BRIGHT FRINGES IN SHADOWS. 



171 



radical tlio cocfHcient 3 ; since the diffarence of paths Be — BA, by wbicli it 

 is produced, is equal to §A, instead of ^X. For the third ^vc prefix 5. Thus 

 the successive distances are — 



^Jiii^. ,.^/("+'^)^ ,.^=7-2<ri±i!H. [12.] [ 



13.] 



The second hyperbola passes at the distance of f A behind the obstnicting edge, 

 and the third at the distance of ^X. Owing- to the great disproportion between 

 the axes, r — ^X, and s/ta — ^/.^, which arc very nearly in the ratio of -/?' I V^^ 

 the curvature is very slight except for a short distance from the vertex ; so that 

 the branches in fact sensibly coincide with their asymptotes. But near the vertex 

 the curvature is very decided. 



ii 



j\r o.aRi:v: 



Fiff. 40. 



From what has been said, it will be 

 evident that when the obstruction 

 presents a salient angle insiead of a 

 straight edge, the fringes will pass 

 round the angle in circular arcs, instead 

 of making an angle also. Ind(X'd, the 

 systems of fringes around such an 

 angle are surfaces of hyperboloids of 

 revolution. In the case of re-eatering 

 angles of 90"^ or less, the fringes cross 

 each other without interference, and 

 are continued up to the edge of the shadow on each side of the angle. These 

 are easily seen to be necessary consequences of the theory of their formation. 

 The annexed figux'c exhibits the phenomena. 



When the obstructing body is large, no f. inges are seen within the shadow. 

 Some light strays beyond the geometrical boundary of the shadow, but it rapidly 

 fades away, and produces no very sensible effects. If, however, a very narrow 

 object be employed, the waves from opposite sides may mingle and interfere. 

 In this case fringes or stripes will be seen wit/tin the shadoio. The light con- 

 cerned in producing these fringes comes from the portion of the main wave 

 which is c'ose to the obstruction, as the more distant parts of each half wave 

 will hold each other in check in the manner heretofore exi)lained. Points being 



taken at a, b, c. a', b' , c', &c., such that the succes- 

 sive differences of distance of these points from B 

 may be ^X, we shall have, on each side, a series of 

 resultant actions alternately positive and negative, as 

 illustrated in the foregoing case, such as — 

 -\-m — m'-\-m" — ml"-\-m"", &:c. 

 And as the effect of any portion of this series, left 

 by cutting oft' terms from the beginning, depends on 

 the sign of the first of the remaining terms — that is 

 to say, as the effect due to the first of these terms 

 exceeds the joint effect of all that follow — it is evident that no part of the main 

 wave can have anything to do with producing fringes in the shadow, ex- 

 cept Qa and Aa' . And since there can be no fringes produced at all, unless 

 the light from both sides reaches the same point, the centre of the shadow, Avhich 

 is equidistant from the edges of the obstacle — that is, from the front of the 

 main wave — must exhibit a light stripe. On each side of this will be found posi- 

 tions, as for instance B', where the distances from A and C differ by half an 

 undulation ; and here the darkness will be complete. At distances a little greater 

 will be found positions where the differences of distance from A and C amount 

 to an entire undulation; and here will be found once more bright stripes. If the 

 object be very narrow, these interferences may occur not only throughout all 



