Popular Science Monthly/Volume 14/March 1879/The Chemical Elements

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THE CHEMICAL ELEMENTS.
By J. NORMAN LOCKYER, F. R. S.

I HAVE recently announced to the Royal Society that, reasoning from the phenomena presented to us in the spectroscope when known compounds are decomposed, I have obtained evidence that the so-called elementary bodies are in reality compound ones.

Although the announcement took this form, the interest taken in science nowadays by the general public is so great that it is apt to travel beyond the record; and, as able editors are not content to wait for what the experimentalist himself has to say, they are often at the mercy of those who, perhaps more from misapprehension than anything else, are prepared to provide columns filled with statements wide of the mark. Nor is this all. If there be a practical side to the work, some "application of science" is brought to the front, and the worker's own view of his labor is twisted out of all truth.

This has happened in my case. The idea of simplifying the elements is connected with the philosopher's stone. The use of the philosopher's stone was to transmute metals; therefore I have been supposed to be "transmuting" metals; and imaginations have been so active in this direction that I am not sure that, when my paper was eventually read at the Royal Society, many were not disappointed that I did not incontinently then and there "transmute" a ton of lead into a ton of gold.

It is in consequence of this general misapprehension of the nature of my work, that I the more willingly meet the wishes of the editor that I should say something about it. The paper itself I need not reproduce, as it has appeared in extenso elsewhere[1]; but there are many points touching both the origin of the views I have advanced and the work which has led up to them, on which I am glad of the opportunity of addressing a wider public.

It is now upward of ten years since I began a series of observations having for their object the determination of the chemical constitution of the atmosphere of the sun. The work done, so far as the number of elementary substances found to exist in it, I summed up in a former article[2]; but the ten years' work had opened up a great number of problems above and beyond the question of the number of elements which exist in the solar atmosphere, because we were dealing with elements under conditions which it is impossible to represent and experiment on here.

In the first place, the temperature of the sun is beyond all definition; secondly, the vapors are not confined; and, thirdly, there is an enormous number of them all mixed together, and free, as it were, to find their own level. Nor is this all. Astronomers have not only determined that the sun is a star, and have approximately fixed his place in nature as regards size and brilliancy, but they have compared the spectrum of this star, this sun of ours, with those of the other bodies which people space, and have thus begun to lay the foundations of a science which we may christen Comparative Stellar Chemistry. Dealing with the knowledge already acquired along this line, we may say roughly that there are four kinds of stars recognizable by their spectra.

We have first the brightest and presumably hottest stars, and of these the spectrum is marvelously simple—so simple, in fact, that we say their atmospheres consist in the main of only two substances—a statement founded on the observation that the lines in the spectra are matched by lines which we see in the spectra of hydrogen and calcium; there are traces of magnesium, and perhaps of sodium too, but the faintness of the indication of these two latter substances only intensifies the unmistakable development of the phenomena by which the existence of the former is indicated.

So much, then, for the first class; now for the second. In this we find our sun. In the spectra of stars of this class, the indications of hydrogen are distinctly enfeebled, the evidences by which the existence of calcium has been traced in stars of the first class are increased in intensity, and, accompanying these changes, we find all simplicity vanished from the spectrum. The sodium and magnesium indications have increased, and a spectrum in which the lines obviously visible may be counted on the fingers is replaced by one of terrific complexity.

The complexity which we meet with in passing from the first class to the second is one brought about by the addition of the lines produced by bodies of chemical substances of moderate atomic weight. The additional complexity observed when we pass from the second stage to the third is brought about by the addition of lines due in the main to bodies of higher atomic weight. And—this is a point of the highest importance—at the third stage the hydrogen, which existed in such abundance in stars of the first class, has now entirely disappeared.

In the last class of stars to which I have referred, the fourth, the lines have given place to fluted bands, at the same time that the light and color of the star indicate that we have almost reached the stage of extinction. These facts have long been familiar to students of solar and stellar physics. Indeed, in a letter written to M. Dumas, December 3, 1873, and printed in the "Comptes Rendus," I thus summarized a memoir which has since appeared in the "Philosophical Transactions":

Il semble que plus une étoile est chaude, plus son spectre est simple, et que les éléments métalliques se font voir dans l'ordre de leurs poids atomiques.[3]

Ainsi nous avons:

1. Des étoiles très-brillantes où nous ne voyons que l'hydrogène en quantité énorme, et le magnésium;

2. Des étoiles plus froides, comma notre soleil, où nous trouvons:

 
Hydrogène + Magnésium + Sodium
Hydrogène + Magnesium + Sodium + Calcium + Fer,. . .;
 

dans ces étoiles, pas de métalloïdes;

3. Des étoiles plus froides encore, dans lesquelles tous les éléments métalliques sont associés, où leurs lignes ne sont plus visibles, et où nous n'avons que les spectres des métalloïdes et des composés.

