WAVE T H E 11 Y 443 These integrals, taken from r=0, are known as Fresnel s integrals; we will denote them by C and S, so that C/ cos ^irv-.dv, S = / sm^irv^.dv . . . (5). Jo Jo 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 J, by a known formula ; and on account of the rapid fluctuation of sign the parts of the range beyond very moderate values of v con tribute but little to the result. Cnockeii- Ascending series for C and S were given by Knockenhauer, and .auer s are readily investigated. Integrating by parts, we find o and, by continuing this process, i.^-Tv" ( ITT , iir ITT r in ITT iir C + lS = e 5 ^--5-^ + 1/ 1 - ( 3 35 357 By separation of real and imaginary parts, C = M cos jTi-y 2 + N sin irv- S = M sin -nv"- - N cos irv~ j where M = -- + 1 3.5 3.5.7.9 S + These series are convergent for all values of v, but are practically useful only when v is small. Gilbert s Expressions suitable for discussion when v is large were obtained ntegrals. by Gilbert. 1 Taking 4i> 2 -M (9), we may write . C + iS = / - (10). Again, by a known formula, Substituting this in (10), and inverting the order of integration, we get 1 ^ dx fu u(t-*) - / c ax V 3 -- Jo dx c u ^ l ~ x - 1
- - (12).
Thus, if we take
H = iV2_/o C = r- G cos u + II sin , S = ^- G sinw-H cosu . (14). The constant parts in (14), viz. i, may be determined by direct integration of (12), or from the observation that by their constitu tion G and II vanish when u = co , coupled with the fact that C and S then assume the value . Comparing the expressions for C, S in terms of M, N, and in terms of G, H, we find that G = J(COSM + sin u) - M, H*=(cos?{-sin) + N . (15), formulae which may be utilized for the calculation of G, II when u (or v} is small. For example, when u = 0, M = 0, N = 0, and con sequently G = H = Jj. Caucliy s Descending series of the semi-convergent class, available for series. 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 / y (j i e V dy = T(q + -%) = ( 2 - i)(*Z ~ f ) 2V ir j and we get in terms of v 1 G = _ -II -- - 1.3.5 1.3. 1 1.3 1.3.5.7 The corresponding values of C and S were originally derived by Cauchy, without the use of Gilbert s integrals, by direct integration by parts. From the series for G and II just obtained it is easy to verify that dll dG = -7ri G, - r -= 7 rrll-l .... (IS). dv dv 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. 22), the nearest point on the . couronne s de I Acad. de Bruxcllts, x..i. 1. See also Vcrdet, Lemons, wave-front, is wholly intercepted, and on the left the integra tion is to be taken from s = CA to s = co. If V be the value of v corresponding to CA, viz. ,
we may write /!"< (19), =>( /" V/v ( /" Jv or, according to our previous notation, (21). Now in the integrals represented by G and H every element No band diminishes as V increases from zero. Hence, as CA increases, viz., as the point Pis more and more deeply immersed in the shadow, the illumi nation continuously decreases, and that without limit. It has long been known from observa tion that there are no bands on the interior side of the shadow of the edge. The law of diminution when V is mode rately large is easily expressed with the aid of the series (16), (17) for G, H. H^irV)- 1 , so that Fig. 22. We have ultimately G = 0, 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 I 2 = (i + Cv) 2 +(i + Sv) 2 ..... (22); and the maxima and minima occur when whence (23). When Y = 0, viz., at the edge of the shadow, I 2 = ; when V = o, I 2 = 2, on the scale adopted. The latter is the intensity due to the uninterrupted wave. The quadrupling of the intensity in passing Position outwards from the edge of the shadow is, however, accompanied by of ex- tluctuations giving rise to bright and dark bands. The position terior of these bands determined by (23) may be very simply expressed bands, when V is large, for then sensibly G = 0, and n being an integer. In terms of 5, we have from (2) S = (f + in)A (25). The first maximum in fact occurs when 5=A- 0046A, and the first minimum when 8 = gA-0016A, 2 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, Hyper- to a band of given order, is by (19) bolic pr< BQ. 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 a;, and rationalizing (26), we have 2ax 2 - Y 2 Aj/ 2 - Y 2 A?/ = , which represents a hyperbola with vertices at and A. From (24), (26) we see that the width of the bands is of the order /{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 to be the apparent magnitude of the source seen from A, cab should be much smaller than the above quantity, or If a be very great in relation to b, the condition becomes of bauds so that if b is. to be moderately great (1 metre), the apparent mag nitude of the sun must be greatly re duced 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 calcu lated by Fresnel. An extract from his results is given in the accom panying table. 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 recent paper by Lommel. 3 2 Verdet, Lefons, 90. 3 "Die Beugungserschelnungen gcradlinig begrenzter Schinnc," AM. buyer. Akad. da- Wiss., ii. Cl., xv. 13d., iii. Auth., 1SSC. V 12 First maximum.. 1-2172 2-7413 Fresnel First minimum... 1-3726 1-5570 table. Second maximum 2-;M49 2 -301)0 Second minimum 2-7302 l-G8(i7 Third maximum. 3-0820 2-3022 Third minimum.. 3-3U13
1-7440
