we find the experimental numbers reproduced with a maximum error of 8 in 1000. This of course is a very large error in investigations connected with wave-numbers of light, but, nevertheless, although our final result is not an accurate representation of the peculiar Mg series, our analysis proves conclusively enough that we have to do with a phenomenon of the nature of nodal subdivision of a vibrating body. The series of numbers marked A, above, proves that the modes of vibration analogous to harmonics possess the harmonic periods 1/3, 1/4, 1/5, 1/6, 1/7, 1/8 within one per cent. 3. A Kinematical Analysis of Balmer's Formula. It will simplify the rest of our work if we now investigate a theory of Balmer's formula in the somewhat more general form given to it by Rydberg for elements other than hydrogen, namely, n=no―B/(m +μ)2. If A is the wave-length in free æther, where the velocity of light is V, then the time of vibration being equal to X/Ň 1 1 = + 1 }, (3) √no−√ B/(m +μ) and 7 appears as the sum of two times. This suggests the following line of thought. Consider the atom simplified to a circle which is to represent the closed path round which a disturbance travels. Let it travel in the two opposite directions with angular velocity for the radius-vector from centre to disturbance, and let another radius-vector, which we may call the reference-vector, travel with angular velocity u, so that the one disturbance-vector has an angular velocity v+u, and the other v-u, relative to the reference-vector. Now the one disturbance meets the reference-vector after time 2π/(v+u), and, if instantaneously reflected with the same velocity, will meet it again after time 2π/(-u), when it is again instantaneously reflected and starts to repeat the The other disturbance would meet the referencevector after time 2π/(-), and on reflexion again after time 2π/(x+u), when the whole motion would be repeated. Thus, after a time movement. 2π {1/(v + u) + 1/(v—u)}, both disturbances would be ready to repeat their movements. The period of the type of uneven vibration thus established is and on comparing this with (3) we get the relations (4) whereby A/V, which may be denoted by 7, appears as the natural period of vibration of the circle when the referencevector is at rest, and also the ratio of the two velocities v and u appears to have a series of values depending on m+u. It may be noticed that this explanation of the origin of Balmer's formula is kinematical, in accordance with Rayleigh's surmise that that formula must indicate kinematic rather than dynamic relations (Phil. Mag. [5] xliv. 1897). Now, as we have already seen, the fundamental period of the simple harmonic vibrations of a circle being taken as 1, its possible periods of vibration are given by 1/(r+p/s), and if we derive these harmonic motions by projecting the motions of the ends of uniformly revolving radius-vectors on fixed diameters in the usual way, the relative values of the periods of revolution of these vectors will be given by 1/(r±p/s), so that we understand equation (6) as giving us the ratio of such angular velocities of vectors as we have just been considering. We must postpone the further consideration of our formulæ till we have considered Rydberg's laws. 4. Rydberg's Laws. In the Phil. Mag. [5] xxix. Rydberg gives a summary of the results arrived at in his chief memoir, published in French by the Swedish Academy (Svensk. Vet. Akad. xxiii.). In the first place he extends Hartley's discovery that certain lines of the spectra can be grouped in pairs or in threes, such that the difference or differences of their wave-numbers is or are the same for all the pairs or threes, and makes this an important principle in helping to pick out from the bewildering number of lines those which form definite series, because the pairs or threes so characterized are members of parallel series. If one of the series can be expressed by a formula n=n ̧-B/(m +μ)2, then in the case of pairs the other is given by n=n ̧+v−B/(m+μ)2, where v is the constant difference of the pairs; and similarly in the case of threes, the two other series are given by increasing n by v1 and v2 the constant differences in the threes, as in the Mg series given at the beginning of Section 2. Rydberg connects the values of v or v1 and v, for different elements by the law that in each group of elements the value of increases in a somewhat quicker proportion than the square of the atomic weight. It will be shown in the next section that the relation between the values of v for different elements of a group is purely numerical and not directly connected with atomic weights. ν In Series are divided into three classes-Diffuse, Sharp, and Principal. The Diffuse and Sharp series consist of pairs or threes of series of the sort just mentioned as specifiable by means of one series and the common difference v or the common differences v1 and v, the adjectives Diffuse and Sharp describing the appearances of the lines in the two classes. the Principal series, which have hitherto been clearly made out only in the alkali metals, the lines are associated in pairs, so that there are two principal series running side by side, but the differences of their corresponding members are no longer constant, the one series cannot be derived from the other by adding a constant to n, but by subtracting a constant from. It may happen, as is probably the case with hydrogen and lithium, that y and the constant to be added to become so small that the usual