4. Plus une étoile est âgée, plus l'hydrogène libte disparaît; sur la terre, nous ne trouvons plus d'hydrogène en liberté.

Il me semble que ces faits sont les preuves de plusieurs idées émises par vous. J'ai pensé que nous pouvions imaginer une "dissociation céleste," qui continue le travail de nos fourneaux, et que le métalloïdes sont des composés qui sont dissociés par la température solaire, pendant que les éléments métalliques monatomiques, dont les poids atomiques sont les moindres, sont précisément ceux qui resistent même à la température des étoiles les plus chaudes.

Before I proceed further I should state that, while observations of the sun have since shown that calcium should be introduced between hydrogen and magnesium for that luminary, Dr. Huggins's photographs have demonstrated the same fact for the stars, so that in the present state of our knowledge, independent of all hypotheses, the facts may be represented as follows:

 
Hottest Stars \begin{matrix}\Bigg\}\end{matrix} Lines[4] of \begin{matrix}\Bigg\{\end{matrix} H + Ca + Mg
Sun H + Ca + Mg + Na + Fe
Cooler Stars — — Mg + Na + Fe + Bi + Hg
Coolest . Fluted bands of — — — — — — \begin{matrix}\Bigg\{\end{matrix} Fluted Spectra of Metals and Metalloids.
 

I have no hesitation in stating my opinion that in this line of facts we have the most important outcome of solar work during the last ten years; and if there were none others in support of them, the conclusion would still stare us in the face that the running down of temperature in a mass of matter which is eventually to form a star is accompanied by a gradually increasing complexity of chemical forms.

This, then, is the result of one branch of the inquiry, which has consisted in a careful chronicling of the spectroscopic phenomena presented to our study by the various stars.

Experimentalists have observed the spectrum of hydrogen, of calcium, etc., in their laboratories, and have compared the bright lines visible in the spectra with the dark ones in the stars, and on this ground they have announced the discovery of calcium in the sun or of hydrogen in Sirius.

In all this work they have taken for granted that in the spectrum thus produced in their laboratories, they have been dealing with the vibration of one unique thing, call it atom, molecule, or what you will; that one unique thing has by its vibrations produced all the lines visible, which they have persistently seen and mapped in each instance.

It is at this point that my recent work comes in, and raises the question whether what has been thus taken for granted is really true. And now that the question is raised, the striking thing about it is that it was not asked long ago.

One reason is this: Time out of mind—or, rather, ever since Nicolas Le Fèvre, who was sent over here by the French King at the request of our English one at the time the Royal Society was established, pointed out that chemistry was the art of separations as well as of transmutations—it has been recognized that, with every increase of temperature, or dissociating power, bodies were separated from each other. In this way Priestley, from his "plomb rouge," separated oxygen, and Davy separated potassium; and as a final result of the labor of generations of chemists, the millionfold chemical complexity of natural bodies in the three kingdoms of nature has been reduced by separations till only some seventy so-called elements are left.

Now this magnificent simplification has been brought about by the employment of moderate temperatures—moderate, that is to say, in comparison with the transcendental dissociating energies of electricity as employed in our modern voltaic arcs and electric sparks.

But, in the observations made during the last thirty years on the spectra of bodies rendered incandescent by electricity, we have actually, though yet scarcely consciously, been employing these transcendental temperatures, and, if it be that this higher grade of heat does what all other lower grades have done, then the spectrum we have observed in each case is not the record of the vibrations of the particular substance with which we have imagined ourselves to be working only, but of all the simpler substances produced by the series, whether short or long, of the "separations" effected.

The question, then, it will be seen, is an appeal to the law of continuity, nothing more and nothing less. Is a temperature higher than any yet applied to act in the same way as each higher temperature, which has been applied, has done? Or is there to be some unexplained break in the uniformity of nature's processes?

