QP2 = (u+a sin sin (0)2
—a cos 0(u—a cos ¢) sin 2w+4a(a—u cos 0) sin 4w.
But if we now suppose that Q lies on the circle u=a cos 41, the middle term vanishes, and we get, correct as far as w4,
QP=(u+asin4i sin (a) ) 1 { a2sin20sin4w .
VVVV 4u
so that
QP—u=asin4sinw+Basin4tan4sin4w . . (9),
in which it is to be noticed that the adjustment necessary to secure the disappearance of sin 2w is sufficient also to destroy the term in sin3 w.
A similar expression can be found for Q'P—Q'A; and thus, if Q'A=v, Q'AO=¢', where v=a cos ¢', we get
QP + PQ'QAAQ' = a sin w(sin 4isin 4i' )
+ 8a sin4w(sin4tan4'+sin0'tan0') . . . (10).
If 4,'=4', the term of the first order vanishes, and the reduction of the difference of path via P and via A to a term of the fourth order proves not only that Q and Q' are conjugate foci, but also that the foci are exempt from the most important term in the aberration. In the present application ¢' is not necessarily equal to 0; but if P correspond to a line upon the grating, the difference of retardations for consecutive positions of P, so far as expressed by the term of the first order, will be equal to = mX (m integral), and therefore without influence, provided
a (sin ¢sin¢') = ma (11),
where a denotes the constant interval between the planes containing the lines. This is the ordinary formula for a reflecting plane grating, and it shows that the spectra are formed in the usual directions. They are here focused (so far as the rays in the primary plane are concerned) upon the circle OQ'A, and the outstanding aberration is of the fourth order.
In order that a large part of the field of view may be in focus at once, it is desirable that the locus of the focused spectrum should be nearly perpendicular to the line of vision. For this purpose Rowland places the eyepiece at 0, so that ¢=o, and then by (ii) the value of 0.' in the mu spectrum is
a sin = =mX (12).
If w now relate to the edge of the grating, on which there are altogether n lines,
na=2a sin w,
and the value of the last term in (1o) becomes Ana sin 3w sin 0' tan
4,',
mnX sin3 w tan (13). This expresses the retardation of the extreme relatively to the central ray, and is to be reckoned positive, whatever may be the signs of w, and 4'. If the semiangular aperture (w) be 1,; 0, and tan 4i' =1, mn might be as great as four millions before the error of phase would reach 4X. If it were desired to use an angular aperture so large that the aberration according to (13) would be injurious, Rowland points out that on his machine there would be no difficulty in applying a remedy by making a slightly variable towards the edges. Or, retaining a constant, we might attain compensation by so polishing the surface as to bring the circumference slightly forward in comparison with the position it would occupy upon a true sphere.
It may be remarked that these calculations apply to the rays in the primary plane only. The image is greatly affected with astigmatism; but this is of little consequence, if y in (8) be small enough. Curvature of the primary focal line having a very injurious effect upon definition, it may be inferred from the excellent performance of these gratings that y is in fact small. Its value does not appear to have been calculated. The other coefficients in (8) vanish in virtue of the symmetry.
The mechanical arrangements for maintaining the focus are of great simplicity. The grating at A and the eyepiece at 0 are rigidly attached to a bar AO, whose ends rest on carriages, moving on rails OQ, AQ at right angles to each other. A tie between the middle point of the rod OA and Q can be used if thought desirable.
The absence of chromatic aberration gives a great advantage in the comparison of overlapping spectra, which Rowland has turned to excellent account in his determinations of the relative wavelen the of lines in the solar spectrum (Phil. Mag., 1887).
For absolute determinations of wavelengths plane gratings are used. It is found (Bell, Phil. Meg., 1887) that the angular measurements present less difficulty than the comparison of the grating interval with the standard metre. There is also some uncertainty as to the actual temperature of the grating when in use. In order to minimize the heating action of the light, it might be submitted to a preliminary prismatic analysis before it reaches the slit of the spectrometer, after the manner of Helmholtz.
In spite of the many improvements introduced by Rowland and249
of the care with which his observations were made, recent workers have come to the conclusion that errors of unexpected amount have crept into his measurements of wavelengths, and there is even a disposition to discard the grating altogether for fundamental work in favour of the socalled " interference methods," as developed by A. A. Michelson, and by C. Fabry and J. B. Perot. The grating would in any case retain its utility for the reference of new lines to standards otherwise fixed. For such standards a relative accuracy of at least one part in a million seems now to be attainable.
. Since the time of Fraunhofer many skilled mechanicians have given their attention to the ruling of gratings. Those of Nobert were employed by A. J. Angstrom in his celebrated researches upon wavelengths. L. M. Rutherfurd introduced into common use the reflection grating, finding that speculum metal was less trying than glass to the diamond point, upon the permanence of which so much depends. In Rowland's dividing engine the screws were prepared by a special process devised by him, and the resulting gratings, plane and concave, have supplied the means for much of the best modern optical work. It would seem, however, that further improvements are not excluded.
