642 Mr. C. G. Darwin on the 



It should be said at once that it seems improbable that 

 any practical use could be made o£ the relations, or even 

 that an example o£ them could be given. For in an example 

 it is only possible to specify the optical quantities by virtue 

 of some regularity in their character, and this very regularity 

 makes them unsuited for thermodynamic expression. 



2. The only process by which the two types of quantity 

 can be related consists in expressing as integrals the whole 

 flow of energy across an area in terms of both, and then 

 identifying the elements of the two integrands. In no other 

 way can a rigorous definition be given for the thermo- 

 dynamic quantities. The modus operandi consists in the 

 repeated application of Fourier integrals. In the present 

 note no pretence is made to mathematical rigour in the 

 deductions, but there can, I think, be no doubt of their 

 validity. The most important step in the process is based 

 on a theorem, due originally to Stokes *_, and used for the 

 present purpose by Rayleigh f . Stokes and Rayleigh both 

 proved this theorem by introducing an exponential factor to 

 help convergence. Here I have not used one, as it would 

 probably be as hard to justify its introduction as to prove 

 rigorously the result without it. To save confusion with a 

 more celebrated theorem, and because Stokes's work dealt 

 with a rather different aspect, I shall call it Rayleiglr's 

 theorem. It will be convenient to exhibit it here first, so as 

 to make its repeated use in the later work easier to follow. 



/*T/2 



Let (j> v = 2\ Z(t) cos 2irvt dt, 



J-V2 



r-T/2 

 <x/r v = 2J Z(t)$m2irvtdt, 



J-T/2 



where Z is a function of t, which is restricted in such a way 

 that 0„ and ty v are continuous, and vanish for large values 

 of v according to some appropriate rule of convergence 

 (the sequel suggests that the necessary condition is 

 <p y> ty v = 0(v~*~ e ), e>0), while they vanish for v = 0. T is so 

 large that for z; = 0(l/T), <jy v and -\jr v are negligible. 



By the principle of the inversion 'of Fourier integrals, if 



Z i=| (<t> v Gos27rvt + ^r v sm27rvt)djj, . . (2'2) 



then Z X = Z in the interval +T/2 and vanishes outside it. 



* Stokes, Edinb. Trans, xx. p. 317 (1852), 

 t Rayleigh, Phil. Mag. xxvii. p. 460 (1889). 



(2-1) 