The definite reason for my asking the question at the present time has been this: The final reduction of four years' work at a special branch of the subject to which I will refer presently, on the assumption that at the temperature of the electric arc we do not get such "simplifications," has landed me in the most helpless confusion, and, if I do not succeed in finding a higher law than that on which I have been working, my four years' work, in this direction at all events, will have been thrown away.

This and other reasons compel me to hold that the answer to the question put is, that what has been taken for granted is, in all probability, not true. But before I proceed to give the reasons for the faith that is in me I must, at the risk of being both technical and tedious when I should wish to be neither, lead up to the understanding of them.

The spectroscope, however simple or complex it may be, is an instrument which allows us to observe the image of the slit through which the light enters it, in the most perfect manner. If the light contains rays of every wave-length, then the images formed by each will be so close together that the spectrum will be continuous, that is, without break. If the light contains only certain wave-lengths, then we shall get certain, and not all, of the possible images of the slit, and the spectrum will be discontinuous.

Again, if we have an extremely complex light-source, let us say a solid and a mixture of gases giving us light, and we allow the light to enter, so to speak, indiscriminately into the spectroscope, then in each part of the spectrum we shall get a summation—a complex record—of the light of the same wave-length proceeding from all the different light-waves. But if by means of a lens we form an image of the light source, so that each particular part shall be impressed in its proper place on the slit-plate, then in the spectrum the different kinds of light will be sorted out.

There is a simple experiment which shows clearly the different results obtained. If we observe the light of a candle with the spectroscope in the ordinary manner, that is, by placing the candle in front of the slit at some little distance from it, we see a band of color—a continuous spectrum—and in one particular part of the band we see a yellow line, and occasionally in the green and in the blue parts of the band other lines are observable. Now, if we throw an image of the candle on to the slit—the slit being horizontal and the image of the candle vertical—we then get three perfectly distinct spectra. We find that the interior of the candle, that is the blue part (best observed at the bottom of the candle), gives us one spectrum, the white part gives us another, while on the outside of the candle, so faint as to be almost invisible to the eye, there is a region which gives us a perfectly distinct spectrum with a line in the yellow. In this way there is no difficulty whatever in determining the coexistence of three light-sources, each with its proper spectrum, in the light of a common candle.

We see in a moment that much the same condition of affairs will be brought about if, instead of using a candle, we use an electric arc, in which the pure vapor of the substance which is being rendered incandescent fills the whole interval between the poles, the number of particles and degree of incandescence being smaller at the sides of the arc. We can throw an image of such an horizontal arc on a vertical slit; the slit will give then the spectrum of a section of the arc at right angles to its length. The vapor which exists farthest from the core of the arc has a much more simple spectrum than that of the core of the arc itself. The spectrum of the core consists of a large number of lines, all of which die out until the part of it farthest from the center gives but one line.

In this way the spectrum of each substance furnishes us with long and short lines, the long lines being common to the more and less intensely heated parts of the arc, and the short lines special to the more heated one. This is the first step.

It has been necessary to enter thus at length into the origin of the terms long and short lines, because almost all the subsequent work which need be referred to now has had for its object the investigation of the phenomena presented by them under different conditions. The first results obtained were as follows:

1. When a metallic vapor was subjected to admixture with another gas or vapor, or to reduced pressure, I found that its spectrum became simplified by the abstraction of the shortest lines and by the thinning of many of the remaining ones. To obtain reduction of pressure, the metals were inclosed in tubes in which a partial vacuum was produced. In all these experiments it was found that the longest lines invariably remained visible longest.[5]

2. When we use metals chemically combined with a metalloid—in other words, when we pass from a metal to one of its salts (I used chlorine)—only the longest lines of the metal remain. The number is large in the case of elements of low atomic weight, and small In the case of elements of high atomic weight, and of twice the atom-fixing power of hydrogen.

3. When we use metals mechanically mixed, only the longest lines of the smallest constituent remain. On this point I must enlarge somewhat by referring to a series of experiments recorded in the "Philosophical Transactions" (1873).

A quantity of the larger constituent, generally from five to ten grammes, was weighed out, the weighing being accurate to the fraction of a milligramme; and the requisite quantity of the smaller constituent was calculated to give, when combined, a mixture of a definite percentage composition by weight (this being more easily obtainable than a percentage composition by volume).