There are various copying processes by which it is possible to reproduce an original ruling in more or less perfection. The earliest is that of Quincke, who coated a glass grating with a chemical silver deposit, subsequently thickened with copper in an electrolytic bath. The metallic plate thus produced formed, when stripped from its support, a reflection grating reproducing many of the characteristics of the original. It is best to commence the electrolytic thickening in a silver acetate bath. At the present time excellent reproductions of Rowland's speculum gratings are on the market (Thorp, Ives, Wallace), prepared, after a suggestion of Sir David Brewster, by coating the original with a varnish, e.g. of celluloid. Much skill is required to secure that the film when stripped shall remain undeformed.
A much easier method, applicable to glass originals, is that of photographic reproduction by contact printing. In several papers dating from 1872, Lord Rayleigh (see Collected Papers, i. 157, 16o, 199, 504; iv. 226) has shown that success may be attained by a variety of processes, including bichromated gelatin and the old bitumen process, and has investigated the effect of imperfect approximation during the exposure between the prepared plate and the original. For many purposes the copies, containing lines up to ro,000 to the inch, are not inferior. It is to be desired that transparent gratings should be obtained from firstclass ruling machines. To save the diamond point it might be possible to use something softer than ordinary glass as the material of the plate.
g. Talbot's Bands.—These very remarkable bands are seen under certain conditions when a tolerably pure spectrum is regarded with the naked eye, or with a telescope, half the aperture being covered by a thin plate, e.g. of glass or mica. The view of the matter taken by the discoverer (Phil. Mag., 1837, 10, p. 364) was that any ray which suffered in traversing the plate a retardation of an odd number of half wavelengths would be extinguished, and that thus the spectrum would be seen interrupted by a number of dark bars. But this explanation cannot be accepted as it stands, being open to the same objection as Arago's theory of stellar scintillation.' It is as far as possible from being true that a body emitting homogeneous light would disappear on merely covering half the aperture of vision with a halfwave plate. Such a conclusion would be in the face of the principle of energy, which teaches plainly that the retardation in question leaves the aggregate brightness unaltered. The actual formation of
On account of inequalities in the atmosphere giving a variable refraction, the light from a star would be irregularly distributed over a screen. The experiment is easily made on a laboratory scale, with a small source of light, the rays from which, in their course towards a rather distant screen, are disturbed by the neighbourhood of a heated body. At a moment when the eye, or objectglass of a telescope; occupies a dark position, the star vanishes. A fraction of a second later the aperture occupies a bright place, and the star reappears. According to this view the chromatic effects depend entirely upon atmospheric dispersion.
the bands comes about in a very curious way, as is shown by a circumstance first observed by Brewster. When the retarding plate is held on the side towards the red of the spectrum, the bands are not seen. Even in the contrary case, the thickness of the plate must not exceed a certain limit, dependent upon the purity of the spectrum. A satisfactory explanation of these bands was first given by Airy (Phil. Trans., 184o, 225; 1841, 1), but we shall here follow the investigation of Sir G. G. Stokes (Phil. Trans., 1848, 227), limiting ourselves, however; to the case where the retarded and unretarded beams are contiguous and of equal width.
The aperture of the unretarded beam may thus be taken to be limited by x = =h, x = o, y = 1, y = +1; and that of the beam retarded by R to be given by x=o, x=h, y=—1, y=+1. For the
For the retarded stream the only difference is that we must subtract R from at, and that the limits of x are o and +h. We thus get for the disturbance at t, ,t, due to this stream
211h f k sin f f f
knl. k h sin 2f sink at—f=R+2h (2)
If we put for shortness ,r for the quantity under the last circular function in (i), the expressions (1), (2) may be put under the forms u sin r, v sin (r—a) respectively; and, if I be the intensity, I will be measured by the sum of the squares of the coefficients of sin r and cos r in the expression
u sin 7+v sin (t —a),
so that
I =u2+v2+2uv cos a,
which becomes on putting for u, v, and a their values, and putting
k~lsin fd E =Q . . (3),
E l
I=Q 4d sinnrf € 2+2cos (27rR—2~fhl (. . (4).
If the subject of examination be a luminous line parallel to ,t, we shall obtain what we require by integrating (4) with respect to ,t from —so to + so . The constant multiplier is of no especial interest so that we may take as applicable to the image of a line
I =
2 sin2'r f 1 1 +cos `\(2R 2Xf Ih1 . . (5).
If R=1a, I vanishes at =0; but the whole illumination, represented by _ I dt, is independent of the value of R. If R=o, I=esin22~ 27th ,in agreement with
§ 3, where a has the meaning
here attached to 2h.
