to be available, would recognise the intervals between these waves, but the light appearing in a spectroscope would seem to fill uniformly the space included between the sodium lines. If throughout the spectrum homogeneous vibrations were distributed at intervals equal to the above, no instrument could tell us that we were not dealing with what is called a continuous spectrum. We are at liberty therefore to assume that all continuous spectra are made up of homogeneous vibrations in such close proximity that we cannot separate them. This is not put forward as a physical theory, but as a method of obtaining an analytical expression of the facts in a simple manner. 21. Intensity. In comparing different radiations in the same medium, we may take the square of the amplitude as a measure of their intensity. As comparative measurements are always made in the same medium, this definition is sufficient for practical purposes. Waves of different wave-lengths can only be compared with each other when their energy is converted into some common type. This is generally effected by absorption, the heat equivalent of the radiations. being compared by the bolometer or thermopile. 22. Velocity of Light. The experimental methods by means of which the velocity of light may be measured are explained in elementary books. Fizeau's method of revolving apertures was used by A. Cornu in a series of experiments to which the highest value must be attached. The final number arrived at for the velocity in vacuo 3004 x 100 cms./second, was a result which is not likely to be in error by more than 3%. Foucault's method of the revolving mirror was used by Michelson, and later by Newcomb in conjunction with Michelson. The final result gave 2.9986 x 100 cms./second. The accidental errors of this method seem considerably smaller than in the method of Fizeau, but certain assumptions on which it rests are not quite free from objection. Professor A. Cornu has published in the Rapports de Physique du Congrès International de Physique, 1900, a very clear discussion of the relative merits of the two methods. His conclusion is, that the arithmetic mean of the above determinations gives us at present the best result, and that the most probable value of velocity of light in c.G.s. units is 3·0013 × 1010. An error of one part in a thousand in the number is quite possible so that for all purposes we may for convenience adopt the simple and easily remembered number 3 x 1010. The velocity of light in empty space will throughout this book be denoted by V. 23. Optical length and optical distance. The optical length of a path is defined as its equivalent in vacuo, two lengths being called equivalent when light occupies the same time in travelling along them. If the path traverses several media, the total optical length is the sum of the optical lengths of all the different parts. Thus if v1, v2, v3, etc. are the velocities of light and 81, 82, 83, etc. the lengths of the paths in the various media, then the optical length is But by Art. 18, if μ1, M2, μ are the refractive indices, μ1 = V/v1; μ2 = V/V2; 3 = V/v3; and hence the optical length of the path is The optical distance between two points is defined to be the shortest optical length of any line, curved, straight, or broken, that can be drawn between them. If both points lie in the same medium, the shortest path is clearly the straight line which joins them, and the optical distance is the length of this line multiplied by the refractive index of the substance. A "ray" is defined to be a path of shortest optical length. In a medium possessing uniform optical properties, a ray passing through two given points, must, by this definition, always be the straight line which joins them. The path of a ray between two points which are situated in different media may be determined as follows: K T RX Fig. 25. Let A and R, Fig. 25, be the two points, and S some point on the surface of separation, which lies in the plane drawn through A and R, perpendicular to the surface. Draw AC perpendicular to AS, and RE perpendicular to SR. From any point T in the plane ASR draw TC parallel to AS, and TE parallel to SR, and construct perpendiculars SH and KT from S and T, on CT and SR respectively. Let the position of S be such that the optical length HT is equal to the optical length SK, then the optical length of CT+ TE is equal to that of AS+ SR. But from inspection of the figure, AT> CT, RT>TE, hence the optical length of the path AT + TR must be greater than that of the path CT + TE which is equal to that of AS+ SR. Students should convince themselves that the same result follows when the point T is taken to lie on the other side of S. It follows that the optical length AS+ SR is smaller than that of any other path joining A and R in the plane of the paper. The condition on which this result depends is that the optical length of HT is equal to that of SK or that if μ1, 2 are the two refractive indices, μ1HT=μ2SK. If 0, and 0, are the angles which AS and SR form with the normal to the surface, the condition reduces to μ1 sin 1 = μ2 sin 02, which is the well-known law of refraction. The rays as defined by us are therefore identical with the rays of geometrical optics. It has been assumed in the above proof, that the path of shortest optical distance lies in the plane which is at right angles to the surface separating the two media. The restriction may be removed by giving to S a small displacement to either side at right angles to that plane, and showing that the optical distances AS and SR are both clearly increased. B C A ray may be drawn between any two points of an optical system, but only a single set of rays belong to one set of wave-fronts. Let HK and H'K' (Fig. 26) represent two wave-fronts of the same disturbance. From a point A on HK, a line may be drawn tracing the shortest optical length between A and any given point