The quantities generally chosen were 10, 5, 1, and 0·1 per cent.

In a few cases, with metals known to have very delicate spectral reactions, a mixture of 0·01 per cent, was prepared.

Observations were then made of the spectrum of each specimen, and the result was recorded in maps in the following manner: First, the pure spectrum of the smallest constituent was observed, and the lines laid down from Thalén's map.

The series thus mapped was as follows:

Tin + Cadmium, percentages of Cd 10, 5, 1, 0·15
Lead + Zinc, " Zn 10, 5, 1, 0·1
Lead + Magnesium, " Mg 10, 1, 0·1, 0·01

The observations showed that the lines of the smallest constituent disappeared as the quantity got less. Although we had here the germs of a quantitative spectrum analysis, the germs only were present, because from the existence of several "critical points," and great variations due to other causes, the results obtained were not constant.

In a subsequent research on the gold-copper alloys used in the coinage, Mr. Roberts, the Chemist of the Mint, and myself were able to show that the shortening in the length of the lines by reduced quantity was such a definite physical effect following upon reduced quantity, that a difference of 110000 part of copper in gold could be detected.

We are now in possession of the facts utilized in the work which has led up to the subject discussed in the present paper.

They have been utilized along two perfectly distinct lines of thought:

(1) They have been used in an attempt to enable us to produce a spectrum of a substance free from lines due to the impurities which are almost always present.

(2) They have been used to indicate the existence and amount of dissociation when acknowledged compounds have been submitted to the action of different and increasing temperatures.

I will deal with (1) first.

The elimination of impurity lines is conducted as follows: The spectrum of the element is first confronted with the spectra of the substances most likely to be present to impurities. This is most conveniently done by photographing the spectra on the same plate one above the other, so that common lines are continuous.

The retention or rejection of lines coincident in two or more spectra is determined by observing in which spectrum the line is thickest; where several elements are mapped at once, all their spectra are confronted on the same plate, as by this means the presence of one of the substances as an impurity in the others can be at once detected.

Lines due to impurities, if any are thus traced, are marked for omission from the map and their true sources recorded, while any line that is observed to vary in length and thickness in the various photographs is at once suspected to be an impurity line, and if traced to such is likewise marked for omission. I give a case.

The two lines H and K (3933 and 3968), assigned both to iron and calcium by Ångström, are proved to belong to calcium in the following way:

a. The lines are well represented in the spectrum of commercial wrought iron, but are absolutely coincident with two thick lines in the spectrum of calcium chloride with which the iron spectrum was confronted.

b. The lines are represented by mere traces in the spectrum of a specimen of pure iron prepared by the late Dr. Matthiessen. Both poles of the lamp were of iron, the lower pole consisting of an ingot of the metal which had been cast in a lime-mold.

c. The lines are altogether absent in a photograph of pure iron, where both poles of the lamp were of the pure metal not cast in lime, and they are likewise absent in a photograph of the spectrum of the Lenarto meteorite.

By eliminating lines due to impurities in the manner just described, a spectrum is at length obtained, of which every line is assignable to the particular element photographed, the same temperature being employed in the case of all the elements observed.

With regard to the second line of work, I should commence by stating that from a beautiful series of researches carried on by several methods, Mitscherlich concluded in 1864 that every compound of the first order, heated to a temperature adequate for the production of light, is not decomposed, hut exhibits a spectrum peculiar to this compound.

In some experiments of my own, communicated to the Royal Society in 1873, I observed:

First. That whether the spectra of iodides, bromides, etc, be observed in the flame or a weak spark, only the longest lines of the metals are visible, showing that only a small quantity of the simple metal is present as a result of partial dissociation, and that by increasing the temperature, and consequently the amount of dissociation, the other lines of the metal appear in the order of their length with each rise of temperature.

Secondly. I convinced myself that while in air, after the first application of heat, the spectra and metallic lines are in the main the same, in hydrogen the spectra are different for each compound, and true metallic lines are represented according to the volatility of the compound, only the very longest lines being visible in the spectrum of the least volatile compound.

Thirdly. I found that with a considerable elevation of temperature the spectrum of the compound faded almost into invisibility.

