Interference-Bands. 205 



if we ascribe a constant value to X, we have the relation 

 between f , rj corresponding to any prescribed values of n — 

 that is, the forms of the interference-bands in homogeneous 

 light. If the light be white, the bands are in general con- 

 fused ; but those points are achromatic for which 



This is a relation between f and rj defining a curve, which we 

 may call the achromatic curve, corresponding in some respects 

 to the achromatic line of former investigations, where n is 

 independent of y. There is, however, a distinction of some 

 importance. When n is a function of f and X only, the 

 achromatic line is also an achromatic band ; that is, n remains 

 constant as we proceed along it. But under the present less 

 restricted conditions n is not constant along (54). The 

 achromatic curve is not an achromatic band ; and, indeed, 

 achromatic bands do not exist in the same development as 

 before. They must be regarded as infinitely short, following 

 the lines n = constant, but existent only at the intersection of 

 these with (54). Practically a small strip surrounding (54) 

 may be regarded as an achromatic region in which are visible 

 short achromatic bands, crossing the strip at an angle de- 

 pendent upon the precise circumstances of the case. 



The application of this theory to the observations of Newton 

 presents no difficulty. The. thickness of the layer of air at the 

 point x, y, measured from the place of closest approach, is 



t = a+b{x 2 +y 2 ); (55) 



and since t = ^nX, the relation of n to x, y, and X is 



±n = aX- l + bX- l (x 2 +f) (56) 



This equation defines the system of bands when the com- 

 bination is viewed directly. The achromatic curve determined 

 by (54) is 



a + b(a?+tf) = 0, 



and is wholly imaginary if a and b are both positive and 

 finite. Only when a = 0, that is when the glasses touch, 

 is there an achromatic point x—0, y = 0. 



When a prism is brought into operation, we may suppose 

 that each homogeneous system is shifted as a whole parallel 

 to x by an amount variable from one homogeneous system to 

 another. If the apparent coordinates be f , rj, we may write 



i = x-f{\), v =y (57) 



Using these in (56), we obtain as the characteristic equation 



