LECTURE V
LIGHT WAVES AS STANDARDS OF LENGTH
In the last lecture it was shown that in many cases the interference fringes could be observed with a very large difference in path—a difference amounting to over 500,000 waves. It may be inferred from this that the wave length, during the transmission of 500,000 or more waves, has remained constant to this degree of accuracy; that is, the waves must be alike to within one part in 500,000. The idea at once suggests itself to use this invariable wave length as a standard of length. The proposition to make use of a light wave for this purpose is, I believe, due to Dr. Gould, who mentioned it some twenty-five years ago. The method proposed by him was to measure the angle of diffraction for some particular radiation—sodium light, for example—with a diffraction grating. If we suppose Fig. 69 to represent, on an enormously magnified scale, the trace of such a grating, then the light for a particular wave length—say one of the sodium lines—which passes through one of the openings in a certain direction, as AB, is retarded, over that which passes through the next adjacent opening, by a constant difference of path; and therefore in the direction AB all the waves, even those which pass through the last of a very large number of such apertures, are in exactly the same phase. There will be then, if we are observing in a spectrum of the first order, as many waves in this distance AB as there are apertures in the distance AC. A diffraction grating is made by ruling upon a glass or a metal surface a great number of very fine lines by a ruling diamond, the number being recorded by the ruling-machine itself, so that there can be no error in determining the number of rulings. This number is usually very large, between 50,000 and 100,000. Since this number of lines is accurately known, we know also the number of spaces in the whole distance AC. This distance can be measured by comparing the two end rulings with an intermediate
FIG. 69 standard of length, which has been compared with the standard yard or the standard meter with as high a degree of accuracy as is possible in mechanical measurements. If, now, we know also the angle ACB, we can calculate the distance AB; and since we know the number of waves in this distance, which is the same as the number of apertures, we have the means of measuring the length of one wave. It will be observed, in making such an absolute determination of wave length by this means, that we have to depend entirely upon the accuracy of the distance between the lines on the grating—a distance which is measured by a screw advancing through a small proportion of its circumference for each line ruled. If the intervals between the lines are not exactly equal, then there will be an error introduced, notwithstanding every precaution taken, which it is extremely difficult, if not impossible, to correct.
Another error may be introduced in making the comparison of the two extreme lines on the grating with the standard decimeter. This error may, roughly, be said to amount to something like one-half a micron, i. e., one-half of one-thousandth of a millimeter. If, then, the entire length of the ruling is fifty millimeters, and the error, say, one ten-thousandth of a millimeter, the wave length may be measured to within one part in 500,000. This is the error upon the supposition that our standard is absolutely correct. But the length of the standard decimeter itself has to be determined by means of microscopic measurements, and since the temperature plays a considerable role, it is difficult to avoid errors very much larger than those due to the microscope. If we combine all these errors, we can probably attain at best an accuracy in all measurements involved of the order of one part in 100,000. Finally, we have to measure the angle ACB, and it is very much more difficult to measure angles than lengths. All these errors—the measurement of the angle, the error in the determination of the distance AC, that in the comparison of the intermediate standard which we use, and that in the distribution of these spaces—may combine in such a way that the total error may amount to very much more than one part in 100,000; it may be one in 20,000 or 30,000. This degree of accuracy, however, is greater than that attained by either of the other two methods which have been proposed for establishing an absolute standard of length.
The first of these proposed standards was the length of the pendulum which vibrates seconds at Paris. Such a pendulum may be obtained by suspending from a knife edge a steel rod upon which a large lens-shaped brass bob is fastened. The steel rod carries another knife edge near the other end, so that the pendulum can be turned over so as to be suspended from this lower knife edge. The pendulum must then be adjusted so that its time of vibration is exactly the same in either position, which can be done with but little difficulty. When such a pendulum vibrates seconds in either position, the distance between the knife edges is the length of a simple seconds pendulum.
We may also construct a simple pendulum by fastening a sphere of metal to the end of a thin, fine wire. It is then necessary to measure the time of oscillation, and the distance between the point of suspension and the center of gravity of the spherical bob. This distance can be measured to a very fair degree of accuracy. Unfortunately the different observations vary among themselves by quantities even greater than the errors of the diffraction method.
