fine Lines or transmitted by narrow Slits. 355 



The two sets of waves advancing along PO, P'O will be 

 supposed to be of equal amplitude ; but we shall require to 

 consider two distinct suppositions as to their phases. In 

 dealing with P the supposition is that the phases are pre- 

 cisely opposed. In this case we obtain from (8) as the 

 complete expression of the secondary waves 



R== — ljL-)$ e -ikr\ 2/B x sin a sin (/>-2B 2 sin 2ct sin 2</> + . . .},. 



. . . (30) 



the term in B disappearing, while the values of B b B 2 are 

 given by (19), (20). 



Each of the two separate solutions here combined, primary 

 and secondary terms included, satisfies the condition R = 

 at the surface of the cylinder and so of course does the 

 aggregate. It is easy to see that the aggregate further 

 satisfies the condition R = along AB, CD where 0, 0' are 

 equal, the contributions from the two solutions being equal 

 and opposite. Hence (30) gives the secondary waves due to 

 the incidence of primary waves along PO upon the reflecting 

 surface ABECD; and the expression for the primary waves 

 themselves is 



J^ _— gik{x cos a+y sin a) Jk(x cos a— y sin a) /0 1 \ 



x being parallel to OD and y parallel to OGr, so that 

 a = rGOS(f>, y = rsin<£. 



In like manner the c* solution may be built up. In this 

 case we have to give the same phase to the two component 

 primaries. Corresponding to the incident 



v»* -— J,k{x cos a+y sin a) i gik(x cos a— y sin a) /Q 0\ 



we have for the secondary disturbance 



c*= -(^V e-^Wo + St'Ci cos a cos $ 



-2C 2 cos2 a cos2(/>+ ...}. . . (33) 



Each solution, consisting of primary and associated secon- 

 dary, satisfies over the surface of the cylinder dc*/dn = 0, 

 dn being an element of the normal. And over the plane 

 part AB, CD the two solutions contribute equal and opposite 

 components to dc*/dn. Hence all the conditions are satisfied 

 for the incidence of waves along PO upon the compound 

 reflecting surface ABECD. 



The problem is now solved for the tw r o principal cases of 

 polarized incident light. If the incident light be unpolarized, 



