WAVE THEORY 433 external luminous rings sufficient to outweigh any improvement due to the diminished diameter of the central area.* ! xi- The maximum brightnesses and the places at which they occur j m of are easily determined with the aid of certain properties of the 3: glit- Bessel s functions. It is known 2 that 1 (17); (18). The maxima of C occur when or by (17) when J 2 (s) = 0. determined, When s has one of the values thus The accompanying table is given by Lominel, 3 in which the first column gives the roots of 3. 2 (z) = 0, and the second and third columns the corresponding values of the functions specified. It appears that the maximum brightness in the first ring is only about -jfo of the brightness at the centre. We will now investigate the total illumination distributed over the area of the circle of radius r. i e- j.ted tensity. We have where Thus z 2*-lJi(*) 4z-aj 1 2() oooooo 5-135C30 8-41723G 11-619857 14-795938 17-959820 + 1-000000 -132279 + -064482 - 040008 + -027919 - -020905 1-000000 017498 004158 001601 000779 000437 (19), (20). Xow by (17), (18) so that and If r, or z, be infinite, J (z), J-^z) vanish, and the whole illumination is expressed by irR 2 , in accordance with the general principle. In any case the proportion of the whole illumination to be found out side the circle of radius r is given by For the dark rings Jj(s) = ; so that the fraction of illumination outside any dark ring is simply J 2 (2). Thus for the first, second, third, and fourth dark rings we get respectively 161, "090, 062, 047, showing that more than i^ths of the whole light is concen trated within the area of the second dark ring. 4 When z is great, the descending series (10) gives solv- wer of >pes. niinous so that the places of maxima and minima occur at equal intervals. The mean brightness varies as z~ 3 (or as r 3 ), and the integral found by multiplying it by zdz and integrating between and oo converges. It may be instructive to contrast this with the case of an in finitely narrow annular aperture, where the brightness is propor tional to J 2 (z). When z is great,
- -/(
The mean brightness varies as z- 1 ; and the integral/" J n 2 (s)zffo J o is not convergent. The efficiency of a telescope is of course intimately connected with the size of the disk by which it represents a mathematical point. The resolving power upon double stars of telescopes of various apertures has been investigated by Dawes and others (OPTICS, vol. xvii. p. 807), with results that agree fairly well with theory. If we integrate the expression (8) for I 2 with respect to 77, we shall obtain a result applicable to a linear luminous source of which the various parts are supposed to act independently. lAiry, Joe. cit. " Thus the magnitude of the central spot is diminished, and the brightness of the rings increased, by covering the centra] parts of the object- glass." - Todhimter s Laplace s Functions, ch. xxxi. " Loc.cit. * Phil. May., March 1881. + . (23), From (19), (20) /*+oo rf| / I 2 drj = : ,/ GO since 77 - = r 2 | 2 . If we write we get d . / J x> JC, Z/(Z- {-) This integral has been investigated by H. Struve, 5 who, calling to his aid various properties of Bessel s functions, shows that /o J 2/? ,7~ o -I /"^TT z/(z z - C 2 ) = ~^ 7 /Q sin(2sinj8)co8 8 rf/J . (25), of which the right-hand member is readily expanded in powers of By means of (24) we may verify that /+OC /*+* 5 / Pcw irR . X ./ CO Contrary to what would naturally be expected, the subject is more easily treated without using the results of the integration with respect to x and y, by taking first of all, as in the investiga tion of Stokes ( 11), the integration with respect to ij. Thus I 2 di?= Limit of 77777" UJJJ cos J and cos 7 ) | (a/ - a-) + 77 (?/ - y) I 2,3 cosy 1 (a/ -a-) (26) ; (27). We have now to consider /3 2 +/c 2 (2/ -?/) 2 // 2 In the integration with respect to y every element vanishes in the limit (0 = 0), unless y = y. If the range of integration for T/ includes the value y, then otherwise it vanishes. The limit of (28) may thus be denoted by A/Y, where Y is the common part of the ranges of integration for y 1 and y corresponding to any values of x and x. Hence (29), YCOS " cos - if, as for the present purpose, the aperture is symmetrical with respect to the axis of y. In the application to the circle we may write i.e., corresponds to the larger of the two abscissae x , x. If we take Y = 2/(R 2 -^ <2 ), and limit the integration to those values of x which are less than x, we should obtain exactly the half of the required result. Thus /*-4-> ~ /" ~ / Hence, writing as before ~ , we get ^/ +C l-d-n = . d j ./ ijr cos 2 j8 sin(2C)3) </ . (30^ J <*> "" ?-/0 in which we may replace o?f/| by d(/(, in agreement with the result obtained by Struve. The integral in (30) may be written in another form. We have " and thus 5 Wied. Ann., xvii. 1008, 1882.
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