ON OPTICAL THEORIES. 199 
by the variation of these angles is omitted, and that, too, with perfect 
justice, for he is only concerned with the effects in the neighbourhood of 
the normal to the primary wave, and the dimensions of the diffracting 
aperture are small compared with the distance between it and the point 
at which the effects are considered, so that the change in either of these 
angles over the whole area of the diffracting area is small. 
Again, it is clear that the effect will be a circular function of r—vt, r 
being the distance between the element and the point at which the dis- 
turbance is sought, and v the velocity of propagation; but the simple 
theory does not indicate the relation between the phase of this circular 
function and that of the function representing the disturbance in the 
original wave. 
§ 2. Both these questions received their complete and final answer in 
the year 1849 from Professor Stokes.' We will quote a few words from the 
introduction to his paper: ‘The object of the first part of the following 
paper is to determine on purely dynamical principles the law of disturb- 
ance in a secondary wave, and that not merely in the neighbourhood of 
the normal to the primary wave, but in all directions. The occurrence of 
the reciprocal of the radius in the coefficient, the acceleration of a 
quarter of an undulation in the phase, and the absolute value of the 
coefficient in the neighbourhood of the normal will thus appear as parti- 
cular results of the general problem.’ 
The equations assumed for the motion are those of an elastic solid in 
the form given by Green— 
dé dé 
= 52 af 2 2 
2 Vet (2 b = (68) 
ee an, 
te. Oye ce mae, ES 
etc., where 7 + ‘iy + 7 
In the preliminary analysis the important general theorem involved in 
the equations 
dV PR ern Py . 
——ds = ——-+— 4 __. |dxdydz = 4nM |. - [(69 
‘enh {NG i, tga) ote, a 
is proved. 
It is then shown that the solution may be written 
E == E, <5 Eo . . . . . (70) 
where dey _ dy 0, ete., | 
dy dx 
dé dn ag i) 
ag ges eae Taare 
dz * dy x dz 
and 
dé dng amas lt 
dy da 
(72) 
dey, day, ly 
da dy da 
‘ Stokes, ‘On the Dynamical Theory of Diffraction, Trans. Camb. Phil. Soc. 
vol. ix. p.1; Math. and Phys. Papers, vol. ii. p. 243. 