The second of these proposed standards was the circumference of the earth measured along a meridian, as it was believed that this distance is probably invariable. There are, however, certain variations in the circumference of the earth, for we know that the earth must be gradually cooling and contracting. The change thus produced is probably exceedingly small, so that the errors in measuring this circumference would not be due so much to this cause as to others inherent to the method of measuring the distance itself. For suppose we determine the latitude of two places, one 45° north of the equator and one 45° south. The difference in latitude of these places can be determined with astronomical precision. The distance between the places is one-fourth of the entire circumference of the earth. This distance must be measured by a system of triangulation—a process which is enormously expensive and requires considerable time and labor; and it is found that the results of these measurements vary among themselves by a quantity even greater than do those reached with the pendulum. So that none of these three methods is capable of furnishing an absolute standard of length.
While it was intended that one meter should be the one forty-millionth of the earth's circumference, in consequence of these variations it was decided that the standard meter should be defined as the arbitrary distance between two lines ruled on a metal bar a little over a meter long, made of an alloy of platinum and irridium. It was made of these two substances principally on account of hardness and durability. In order to bring the metal as nearly as possible to what was termed its "permanent condition," these bars were subjected to all sorts of treatment and maltreatment. The originals were cast and recast a great many times, and afterward they were cooled—a process which took several months.
After such treatment it is believed that the alteration in length of these bars will be exceedingly small, if anything at all. But, as a matter of fact, it is practically impossible to determine such small alterations, because, while there have been a number of copies made from this fundamental standard, these copies are all made of the same metal as the original; hence, if there were any change in the original, there would probably be similar changes in all the copies simultaneously, and it would therefore be impossible to detect the change. The extreme variation, however, must be of the order of one-thousandth of a millimeter or less in the whole distance of 1,000 millimeters.
The question rightly arises then: Why require any other standard, since this is known to be so accurate? The answer is that the requirements of scientific measurement are growing more and more rigorous every year. A hundred years ago a measurement made to within one-thousandth of an inch was considered rather phenomenal. Now it is one of the modern requirements in the most accurate machine work. At present a few measurements are relied upon to within one ten-thousandth of an inch. There are cases in which an accuracy of one-millionth of an inch has been attained, and it is even possible to detect differences of one five-millionth of an inch. Past experience indicates that we are merely anticipating the requirements of the not too distant future in producing means for the determination of such small quantities. Again, in order that the results of scientific work already completed, or shortly to be completed, may be compared and checked with those of subsequent researches, it is essential that the units and standards employed should have the same meaning then as now, and, therefore, that such standards should be capable of being reproduced with the highest attainable order of accuracy. We may, perhaps, say that the limit of such attainable accuracy is the accuracy with which two of the standards can be compared, and this is, roughly speaking, about one-half of a micron—some say as small as three-tenths of a micron. For such work neither of the three methods described above of producing a standard is sufficiently accurate. As before stated, the results obtained by them vary among themselves by quantities of the order of one part in 50,000 to one part in 20,000. Since the whole meter is 1,000,000 microns, an order of accuracy of one-half of a micron, which can be obtained with a microscope, would mean one part in 2,000,000, which is far beyond the possibilities of any of the three methods proposed.
We now turn to the interference method. Some preliminary experiments showed that there were possibilities in this method. The fact to which we have just drawn attention—namely, that the wave lengths are the same to at least one part in 500,000—looks particularly promising and leads us to believe that, if we had the proper means of using the waves and of multiplying them up to moderately long distances, without multiplying the errors, they could be used as a standard of length which would meet the requirement. This requirement is that a sufficient number of waves shall produce a length which may be reproduced with such a degree of accuracy that the difference between the new standard and the one now serving as the standard cannot be detected by the microscope.
The process is, in principle, an ideally simple one, and consists in counting the number of waves in a given distance. However, in counting such an enormous number, of the order of several hundred thousands, one is liable to make a blunder—not an error in a scientific sense, but a blunder. Of course, ultimately, this would be detected by the process of repetition.
The investigation, in a concrete form, presents a number of interesting points, involving problems of construction of a unique character which had to be solved before the process could be said to be perfectly successful.
The construction and operation of the apparatus will be much more readily understood if we first dwell a little upon the conditions that are to be fulfilled. Suppose, for illustration, that it is required to find the distance between two-mile posts on a railroad track. The most convenient method for measuring such a distance would be by a hundred-foot steel tape stretched by a known stretching force and applied to the steel rails. The rails are mentioned simply in order that there should not be any sag of the tape which would introduce still another error. The zero mark of the tape being placed against a mark on the rail which serves as the starting-point, a second mark is made on the rail opposite the hundred-foot mark of the tape. The tape is then placed in position a second time with one end on the second mark, and a third mark is placed at the farther end; and so on indefinitely. This is the first process. By it we have divided the mile into the nearest whole number of hundred-foot spaces. Then we measure the fractions.
