448 Lord Rayleigh on the Light emitted from a 



Taking 1} 2 as coordinates of a representative point, fig. 3 r 

 the sides of the square OACB are 27r. Along the diagonal 



Fig. 3. 





G r 



' / 





/ / 





' / ' 



E 



/ / s 





/ / / 





/ / / 





/ ' 





/ s 





e 2 



B 



F 

 



OC, 1 and 2 are equal. If DE, FG be drawn parallel to OC, 

 so that OD, OF are equal to 8, the prohibited region is that 

 part of the square lying between these lines. Our integra- 

 tions are to be extended over the remainder, viz. the 

 triangles FBG, DAE, and every point, or rather every 

 infinitely small region of given area, is to be regarded as 

 equally probable. Evidently it suffices to consider one 

 triangle, say the upper one, where 2 > 6\. 

 For the denominator in (40) we have 



^d0 1 d0 2 = area of triangle FBG = 1(2tt-S) 2 . 



In the double integral containing the cosine, let us take first 

 the integration with respect to 2 , for which the limits are 

 0i + 8 and 2tt. We have 



.(; 



+5 



cos (0 2 — l )d0 2 = cos 8 — 1 — (27T— 8) sin 8 ; 



and since the limits for 1 are and 2tt — 8, we get as the 

 expectation of intensity 



1— cos 8 + {2tt — 8) sin 8 



2-4 



(41) 



(2tz— S) 2 ' ' ' 



If 8 2 be neglected, this reduces to 



2(1 -8/w) (42) 



If 8=7T, we have 2(1 — 4/7r 2 ) ; and if 8 = 2tt, we have 4, the 

 only available situations being #i = 0, 2 = 2tt, equivalent to 

 phase identity. 



This treatment might perhaps be extended to a greater 

 value, or even to the general (integral) value, of n; but I 

 content myself with the simplifying supposition that 8 is 

 very small. 



In (40) the integration with respect to n supposes 



