ion just described is the simplest, but it is
only one of an indefinite number that might be proposed, and which are
all equally legitimate, so long as the question is regarded as a
merely mathematical one, without reference to the physical properties
of actual screens. If, instead of supposing the _motion_ at dS to be
that of the primary wave, and to be zero elsewhere, we suppose the
_force_ operative over the element dS of the lamina to be that
corresponding to the primary wave, and to vanish elsewhere, we obtain
a secondary wave following quite a different law. In this case the
motion in different directions varies as cos[theta], vanishing at
right angles to the direction of propagation of the primary wave. Here
again, on integration over the entire lamina, the aggregate effect of
the secondary waves is necessarily the same as that of the primary.

In order to apply these ideas to the investigation of the secondary
wave of light, we require the solution of a problem, first treated by
Stokes, viz. the determination of the motion in an infinitely extended
elastic solid due to a locally applied periodic force. If we suppose
that the force impressed upon the element of mass D dx dy dz is

DZ dx dy dz,

being everywhere parallel to the axis of Z, the only change required
in our equations (1), (2) is the addition of the term Z to the second
member of the third equation (2). In the forced vibration, now under
consideration, Z, and the quantities [xi], [eta], [zeta], [delta]
expressing the resulting motion, are to be supposed proportional to
e^int, where i = [sqrt](-1), and n = 2[pi]/[tau], [tau] being the
periodic time. Under these circumstances the double differentiation
with respect to t of any quantity is equivalent to multiplication by
the factor -n^2, and thus our equations take the form

d[delta] \
(b^2[Delta]^2 + n^2)[xi] + (a^2 - b^2) -------- = 0 |
dx |
|
d[delta] |
(b^2[Delta]^2 + n^2)[eta] + (a^2 - b^2) -------- = 0 > (7).
dx |
|
d[delta] |
(b^2[Delta]^2 + n^2)[