These results enable us to make the following statement:

A compound body, such as a salt of calcium, has as definite a spectrum as that given by the so-called elements; but while the spectrum of the metallic element itself consists of lines, the number and thickness of some of which increase with increased quantity, the spectrum of the compound consists in the main of channeled spaces and bands, which increase in like manner.

In short, the molecules of a simple body and a compound one are affected in the same manner by quantity in so far as their spectra are concerned; in other words, both spectra have their long and short lines, the lines in the spectrum of the element being represented by bands or fluted lines in the spectrum of the compound; and in each case the greatest simplicity of the spectrum depends upon the smallest quantity, and the greatest complexity upon the greatest.

The heat required to act upon such a compound as a salt of calcium, so as to render its spectrum visible, dissociates the compound according to its volatility; the number of true metallic lines which thus appear is a measure of the quantity of the metal resulting from the dissociation, and as the metal lines increase in number, the compound bands thin out.

These results bring us face to face with the subject-matter of the recent work.

First with regard to impurity elimination. I find that, although the method is good for detecting and eliminating impurities, there are still short-line coincidences between metals which are pure.

This "higher law" has come out in the following manner:

For the last four years I have been engaged upon the preparation of a map of the solar spectrum on a large scale, the work including a comparison of the Fraunhofer lines with those visible in the spectrum of the vapor of each of the metallic elements in the electric arc.

To give an idea of the thoroughness of the work, at all events in intention, I may state that the complete spectrum of the sun, on the scale of the working map, will be half a furlong long; that to map the metallic lines and purify the spectra in the manner described, more than 100,000 observations have been made and about 2,000 photographs taken.

In some of these photographs we have vapors compared with the sun; in others vapors compared with each other; and others again have been taken to show which lines are long and which are short in the spectra.

A rigorous application of the system of impurity elimination formed, of course, a large part of the work.

The final reduction of the photographs of all the metallic elements in the region 39-40—a reduction I began in the early part of last year—summarized all the observations of metallic spectra compared with the Fraunhofer lines accumulated during the whole period of observation, and all the results of the impurity elimination.

Now this reduction has shown me that the hypothesis that identical lines in different spectra are due to impurities is not sufficient. I show in detail in the paper the hopeless confusion in which I have been landed.

I find short-line coincidences between many metals the impurities of which have been eliminated, or in which the freedom from mutual impurity has been demonstrated by the absence of the longest lines.

The explanation of this result on the hypothesis that the elements are elementary does not lie on the surface, but it does on the assumption that they are compounds and behave like them.

This is the first point. We now pass from the results brought about at the same temperature with different substances to those observed at different temperatures with the same substance.

I find that when the temperature is greatly varied, the elements behave spectroscopically exactly as compound bodies do, as we have already seen. New lines are developed with increasing temperatures, and others fade in precisely the same way as the metallic lines made their appearance in the salts at the expense of the latter, which faded too.

In short, the observations and reasoning which I formerly employed to show how acknowledged compounds behaved in the spectroscope are now seen to indicate the compound nature of the chemical elements themselves.

In a paper communicated to the Royal Society in 1874, referring, among other matters, to the reversal of some lines in the solar spectrum, I remarked:

It is obvious that greater attention will have to be given to the precise character as well as to the position of each of the Fraunhofer lines, in the thickness of which I have already observed several anomalies. I may refer more particularly at present to the two H lines 3933 and 3968 belonging to calcium, which are much thicker in all photographs of the solar spectrum (I might have added that they were by far the thickest lines in the solar spectrum) than the largest calcium line of this region (4226·3), this latter being invariably thicker than the H lines in all photographs of the calcium spectrum, and remaining, moreover, visible in the spectrum of substances containing calcium in such small quantities as not to show any traces of the H lines.

How far this and similar variations between photographic records and the solar spectrum are due to causes incident to the photographic record itself, or to variations in the intensities of the various molecular vibrations under solar and terrestrial conditions, are questions which up to the present time I have been unable to discuss.

The progress of the work has shown that the differences here indicated are not exceptions, but are truly typical when the minute anatomy of the solar spectrum is studied.