The expression (5) gives the illumination at t due to that part of the complete image whose geometrical focus is at t=o, the retardation for this component being R. Since we have now to integrate for the whole illumination at a particular point 0 due to all the components which have their foci in its neighbourhood, we may conveniently regard 0 as origin. t is then the coordinate relatively to 0 of any focal point 0' for which the retardation is R; and the required result is obtained by simply integrating (5) with respect to l; from —co to +oo. To each value oft corresponds a different value of a, and (in consequence of the dispersing power of the plate) of R. The variation of a may, however, be neglected in the integration, except in airR/a, where a small variation of X r entails a comparatively large alteration of phase. If we write
p=27rR/X . . . (6),
we must regard p as a function of f, and we may take with sufficient approximation under any ordinary circumstances
p=P +aE (7),
where p' denotes the value of p at 0, and a is a constant, which is positive when the retarding plate is held at the side on which the blue of the spectrum is seen. The possibility of dark bands depends upon a being positive. Only in this case can
cos{p +(a—27Th/af)E}
retain the constant value 1 throughout the integration, and then only when
dark bands, and the second marks their situation, which is the same as that determined by the imperfect theory.
The integration can be effected without much difficulty. For the first term in (5) the evaluation is effected at once by a known formula. In the second term if we observe that
cos (pi +(a—27Th/Xf) } =cos{p'—g1 } cos p' cos glt+sin p' sin git;,
we see that the second part vanishes when integrated, and that the remaining integral is of the form
+is
w = f sin2hl cos gtl e ,
where
h1=7h/Xf, gi=—27rh/Xf . . . (10).
By differentiation with respect to g1 it may be proved that
27rh1+2 cos pie, is thus
I'=27rh1 (11),
when gi numerically exceeds 2h1; and, when g1 lies between =2h1, I=7r{2hid(2hi— 1g12) cosp'} . . (12).
It appears therefore that there are no bands at all unless a Iles between o and +4h1, and that within these limits the best bands are formed at the middle of the range when a=2h1. The formation of bands thus requires that the retarding plate be held upon the side already specified, so that a be positive; and that the thickness of the plate (to which a is proportional) do not exceed a certain limit, which we may call 2To. At the best thickness To the bands are black, and not otherwise.
The linear width of the band (e) is the increment of t which alters p by 27, so that
e=27r/a .
With the best thickness
so that in this case
e=xf/h . . . . (15). The bands are thus of the same width as those due to two infinitely narrow apertures coincident with the central lines of the retarded and unretarded streams, the subject of examination being itself a fine luminous line.
If it be desired to see a given number of bands in the whole or in any part of the spectrum, the thickness of the retarding plate is thereby determined, independently of all other considerations. But in order that the bands may be really visible, and still more in order that they may be black, another condition. must be satisfied. It is necessary that the aperture of the pupil be accommodated to the angular extent of the spectrum, or reciprocally. Black bands will be too fine to be well seen unless the aperture (2h) of the pupil be somewhat contracted. Onetwentieth to onefiftieth of an inch is suitable. The aperture and the number of bands being both fixed, the condition of blackness determines the angular magnitude of a band and of the spectrum. The use of a grating is very convenient, for not only are there several spectra in view at the same time, but the dispersion can be varied continuously by sloping the grating. The slits may be cut out of tinplate, and half covered by mica or " microscopic glass," held in position by a little cement.
If a telescope be employed there is a distinction to be observed, according as the halfcovered aperture is between the eye and the ocular, or in front of the objectglass. In the former case the function of the telescope is simply to increase the dispersion, and the formation of the bands is of course independent of the particular manner in which the dispersion arises. If, however, the halfcovered aperture be in front of the objectglass, the phenomenon is magnified as a whole, and the desirable relation between the (unmagnified) dispersion and the aperture is the same as without the telescope. There appears to be no further advantage in the use of a telescope than the increased facility of accommodation, and for this of course a very low power suffices.
The original investigation of Stokes, here briefly sketched, extends also to the case where the streams are of unequal width h, k, and are separated by an interval 2g. In the case of unequal width the bands cannot be black; but if h=k, the finiteness of 2g does not preclude the formation of black bands.
The theory of Talbot's bands with a halfcovered circular aperture has been considered by H. Struve (St Peters. Trans., 1883, 31, No. 1).
The subject of " Talbot's bands " has been treated in a very instructive manner by A. Schuster (Phil. Meg., 1904), whose point of view offers the great advantage of affording an instantaneous explanation of the peculiarity noticed by Brewster. A plane pulse, i.e. a disturbance limited to an infinitely thin slice of the medium, is supposed to fall upon a parallel grating, which again may
former (1) § 3 gives
f+t
7~f,l t sin k 1 at — f } x f y't dxdy
2lh knl lea h = ]If ' knl sin f . kph sin 2ff . sink at—f f
on integration and reduction.
and
=27rh/Xf (8)
cosp'=—1 . . . . (9).
The first of these equations is the condition for the formation of
w =0 from g1 = —ao to gi = 2h1,
w=47r(2h1+g1) from g1=—2k1 to g1=0,
w=17r(2h1—g1) from gi = 0 to gi=2h1,
w=0 from g1=2h1 to g1=co.
