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



Z, m are small and are measured with reference to some 

 standard direction in the conglomerate, not necessarily the 

 mean direction of the blocks. The intensity of the beam 

 reflected by a single block is then given by (4 '5) provided 

 that u— I is put for u } v + l for v, and ty — 2m tan 6 for -x/r. 

 To specify the distribution of the blocks let 



VF(W, l,m)dWdldm .... (5-1) 



be the number of blocks in the volume V of the con- 

 glomerate which are themselves of volumes between 

 W and W + dW and have normals between I, m and 

 I + dl, m -j- dm. It follows that 



00 



(dW ((did 



mWF(W, Z,m) = l. . . (5*2) 



The intensity of reflexion in the direction v, yjr is therefore 

 obtained by taking (4*5) for a volume W (modified as 

 above), multiplying it by (5*1) and integrating over all 

 values of W, I, m that occur. To obtain the reflected 

 power we multiply by r 2 cos dv dty and integrate over 

 all values of v, ijr. The result is 



00 



JN 2 / 2 cos 6 . Y It dv dyjr \ F(W, I, m)dW dl dm \ dV dY' 



— CO 



expik{(x — a:')(u — v — 21) sin^+ (y—y')^ cos 6 — 2m sin 6) 



-r(z-z[)(u + v)GOsd\. . . . (5-3) 



This is a function of the angle of incidence, that is of u, and 

 the fact that we are not to integrate for u alters the proce- 

 dure. We must use the assumption that the diffraction 

 pattern of each block extends over a much narrower angle 

 than the distribution of the blocks. Now, if the shape of the 

 blocks were known, it would be possible to carry out the six 

 last integrations, and, regarding the result as a function of /, 

 the I integrand would then consist of the product of two 

 functions, one of which vanishes except for a narrow peak. 

 It would then be correct to substitute in the rest of the 

 integrand the value of I given by the maximum of the narrow 

 peak — that is, to substitute i(u — v) for / in F. We may 

 make the same change, even when the order of integrations 

 is altered so as to do that for / first. The same argument 

 also applies for ijr and m. If ^ oo , m^ denote quantities which 



