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of the luminous origin and of the screen-are fully explained on these hypotheses. It is easy to infer from them that, as the position of the screen is varied, the successive points of the same fringe are not in a right line, but form an hyperbola; and that, when the distance of the luminous origin is lessened, the inclination of these hyperbolic branches (considered as coincident with their asymptots) augments, and the fringes dilate.

The theory of Young, however, did not bear a closer comparison with facts. If the exterior fringes arose from the interference of the direct light with that obliquely reflected from the edge of the obstacle, it would follow that the intensity of the light in them should depend on the extent and curvature of the edge. Fresnel found, on the contrary, that the fringes were wholly independent of the form of the diffracting edge; the fringes formed by the back and by the edge of a razor, for example, being precisely alike in every respect. In the other cases of diffraction also, he perceived that the rays grazing the edge of the body were not the only rays concerned in the production of the fringes; but that the light which passed by these edges at sensible distances was also deviated, and concurred in their formation. Fresnel was thus forced to seek a broader foundation for his theory.

(99) In this theory the phenomena of diffraction are ascribed to the interference of the partial, or secondary waves, which are separated from the grand wave by the interposition of the obstacle. In applying this principle, Fresnel supposes the surface of the wave, when it reaches the obstacle, to be subdivided into an indefinite number of equal portions. Each of these portions may, by the principle of Huygens, be considered as the centre of a system of partial waves; and the mathematical laws of interference enabled him to compute the resultant of all these systems at any given point. This resultant vibration, Fresnel has shown, is in general expressed

by means of two integrals, which are to be taken within limits determined by the particular nature of the problem. Its square is the measure of the intensity of the light; and it is found that its value has several maxima and minima, which correspond to the intensities of the light in the bright and dark bands.

The problem of diffraction was thus completely solved; and its laws derived from the two principles to which the laws of reflexion and refraction are themselves referred,—the principle of Interference and the principle of Huygens. It only remained to apply the solution to the principal cases, and to compare the results with those of observation. The cases of diffraction selected by Fresnel are those whose laws have been already explained; viz. the phenomena produced-1, by a single straight edge; 2, by an aperture terminated by parallel straight edges; and 3, by a narrow opaque body of the same form. The agreement of observation and theory is so complete, that the computed places of the several bands seldom differ from those observed by more than the 100th part of a millimetre.

(100) The general circumstances of these phenomena may be deduced by very simple considerations from the principles already laid down; although the complete development of these principles demands the aid of a complicated analysis. Thus, in the case of the fringes produced by a single

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edge, let O be the luminous origin, MaN a diverging wave, and R any point at which the illumination is sought. From

this point, as centre, let a circle be described, touching the circle MaN in a, and let the lines Rob, Rec, &c., be drawn in such a manner that the intercepts bb', cc', dd', &c., are equal respectively to one, two, three, &c. semi-undulations. The effect produced at the point R is then, by the principle of Huygens, the sum of the effects produced by each of the portions ab, bc, cd, &c., separately. But, the distances of these consecutive portions from the point R differing by half a wave, their effects will be opposed at that point; so that, if m denote the intensity of the light sent from the portion ab, m' that from bc, &c.—the light sent from the indefinite wave, aM or aN, being taken as unity—the actual light which reaches the point R will be 1, 1 + m, 1 + m − m', 1 + m − m' +m", &c., according as the obstacle is placed at the point a, b, c, d, &c. And the intensity of the light when the obstacle is altogether withdrawn is

= 2.

1 + m m' + m" - m"" + &c. Now, as the terms of this series are continually decreasing, and are affected alternately with opposite signs, it is manifest that if we stop at any term, the sign of the remainder will be the same as that of its first term, and therefore alternately positive and negative. Accordingly the intensities, 1+ m, 1+ m - m', 1 + m − m' + m", &c., are alternately greater and less than 2; and the intensity of the light sent to the point R is alternately greater and less than when no obstacle is interposed.

They

It will be easily understood, from this general explanation, in what manner the magnitude of the fringes depends on the length of the wave, on the distance of the luminous origin from the obstacle, and on the distance of the screen. must be broadest in red light, and narrowest in violet light; and in white or compound light, the diffracted bands of different colours will occupy different positions, so as to form a succession of iris-coloured bands having the violet or blue inside, and the red without. After a few successions these

bands wholly disappear, owing to the superposition of bands of different colours.

(101) It is easy to compute the relative places of the same fringe, for different positions of the luminous point, and of the

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and let QR be the screen, and R the place of a fringe of any given order. Then, in order that this point should belong to the same fringe, for every distance of the luminous origin and of the screen, it is only necessary that the interval of retardation, RP – RA, of the central and marginal parts of the wave should be constant. For in this case the whole wave, AP, may be divided into a given number of parts, such that the difference of the distances of the successive points of division from the point R shall be constant; and therefore the effective wave consists of the same number of elementary portions in the same relative state as to interference.

Now, denoting OP by a, PQ by b, and QR by x, we have

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q. p., since x is very small in comparison with b. Similarly,

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But, by the condition of the question, this difference is a

constant quantity; and denoting this constant by d, we have

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When b varies, a remaining unaltered,—i. e. when the position of the screen is varied,-the value of x is the ordinate

X

of an hyperbola whose abscissa is b; so that the successive points of the same fringe belong to an hyperbola, whose summit is the edge of the obstacle.

(102) The interior fringes formed in the shadow of a narrow opaque body arise, it has been said, from the interference

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of the two portions of the wave which pass by the edges on either side. Let PP' be the section of the opaque body, Pe and P'c' the two portions of the diverging wave which has just reached its edges, and R any point of the shadow. Then, if these portions be divided in the points a, b, c, &c., a', b', c', &c., in such a manner, that the difference of the distances of any two consecutive points from the point R is equal to halfan undulation, the elementary wave sent from each portion will be in complete discordance with those sent from the two adjacent portions; so that, if the several portions be equal, they will neutralize one another's effects at the point R, with the exception of the extreme portions, Pa, P'a', the halves of which next the edges remain uncompensated.

Now the arcs Pa, ab, bc, &c., are very nearly equal, when the lines drawn from their extremities to the point R are sufficiently inclined to the normal,- or, in other words, when this point is sufficiently removed from the edge of the geometric shadow. In this case, then, the only efficacious parts of the wave are the halves of the extreme portions, Pa and P'a'; and the intensity of the light at the point R will be de

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