pass on without resolution, and the integrated effect of all the
secondary waves (S 2). The occurrence of factors such as sin [phi], or
1/2(1 + cos [theta]), in the expression of the secondary wave has no
influence upon the result of the integration, the effects of all the
elements for which the factors differ appreciably from unity being
destroyed by mutual interference.

The choice between various methods of resolution, all mathematically
admissible, would be guided by physical considerations respecting the
mode of action of obstacles. Thus, to refer again to the acoustical
analogue in which plane waves are incident upon a perforated rigid
screen, the circumstances of the case are best represented by the
first method of resolution, leading to symmetrical secondary waves, in
which the normal motion is supposed to be zero over the unperforated
parts. Indeed, if the aperture is very small, this method gives the
correct result, save as to a constant factor. In like manner our
present law (20) would apply to the kind of obstruction that would be
caused by an actual physical division of the elastic medium, extending
over the whole of the area supposed to be occupied by the intercepting
screen, but of course not extending to the parts supposed to be
perforated.

On the electromagnetic theory, the problem of diffraction becomes
definite when the properties of the obstacle are laid down. The
simplest supposition is that the material composing the obstacle is
perfectly conducting, i.e. perfectly reflecting. On this basis A. J.
W. Sommerfeld (_Math. Ann._, 1895, 47, p. 317), with great
mathematical skill, has solved the problem of the shadow thrown by a
semi-infinite plane screen. A simplified exposition has been given by
Horace Lamb (_Proc. Lond. Math. Soc._, 1906, 4, p. 190). It appears
that Fresnel's results, although based on an imperfect theory, require
only insignificant corrections. Problems not limited to two
dimensions, such for example as the shadow of a circular disk, present
great difficulties, and have not hitherto been treated by a rigorous
method; but there is no reason to suppose that Fresnel's results would
be departed from materially. (R.)

FOOTNOTES:

[1] The descending series for J0(z) appears to have been first given
by Sir W. Hamilton in a memoir on "Fluctuating Functions," _Roy.
Irish Trans._, 1840.

[2] Airy, loc.