OF THE RAYS OF LIGHT. 151 



n being any number of the natural series 0, 1, 2, 3, &c. Bright 

 fringes, therefore, will be formed at all the points included in the 

 former equation, and dark ones at the points corresponding to the 

 latter. 



A 



B 



A. 



M 







Let OP be the reflector; OM the screen placed in contact with it, 

 and perpendicular to its plane ; and let A be the luminous origin, 

 and A its reflected image at an equal distance below the line OB. 

 Then, if M be any point whose illumination is required, 8 = AM, 

 X = AM. 



Now if AB be denoted by p, BO by d, and OM by v, it is 

 obvious that 



& = ffl + ( p _ ^ %* = d 2 +(p + z) 2 . 



Hence, approximately, 



and therefore 



the angle AOB being denoted by a. Hence the general expres- 

 sion of the intensity of the light, at any point M, is 



/ #\ 



A* = a 2 + 2aa cos UTT tan a v + a*. 



Again, substituting for S' - S its value just found, we see that 

 the successive fringes will be formed at the distances given by the 



formula 



x = \ m X cotan a ; 



in which m is any number of the natural series, its even values 

 giving the places of the bright fringes, and its odd values those of 

 the dark ones. Accordingly, the bright fringes are formed at t 

 distances 0, 21, 4/, &o., and the dark ones at the distances inter- 

 mediate, /, 3/, 5/, &c., / being equal to A cotan a : the successive 

 fringes, therefore, are equidistant. It is obvious that the angle 



