808 Mr. C. G. Darwin on the Reflexion of 



the integrations change variables 

 the latter will all go from -co to 



du dv dyjr = dF dG d~K cosec 20 sec 0, 



To perform the integrations change variables from u, v\ ^r 

 to F, G, H ; the latter will all go from -co to go , and 



and so 



err f^ 



Eo)/J=N2/ 2 cosec 20 U\dFdGdR\ dY dY' 



CO 



This expression can be evaluated by an inversion of the 

 order of integration ; I shall not attempt to justify the pro- 

 cess rigorously. First, take the F, G, H integrations between 

 the large limits +F , etc. Then 



<J CO 7 



Efi)/J = N 2 / 2 cosec 20(2/ky Lt I dY dY' 



sinkF^(x-x') sinkG^jy -y') sin kK^jz-z') 

 x — x ! y — y 1 z — z' 



Now in the x' integration, which is to follow next, the 

 presence of F^ implies that the only important part is near 

 x' = x. Similarly, for y' and z'. Hence it will be valid to 

 take these three integrations over all space instead of only 

 over the crystal, for the parts outside will contribute 

 nothing. We now have 



J" 



'sin A F^ (#■«-#') , , , 



-ax —7T, etc. 





The final three integrations then simply yield 



Ea>/J = N 2 / 2 cosec 20{2/k) z . tt 3 . V. 



Now 2ir/k is the wave-length X. Also we shall adopt the 

 notation of B.J.B. ii. and write 



Q = N 2 / 2 A, 3 cosec 2<9 (4*7) 



Then 



Er»/J = QY (4-8) 



Q is of the dimensions of the reciprocal of a length. This 

 equation is the same as B.J.B. i. p. 326 (4). A little care is 

 needed in considering it, because its physical dimensions are 

 different from those of other equations which will occur 

 later, though it is similar to them in appearance. 



The factor Q will include the special peculiarities of the 

 crystal, such as the weak (1, 1, 1) reflexion of rock-salt. 