C on H'K'. By altering the position of C, its optical distance from A changes, and some point may be found on H'K' for which that optical distance is least. Let B be that point. The path of shortest optical length between A and B is one ray of the system which belongs to the two wave-fronts. We may similarly trace a ray Fig. 26. satisfying the same conditions from every point P on HK to a corresponding point Q on H'K', and thus obtain the system of rays belonging to a given system of wave-fronts. P K K Q If the medium is homogeneous, the rays must be straight lines. In a number of separate media, each being homogeneous, the system of rays is made up of a system of straight lines, which will in general change in direction when passing from one medium to another. A B If the medium is isotropic, so that one wave-front may be obtained from another by Huygens' construction, as explained in Art. 17, the system of rays intersects the system of wavefronts at right angles. This is proved by considering two points, A1, A2, on a wave-front HK. Every other wavefront H'K' will be a tangent surface to two spheres, drawn with the same radius round A1 and A2 as centres, so that if B1, B2, are the two points of contact, A1B1 and A,B must be at right angles to H'K'. This being so, A,B, is necessarily longer than A,B1, provided that A is sufficiently near to A1. Hence all points on HK which are near A2 K Fig. 27. B2 K' 1 A, are further from B, than A1, and therefore the sphere which is drawn through A, round B1 as centre, cannot intersect, but must touch the surface HK. A, B, stands therefore at right angles both to HK and to H'K'. If the medium is isotropic, but not homogeneous, as e.g. the air surrounding the earth, which varies in density and temperature, the course of a ray may be curved, but the above proof still holds if we take HK, H'K' to lie near each other, and hence the rays are in this case also at right angles to the wave-fronts. It also follows from Huygens' construction that the optical length from one wave-front to another is the same when measured along different rays. We shall call this length the optical distance between the two wave-fronts. To illustrate the use which may be made of these propositions, we may deduce the well-known formula connecting the position of a small object with that of its image formed by a lens. If waves spread out from a point source at P, the wave-fronts are spheres with the point as centre. If these wave-fronts, after passing H K MCN Fig. 28. K through the lens, are spheres with Qas centre, the wave-fronts will gradually contract until the energy of the waves is concentrated at Q. (This is not quite correct, owing to the fact that the wavefronts after emergence are not complete spheres, but this does not affect the argument.) The optical length from P to any point on HK is the same, and also the optical length from any point on H'K' to Q. It has been proved above that the optical distance from HK to H'K' is the same when measured along any ray, hence the optical distance from an object to its image is the same along all rays. PSQ and PMNQ are clearly lines satisfying all conditions laid down for the rays belonging to the system. If μ is the refractive index of the lens, the equality of optical lengths leads to the equation or Also PS+SQ=PM+μMN+ NQ=PQ+(-1) MN, if the angle SPC is so small that its square may be neglected. If MN and SC are expressed in terms of the radii of curvature of the surfaces of the lens, we obtain the well-known relation between the position of object and image. 24. Fermat's Principle and its application. Fermat (1608 to 1668) making use of the argument that Nature could not be wasteful, and was bound for this reason to cause the rays of light to travel between two points in the shortest time possible, was able to deduce from this proposition the laws of reflexion and refraction. Though we do not now attach any weight to the premiss, we accept the conclusion. "Fermat's Principle," as it is called, may serve as a connecting link between the waves of the undulatory theory, and the rays of Geometrical Optics, and often gives us a powerful method of dealing quickly with otherwise complicated problems. The ray being defined as the path of the shortest optical length, Fermat's principle requires no proof, but what must now be proved, and has been proved above, is, that the course of the rays so defined leads to the correct construction according to laws of geometrical optics. The importance of the property of minimum optical length lies in the fact that it enables us often to determine optical distances with sufficient accuracy when the course of the rays is only approximately known. That the optical length is the same when measured along a ray or a line infinitely near the ray, follows from the minimum property, but in view of the importance of the proposition, it may be more formally proved thus: H' B HK, H'K' being wave-fronts, let APB be a ray belonging to the system. Let AQB be a line lying near APB along its whole course, in such a way that their distance apart ST at any point S may be expressed in terms of the position of S, and the separation PQ at some definite point Q. Writing PQa, the difference in the optical length of AQB and APB must then be expressible in terms of a, and if a is small, must be capable of expansion in a series proceeding by powers of a as e.g. Fig. 29. h1a + h2a2 + ha3 + If the product ha were negative and a were made to diminish in magnitude, so that the higher powers ultimately vanish, it would follow that AQB is shorter than APB, which is contrary to the supposition that APB is a path of minimum length. Hence ha cannot be negative, but as nothing limits the sign of a, it follows that h1 must be zero. We also see that the difference in length between APB and AQB can ultimately only depend on the square of PQ, hence we may conclude |