112 DIFFRACTION. 



given order. Then, in order that this point should belong to 

 the same fringe, for every distance of the luminous origin and 

 of the screen, it is only necessary that the interval of retarda- 

 tion, EP - EA, of the central and marginal parts of the wave 

 should be constant. For in this case the whole wave, AP, 

 may be divided into a given number of parts, such that the 

 difference of the distances of the successive points of division 

 from the point E shall be constant ; and therefore the effec- 

 tive wave consists of the same number of elementary portions 

 in the same relative state as to interference. 



Now, denoting OP by a, PQ, by 6, and QE by x, we 

 have 



EP = 

 since x is very small in comparison with b. Similarly, 



BO = V(a + b)* + r* = a + b + 2(a + b}' 



so thatEA = b + . , and EP - EA = J 



2(a + b) \b a + b 



But, by the condition of the question, this difference is a 

 constant quantity ; and denoting this constant by S, we have 



b a + b 



l2S(a + b 



/ - 

 \ a 



When b varies, a remaining unaltered, i. e. when the po- 

 sition of the screen is varied, the value of x is the ordinate 

 of an hyperbola whose abscissa is b ; so that the successive 

 points of the same fringe belong to an hyperbola, whose sum- 

 mit is the edge of the obstacle. 



(127) The interior fringes formed in the shadow of a nar- 

 row opaque body arise, it has been said, from the interference 



