W A V E T H E 11 Y atial uase of cond- y aves. the law of the secondary wave, by comp.-iring the result of the integration with that obtained by supposing the primary wave to pass on to P without resolution. Now as to the phase of the secondary wave, it might appeal- natural to suppose that it starts from any point Q with the phase of the primary wave, so that on arrival at P it is retarded by the amount corresponding to QP. But a little consideration will prove that in that case the series of secondary waves could not reconstitute the primary wave. For the aggregate effect of the secondary waves is the half of that of the first Huygens zone, and it is the central element only of that zone for which the distance to be travelled is equal to r. Let us conceive the zone in question to be divided into infinitesimal rings of equal area. The effects due to each of these rings are equal in amplitude and of phase ranging uniformly over half a complete period. The phase of the resultant is midway between those of the extreme elements, that is to say, a quarter of a period behind that due to tlie element at the centre of the circle. It is accordingly necessary to suppose that the secondary waves start with a phase one-quarter of a period in advance of that of the primary wave at the surface of resolution. .ictor Further, it is evident that account must be taken of the variation r of phase in estimating the magnitude of the effect at P of the first cond- zone. The middle element alone contributes without deduction ; y the effect of every other must be found by introduction of a resolv- ave. ing factor, equal to cos0, if 6 represent the difference of phase between this element and the resultant. Accordingly, the ampli tude of the resultant will be less than if all its components had the same phase, in the ratio or 2:Tr. Xow 2 area /7r = 2Ar ; so that, in order to reconcile the amplitude of the primary wave (taken as unity) with the half effect of the first zone, the amplitude, at distance r, of the secondarv wave emitted from the element of area dS must be taken to be ...... (1). Ar By this expression, in conjunction with the quarter-period accelera tion of phase, the law of the secondary wave is determined. That the amplitude of the secondary wave should vary as r~ l was to be expected from considerations respecting energy ; but the occurrence of the factor A" 1 , and the acceleration of phase, have sometimes been regarded as mysterious. It may be well therefore to remember that precisely these laws apply to a secondary wave of sound, which can be investigated upon the strictest mechanical principles. The recomposition of the secondary waves may also be treated analytically. If the primary wave at be cos K(tt, the effect of the secondary wave proceeding from the element dS at Q is dS dS . . . . cosK(at- p + l}= sin K(< - />) . Ap Ap If dS = 2ir.) i d.i , we have for the whole effect 2?r /" x sin K(at- Ayo p or, since xdx = pdp, K = 2n-/A, In order to obtain the effect of the primary wave, as retarded by traversing the distance r, viz., cos K (at r), it is necessary to suppose that the integrated term vanishes at the upper limit. And it is important to notice that without some further understanding the integral is really ambiguous. According to the assumed law of the secondary wave, the result must actually depend upon the precise radius of the outer boundary of the region of integration, supposed to be exactly circular. This case is, however, at most very special and exceptional. We may usually suppose that a large number of the outer rings are incomplete, so that the in tegrated term at the upper limit may properly be taken to vanish. If a formal proof be desired, it may be obtained by introducing into the integral a factor such as e "P,in which h is ultimately made to diminish without limit. When the primary wave is plane, the area of the first Huygens /.one isirr, and, since the secondary waves vary as r~ l , the intensity is independent of r, as of course it should be. If, however, the prim ary wave be spherical, and of radius a at the wave-front of resolu tion, then we know that at a distance r further on the amplitude of the primary wave will be diminished in the ratio a:(r + n). This may be regarded as a consequence of the altered area of the first Huygens zone. For, if x be its radius, we have so that n earl v. Since the distance to Lc travelled by the secondary waves is still /, we sec how the effect of the first zone, and therefore of the whole series is proportional ioa/(a + r). In like manner may be treated other cases, such as that of a primary wave-front of un equal principal curvatures. The general explanation of the formation of shadows may also be conveniently based upon Huygens s zones. If the point under consideration be so far away from the geometrical shadow that a large number of the earlier zones are complete, then the illumina tion, determined sensibly by the first zone, is the same as if there were no