. (1), The integrated intensity, I', or
a=27rh/Xf .
be regarded as formed of infinitely thin wires, or infinitely narrow lines traced upon glass. The secondary pulses diverted by the ruling fall upon an objectglass as usual, and on arrival at the focus constitute a procession equally spaced in time, the interval between consecutive members depending upon the obliquity. If a retarding plate be now inserted so as to operate upon the pulses which come from one side of the grating, while leaving the remainder unaffected, we have to consider what happens at the focal point chosen. A full discussion would call for the formal application of Fourier's theorem, but some conclusions of importance are almost obvious.
3reviously to the introduction of the plate we have an effect corresponding to wavelengths closely grouped around the principal wavelength, viz. a sin 4i, where a is the gratinginterval and 4, the obliquity, the closeness of the grouping increasing with the number of intervals. In addition to these wavelengths there are other groups centred round the wavelengths which are submultiples of the principal one—the overlapping spectra of the second and higher orders. Suppose now that the plate is introduced so as to cover half the aperture and that it retards those pulses which would otherwise arrive first. The consequences must depend upon the amount of the retardation. As this increases from zero, the two processions which correspond to the two halves of the aperture begin to overlap, and the overlapping gradually increases until there is almost complete superposition. The stage upon which we will fix our attention is that where the one procession bisects the intervals between the other, so that a new simple procession is constituted, containing the same number of members as before the insertion of the plate, but now spaced at intervals only half as great. It is evident that the effect at the focal point is the obliteration of the first and other spectra of odd order, so that as regards the spectrum of the first order we may consider that the two beams interfere. The formation of black bands is thus explained, and it requires that the plate be introduced upon one particular side, and that the amount of the retardation be adjusted to a particular value. If the retardation be too little, the overlapping of the processions is incomplete, so that besides the procession of half period there are residues of the original processions of full period. The same thing occurs if the retardation be too great. If it exceed the double of the value necessary for black bands, there is again no overlapping and consequently no interference. If the plate be introduced upon the other side, so as to retard the procession originally in arrear, there is no overlapping, whatever may be the amount of retardation. In this way the principal features of the phenomenon are accounted for, and Schuster has shown further how to extend the results to spectra having their origin in prisms instead of gratings.
io. Diffraction when the Source of Light is not seen in Focus.
—The phenomena to be considered under this head are of less importance than those investigated by Fraunhofer, and will be treated in less detail; but in view of their historical interest and of the ease with which many of the experiments may be tried, some account of their theory cannot be omitted. One or two examples have already attracted our attention when considering Fresnel's zones, viz. the shadow of a circular disk and of a screen circularly perforated.
Fresnel commenced his researches with an examination of the fringes, external and internal, which accompany the shadow of a narrow opaque strip, such as a wire. As a source of light he used sunshine passing through a very small hole perforated in a metal plate, or condensed by a lens of short focus. In the absence of a heliostat the latter was the more convenient. Following, unknown to himself, in the footsteps of Young, he deduced the principle of interference from the circumstance that the darkness of the interior bands requires the cooperation of light from both sides of the obstacle. At first, too, he followed Young in the view that the exterior bands are the result of interference between the direct light and that reflected from the edge of the obstacle, but he soon discovered that the character of the edge—e.g. whether it was the cutting edge or the back of a razor—made no material difference, and was thus led to the conclusion that the explanation of these phenomena requires nothing more than the application of Huygens's principle to the unobstructed parts of the wave. In observing the bands he received them at first upon a screen of finely ground glass, upon which a magnifying lens was focused; but it soon appeared that the ground glass could be dispensed with, the diffraction pattern being viewed in the same way as the image formed by the objectglass of a telescope is viewed through the eyepiece. This simplification was attended by a great saving of light, allowing measures to be taken such as would otherwise have presented great difficulties.
In theoretical investigations these problems are usually treated as of two dimensions only, everything being referred to the planepassing through the luminous point and perpendicular to the diffracting edges, supposed to be straight and parallel. In strictness this idea is appropriate only when the source is a luminous line, emitting cylindrical waves, such as might be obtained from a luminous point with the aid of a cylindrical lens. When, in order to apply Huygens's principle, the wave is supposed to be broken up, the phase is the same at every element of the surface of resolution which lies upon a line perpendicular to the plane of reference, and thus the effect of the whole line, or rather infinitesimal strip, is related in a constant manner to that of the element which lies in the plane of reference, and may be considered to be represented thereby. The same method of representation is applicable to spherical waves, issuing from a point, if the radius of curvature be large; for, although there is variation of phase along the length of the infinitesimal strip, the whole effect depends practically upon that of the central parts where the phase is sensibly constant.'
In fig. 17 APQ is the arc of the circle representative of the wavefront of resolution, the centre being at 0, and the radius OA being equal to a. B is the point at which the effect is required, distant a+b from 0, so that AB =b, AP=s, PQ=ds.
Taking as the standard phase that of the secondary wave from A, we may represent the effect of PQ by
cos 2r (t) .ds,
where = B? —AP is the retardation at B of the wave from P relatively to that from A.
