Page:EB1911 - Volume 08.djvu/269

by L. P. Gilbert (Mem. cour. de l’Acad. de Bruxelles, 31, p. 1). Taking

1/2πv2＝u

(9),

we may write

$\mathrm {C} +i\mathrm {S} ={\frac {1}{{\sqrt {}}(2\pi )}}\int _{0}^{u}{\frac {e^{iu}du}{{\sqrt {}}u}}$

(10).

Again, by a known formula,

${\frac {1}{{\sqrt {}}u}}={\frac {1}{{\sqrt {}}\pi }}\int _{0}^{\infty }{\frac {e^{-ux}dx}{{\sqrt {}}x}}$

(11).

Substituting this in (10), and inverting the order of integration, we get

$\mathrm {C} +i\mathrm {S} ={\frac {1}{\pi {\sqrt {}}2}}\int _{0}^{\infty }{\frac {dx}{{\sqrt {}}x}}\int _{0}^{u}e^{u(i-x)}dx={\frac {1}{\pi {\sqrt {}}2}}\int _{0}^{\infty }{\frac {dx}{{\sqrt {}}x}}{\frac {e^{u(i-x)}-1}{i-x}}$

(12).

Thus, if we take

$\mathrm {G} ={\frac {1}{\pi {\sqrt {}}2}}\int _{0}^{\infty }{\frac {e^{-ux}{\sqrt {}}x{.}dx}{1+x^{2}}},\mathrm {H} ={\frac {1}{\pi {\sqrt {}}2}}\int _{0}^{\infty }{\frac {e^{-ux}dx}{{\sqrt {}}x{.}(1+x^{2})}}$

(13),

C＝1/2 − G cos u + H sin u,   S＝1/2 − G sin u − H cos u

(14).

The constant parts in (14), viz. 1/2, may be determined by direct integration of (12), or from the observation that by their constitution G and H vanish when u = ∞, coupled with the fact that C and S then assume the value 1/2.

Comparing the expressions for C, S in terms of M, N, and in terms of G, H, we find that

G＝1/2 (cos u + sin u) − M,   H＝1/2 (cos u − sin u) + N

(15),

formulae which may be utilized for the calculation of G, H when u (or v) is small. For example, when u = 0, M = 0, N = 0, and consequently G = H = 1/2.

Descending series of the semi-convergent class, available for numerical calculation when u is moderately large, can be obtained from (12) by writing x = uy, and expanding the denominator in powers of y. The integration of the several terms may then be effected by the formula

$\int _{0}^{\infty }e^{-y}y^{q-{\frac {1}{2}}}dy=\Gamma (q+{\tfrac {1}{2}})=(q-{\tfrac {1}{2}})(q-{\tfrac {3}{2}})\ldots {\tfrac {1}{2}}{\sqrt {}}\pi ;$

and we get in terms of v

$\mathrm {G} ={\frac {1}{\pi ^{2}v^{3}}}-{\frac {1{.}3{.}5}{\pi ^{4}v^{7}}}+{\frac {1{.}3{.}5{.}7{.}9}{\pi ^{6}v^{11}}}-\ldots$

(16),

$\mathrm {H} ={\frac {1}{\pi v}}-{\frac {1{.}3}{\pi ^{3}v^{5}}}+{\frac {1{.}3{.}5{.}7}{\pi ^{5}v^{9}}}-\ldots$

(17).

The corresponding values of C and S were originally derived by A. L. Cauchy, without the use of Gilbert’s integrals, by direct integration by parts.

From the series for G and H just obtained it is easy to verify that

${\frac {d\mathrm {H} }{dv}}=-\pi v\mathrm {G} ,\qquad \qquad {\frac {d\mathrm {G} }{dv}}=\pi v\mathrm {H} -1$

(18).

We now proceed to consider more particularly the distribution of light upon a screen PBQ near the shadow of a straight edge A. At a point P within the geometrical shadow of the obstacle, the half of the wave to the right of C (fig. 18), the nearest point on the wave-front, is wholly intercepted, and on the left the integration is to be taken from s = CA to s = ∞. If V be the value of v corresponding to CA, viz.

$\mathrm {V} ={\sqrt {\big .}}\left\{{\frac {2(a+b)}{ab\lambda }}\right\}{.}\mathrm {CA} ,$

(19).

we may write

$\mathrm {I} ^{2}=\left(\int _{v}^{\infty }\cos {\tfrac {1}{2}}\pi v^{2}{.}dv\right)^{2}+\left(\int _{v}^{\infty }\sin {\tfrac {1}{2}}\pi v^{2}{.}dv\right)^{2}$

(20),

or, according to our previous notation,

I2＝(1/2 − Cv)2 + (1/2 − Sv)2＝G2 + H2

(21).

Fig. 18.

