202 Lord Rayleigh on Achromatic 



entire spectrum. The formation of the achromatic band, 

 or rather central black bar, depends indeed npon precisely 

 the same principles as the fictitious shifting of the centre of a 

 system of Fresnel's bands when viewed through a prism. 



In this example the formation of visible rings at unusual 

 thicknesses is easily understood ; but it gives no explanation 

 of the increased numbers observed by Newton. The width of 

 the bands for any colour is proportional to X, as well after the 

 displacement by the prism as before. The manner of over- 

 lapping of two systems whose nth bars have been brought 

 to coincidence is unaltered ; so that the succession of colours 

 in white light, and the number of perceptible bands, is much 

 as usual. 



In order that there may be an achromatic system of bands, 

 it is necessary that the width of the bands near the centre be 

 the same for the various colours. As we have seen, this con- 

 dition cannot be satisfied when the plate is a true wedge ; for 

 then the width for each colour is proportional to X. If, how- 

 ever, the surfaces bounding the plate be curved, the width for 

 each colour varies at different parts of the plate, and it is 

 possible that the blue bands from one part, when seen through 

 the prism, may fit the red bands from another part of the 

 plate. Of course, when no prism is used, the sequence of 

 colours is the same whether the boundaries of the plate be 

 straight or curved. 



For simplicity we will first suppose that the surfaces are 

 still cylindrical, so that the thickness is a function of but one 

 coordinate x, measured in the direction of refraction. If we 

 choose the point of nearest approach as the origin of x, the 

 thickness may be taken to be 



" t = a + bx 2 , (38) 



a being thus the least distance between the surfaces. The 

 black of the nth. order for wave-length X occurs when 



|nX = a + bx 2 ; (39) 



so that the width (8x) of the band at this place (x) is given by 

 ^X = 2bx 8x, 



01 Bx = X/iJba. ....... (40) 



Substituting for x from (38), we obtain, as the width of the 

 band of nth order for any colour, 



8 * = 4:s/b.s/(in\-a) ( 41) 



It will be seen that, while at a given part of the plate the 