Now
so that, if we write S=(a+b)s2/tab . .. (1),
X ab7i 2
the effect at B is
1 2 a+b ' cos2Ttt f cos 4~rv2.dv+sin 2Tt f sin' rv2.dv (3),
the limits of integration n depending upon the disposition of the diffracting edges. When a, b, A are regarded as constant, the first
factor may be omitted,—as indeed should be done for consistency's sake, inasmuch as other factors of the same nature have been omitted already.
The intensity I2, the quantity with which we are principally concerned, may thus be expressed
I2= { fcosiirv2.dvfsini,2rv2.dv12 . . (4).
These integrals, taken from v=o, are (( known as Fresnel's integrals; we will denote them by C and S, so that
C =is cos lirv2.dv, S = f 'sin44irv2.dv . . . (5).
When the upper limit is infinity, so that the limits correspond to the inclusion of half the primary wave, C and S are both equal to 1, by a known formula; and on account of the rapid fluctuation of sign the parts of the range beyond very moderate values of v contribute but little to the result.
Ascending series for C and S were given by K. W. Knockenhauer, and are readily investigated. Integrating by parts, we find
o ~v i.i'"'2 f
i.l+rv2 v i.;,rv2
e dv =e e dvi; C+i„ —Jo
and, by continuing this process,
C sS=eZ.3,rv v_3 v3123 sr —i3 25 27v7} .
By separation of real and imaginary parts,
C=M cos ziv2+N sin i,rv2 S = M sin .,rv2 — N cos 4,rv2
where
M=v—'revs I 17V9 — (7),
1 3.5 3.5.7.9
_rv2_ ir2V7 ,t1Vii
N 1.3 1.3.5.7+1.3.5.7.9.11 — . . . '(8)'
These series are convergent for all values of v, but are practically useful only when v is small .
Expressions suitable for discussion when v is large were obtained
In experiment a line of light is sometimes substituted for a point in order to increase the illumination. The various parts of the line are here independent sources, and should be treated accordingly. To assume a cylindrical form of primary wave would be justifiable only when there is synchronism among the secondary waves issuing from the various centres.
. . . . (6),
by L. P. Gilbert (Mem. tour. de l'Acad. de Bruxelles, 31, p: i). Taking
arv2=u . . (9),
we may write
u iu
C+iS=T 2r) U   . (10).
Again, by a known formula,
1 _ 1 °°e"Zdx
du J2Jo vx
Substituting this in (io), and inverting the order of integration, we get
Thus, if we take
G Mrs/ 1 ('x.dx H = 1 (' eu'dx (13)
2Jo 1+x2 ' rv2)o'Ix.(1+x2)' '
C = fG cos u+H sin u, S = sG sin uH cos u . (14).
The constant parts in (14), viz. may be determined by direct integration of (12), or from the observation that by their constitution G and H vanish when u=oo, coupled with the fact that C and S then assume the value 1.
Comparing the expressions for C, S in terms of M, N, and in terms of G, H, we find that
G = 1 (cos u+sin u)M, H = i (cos usin u) +N . (15),
formulae which may be utilized for the calculation of G, H when u (or v) is small. For example, when u=o, M =o, N=o, and consequently G=H=1.
Descending series of the semiconvergent 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
'0 vz
o e y dymP(4+z)=(4` z)(41) ... i^hr; and we get in terms of v
G 1 1.3.5 1.3.5.7.9
. (16),
r2v3 rav7 + r6v11
H 1 1_3 1.3.5.7 . . (17).
ry r3v6+ rbv°
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
dH=rvG, d =H1 (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 wavefront, is wholly intercepted, and on the left the integration is to be taken from s=CA to s=co. If V be the value of v corresponding to CA, viz.
V V 2(ab b) .CA, (19),
I2= U cos 2rv2.dv) 2+ U sin arv2.dv> 2 (20), or, according to our previous notation,
12=(7—Cv)1+(2—Sy)2=G2+H2 . . . (21).
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 0 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 = (rV)1, so that
I2=1/r2V2
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
we may write
more than half the primary wave. The intensity may be expressed by
I2=(a+Cv)2+(i+Sv)2 . . (22); and the maxima and minima occur when
(a+cy)av+(4+sy)av=0,
whence
sin arV2+cosarV2=G . . . . (23).
When V=o, viz. at the edge of the shadow, I2=1; when V=co, 12=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 =o, and
BQ=aa b AD=V~ b~(2a b) . . (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 coordinates 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 — V2Xy2  V2aXy = o,
which represents a hyperbola with vertices at 0 and A.