obstruction at all. If, on the other hand, the point be well immersed in the geometrical shadow, the earlier zones are alto gether missing, and, instead of a series of terms beginning with finite numerical magnitude and gradually diminishing to zero, we have now to deal with one of which the terms diminish to zero at loth ends. The sum of such a series is very approximately zero, each term being neutralized by the halves of its immediate neigh bours, which are of the opposite sign. The question of light or darkness then depends upon whether the series begins or ends abruptly. With few exceptions, abruptness can occur only in the presence of the first term, viz., when the secondary wave of least retardation is unobstructed, or when a ray passes through the point under consideration. According to the undulatory theory the light cannot be regarded strictly as travelling along a ray ; but the existence of an unobstructed ray implies that the system of Huygens s zones can be commenced, and, if a large number of these zones are fully developed and do not terminate abruptly, the illumination is unaffected by the neighbourhood of obstacles. Intermediate cases in which a few zones only are formed belong especially to the province of diffraction. An interesting exception to the general rule that full brightness requires the existence of the first zone occurs when the obstacle assumes the form of a small circular disk parallel to the plane of the incident waves. In the earlier half of the 18th century 1 Delisle found that the centre of the circular shadow was occupied by a bright point of light, but the observation passed into oblivion until Poisson brought forward as an objection to Fresnel s theory that it required at the centre of a circular shadow a point as bright as if no obstacle were intervening. If we conceive the primary wave to be broken up at the plane of the disk, a system of Huygens s zones can be constructed which begin from the circumference ; and the first zone external to the disk plays the part ordinarily taken by the centre of the entire system. The whole effect is the half of that of the first existing zone, and this is sensibly the same as if there were no obstruction. When light passes through a small circular or annular aperture, the illumination at any point along the axis depends upon the precise relation between the aperture and the distance from it at which the point is taken. If, as in the last paragraph, we imagine a system of zones to be drawn commencing from the inner circular boundary of the aperture, the question turns upon the manner in which the series terminates at the outer boundary. If the aperture be such as to fit exactly an integral number of zones, the aggregate effect may be regarded as the half of those due to the first and last zones. If the number of zones be even, the action of the first and last zones are antagonistic, and there is complete darkness at the point. If on the other hand the number of zones be odd, the effects con spire ; and the illumination (proportional to the square of the ampli tude) is four times as great as if there were no obstruction at all. The process of augmenting the resultant illumination at a particular point by stopping some of the secondary rays may be carried much further. 2 By the aid of photography it is easy to prepare a plate, transparent where the zones of odd order fall, and opaque where those of even order fall. Such a plate has the power of a condensing lens, and gives an illumination out of all propor tion to what could be obtained without it. An even greater effect (fourfold) would be attained if it were possible to provide that the stoppage of the light from the alternate zones were replaced by a phase-reversal without loss of amplitude. In such experiments the narrowness of the zones renders necessary a pretty close approximation to the geometrical conditions. Thus in the case of the circular disk, equidistant (r) from the source of light and from the screen upon which the shadow is observed, the width of the first exterior zone is given by Shadows. Poisson s problem. Circular or annular aperture. Soret s experi ment. Degree of accuracy required. 2x being the diameter of the disk. If 2r=1000 cm., 2x = l cm., A = 6xlO- 5 cm., then dx= QQl5 cm. Hence, in order that this zone may be perfectly formed, there should be no error in the cir cumference of the order of 001 cm. 3 The experiment succeeds in a dark room of the length above mentioned, with a threepenny bit (supported by three threads) as obstacle, the origin of light being a small needle hole in a plate of tin, through which the sun s rays shine horizontally after reflexion from an external mirror. In the absence of a heliostat it is more convenient to obtain a point of light with the aid of a lens of short focus. i Yerdet, Lemons tTOftique Physique, i. (Hi. 2 Soret, Pogg. Ann., elvl. p. 90, 187.",. 3 It is easy to sec that the radius of the bright spot is of the same order of
magnitude.