WAVE THEORY The integral is thus expressible by means of the function K^ 1 and we have rffyT^IWil-iB C- dfK^SO . (32). The ascending series for K 1 (z) is 2 ( J z 5 and this is always convergent. The descending semi-convergent series is the same as those which occur in the series within braces bein the expression of the function When " (or |) is very great, so that the intensity of the image of a luminous line is ultimately inversely as the square of the distance from the central axis, or geometrical image. As is evident from its composition, the intensity remains finite for all values of ; it is, however, subject to fluctuations presenting { Intensity. On the axis itself. o-oo 1 First minimum.... 3-55 A First maximum... 4-65 A Second minimum. 6-80 ih Second maximum 8-00 A Third minimum... 9-60 rii Third maximum.. 11-00
Fourth minimum 1320 using apparently the method of quadratures. The results are also exhibited by M. Andre in the form of a curve, of which fig. 6 is a copy. It will be seen that the distribution of brightness does not differ greatly from that due to a rectangular aperture whose width (perpendicular to the luminous line) is equal to the diameter of the circular aperture. It will be in- Double structive to examine the image of a double line, Hue. whose components present an interval correspond ing to C=TT, an( l to compare the result with that already found for a rectangular aperture (- 11). We may consider the brightness at distance pro portional to 1 2 2 C 2 2 4 (* 4 L (fl=p73-jT^r5 + l2 : i3 2 <5 a i7 - In the compound image the illumination at the geometrical focus of one of the luminous lines is represented by L(0) + L0r); and the illumination midway between the geo metrical images of the two lines 2L(i). Fi e- 6 - We find by actual calculation from the series, L(ir)= - 0164, L(in-) = 1671, L(0) = -3333, so that and )=-3497, 2L(ir)= 3342, 2L(4) 7r) = -955. Compari- The corresponding number for the rectangular aperture was 811 ; son with so that, as might have been expected, the resolving power of the rect- circular aperture is distinctly less than that of the rectangular angular aperture of equal width. Hence a telescope will not resolve a aperture, double line unless the angular interval between them decidedly exceeds that subtended by the wave-length of light at a distance equal to the diameter of the object-glass. Experiment shows that resolution begins when the angular interval is about a tenth part greater than that mentioned. Uniform If we integrate (30) with respect to between the limits- oo and field of + co we obtain irR 2 , as has already been remarked. This represents light ter- the whole illumination over the focal plane due to a radiant point urinated whose image is at 0, or, reciprocally, the illumination at (the same by as at any other point) due to an infinitely extended luminous area. straight If we take the integration from (supposed positive) to =o we get edge. the illumination at due to a uniform luminous area extending over this region, that is to say, the illumination at a point situated at distance outside the border of the geometrical image of a large uniform area. If the point is supposed to be inside the geometri cal image and at a distance from its edge, we are to take the in- Theory of Sound, 2 Ann. d. V Ecole Normale, v. p. 310, 1876. tegration from -co to |. Thus, if we choose the scale of intensities so that the full intensity is unity, then the intensity at a distance corresponding to + (outside the geometrical image) may be repre sented by 3E ( + 0> aut l that at a distance - f by 1E( - ), where (36). 2 ITT Jo This is the result obtained by Struve, who gives the following Struve i series for (). results. The ascending series, obtained at once by integration from (33), is 3T//*_] _ yi nn-l^ >t "H ^" &" /Q7 M5J i -- ^ *) Oni 1 V2 Q2 p.., fftm t 1V2 . . (oi j. Thus "at great distances from the edge of the geometrical image the intensity is inversely proportional to the distance, and to the radius of the object-glass." The following table, abbreviated from that given by Struve, will serve to calculate the enlargement of an image due to diffraction in any case that may arise. ? !()
KO f K) o-o 5000 2 5 0765 7 0293 0-5 3678 3-0 0630 9-0 0222 I/O 2521 4-0 0528 11-0 0186 1-5 1642 5-0 0410 15-0 0135 2-0 1073 6-0 0328 It may perhaps have struck the reader that there is some want Integra of rigour in our treatment of (30) when we integrate it over the tion of whole focal plane of , TJ, inasmuch as in the proof of the formulaj intensi and i) are supposed to be small. The inconsistency becomes very further apparent when we observe that according to the formula; there is con- no limit to the relative retardation of secondary waves coming from sidered various parts of the aperture, whereas in reality this retardation could never exceed the longest line capable of being drawn within the aperture. It will be worth while to consider this point a little further, although our limits forbid an extended treatment. The formula becomes rigorous if we regard it as giving the illumination on the surface of a sphere of very large radius /, in a direction such that =/sin cos , rj =/sin 6 sin ; it may then be written 12 = x - ay - 2 j /2/7"cos K { (a! - x) sin cos + (y -y}sm6& <pdxdydx dy . The whole intensity over the infinite hemisphere is given by (38). According to the plan formerly adopted, we postpone the integra tion with respect to x, y, x , y , and take first that with respect to 6 and 0. Thus for a single pair of elements of area dxdy, dx dy 1 we have to consider yycos K { (x - x} sin cos (j> + (y - y} sin 0sin <j> j- sin 9 de d<j> , or, if we write
/ Jo cos(/crsin0cosc&)sine dO Now it may be proved (e.g., by expansion in powers of K? - ) that " ~"cos( K r sin cos 0) sin Ode cfy> = 27r ~ (39); o and thus r being the distance between the two elements of area dx dy, dx dy . In the case of a circular area of radius R, we have 3 ffff dxdydx dy = J,(2 K R) -- and thus When :R=oo, (41).
Theory of Sou/id, 302.