From (24), (26) we see that the width of the bands is of the order J lbX(a+b)lal. 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 w be the apparent magnitude of the source seen from A, cob should be much smaller than the above quantity, or
u<,I {X(a+ b)/abl . . . . (27). If a be very great in relation to b, the condition becomes
co< / (X/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 2.3o22
Third minimum . 3.3913 1'7440
A very thorough investigation of this and other related questions, accompanied by fully workedout tables of the functions concerned, will be found in a paper by E. Lommel (Abh. bayer. Akad. d. Wiss. H. Cl., 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 coordinates (x, y) of a point. Such a curve is shown in fig. 19, where, according to the definition (5) of C, S,
x=J Dcos 2rv2.dv, y= f sin 7rv2.dv . (29).
The origin of coordinates 0 corresponds to v =o; and the asymptotic points J, J', round which the curve revolves in an everclosing spiral, correspond to v = = o0
The intrinsic equation, expressing the relation between the arc o (measured from 0) and the inclination ¢ of the tangent at any points to the axis of x, assumes a very simple form. For
dx = cos 7rv2.dv, dy = sin 4rv2.dv;
a.= f I (dx2+dye) =v, (30),
~=tan1(dy/dx)=72 (31).
(11).
fuu(iz)
C +i S=rr~
1 2 0 xj e dx
_ 1 ('dx e"('s)1
'xJ2 Jo ~lx ix
. (12).
arV2=9r+nr ..
n being an integer. In terms of 8, we have from (2)
8=(1+zn)x. .
(24) (25).
The first maximum in fact occurs when 5=ja•oo46a, and the first minimum when S =$a•ooi6a, 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)
so that
independent of special views as to the nature of the aether, at least in its main features; for in the absence of a more complete ' foundation it is impossible to treat rigorously the mode of action of a solid obstacle such as a screen. But, without entering upon matters of this kind, we may inquire in what manner a primary wave may be resolved into elementary secondary waves, and in particular as to the law of intensity and polarization in a secondary wave as dependent upon its direction of propagation, and upon the character as regards polarization of the primary wave. This question was treated by Stokes in his " Dynamical Theory of Diffraction " (Camb. Phil. Trans., 1849) on the basis of the elastic solid theory.
Let x, y, z be the coordinates of any particle of the medium in its natural state, and f, ,t, i the displacements of the same particle at the end of time t, measured in the directions of the three axes respectively. Then the first of the equations of motion may be put
Accordingly,
and for the curvature,
Cornu remarks that this equation suffices to determine the general
character of the curve. For the osculating circle at any point
includes the whole of the
curve which lies beyond;
and the successive con
volutions envelop one an
other without intersection.
The utility. of the curve depends upon the fact that the elements of arc represent, in amplitude and phase, the component vibrations due to the corresponding portions of the primary wavefront. For by (30) da'=dv, and by (2) dv is proportional to ds. Moreover by (2) and (31) the retardation of phase of the elementary vibration
Fin 1q. from PQ (fig. 17) is 2,rs/A,
or 0. Hence, in accordance
with the rule for compounding vector quantities, the resultant vibration at B, due to any finite part of the primary wave, is represented in amplitude and phase by the chord joining the extremities of the corresponding arc (,72—01).
In applying the curve in special cases of diffraction to exhibit the effect at any point P (fig. 18) the centre of the curve 0 is to be considered to correspond to that point C of the primary wavefront which lies nearest to P. The operative part, or parts, of the curve are of course those which represent the unobstructed portions of the primary wave.
Let us reconsider, following Cornu, the diffraction of a screen unlimited on one side, and on the other terminated by a straight edge. On the illuminated side, at a distance from the shadow, the vibration is represented by IJ'. The coordinates of J, J' being (z, 1), (—%, —z), I2 is 2; and the phase is a period in arrear of that of the element at O. As the point under contemplation is supposed to approach the shadow, the vibration is represented by the chord drawn from J to a point on the other half of the curve, which travels inwards from J' towards O. The amplitude is thus subject to fluctuations, which increase as the shadow is approached. At the point 0 the intensity is onequarter of that of the entire wave, and after this point is passed, that is, when we have entered the geometrical shadow, the intensity falls off gradually to zero, without fluctuations. The whole progress of the phenomenon is thus exhibited to the eye in a very instructive manner.
We will next suppose that the light is transmitted by a slit, and inquire what is the effect of varying the width of the slit upon the illumination at the projection of its centre. Under these circumstances the arc to be considered is bisected at 0, and its length is proportional to the width of the slit. It is easy to see that the length of the chord (which passes in all cases through 0) increases to a maximum near the place where the phaseretardation is i of a period, then diminishes to a minimum when the retardation is about 8 of a period, and so on.
If the slit is of constant width and we require the illumination at various points on the screen behind it, we must regard the arc of the curve as of constant length. The intensity is then, as always, represented by the square of the length of the chord. If the slit be narrow, so that the arc is short, the intensity is constant over a wide range, and does not fall off to an important extent until the discrepancy of the extreme phases reaches about a quarter of a period.
