604 LIGHT Wave- motion 111 homo ~ geneous isotropic medium. Here we have simply a suc- cession of concentric spherical wave-fronts, their radii differing by one or more whole wave-lengths. The dis- turbance in any portion of one of these fronts is propagated medium, radially. But we may consider it from a different point of view, as hinted in (d) above. Simple as this particular case is, the reader will probably find that it will greatly assist him in understanding the more complex ones which follow. Every disturbed portion of the medium may be looked upon as a centre of disturbance from which a new set of spherical waves is constantly spreading. Take then, as common radius, the space described by a disturbance in any very short interval ; and, with centres at every point of any one wave -front, describe a series of spheres. The ultimate intersections of these spheres will lie on a surface which is the envelop of them all. In the case considered, it is obviously a sphere whose radius exceeds that of the wave-front from which we started by the common radius of the set of spheres. This is shown in a central section in fig. 28 below, which suffices to prove that we arrive by this mode of construction at the result which we know in this simple case to be the correct one. It will be seen that the centres of the construction-spheres lie on a certain part of one wave-front, while their ultimate intersections lie on the corresponding part of the future wave-front. This holds for spheres of all radii, and for continually increas ing radii shows that a plane wave moves perpendicularly to its front. This is so important a part of Huygens s work that we give it in his own words (Traite de la Lumiere, 1690, pp. 18-20): " Pour venir aux proprietez de la lumiere ; remarquons premiere- ment que chaque partie d onde doit s etendre en sorte, que les extremitez soient tousjours comprises entre les mesmes lignes droites tirees du point lumineux. Ainsi la partie de 1 onde BG, ayant le point lumineux A pour centre, s etendra en 1 arc CE, termine par les droites ABC, AGE. Car bien que les ondes particulieres, pro- duites par les particules que comprend 1 espace CAE, se repandent aussi liors de cet espace, toutesfois elles ne concourent point en mesine instant, a composer ensemble une onde qui termine le mouve- ment, que precisement dans la circonference CE, qui est leur tangente commune. " Et d icy Ton voit la raison pourquoy la lumiere, a moins que ses rayons ne soient reflechis ou rompus, ne se repand que par des lignes droites, en sorte qu elle n eclaire aucun objet que quand le chomin depuis sa source jusqu a cet objet est ouvert suivant de telles lignes. Car si, par exemple, il y avoit une ouverture BG, bornee par des corps opaques BH, GI ; 1 onde de lumiere qui sort du point A sera tousjours terminee par les droites AC, AE, comme il vient d estre de- monstre : les parties des ondes particulieres, qui s etendent liors de 1 espace ACE, estant trop foibles pour y produire de la lumiere. " Or quelque petite que nous fassions 1 ouverture BG, la raison est tousjours la mesme pour y faire passer la lumiere entre des lignes droites ; parce que cette ouverture est tousjours assez grande pour contenir un grand nombre de particules de la matiere etheree, qui sont d une petitesse inconcevable ; de sorte qu il paroit que chaque petite partie d onde s avance necessairement suivant la ligne droite qui vient du point luisant. Et c est ainsi que Ton peut prendre des rayons de lumiere comme si c estoient des lignes droites. " II paroit an reste, par ce qui a este remarque touchant la foiblesse des ondes particulieres, qu il n est pas necessaire que toutes les particules de 1 Ether soient egales entre elles, quoique Tegalite soit plus propre a la propagation du mouvement. Car il est vray que 1 inegalite fera qu une particule, en poussant une autre plus grande, fasse effort pour reculer avec une partie de son mouvement, mais il ne s engendrera de cela que quelques ondes particulieres en arriere vers le point lumineux, incapables de faire de la lumiere : & uon pas d onde composee de plusieurs, comme estoit CE. "Une autre, et des plus merveilleuses proprietez de la lumiere est que, quand il en vient de divers costez, ou mesme d opposez, elles font leur efifet 1 une a travers 1 antre sans aucun empechement. D ou vient aussi que par une mesme ouverture plusieurs spectateurs peu- vent voir tout a la fois des objets differens, et que deux personnes se voyent en mesme instant les yeux 1 un de 1 autre. Or suivant ce qui a este explique de 1 action de la lumiere, et comment ses ondes ne se detruisent point, ny ne s interrompent les unes les autres quand elles se croisent, ces effets que je viens de dire sont aisez a concevoir. Qui ne le sont nullemeut a mon avis selon 1 opinion de Des-Cartes, qui fait consister la lumiere dans une pression continuelle, qui ne fait que tendre au mouvement. Car cette pression ne pouvant agir tout a la fois des deux costez opposez, contre des corps qui n ont aucune inclination a s approcher ; il est impossible de comprendre ce que je viens de dire de deux personnes qui se voyent les yeux mutuellement, ni comment deux flambeaux se puissent eclairer l un 1 autre." We will now, for the purposes of this elementary article, assume that something similar holds in all cases, and will not trouble ourselves with the fact that our construc tion, if fully carried out, would indicate a retrograding wave as well as a progressive one. The obvious fact that a solitary wave can be propagated in water, or along a stretched string, may assist the reader in taking the bold step which we have proposed to him. And we will also assume that this mode of representation leads to correct results even when we do not choose a wave-front as the locus of the centres of disturbance, that in fact we may choose for our purpose any surface through which the rays pass, provided always that the radii of the spheres are so chosen that the length of each ray from some definite wave-front to the centre of the sphere, together with the radius of that sphere, always corresponds to a path described in a given time. We are now prepared to explain the reflexion of light, and we need do so for a plane reflecting surface alone, bscause the length of a wave, as we shall soon see, is an almost vanishing quantity in comparison with the radius of curvature of any artificial mirror, be it even the smallest visible drop of mercury. Let a plane wave-front be approaching a plane mirror, and at any instant let fig. 29 represent a section by a plane perpendicular to each, cutting the wave-front in AB and the mirror in AC. From what has been already said, the motion of every part of AB is perpendicular to that line, and in the plane of the figure. During the time that the disturbance at B takes to reach C, the disturbance which had reached A will have (in part, for there is usually a refracted part also) spread back into the medium in the form of a spheri cal wave whose radius, AD, is equal to BC. Its section is of course Fig. 29. a circle. That from any other point P will have reached Q, and then (in part) diverged into a spherkal wave whose centre is Q and radius QT( = QT ) = BC - PQ. Obviously all the circles which can be thus drawn ultimately intersect in the straight line CD. This is a section of the reflected wave-front. A plane wave, therefore, remains a plane wave after reflexion, each part of it obviously moves in the plane of incidence, and the similarity of the triangles ABC and CDA proves the equality of the angles of incidence and reflexion, for the ray is everywhere perpendicular to the wave-front. It is to be particularly noted that this is in dependent of the velocity of the light, so that all rays are reflected alike. In this, as in the preceding and the im mediately following instances, the diagram has been taken (with but slight change) from Huygens. This being true of any plane wave-front, large or small in area, is necessarily also true of any wave-front of finite u,ulu to T < plana
