PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 33 



ultimately zero, and the limit of the above expression will be rigorously zero. Hence we get 

 in the limit 



£ = c sin — (bt - p), or = 0, 

 • A 



according as p is positive or negative. Hence the disturbance continually transmitted 



across the plane P produces the same disturbance in front of that plane as if the wave had 



not been broken up, and does not produce any back wave, which is what it was required to 



verify. 



It may be objected that the supposition that the perimeter of A is free from circular 



arcs having 0' for centre is an arbitrary restriction. The reply to this objection is, that we 



have no right to assume that the disturbance at O which corresponds to an area A approaches 



in all cases to a limit as A expands, remaining similar to itself. All we have a right to assert 



a priori is, that if it approaches a limit that limit must be. the disturbance which would exist 



if the wave had not been broken up. 



It is hardly necessary to observe that the more general formula (45) might have been 

 treated in precisely the same way as (46). 



35. In the third Volume of the Cambridge Mathematical Journal, p. 46, will be found 

 a short paper by Mr. Archibald Smith, of which the object is to determine the intensity in a 

 secondary wave of light. In this paper the author supposes the intensity at a given distance 

 the same in all directions, and assumes the coefficient of vibration to vary, in a given direction, 

 inversely as the radius of the secondary wave. The intensity is determined on the principle 

 that when an infinite plane wave is conceived to be broken up, the aggregate effect of the 

 secondary waves must be the same as that of the primary wave. In the investigation, the 

 difference of direction of the vibrations corresponding to the various secondary waves which 

 agitate a given point is not taken into account, and moreover a term which appears under the 

 form cos co is assumed to vanish. The correctness of the result arrived at by the latter 

 assumption maybe shewn by considerations similar to those which have just been developed. 

 If we suppose the distance from the primary wave of the point which is agitated by the 

 secondary waves to be large in comparison with \, it is only those secondary waves which reach 

 the point in question in a direction nearly coinciding with the normal to the primary wave 

 that produce a sensible effect, since the others neutralize each other at that point by inter- 

 ference. Hence the result will be true for a direction nearly coinciding with the normal to 

 the primary wave, independently of the truth of the assumption that the disturbance in a 

 secondary wave is equal in all directions, and notwithstanding the neglect of the mutual 

 inclination of the directions of the disturbances corresponding to the various secondary waves. 

 Accordingly, when the direction considered is nearly that of the normal to the primary wave, 

 cos# and sin <p in (46) are each nearly equal to 1, so that the coefficient of the circular function 

 becomes cdS (\r) -1 , nearly, and in passing from the primary to the secondary waves it is 

 necessary to accelerate the phase by a quarter of an undulation. This agrees with Mr. Smith's 

 results. 



The same subject has been treated by Professor Kelland in a memoir On the Theoretical 

 Vol. IX. Part I. 5 