We have hitherto supposed that the shadow of a diffracting obstacle is received upon a diffusing screen, or, which comes to nearly the same thing, is observed with an eyepiece. If the eye, provided if necessary with a perforated plate in order to reduce the aperture, be situated inside the shadow at a place where the illumination is still sensible, and be focused upon the diffracting edge, the light which it receives will appear to come from the neighbourhood of the edge, and will present the effect of a silver lining. This is doubtless the explanation of a pretty optical phenomenon, seen in Switzerland, when the sun rises from behind distant trees standing on the summit of a mountain." I
11. Dynamical Theory of Diffraction.—The explanation of diffraction phenomena given by Fresnel and his followers is
1 H. Necker (Phil. Mag., November 1832) ; Fox Talbot (Phil.Mag., June 1833). " When the sun is about to emerge . . . . every branch and leaf is lighted up with a silvery lustre of indescribable beauty... . The birds, as Mr Necker very truly describes, appear like flying brilliant sparks." Talbot ascribes the appearance to diffraction; and he recommends the use of a telescope.under the form
d2 __ \d2 d2 d21 2 d /dg d,1 dP)
de dxz+dy+dz2 1(a b2)dx dx+dy+dz '
where a' and b2 denote the two arbitrary constants. Put for shortness
d +dn+d
dx dy dz — .
and represent by v the quantity multiplied by b2. According to this notation, the three equations of motion are s
ate =b2v2f+(a2b2)ax
d2 = btv2n+(a2—b2)dy
z
dt2—b2v21+(as b2)
It is to be observed that s denotes the dilatation of volume of the element situated at (x, y, z). In the limiting case in which the medium is regarded as absolutely incompressible a vanishes; but, in order that equations (2) may preserve their generality, we must suppose a at the same time to become infinite, and replace a28 by a new function of the coordinates.
These equations simplify very much in their application to plane waves. If the ray be parallel to OX, and the direction of vibration parallel to OZ, we have E=o, ,t=o, while is a function of x and t only. Equation (1) and the first pair of equations (2) are thus satisfied identically. The third equation gives
z
d2i='dx2 . (3),
of which the solution is
l =f(bt—x) (4),
where f is an arbitrary function.
The question as to the law of the secondary waves is thus answered by Stokes. "Let s=o, n =o, i=f(bt—x) be the displacements corresponding to the incident light; let 0, be any point in the plane P (of the wavefront), dS an element of that plane adjacent to Oi; and consider the disturbance due to that portion only of the incident disturbance which passes continually across dS. Let 0 be any point in the medium situated at a distance from the point 01 which is large in comparison with the length of a wave; let 010 =r, and let this line make an angle 0 with the direction of propagation of the incident light, or the axis of x, and ¢ with the direction of vibration, or axis of z. Then the displacement at 0 will take place in a direction perpendicular to 010, and lying in the plane ZOIO; and, if r be the displacement at 0, reckoned positive in the direction nearest to that in which the incident vibrations are reckoned positive,
dS
r'=ar(1+cos B) sin ,y) f'
4 (bt—r). In particular, if
f(bt—x) =emit 2sr (bt—x)
we shall. have
cdS 27r
r'=2~r(1+cos0)sin4cos T (bt—r)
It is then verified that, after integration with respect to dS, (6) gives the same disturbance as if the primary wave had been supposed to pass on unbroken.
The occurrence of sin 4, as a factor in (6) shows that the relative intensities of the primary light and of that diffracted in the directions depend upon the condition of the former as regards polarization. If the direction of primary vibration be perpendicular to the plane of diffraction (containing both primary and secondary rays), sin 0=1; but, if the primary vibration be in the plane of diffraction, sin its= cos B. This result was employed by Stokes as a criterion of the direction of vibration; and his experiments, conducted with gratings, led him to the conclusion that the vibrations
do/du = ao
(32) ;
(33) .
. (1),
• (5), (6)."
of polarized light are executed in a direction perpendicular to the plane of polarization.
The factor (i+cos 0) shows in what manner the secondary disturbance depends upon the direction in which it is'propagated with respect to the front of the primary wave.
If, as suffices for all practical purposes, we limit the application of the formulae to points in advance of the plane at which the wave is supposed to be broken up, we may use simpler methods of resolution than that above considered. It appears indeed that the purely mathematical question has no definite answer. In illustration of this the analogous problem for sound may be referred to. Imagine a flexible lamina to be introduced so as to coincide with the plane at which resolution is to be effected. The introduction of the lamina (supposed to be devoid of inertia) will make no difference to the propagation of plane parallel sonorous waves through the position which it occupies. At every point the motion of the lamina will be the same as would have occurred in its absence, the pressure of the waves impinging from behind being just what is required to generate the waves in front. Now it is evident that the aerial motion in front of the lamina is determined by what happens at the lamina without regard to the cause of the motion there existing. Whether the necessary forces are due to aerial pressures acting on the rear, or to forces directly impressed from without, is a matter of indifference. The conception of the lamina leads immediately to two schemes, according to which a primary wave may be supposed to be broken up. In the first of these the element dS, the effect of which is to be estimated, is supposed to execute its actual motion, while every other element of the plane lamina is maintained at rest. The resulting aerial motion in front is readily calculated (see Rayleigh, Theory of Sound, § 278) ; it is symmetrical with respect to the origin, i.e. independent of B. When the secondary disturbance thus obtained is integrated with respect to dS over the entire plane of the lamina, the result is necessarily the same as would have been obtained had the primary wave been supposed to pass on without resolution, for this is precisely the motion generated when every element of the lamina vibrates with a common motion, equal to that attributed to dS. The only assumption here involved is the evidently legitimate one that, when two systems of variously distributed motion at the lamina are superposed, the corresponding motions in front are superposed also.