Kirchhoff, indeed, as early as 1869 seems to have got a glimpse of the same thing, for in his memorable paper, which may justly be regarded as the basis of all subsequent work, he is careful to state that the sixty iron lines in the sun, to which he referred, only agree "as a rule" in intensity with those observed in the electric spark. Those who have given an account of his work have not always been so cautious. Indeed, I find Professor Roscoe[6] running far beyond the record in the following sentence:

In order to map and determine the positions of the bright lines found in the electric spectra of the various metals, Kirchhoff, as I have already stated, employed the dark lines in the solar spectrum as his guides. Judge of his astonishment when he observed that dark solar lines occur in positions connected with those of all the bright iron lines! Exactly as the sodium lines were identical with Fraunhofer's lines, so for each of the iron lines, of which Kirchhoff and Ångström have mapped no less than 460, a dark solar line was seen to correspond. Not only had each line its dark representative in the solar spectrum, but the breadth and degree of shade of the two sets of lines were seen to agree in the most perfect manner, the brightest iron lines corresponding to the darkest solar lines.

This statement was made to prove the absolutely identical nature of the iron vapor in the sun's atmosphere and in the electric spark. As the statement is not true, the vapors can hardly be identical.

Such, then, is the reasoning on which I base the two counts in the indictment against the simple nature of the elementary bodies.

First, the common lines visible in the spectra of different elements at high identical temperatures point to a common origin. Secondly, the different lines visible in the spectra of the same substance at high and low temperatures indicate that at high temperatures dissociation goes on as continuously as it is generally recognized to do at all lower temperatures.

In my paper I attempt to show that if we grant that the highest temperatures produce common bases—in other words, if the elements are really compounds—all the phenomena so difficult to account for on the received hypothesis find a simple and sufficient explanation. And, with regard to the second count, I discuss the cases of calcium, iron, lithium, and hydrogen. I might have brought, and shall subsequently bring, other cases forward. In all these I show that the lines most strongly developed at the highest temperatures are precisely those which are seen almost alone in the spectra of the hottest stars, and which are most obviously present in the spectrum of our own sun. Now, if it be true that the temperature of the arc breaks up the elements, then the higher temperature of the sun should do this in a still more effective manner. Here, then, we have a test.

I have put this question to the sun, and I have sent in a second paper to the Royal Society embodying a preliminary discussion of Professor Young's work at Sherman, Tacchini's observations, and my own. In this paper I state my grounds for the belief that all the solar phenomena we have been watching with our spectroscopes for the last ten years can not be explained on the existing-hypothesis, and that they are simply and sufficiently accounted for by supposing that primordial atoms are associated in the corona and dissociated in the reversing layer.

In this way the vertical currents in the solar atmosphere, both ascending and descending, the intense absorption in sun-spots, their association with the faculæ, and the apparently continuous spectrum of the corona, and its structure, find an easy solution.

We are yet as far as ever from a demonstration of the cause of the variation in the temperature of the sun; but the excess of so-called calcium with minimum sun-spots, and excess of so-called hydrogen with maximum sun-spots, follow naturally from the hypothesis, and afford indications that the temperature of the hottest region in the sun closely approximates to that of the reversing layer in stars of the type of Sirius and a Lyræ.

 
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  1. "American Journal of Science and Arts."
  2. Printed in the "Popular Science Monthly Supplement" for August, 1878.
  3. The old system of atomic weights was the one referred to.
  4. Symbols are used here to save space. H = Hydrogen, Ca = Calcium, Mg = Magnesium, Na = Sodium, Fe = Iron, Bi = Bismuth, Hg = Mercury.
  5. In the case of zinc the effect of these circumstances was very marked, and they may be given as a sample of the phenomena generally observed. When the pressure-gauge connected with a Sprengel pump stood at from 35 to 40 millimetres, the spectrum at the part observed was normal, except that the two lines 4924 and 4911 (both of which, when the spectrum is observed under the normal pressure, are lines with thick wings) were considerably reduced in width. On the pump being started these lines rapidly decreased in length, as did the line at 4679—4810 and 4721 being almost unaffected; at last the two at 4924 and 4911 vanished, as did 4679, and appeared only at intervals as spots on the poles, the two 4810 and 4721 remaining little changed in length, though much in brilliancy. This experiment was repeated four times, and on each occasion the gauge was found to be almost at the same point, viz.:
     
    1st observation, when the lines 4924 and 4911 were gone, the gauge
    stood at 30 millimetres.
    2d " " " " 29 "
    3d " " " " 29 "
    4th " " " " 31 "
  6. "Spectrum Analysis," third edition, p. 240.