Now in the integrals represented by G and H every element diminishes as V increases from zero. Hence, as CA increases, viz. as the point P is more and more deeply immersed in the shadow, the illumination continuously decreases, and that without limit. It has long been known from observation that there are no bands on the interior side of the shadow of the edge.

The law of diminution when V is moderately large is easily expressed with the aid of the series (16), (17) for G, H. We have ultimately G = 0, H = (πV)⁻¹, so that

I²＝1/π²V²,

or the illumination is inversely as the square of the distance from the shadow of the edge.

For a point Q outside the shadow the integration extends over more than half the primary wave. The intensity may be expressed by

I2＝(1/2 + Cv)2 + (1/2 + Sv)2

(22);

and the maxima and minima occur when

$({\tfrac {1}{2}}+\mathrm {C} _{v}){\frac {d\mathrm {C} }{d\mathrm {V} }}+({\tfrac {1}{2}}+\mathrm {S} _{v}){\frac {d\mathrm {S} }{d\mathrm {V} }}=0,$

whence

sin 1/2πV2 + cos 1/2πV2＝G

(23).

When V = 0, viz. at the edge of the shadow, I² = 1/2; when V = ∞, I² = 2, on the scale adopted. The latter is the intensity due to the uninterrupted wave. The quadrupling of the intensity in passing outwards from the edge of the shadow is, however, accompanied by fluctuations giving rise to bright and dark bands. The position of these bands determined by (23) may be very simply expressed when V is large, for then sensibly G = 0, and

1/2πV2＝3/4π + nπ

(24),

n being an integer. In terms of δ, we have from (2)

δ＝(3/8 + 1/2n)λ

(25).

The first maximum in fact occurs when δ = 3/8λ −·0046λ, and the first minimum when δ = 7/8λ −·0016λ, the corrections being readily obtainable from a table of G by substitution of the approximate value of V.

The position of Q corresponding to a given value of V, that is, to a band of given order, is by (19)

$\mathrm {BQ} ={\frac {a+b}{a}}\mathrm {AD} =\mathrm {V} {\sqrt {\big .}}\left\{{\frac {b\lambda (a+b)}{2a}}\right\}$

(26).

By means of this expression we may trace the locus of a band of given order as b varies. With sufficient approximation we may regard BQ and b as rectangular co-ordinates of Q. Denoting them by x, y, so that AB is axis of y and a perpendicular through A the axis of x, and rationalizing (26), we have

2ax² − V²λy² − V²aλy＝0,

which represents a hyperbola with vertices at O and A.

From (24), (26) we see that the width of the bands is of the order √{bλ(a + b)/a}. From this we may infer the limitation upon the width of the source of light, in order that the bands may be properly formed. If ω be the apparent magnitude of the source seen from A, ωb should be much smaller than the above quantity, or

ω < √λ(a + b)/ab

(27).

If a be very great in relation to b, the condition becomes

ω < √(λ / b)

(28).

so that if b is to be moderately great (1 metre), the apparent magnitude of the sun must be greatly reduced before it can be used as a source. The values of V for the maxima and minima of intensity, and the magnitudes of the latter, were calculated by Fresnel. An extract from his results is given in the accompanying table.

	V	I2
First maximum	1·2172	2·7413
First minimum	1·8726	1·5570
Second maximum	2·3449	2·3990
Second minimum	2·7392	1·6867
Third maximum	3·0820	2·3022
Third minimum	3·3913	1·7440

A very thorough investigation of this and other related questions, accompanied by fully worked-out tables of the functions concerned, will be found in a paper by E. Lommel (Abh. bayer. Akad. d. Wiss. II. CI., 15, Bd., iii. Abth., 1886).

When the functions C and S have once been calculated, the discussion of various diffraction problems is much facilitated by the idea, due to M. A. Cornu (Journ. de Phys., 1874, 3, p. 1; a similar suggestion was made independently by G. F. Fitzgerald), of exhibiting as a curve the relationship between C and S, considered as the rectangular co-ordinates (x, y) of a point. Such a curve is shown in fig. 19, where, according to the definition (5) of C, S,

$x=\int _{0}^{v}\cos {\tfrac {1}{2}}\pi v^{2}{.}dv,\qquad y=\int _{0}^{v}\sin {\tfrac {1}{2}}\pi v^{2}{.}dv$

(29).

The origin of co-ordinates O corresponds to v = 0; and the asymptotic points J, J′, round which the curve revolves in an ever-closing spiral, correspond to v = ±∞.

The intrinsic equation, expressing the relation between the arc σ (measured from O) and the inclination φ of the tangent at any points to the axis of x, assumes a very simple form. For

dx＝cos 1/2πv²·dv,   dy＝sin 1/2πv²·dv;

so that

σ＝∫ √(dx2 + dy2)＝v,

(30),

φ＝tan−1(dy/dx)＝1/2πv2

(31).