The second operation consists in verifying the length of the steel tape, which we must do by comparing it with a standard yard or foot by the same stepping-off process.
The process of measuring the meter in light waves is essentially the same as that described above, the meter answering to the distance of a mile of track, and the hundred-foot tape corresponding to a considerably smaller distance. This smaller distance is what I have termed an "intermediate standard." There is in this latter case the additional operation of finding the number of light waves in the intermediate standard; so that, in reality, there are three distinct processes to be considered.
In the first operation it is evident that, if an error is committed whenever we lift the tape and place it down again, the smaller the number of times we lift it and place it down, the smaller the total error produced; hence, one of the essential conditions of our apparatus would be to make this small standard as long as possible. The length of the intermediate standard is, however, limited by the distance at which we can observe interference fringes. The limit, as was stated in the last lecture, is reached when this distance is of the order of several hundred thousand waves. At this distance the interference fringes are rather faint, and it seemed better for such determinations not to make use of the extreme distance, but of such a smaller distance as would insure distinct interference fringes. It was found convenient to use, as the maximum length of the intermediate standard, one decimeter. The number of light waves in the difference of path (which is twice the actual distance, because the light is reflected back) would be something of the order of three or four hundred thousand waves. With such a difference of path we can still see interference fringes comparatively clearly, if we choose the radiating substance properly.
The investigations described in the last lecture showed that the radiations emitted by quite a number of the substances which were examined were more or less highly complex. One remarkable exception, however, was found in the red radiation of cadmium vapor. This particular radiation proved to be almost ideally homogeneous, i. e., to consist very nearly of a series of simple harmonic vibrations. This radiation was therefore eminently suited to the purpose, and was adopted as the standard wave length.
Most substances produce a more or less complicated spectrum involving quite a number of lines, but in the case of cadmium vapor, though there are three different radiations, these three are all so nearly homogeneous that each one can be used; and the complexity of the spectrum is in this case an advantage, as will be shown below. To produce the cadmium radiation, metallic cadmium is placed in a glass tube which contains two aluminum electrodes. The tube is then connected by glass tubing with an air pump and exhausted of air. The tube is also heated so as to drive off all residual gas and vapor, and when the required degree of exhaustion is reached, it is hermetically sealed and in condition to use. The cadmium is not very volatile, and at ordinary temperatures we should see scarcely anything of the cadmium light when the electric discharge passes. The tube is therefore placed in a metal box, as shown in Fig. 60, which is furnished with a window of mica and has a thermometer introduced into one side. If the box be heated by a Bunsen burner to a temperature in the neighborhood of 300° C, the cadmium vapor fills the tube, and can then be rendered luminous by the passage of the electric spark.
Now, it is found most convenient not to make this first intermediate standard in the form of a bar like the standard meter, with two lines drawn upon it; for then we should introduce errors of the microscope at every reading, and these errors would be added together. Thus, since this is one-tenth of the whole meter, we might have, in adding up, ten times the error of the microscope, which we said was of the order of one-half a micron; we could thus have, in the end, an error of five microns. The interference method gives us the means of multiplying the length of the intermediate standard with the slightest possible error, amounting, perhaps, to one-twentieth of a micron; in some cases a little less. If two plane surfaces be parallel to one another and a given distance apart, then, with the interferometer, we may locate the position of either one of these surfaces by means of the interference
FIG. 70 fringes in white light to within one-twentieth of a fringe, which means one-fortieth of a wave, or one-eightieth of a micron. It has been found most convenient to use glass surfaces very carefully polished and made as nearly plane as possible, and silvered on the front. The two surfaces are mounted on a brass casting, and carefully adjusted so as to be as nearly parallel as possible, so that it does not matter what part of the surface is used. This parallelism of the two surfaces must be arranged with extraordinary accuracy; the greatest deviation from true parallelism must be of the order of one-half of a fringe, which would be one-fourth of a wave length, or one-eighth of a micron. Since the width of the surface is something like two centimeters, the allowable angle between the two surfaces is something like one part in two hundred thousand.
