572 ON THE LIGHT EMITTED FROM A [436
So far positions of the first source near the ends of the line have been excluded. If the neglect of these positions can be justified, (20) reduces to 2 simply.
It is not difficult to see that the suggested simplification is admissible under the conditions contemplated. If SB, so be the distances of the two sources from one end of the line, the question is as to the value of
doc
where the integration with respect to x' may be taken first. Let X denote a length large in comparison with X, but at the same time small in comparison with /. If as lie between X and I — X, the integral with respect to x may be identified with jrjkl, and neglected, as we have seen. We have still to include the ranges from x = 0 to os = X, and from x = I — X to x = I, of which it suffices to consider the former. The range for x may be divided into two parts, from 0 to x, and from as to I. For the latter we may take
• da)' sin k (x — x) _ TT
so that this part yields finally after integration with respect to as,
X 7T
J'W................................
As regards the former part, we observe that since {Hsinfl can never exceed unity,
; das' sin k (a/ — x) x
in which again x<X. The result of the second integration leaves us with a quantity less than Xzjl\ The anomalous part, both ends included, is less than
which is small in comparison with the principal part*, of the order Tr/kl and itself negligible. We conclude that here again the mean intensity in a great number of trials is 2 simply. It may be remarked that this would not apply to the mean intensity in a specified direction, as we may see from the case where the initial phases are the same. In a direction perpendicular to the line on which the sources lie, the phases on arrival are always in agreement, and the intensity is 4, wherever upon the line the sources may be situated. The conclusion involves the mean in all directions, as well as the mean of a large number of trials.
Under a certain restriction this argument may be extended to a large number n of unit sources, since it applies to every term under the summation
[* Provided that Xjl is small compared with \/X; if these ratios are of the same order, (25) is comparable with ir/hl. W. F. S.]t, kl being by supposition a large quantity. write