The method of resolution just described is the simplest, but it is only one of an indefinite number that might be proposed, and which are all equally legitimate, so long as the question is regarded as a merely mathematical one, without reference to the physical properties of actual screens. If, instead of supposing the motion at dS to be that of the primary wave, and to be zero elsewhere, we suppose the force operative over the element dS of the lamina to be that corresponding to the primary wave, and to vanish elsewhere, we obtain a secondary wave following quite a different law. In this case the motion in different directions varies as cos0, vanishing at right angles to the direction of propagation of the primary wave. Here again, on integration over the entire lamina, the aggregate effect of the secondary waves is necessarily the same as that of the primary.
In order to apply these ideas to the investigation of the secondary wave of light, we require the solution of a problem, first treated by Stokes, viz. the determination of the motion in an infinitely extended elastic solid due to a locally applied periodic force. If we suppose that the force impressed upon the element of mass D dx dy dz is
being everywhere parallel to the axis of Z, the only change required in our equations (I), (2) is the addition of the term Z to the second member of the third equation (2). In the forced vibration, now under consideration, Z, and the quantities , ,t, expressing the resulting motion, are to be supposed proportional to ei^i, where i=s/ (1), and n=27/r, r being the periodic time. Under these circumstances the double differentiation with respect to t of any quantity is equivalent to multiplication by the factorn2, and thus our equations take the form
(bkv2+n2) +(a2  b2) ax =0
(bed°+n2)'n+(a2b2)dY=0  . (7).
(bed2+n2)3'+(a2b2)dz = Z
It will now be convenient to introduce the quantities al, a2, a2, which express the rotations of the elements of the medium round axes parallel to those of coordinates, in accordance with the equations
dE do do d~ dr d (8)
In wiyax, ffi1 ;
—Ty, m2=~dz . terms of these we obtain from (7), by differentiation and subtraction,
(b2v2+n2)m2 =0
(b2172+n2)'' 1=dZ/dy . (9).
(lb2V2+n2)=2 = dZ/dxThe first of equations (9) gives
1,8=0 (10).
For al we have
Z e kr
''1=41E27 r dx dy dz . . . (11),
where r is the distance between the element dx dy dz and the point where a1 is estimated, and
k=n/b=2u/a. .. . (12),
a being the wavelength.
(This solution may be verified in the same manner as Poisson's theorem, in which k=o.)
We will now introduce the supposition that the force Z acts only within a small space of volume T, situated at (x, y, z), and for simplicity suppose that it is at the origin of coordinates that the rotations are to be estimated. Integrating by parts in (u), we get
f(e 'k' dZ a ] _ ( d (e ik*
J r dy Z r J J zay ` r) dY,
in which the integrated terms at the limits vanish, Z being finite only within the region T. Thus
((( i~1=4bz.IJJ Zdy (e r) dx dy dz.
Since the dimensions of T are supposed to be very small in com
parison with a, the factor dy (e y) is sensibly constant; so that,
if Z stand for the mean value of Z over the volume T, we may write
TZ Z d e'{1r
Zrl
4rb2'rd r ) In like manner we find
TZ x d e'k'
2 4rb2'r'dr r
From (io), (13), (14) we see that, as might have been expected, the rotation at any point is about an axis perpendicular both to the direction of the force and to the line joining the point to the source of disturbance. If the resultant rotation be s, , we have
TZ ' (x2+y2) d e ikr TZsin4, d e *
41rb2 ' r ' dr t r)  4ab2 dr ( r '
q, denoting the angle between r and z. In differentiating eikr/r with respect to r, we may neglect the term divided by r2 as altogether insensible, kr being an exceedingly great quantity at any moderate distance from the origin of disturbance. Thus
ik.TZsin4 e='kr o=
4ab2 r
which completely determines the rotation at any point. For a disturbing force of given integral magnitude it is seen to be everywhere about an axis perpendicular to r and the direction of the force, and in magnitude dependent only upon the angle (0) between these two directions and upon the distance (r).
The intensity of light is, however, more usually expressed in terms of the actual displacement in the plane of the wave. This displacement, which we may denote by f , is in the plane containing z and r, and perpendicular to the latter. Its connexion with a is expressed by a =dg'/dr; so that
L,=TZsin¢ e'(°'kr>
41rb2 ' r
where the factor is restored.
Retaining only the real part of (16), we find, as the result of a local application of force equal to
End of Article: QP2 

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