid ambiguity, we will 

 introduce the factor e"^*^, where a is a small positive quantity. 

 The positive sign in the alternative is to be taken when a; is 



* EdinlD. Trans, xx. p. 317 (1853) ; see also Enc. Brit. t. xxiv. p. 431. 



where 



