g good when = 0, in which case it agrees with (31). Let To be the value of the thickness T for which g=0. Then T=0 corresponds to =−(h+k), T=T, to g=0, and T=2T, to g=h+k; and for values of T equidistant from To, the values of gare equal in magnitude but of opposite signs. Hence, provided T be less than 27, there are dark and bright bands formed, the vividness of the bands being so much the greater as T is more nearly equal to T, for which particular value the minima are absolutely black. "Secondly, suppose the breadths h, k of the two streams to be equal as before, but suppose the streams separated by an interval 2g; then the only difference is that g=− (h + k) corresponds to a positive value, T, suppose, of T. If T be less then T2, or greater than 2T, T, there are no bands; but if T lie between T, and 2TT, bands are formed, which are most vivid when T-To, in which case the minima are perfectly black. 2 'Thirdly, suppose the breadths h, k of the interfering streams unequal, and suppose, as before, that the streams are separated by an interval 2g; then g=-(h+k) corresponds to a positive value, T suppose, of T: a=-(h~k) corresponds to another positive value, T1 suppose, of T, T, lying between T, and T., To being as before the value of T which gives g=0. As T increases from To, g becomes positive and increases from 0, and becomes equal to h-k when T = 2T, T1, and to h+k when T = 2T, - T. When T<T2 there are no bands. As T increases to T, bands become visible, and increase in vividness till TT, when the ratio of the minimum intensity to the maximum becomes that of h-k to h+3k, or of k-h to k + 3h, according as h> or <k. As T increases to 2T - T1, the vividness of the bands remains unchanged; and as T increases from 27 - T1 to 2T - T2, the vividness decreases by the same steps as it increased. When T-2T, -T, the bands cease to exist, and no bands are formed for a greater value of T. 0 1 The particular thickness To may be conveniently called the best thickness. This term is to a certain extent conventional, since when hand k are unequal the thickness may range from T1 to 2T-T, without any change being produced in the vividness of the bands. The best thickness is determined by the equation RESOLVING POWER OF OPTICAL INSTRUMENTS. 63 Now in passing from one band to its consecutive, p changes by 2π, and p by e, if e be the linear breadth of a band; and for this small change of p we may suppose the changes in p and & proportional, or put -dp/dp=2π/e. Hence the best aperture for a given thickness is that for which 4g + h + k = 2xƒ/e. If g = 0, and k=h, this equation becomes h = λf/e." The theory of Talbot's bands with a half covered circular aperture has been discussed by H. Struve'. Resolving Power of Optical Instruments. 52. When a distant object is viewed through a telescope, an image of the object is formed at the focus of the object-glass which is magnified by the eye-glass; and in order that the object should appear well defined, it is necessary that each point of it should form a sharp image. The indefiniteness which is sometimes observed in images is partly due to aberration; this however can in great measure be got rid of by proper optical appliances, but there is another cause, viz. diffraction, which also produces indefiniteness, as we shall proceed to show. If we suppose that the aperture of the telescope is a rectangle, it appears from § 43, that the intensity at the focal point is equal to (abλƒ), and therefore increases as the dimensions of the aperture increase; on the other hand the distances between the dark lines parallel to x and y are respectively equal to λf/a and xf/b, and therefore diminish as a and b increase; accordingly the diffraction pattern becomes almost invisible as the aperture increases, and the bright central spot alone remains. The effect of a large aperture is consequently to diminish the effect of diffraction, and to increase the definition of an image. When two very distant objects, such as a double star, are viewed by the naked eye, the two objects are undistinguishable from one another, and only one object appears to be visible. If however the two objects are viewed through a telescope, it frequently happens that both objects are seen, owing to the fact that the telescope is able to separate or resolve them; and it 1 St Petersburg Trans, vol. xxxI. No. 1, 1883. might at first sight appear, that a telescope of sufficient power would be capable of resolving two objects however distant they might be. This however is not the case, owing to the fact that the finiteness of the wave-length of light, coupled with the impossibility of constructing telescopes of indefinitely large dimensions, impose a limit to the resolving powers of the latter. According to geometrical optics, an image of each double star will be formed at two points which very nearly coincide with the principal focus of the object-glass; but physical optics shows that two diffraction patterns will be formed, whose centres are the geometrical images of each star. If the two diffraction patterns overlap to such an extent, that the appearance consists of a patch of light of variable intensity in which the two central bright spots are undistinguishable, the double star will not be resolved; but if the two patterns do not overlap to such an extent as to make the central spots undistinguishable, the double star will be resolved into its two components. 53. In order to investigate this question mathematically, and at the same time to simplify the analysis as much as possible, we shall suppose that the light from each star consists of plane waves which make an angle with the plane ay; and we shall investigate the intensity at points on the axis of x. If -(Ar)- sin x (Vt-r) dS denote the displacement produced at P, by the element of the wave which is situated at the centre C of the aperture, the displacement produced by any other element will be which gives the position of the central bright spot; and the first minimum, which occurs on the negative side of this point, is given by The intensity due to the other component of the double star will be obtained by changing & into - §; accordingly, the greatest maximum will occur when = {", where Let us now suppose that the first minimum of the diffraction pattern due to the left-hand component of the double star, coincides with the greatest maximum of the right-hand component; then '=', whence by (35) and (36) By (33), the value of the intensity at either of the bright points is ab/Xf; and the intensity of either component at =0 is by (37); whence the ratio of the intensity at the middle point to that of either of the bright points is equal to 8/772 = ·8106. It thus appears that the brightness midway between the two geometrical images, is about 4ths of the brightness of the images themselves; and from experiment it appears that this is about the limit at which there could be any decided appearance of resolution. Now 20 is the angle which the components of the double star subtend at the place of observation; and since by (38) 20=λ/a, B. O. 5 we see that an object cannot be resolved, unless its components subtend at the place of observation, an angle which exceeds that subtended by the wave-length of light, at a distance equal to the breadth of the aperture. If the distance of the object be such that = λ/a, it appears from (35) that = 0; there is accordingly a dark band at the middle point of the two images, which is more than sufficient for resolution. 54. We shall now consider the resolving power of a telescope having a circular aperture. Let the axis of έ be drawn perpendicularly to the line of intersection of the fronts of the waves with the plane of the aperture. Then at points on the axis of §, the intensity due to that component of a double star, which lies on the left-hand side will be and the integration extends over the area of the aperture. From these expressions we see that S = 0, and 2πC where p = 2π (-ƒsin 0)/f, and c is the radius of the aperture. The greatest maximum occurs when p=0, or = fsin 0, which gives the central spot; and the intensity at this point is equal to The first dark band to the left of the central spot occurs when J, (pc)= 0, or pc/π-1.2197; in which case §' =ƒsin 0 - (ƒλ/c) × ·6098. If" be the distance of the central spot due to the right-hand component, |