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η

by taking the quantity under the circular function in place of ʼn for the independent variable. Now it is known that the value of the last integral is π, as will also presently appear, and therefore we have for the intensity I at any point,

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2

I= "ph)

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which is independent of q', as of course it ought to be.

13. Suppose now that instead of a line of homogeneous light we have a line of white light, the component parts of which have been separated, whether by refraction or by diffraction is immaterial, so that the different colours occupy different angular positions in the field of view. Let BB be the illumination on the object-glass due to a length of the line which subtends the small angle B, and to a portion of the spectrum which subtends the small angle at the centre of the object-glass. In the axis of x take a new origin O", and let §, p' be the abscissæ of O', M reckoned from O", so that p = p' - §. In order that (12) may express the intensity at M due to an elementary portion of the spectrum, we must replace A by Bdy, or Bf-1de; and in order to find the aggregate illumination at M, we must integrate so as to include all values of § which are sufficiently near to p' to contribute sensibly to the illumination at M. It would not have been correct to integrate using the displacement instead of the intensity, because the different colours cannot interfere. Suppose the angular extent, in the direction of x, of the system of diffraction bands which would be seen with homogeneous light, or at least the angular extent of the brighter part of the system, to be small compared with that of the spectrum. Then we may neglect the variations of B and of λ in the integration, considering only those of § and p, and we may suppose the changes of p proportional to those of §; and we may moreover suppose the limits of έ to be ∞ and +∞. Let p' be the value of p, and that of dp/d, when = p', so that we may w §

put p = p' +☎ (p' — §); and take p instead of § for the independent variable. Then putting for shortness

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2331 ["_{sin3 h,p+sin2 k,p + 2 sin h,p. sink,p. cos (p ́ — 9,p)

2BN

I=

под

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Now

Similarly,

[_sink,p.

Moreover, if we replace

81

dp

2

p*
Ρ

=

cos (p'-g,p) by cos p'. cos g,p+ sin p'. sin g‚p,

dp

p2

the integral containing sin p' will disappear, because the positive and negative elements will destroy each other, and we have only to find w, where

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Now we get by differentiating under the integral sign,

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= 1 ) __ {sin (g, + h, + k) p + sin (g, − h, − k) p

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dp

P

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according as s is positive or negative. If then we use F (s) to denote a discontinuous function of s which is equal to +1 or 1 according as s is positive or negative, we get

dw П

=

Z (F(g,+h,+ k)+F(g,−h,−k) − F(g,+h,−k)—F (g,+k,−h,)}.

dg, 4

This equation gives

dw

dg,

= 0, from g,= ∞ to g, = − (h,+k,)

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Now w vanishes when g, is infinite, on account of the fluctuation of the factor cos gp under the integral sign, whence we get by integrating the value of dw/dg, given above, and correcting the integral so as to vanish for g,= ∞,

w= 0, from g,= ∞ to g,=

π

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(h, k) ;

w= (h,+k,+g,), from g1 =-
:- (h,+ k) to g1 = — (h, ~ k) ;

2

g,

·w = πk, ог=πh, (according as h,>k, or h ̧<k,)

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= 1⁄2 (h, + k, — 9,), from g, = h' ~ k, to g, = h, + k, ;

w=

2

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Substituting in the expression for the intensity, and putting in (13) g,πg'f, so that

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when the numerical value of g' exceeds h+k;

2 Bl
fa

.........(16),

I= {h + k + (h + k − √ g3) cos p'}.........................

when the numerical value of g' lies between h+k and h - k;

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according as h or k is the smaller of the two, when the numerical value of g' is less than hk.

The discontinuity of the law of intensity is very remarkable.

By supposing g,=0, k,=h, in the expression for w, and observing that these suppositions reduce w to

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a result already employed. This result would of course have been obtained more readily by differentiating with respect to h.

14. The preceding investigation will apply, with a very trifling modification, to Sir David Brewster's experiment, in which the retarding plate, instead of being placed in front of the object-glass of a telescope, is held close to the eye. In this case the eye itself takes the place of the telescope; and if we suppose the whole refraction to take place at the surface of the cornea, which will not be far from the truth, we must replace ƒ by the diameter of the eye, and by the angular extent of the portion of the spectrum considered, diminished in the ratio of m to 1, m being the refractive index of the cornea. When a telescope is used in this experiment, the retarding plate being still held close to the eye, it is still the naked eye, and not the telescope, which must be assimilated to the telescope considered in the investigation; the only difference is that must be taken to refer to the magnified, and not the unmagnified spectrum.

Let the axis of x be always reckoned positive in the direction in which the blue end of the spectrum is seen, so that in the image formed at the focus of the object-glass or on the retina, according as the retarding plate is placed in front of the objectglass or in front of the eye, the blue is to the negative side of the red. Although the plate has been supposed at the positive side, there will thus be no loss of generality, for should the plate be at the negative side it will only be requisite to change the sign of p.

First, suppose p to decrease algebraically in passing from the red to the blue. This will be the case in Sir David Brewster's experiment when the retarding plate is held at the side on which the red is seen. It will be the case in Professor Powell's experiment when the first of the arrangements mentioned in Art. 2 is employed, and the value of N in the table of differences mentioned

in Art. 5 is positive, or when the second arrangement is employed and N is negative. In this case is negative, and therefore g' <- (h+k), and therefore (15) is the expression for the intensity. This expression indicates a uniform intensity, so that there are no bands at all.

Secondly, suppose p to increase algebraically in passing from the red to the blue. This will be the case in Sir David Brewster's experiment when the retarding plate is held at the side on which the blue is seen. It will be the case in Professor Powell's experiment when the first arrangement is employed and N is negative, or when the second arrangement is employed and N is positive. In this case is positive; and since varies as the thickness of the plate, g' may be made to assume any value from − (4g+h+ k) to+by altering the thickness of the plate. Hence, provided the thickness lie within certain limits, the expression for the intensity will be (16) or (17). Since these expressions have the same form as (1), the magnitude only of the coefficient of cos p', as compared with the constant term, being different, it is evident that the number of bands and the places of the minima are given correctly by the imperfect theory considered in Section I.

=

=

15. The plate being placed as in the preceding paragraph, suppose first that the breadths h, k of the interfering streams are equal, and that the streams are contiguous, so that g=0. Then the expression (17) may be dispensed with, since it only holds good when g' 0, in which case it agrees with (16). Let T, be the value of the thickness T for which g': 0. Then T=0 corresponds to g'(h+k), T= T, to g'=0, and T=2T, to g'=h+k; and for values of T' equidistant from T,, the values of g' are equal in magnitude but of opposite signs. Hence, provided T be less than 27, there are dark and bright bands formed, the vividness of the bands being so much the greater as T is more nearly equal to To, for which particular value the minima are absolutely black.

Secondly, suppose the breadths h, k of the two streams to be equal as before, but suppose the streams separated by an interval 2g; then the only difference is that g'(h + k) corresponds to a positive value, T, suppose, of T. If T be less than T2, or greater than 2T-T,, there are no bands; but if T lie between T, and 2T-T, bands are formed, which are most vivid when T-T,, in which case the minima are perfectly black.

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