174 ON THE PASSAGE OF WAVES THROUGH [
As we have seen already, the tabulated quantity when kb is very si takes the form y -f log (ikb/ty, or logkb-0'8091 + l'5708i, whatever may In:-, value of a. In this case the condition (17) can be completely satisfied v W = A (b3 - y2)-*, A being chosen suitably. When kb is finite, (17) cat) longer be satisfied for all values of a. But when kb = |, or even when kb •• the tabulated number does not vary greatly with a and we may consider ( to be approximately satisfied if we make in the first case
TT(- 1-4123 + r4759i)4=-l, ................... (57
and in the second,
TT (_ 0-6432 + 1 '2268 i) A =-1 .................... (5H
The value of ^, applicable to a point at a distance directly in front of aperture, is then, as in (16),
In order to obtain a better approximation we require the aid of a KCC<I solution with a different form of M/1. When this is introduced, as an additi to the first solution and again with an arbitrary constant multiplier, it v enable us to satisfy (17) for two distinct values of «, that is of ?/, arid tli with tolerable accuracy over the whole range from cos a = 0 to cos a = + Theoretically, of course, the process could be carried further so as to satij (17) for any number of assigned values of cos a.
As the second solution we will take simply M> = ij so that the loft-hai member of (17) is
D(kr)dr.
If we omit k, which may always be restored by consideration of horn geneity, we have
_
'. .3 22.42.5
+ the same expression with the sign of 77 changed. The leading term in (60) is thus kb = 2