644 Mr. C. G. Darwin on the 



so as to give the total flow of energy through a given 

 aperture in a given time. By successive applications of 

 Rayleigh's theorem the result can be brought to an integral 

 of the same form as the corresponding thermodynamic 

 expression, and the interrelation is then given by equating 

 the two integrands. 



Let the optical field be given by X, Y, Z ; a, ft, 7, which 

 are arbitrary functions of x, y, z, t, subject to their satisfying 

 the electromagnetic equations and to suitable conditions as 

 to continuity and convergence. We shall evaluate the total 

 radiation crossing the yz plane between y=+p/2 and 

 z= +q/2 during the time between t= +T/2. Here p and q 

 are to be so large that they include many wave-lengths of 

 the types of waves that occur, and the same must be true 

 for T and the periods. On the other hand, they must be 

 sufficiently small to allow for the gross variations of radiation 

 in the different parts of space. These limitations are like 

 tho*e which are imposed in the kinetic theory of gases. 

 More precisely the functions X, Y, Z ; a, ft, 7 are such that 

 the averages to be constructed shall be independent of p, q, 

 and T. It should be said that this independence will not be 

 apparent formally, but will occur in the same way as in 

 (2-3). 



4. We proceed to consider how the radiation is to be 

 analysed into plane waves. A typical plane wave is described 

 by the expression 



S =gi s cos t -j- (j 2 s sin t, 

 where 



r = 2irv{t — (xsm 6 cos <p + y sin #sin(£ + : cos 0)/c}, (4*1) 



while S and <7 S represent in turn each of the components of 

 electric and magnetic force according to the rule 



<j x = U cos cos (f> — Y sin <$> g a = —\] sin.<£ — Y cos 6 cos </> ^ 



g Y = U cos 6 sin cf> 4- V cos (/> <f = U cos $ — Y cos 6 sin <£ \ . (4'2) 



# z =-Usin0 jf/ v = Ysin(9 ) 



Here U corresponds to the wave polarized with electric 

 force in a plane through the z axis and Y to that in the 

 perpendicular direction. 



The arbitrary field can be represented by taking four 

 quantities U l5 Y l5 U 2 , Y 2 as functions of v, 0, <fi and expressing 

 any of the forces as 



S = I dv \ sin 6 dd 1 'd$(ffi* cos r + g£ sin t) . 



Jo Jo Jo 



