648 On the Theory of Radiation. 



really it would be independent, so here K v appears to involve 

 p, g, T. It also involves the factor sin 6 cos <j> and at first 

 sight it seems that this should have disappeared in the course 

 of the calculations, since it arises from the cose in (6*1) and 

 simply means that the cone of rays may be cut by a plane in 

 any direction to give the aperture. But, in effect, this 

 corresponds to taking a new aperture — that is, new values 

 for p and q, — and as it is not possible to verify that such a 

 change is immaterial even when the plane of the aperture is 

 unchanged, a fortiori, it will not be possible when the plane 

 is changed as well. We can only say that if a real radiation 

 field were specified in the optical manner, and if the calcu- 

 lations of §§4, 5, and 6 were applied to it, we should find 

 that the resulting K^ was independent of the time and of 

 both the area and direction of the aperture. 



If the values of U and Y aro written in from (4*6) and 

 (4'4), we have in full 



K = 



i v 2 cp> 2 Cii 2 r T / 2 cvi 2 c* 2 



^ To— 1 dy\ dz\ dt\ dy'\ 



l V COS QbTTCj-pfi 'J - q/2 J-T/2 J-^/2 J-g/S 



p#T sin 6 cos <£ 8ttc ( 



JT/2 

 dt' cos 2irv{{t — 1')~ [(y-y) sin0sin0 + (s— /)cos£]/c} 



x ( — Z cos <f> + /3 sin + y cos sin $) 



x ( — Z' cos </> + & sin 6 + y' cos 6 sin 0), . (6'4) 



where Z' 7 etc. are the same as Z, etc., taken with arguments 

 ?/, z', t f . For K v ' the brackets are 



(7 cos <£ + Y sin 6 + Z cos 6 sin <£), etc. 



and otherwise the expression is the same. The relation (6*4) 

 can be put into other forms by virtue of (4*7). A parti- 

 cularly simple case is that of the radiation going normally 

 from the aperture, which is found by putting = 1712, <£ = (). 

 Kf> z signifies the K v of the ray going along the % axis and 

 polarized with electric force in the xz plane. 



1 „2 r*pl2 f*ql2 r*T/2 f*p/2 Pq/2 



K*<*=-^7r-l dy\ dz\ dt\ dy f \ dz' 



pqT Sttc}_ p/2 J y ql2 J_ T/2 J- p/ 2 J J-. q /2 



/'T/2 



I dt' cos2>irv(t-t')(/3-Z)(p'-Z'). (6-5) 



J-T/2 



For K Il? the last two factors must be replaced by (7 + Y) 



(y'+y'). 



This completes the problem of formally expressing the 

 thermodynamic specification of the field in terms of the 

 optical. 



