Electromagnetic Theory of Light. 91 



condition be satisfied, there is no further limitation upon the 

 size of the obstacle. In the case where the secondary ray 

 forms the prolongation of the primary, or deviates sufficiently 

 little from this direction, the exponential in (44) reduces to 

 unity, signifying that every element of the obstacle acts alike, 

 any retardation of phase at starting due to situation along the 

 primary ray being balanced by an acceleration corresponding 

 to a less distance to be travelled along the secondary ray. At 

 a greater or less obliquity, according to the size of the obstacle, 

 opposition of phase sets in; and at still greater obliquities the 

 resultant can be found only by an exact integration. Its in- 

 tensity is then less, and generally much less, than in the first- 

 case — a conclusion abundantly borne out by observation. 



The simplest example of this kind is that afforded by an 

 infinite cylinder (e. g. a fine spider-line), on which the light 

 impinges perpendicularly to the axis, so that every thing 

 takes places in two dimensions. This case is indeed not 

 strictly covered by the preceding formulae, on account of the 

 infinite extension of the region of disturbance; but a moment's 

 consideration will make it clear that each elementary column 

 here acts according to the laws already described — that is to 

 say, gives rise to a component disturbance whose phase is de- 

 termined by the situation of the element along the primary 

 and secondary rays. If the angle between the two rays be 

 called %, we have to consider the value of 



?C e ik(x+x cos x+V sin \) d x dv. 



Introducing polar coordinates r, #, we find 



(B + OBGOsx+y sin% = 2r cos \% cos (6— \ %) ; 

 so that the integral 



—^e ikr ' 2cos ^' coae rdrd0 



n27T 

 {cos (2kr cos J % cos 0) 



+ i sin (2kr cos -§-% cos 6) } r dr cW 



= 2tt ( a j (2kr cos J %) r dr, (45) 



J denoting the Bessel's function of zero order. 



The integration with respect to r indicated in (45) can be 

 effected by known properties of Bessel's functions; and the 

 result is expressible by a function of the first order. We get 



, TO J,(2fa»cosi X ); (46) 



kcos^x 



