WAVE THEORY 439 ow- ,nd s mcave ratings to a deviation from parallelism, causing the interval to alter gradu ally as we pass along the lines (fig. 16). The error thus arising may be compensated by a rotation of the object-glass about one of the diameters y= x. The term in o?y corresponds to a deviation from parallelism in the same direction on both sides of the central line (tig. 17); and that in y 3 would be caused by a curvature such that there is a point of inflexion at the middle of each line (fig. 18). All the errors, except that depending on a, and especially those depending on 7 and 5, can be diminished, without loss of resolving power, by contracting the vertical aperture. A linear error in the spacing, and a general curvature of the lines, are eliminated in the ordinary use of a grating. The explanation of the difference of focus upon the two sides as due to unequal spacing was verified by Cornu upon gratings pur posely constructed with an increasing interval. He has also shown how to rule a plane surface with lines so disposed that the grating shall of itself give well-focused spectra. 1 A similar idea appears to have guided Rowland to his brilliant invention of concave gratings, by which spectra can be photo graphed without any further optical appli ance. In these instruments the lines are ruled upon a spherical surface of speculum metal, and mark the intersections of the surface by a system of parallel and equidis tant planes, of which the middle member passes through the centre of the sphere. If we consider for the present only the primary plane of symmetry, the figure is reduced to two dimensions. Let AP (fig. 19) represent the surface of the grating, being the centre of the circle. Then, if Q be any radiant point and Q its image (primary focus) in the spherical mirror AP, we have 1 1 _ 2 cos <ft v 1 u a where 1 = AQ , w = AQ, = OA, = angle of incidence QAO, equal to the angle of reflexion Q AO. 2 If Q be on the circle described upon OA as diameter, so that M = acosc/>, then Q lies also upon the same circle ; and in this case it follows from the symmetry that the unsymmetrical aberration (depending upon a) vanishes. This disposition is adopted in Rowland s instrument ; only, in addition to the central image formed at the angle < = <, there are a series of spectra with various values of </> , but all disposed upon the same circle. Rowland s investigation is contained in the paper already referred to ; but the following account of the theory is in the form adopted by Glazebrook. 3 In order to find the difference of optical distances between the courses QAQ , QPQ , we have to express QP - QA, PQ - AQ . To find the former, we have, if OAQ = <ft, AOP = co, QP 2 = tt 2 + 4a" sin"^ca 4ai4sin|wsin(^co - <ft) = ( + a.sin</>sin o>) 2 - a 2 sin 2 0sin 2 co + 4asin 2 ^co (a- MCOSC/>) . Now as far as co 4 4 sin 2 J co = sin 2 co + 1 sin 4 co , and thus to the same order QP 2 = (u + sin <ft sin co) 2 - acos(f> (-acoscft)sin- > co + a (a - u cos eft) sin 4 co . But if we now suppose that Q lies on the circle u = acos<p, the middle term vanishes, and we get, correct as far as co 4 , so that QP-w = asin0sin co + g asin<fttanc/>sin 4 co . . . (9), in which it is to be noticed that the adjustment necessary to secure the disappearance of sin 2 co is sufficient also to destroy the term in sin 3 co. A similar expression can be found for Q P - Q A ; and thus, if Q A = r, Q AO = <ft , where ii = acos0 , we get QP + PQ - QA - AQ = a sin co (sin <j> - sin $ ) + ^ sin 4 a> (sin <ft tan eft + sine/ tan eft ) . . . (10). If </> = cft, 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 eft is not necessarily equal to eft ; but if P correspond to a line upon the grating, the difference of retarda tions for consecutive positions of P, so far as expressed by the term of the first order, will be equal to : fm (m integral), and therefore without influence, provided <r (sin<ft-sin<ft ) = ^fm ...... (11),
- The ruling required is evidently that which would be marked out by inter
ference bands, supposing sources of light of the prescribed wave-length to be situated at the radiant point and at the desired image. 2 Tliis formula maybe obtained as in OPTICS, vol. xvil. p. 800, equation (!!), and may indeed be derived from that equation by writing <f> = <6, u. l 3 Phil. Mag., June 1883, Nov. 1883. where a denotes the constant interval between the planes contain ing 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 out standing 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 eye-piece at 0, so that cft = 0, and then by (11) the value of <ft in the m* 1 spectrum is <rsin(ft = m ....... (12). If co now relate to the edge of the grating, on which there are Aber- altogether n lines, ration. and the value of the last term in (10) becomes w sin eft tan eft , T V?i?zAsin 3 cotaneft (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 co and eft . If the semi-angular aperture (co) be T ^ T , and tan eft = l, inn might be as great as four millions before the error of phase would reach ^A. 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 & slightly variable towards the edges. Or, retaining a constant, we might attain compensation by so polish ing 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 astig matism; but this is of little consequence, if 7 in (8) be small enough. Curvature of the primary focal line having a very in jurious effect upon definition, it may be inferred from the excellent performance of these gratings that 7 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 Rowland s great simplicity. The grating at A and the eye-piece at are mechanics rigidly attached to a bar AO, whose ends rest on carriages, moving arrange- on rails OQ, AQ at right angles to each other. A tie between C ments. 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 wave lengths of lines in the solar spectrum. 4 For absolute determinations of wave-lengths plane gratings are Absolute used. It is found 5 that the angular measurements present less measure- difficulty than the comparison of the grating interval with the ments. 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 Von Helmholtz (OPTICS, vol. xvii. p. 802). Bell found further that it is necessary to submit the gratings to calibration, and not to rest satisfied with a knowledge of the num ber of lines and of the total width. It not unfrequently happens that near the beginning of the ruling the interval is anomalous. If the width of this region be small, it has scarcely any effect upon the angular measurements, and should be left out of account in estimating the effective interval. 15. Theory of Corrugated Waves. The theory of gratings is usually given in a form applicable only to the case where the alternate parts are transparent and opaque. Even then it is very improbable that the process of simply including the transparent parts and excluding the opaque parts in the integrations of 11 gives an accurate result. The condition of things in actual gratings is much more complicated, and all that can with confidence be asserted is the approxi mate periodicity in the interval a. The problem thus presents itself to determine the course of events on the further side of the plane s = when the amplitude and phase over that plane are periodic functions of x; and the first step in the solution would naturally be to determine the effect corresponding to the infinitesimal strip ydx over which the amplitude and phase are constant. In fig. 20 QQ represents the strip in question, of which the effect is to be estimated at P(0, 0, z); QR = ?/, RP=r, If we assume the law of secondary wave determined in 10 so Fig. 20. Wave diverg ing in two dimen sions. < Phil. Mag., March 1887.
Bell, Phil. May., March 1887.