ON OPTICAL THEORIES. 201 
The general integral is then applied to two cases, which must be care- 
fully distinguished from each other. In the first case, suppose that a 
periodic force acting parallel to a fixed direction acts throughout a given 
element of volume in the medium. Let the plane of zz contain the fixed 
direction, and let the axis of « make an angle a with it. Let D be the 
density, and T the volume of the element, and let (DT)~—'/(#)dt be the 
velocity communicated to it in time d¢. 
Then 
) 
=. COs & -") cos a [2, er 
aa ia! a, e 2a Dr? Aer 
n=0 Lis ae ie 
r 
. i r sina f{¢ 
i= pope (! — 5) aap [eee dat | 
a 
Now, we have seen that in the ether the ratio a/b is probably very large, 
hence the first term in £, on which the normal vibrations depend, is pro- 
bably very small compared with the first term in. The molecules of 
‘am incandescent body may be looked upon, at least very approximately, 
as centres of disturbing forces, and the above equations show us how it 
is that from such centres transverse vibrations only are propagated. 
Tf the ether be absolutely incompressible, so that a /b is infinite, then 
longitudinal vibration would be impossible. 
Suppose, now, the first term in & omitted, and put f(t) =c sin 2rbt /A, 
Then for the most important term we have— 
csina . Qqr 
= —| bt — ; ; : 
iat? x (2 r) (78) 
‘ 
and the first term in £ is of the order X /xr compared with the leading 
term in. Hence, except at distances from the source which are com- 
parable with the wave length, the terms in £ may be neglected, and the 
motion is strictly transverse. 
This solution applies to the case of an element of volume vibrating in 
any given manner and emitting light into the surrounding space. Every- 
thing is symmetrical around the direction of vibration of the element of 
volume. It does not apply, as has been supposed by some writers, to 
the problem of diffraction ; for in this case we have a train of waves being 
ai through an aperture, and producing disturbance in the medium 
ond. 
_Let us suppose the aperture to be plane, and that plane waves are 
being propagated through it in the direction of its normal; take 
this for the axis of «, the plane of the aperture being « = 0, and the 
_ axis of z the direction of vibration. Let O, bea point in the aperture, 
: and consider the disturbance propagated in a small interval of time 7, 
across an element dS, at O,. This disturbance occupies a film of thick- 
ness br, and consists of a displacement f(bt’) and a velocity bf’(bt'). 
Thus, for a point O, at a distance r from O,, and at a time ¢, given by 
t=? + 1/b, the initial disturbance is the above displacement and velocity 
extending over a volume brdS about O; and if 1, m, n are the direction 
