346 Lord Hayleigh : Effect of a 
overlapping 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 
condition cannot be satisfied when the plate is a true wedge ; 
for then the width for each colour is proportional to X. If, 
however, 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." 
In the paper f from which the above extracts are taken, 
the question was further discussed, and it appeared that the 
bands formed by cylindrical or spherical surfaces could be 
made achromatic, so far as small variations of A, are con- 
cerned, but only under the condition that there be a finite 
separation of the surfaces at the place of nearest approach. 
If a denote the smallest distance, the region of the nth band 
may form an achromatic system if 
a==±n\ (1) 
At the time pressure of other work prevented my examining 
the question experimentally. Recently I have returned to 
it and I propose now to record some observations and also 
to put the theory into a slightly different form more con- 
venient for comparison with observation. 
For the present purpose it suffices to treat the surfaces as 
cylindrical, so that the thickness is a function of but one 
coordinate x, measured along the surfaces in the direction 
of the refraction. The investigation applies also to spherical 
surfaces if we limit ourselves to to points lying upon that 
diameter of the circular rings which is parallel to the re- 
fraction J. If we choose the point of nearest approach as 
the origin of x, the thickness may be taken to be 
t = a + ku 2 , (2) 
* Strictly speaking the number of visible bands is doubled, inasmucli 
as they are now formed on both sides of the achromatic band. 
f " On Achromatic Interference Bands," Phil. Mas:, xxvii. pp. 77, 189, 
1889 ; ' Scientific Papers,' iii. p. 313. 
X In the paper referred to the general theovy of curved achromatic 
bands is considered at lenath. 