spectroscopic appliances do not resolve the pairs of lines, that is to say that the two diffuse series may coincide and likewise the two principal series. also the more ν The series can be conveniently named in the following manner:-The Diffuse or Sharp series which contains no is the first Diffuse or Sharp series, that which contains n+v or " the second, and that containing n+v, the third. Amongst Principal series, the stronger of the two, which is refrangible, is called the first, and the other the second. These classifications are expressed by Rydberg in a convenient notation which can be most easily explained by an example or two. X(D17) means that line for which m=7 in the first Diffuse series of the element X; Y(S,4) means that line for which m=4 in the second Sharp series of the element Y, and so on. Between the Diffuse and Sharp series of the same order (first, second, or third) there is always this relation, namely, that they have the same value of no, so that they differ only in their value of μ, Rydberg assuming that a single value of B holds not only for all the series of a given element, but also for all elements. These facts can be conveniently summarized simply by defining a notation to express them as follows:-In the two Principal series n, has the one value,n, and has the value in the first and in the second. The μ Diffuse and Sharp series taken together may be called the Usual series in contrast to the Principal series, which have been clearly shown only in the Alkali spectra. In the first Diffuse and the first Sharp series the value of n is nearly the same and will be denoted by ; but if a difference has to be expressed, am and will be used. Similarly in the second Diffuse and second Sharp series n becomes unv1, and in the third it is un+v2. In all the Diffuse series μ has the value a, and in all the Sharp series pl. The most important of Rydberg's laws is that the difference between the values of n for a Principal series and for a Usual series of the same order is equal to the wave-number of that line which is the first member of the Principal series. In symbols this is probably we can write similarly ? ̧+v=B/(1+pμq). Rydberg gives the data establishing this law in Wied. Ann. lviii. p. 674 (1896). He holds that there is a reciprocal relation to go along with (8), namely, but this has not yet been as well established as (8). Rydberg compresses (8) and (9) into the following terse expression:The wave-numbers of the Principal and Sharp series are given by the formula where for the Principal series m has always the value 1 in the first term on the right, while in the second term it has any integral value; and for the Sharp series m has the fixed value 1 in the second term but any integraì value in the first. Thus a Principal and a Sharp series may be regarded as having their first line in common and as being branches of a single series. The value of Rydberg's discoveries can be best appreciated by following his reasonings concerning the hydrogen spectra. If we write out the wave-numbers for Balmer's series in hydrogen and take the differences of the successive members, we get a series of numbers almost the same as the corresponding differences in the lithium Diffuse series and in the sodium Diffuse series, so that probably the Balmer series in hydrogen is a Diffuse series, and B for Li and Na must be nearly the same as for H, and μ in the Li and Na Diffuse series is 0 or an integer as in the Balmer hydrogen series. Rydberg takes Pickering's series with u=5 to be the Sharp series in hydrogen, whose formula is therefore n = 109675 (1+1) 2 (m + ·5)" and therefore by Rydberg's law the Principal series for H must be n = 109675 1 1 With m=1 this gives for the wave-length of the first line common to both Sharp and Principal series the value 4687-88, almost identical with the 4688 found by Maury and Pickering in certain star spectra for a line which exceeds in intensity all the known hydrogen lines in the same spectra. In theoretical spectrum analysis this discovery of the chief line in the hydrogen spectrum is a worthy analogue to the great astronomical achievement of the discovery of Neptune. As Rydberg points out, we should by analogy expect series 2 in which while my has the fixed value 2, m, has all possible integral values, while my is fixed at 3, m, has all integral values, and so on, and vice versa. Corresponding to the co modes of vibration of a simple system we seem to have modes of vibration in spectra, as we should infer from our idea that we have two things whose relative motion causes radiation. For if our reference-vector like the disturbancevector rotates in opposite directions and rotates with the angular velocities 1, 2, 3, ... n so that the projection of its end gives a Fourier series of simple harmonic motions of periods 1, 2, .............. 1/n, then each possible motion of the disturbance-vector can be combined with a motion of the referencevector to give a relative motion of the type our analysis shows to be the cause of the structure of spectra. If, instead of a Fourier series with a general frequency n, we take our more general series giving all possible undertones and overtones with the general frequency r±p/s, and if we ascribe to both the disturbance-vector and the reference-vector the 3 modes of vibration given by this, then there will be relative motions connected with a spectrum. Thus our conception ist capable of explaining the great complexity of spectra. If we compare (6) and (9) we have 6 (12) |