which the theory has
gained its greatest triumphs. The principle employed in these
investigations is due to C. Huygens, and may be thus formulated. If
round the origin of waves an ideal closed surface be drawn, the whole
action of the waves in the region beyond may be regarded as due to the
motion continually propagated across the various elements of this
surface. The wave motion due to any element of the surface is called a
_secondary_ wave, and in estimating the total effect regard must be paid
to the phases as well as the amplitudes of the components. It is usually
convenient to choose as the surface of resolution a _wave-front_, i.e. a
surface at which the primary vibrations are in one phase. Any obscurity
that may hang over Huygens's principle is due mainly to the
indefiniteness of thought and expression which we must be content to put
up with if we wish to avoid pledging ourselves as to the character of
the vibrations. In the application to sound, where we know what we are
dealing with, the matter is simple enough in principle, although
mathematical difficulties would often stand in the way of the
calculations we might wish to make. The ideal surface of resolution may
be there regarded as a flexible lamina; and we know that, if by forces
locally applied every element of the lamina be made to move normally to
itself exactly as the air at that place does, the external aerial motion
is fully determined. By the principle of superposition the whole effect
may be found by integration of the partial effects due to each element
of the surface, the other elements remaining at rest.

We will now consider in detail the important case in which uniform
plane waves are resolved at a surface coincident with a wave-front
(OQ). We imagine a wave-front divided into elementary rings or
zones--often named after Huygens, but better after Fresnel--by spheres
described round P (the point at which the aggregate effect is to be
estimated), the first sphere, touching the plane at O, with a radius
equal to PO, and the succeeding spheres with radii increasing at each
step by 1/2[lambda]. There are thus marked out a series of circles,
whose radii x are given by x^2 + r^2 = (r + 1/2n[lambda])^2, or x^2 =
n[lambda]r nearly; so that the rings are at first of nearly equal
area. Now the effect upon P of each element of the plane is
proportional to its area; but it depends also upon the distance from
P, and possibly upon