178 ON THE PASSAGE OF WAVES THROUGH [375
The supplementary solutions ^, equal to <p — %, may be written
where M/w, ^ are functions of y, and the integrations are over the aperture. D as a function of r is given by (13), and r, denoting the distance between dy and the point (x, 17),' at which ^m, typ are estimated, is equal to ^{3? + (y- v))2}. The form (71) secures that on the walls ^m = ^P~ 0, so bhat the condition of evanescence there, already satisfied by %, is not disturbed. It remains to satisfy over the aperture
m = ^p, -2ik + d'tym/da; = dilrPldas ............. (72)
The first of these is satisfied if XFTO= - "Vp, so that t/rm and ^p are equal at any pair of corresponding points on the two sides. The values of d-^rm/d,v, dtyp/dx are then opposite, and the remaining condition is also satisfied if
dfym/dx = ik, dtyp/d(v = — ik ................... (73)
At a distance, and if the slit is very narrow, dDjdx may be removed from under the integral sign, so that
-ikr />7K\
6 ........................... (7S)
And even if kb be not small, (74) remains applicable if the distant point be directly in front of the slit, so that as — r. For such a point
(76>
There is a simple relation, analogous to (68), between the value of Wp at any point (77) of the aperture and that of fyp at the same point. For in the application of (71) only those elements of the integral contribute which lie infinitely near the point where ^ is to be estimated, and for these dD[doc = xjr\ The evaluation is effected by considering in the first instance a point for which x is finite and afterwards passing to the limit. Thus
It remains to find, if possible, a form for >PP, or ^, which shall make d^!fPldx constant over the aperture, as required by (73). In my former paper, dealing with the case where kb is very small, it was shown that knownrious phases, of which due account must !><• takem The disturbance is no longer circularly symmetrical as in (16). But if, as is usual in observations with the microscope, we restrict ourselves to