A section of the intermediate standard we have been describing is represented in Fig. 70. The two glass surfaces are about two centimeters square and silvered on their front surfaces, which are very nearly true planes. Their rear surfaces press against three small pins. These are adjusted for parallelism by filing until the requisite degree of accuracy is obtained. The parallelism cannot be made altogether perfect, and, as a matter of fact, in some cases the error may amount to as much as one-tenth of a micron or more.
Fig. 71 represents a perspective view of the same thing.
FIG. 71 In this figure the intermediate standard rests on a carriage by means of which it may be moved as necessary for the purpose of comparing it with the whole meter. In making this comparison the surfaces must be parallel to the mirror which serves as a reference plane in the interferometer. The parallelism in this case must be of the same order of accuracy as that between the surfaces themselves. The adjustment is made by the screws at the rear, one of which turns the whole standard about a vertical axis and the other about a horizontal one.
In determining the number of waves in the meter, the first operation is to find the number of whole waves in this intermediate standard. It can readily be conceived that the counting of something like 300,000 waves would be no small matter; in fact, a little calculation would show that, if we counted two per second, it would take over forty hours to make the count. Probably a number of methods will suggest themselves of making such a process of counting automatic, Indeed, several experiments have been made, and with some promise of success; but the possibility of skipping over one fringe, through some accident, is serious. It was therefore thought desirable to use another process, very much longer and more tedious, but very much surer. This process consists in dividing the distances to be measured into a very much smaller number of parts, so that the distances to be measured in waves would be very much smaller. Thus a distance of ten centimeters contains 300,000 waves; half of this distance would contain 150,000. If we go on dividing in this way, until we get to the last one of nine such steps, we reach an intermediate standard whose length is something of the order of one-half millimeter. The total number of waves in this standard is about 1,200, and this number it is a comparatively simple matter to count.
The method of proceeding in counting these fringes is the same as that described above. The reference plane, as we will call the movable mirror in the interferometer, is moved gradually from coincidence with the first surface to coincidence with the second, and the fringes which pass are counted. Such a count was made for the three standard radiations, namely, the red, green, and blue of cadmium vapor. The result was 1,212.37 for red, 1,534.79 for green, and 1,626.18 for blue. Now, an important point is that we can measure these fractions with an extraordinary degree of accuracy; so the second decimal place is probably correct to within two or three units. The whole number we know to be correct by repeating the count and getting the same result. Having thus obtained this number, including also the fractions of waves on the shorter standard to a very close approximation, we compare it with the second, which is, approximately, twice as long. This comparison gives us, without further counting, the whole number of waves in the second standard by multiplying the number in the first by two. We have the same possibility of measuring fractions on the second standard, and so can determine the number of waves in its length with an equal degree of accuracy.
I will give the description of this process somewhat more in detail. In Fig. 72 mm' represents the first or the shorter
FIG. 72 standard viewed from above. This standard rests on a carriage which can be moved with a screw. The second standard nn' is twice as long as the first, and is placed as close as possible to the first and rigidly connected with some part of the frame. The mirror d is the reference plane.[1] The two front mirrors of the two standards are adjusted to give fringes in white light with the reference plane. The central fringe in the white-light system is black; the others are colored. Hence we can always distinguish the central fringe. When the central fringes occur in the same relative position upon the two front mirrors m and n, then these two surfaces are exactly in the same plane. Now, if we move the reference plane backward through the length of the shorter standard, its surface will coincide with the mirror m', and at this instant fringes in white light will appear. Thus we have the means of knowing when the reference plane has been moved the length of the first standard to an order of accuracy of one-tenth or one-twentieth of a fringe.
The next process is to move the first standard backward through the same distance. Then the white-light fringes will again appear on the front mirror m. Finally we move the reference plane again through the same distance and, if the second standard is twice as long as the first, we get interference fringes on the two rear mirrors of the two intermediate standards. If there is any difference, then the central fringe of the white-light system will not be in the same position on both mirrors, and we shall know that one is twice as long as the other less, say, two fringes, which would mean less one-half micron. In this way we can tell whether one is exactly twice as long as the other or not; and if not, we can determine the difference to within a very small fraction of a wave.
When we multiply the number of waves in the first standard by two, any error in the fractional excess is, of course, also multiplied by two. So the fraction of a wave which must be added to the second number is uncertain. If we observe the fringes produced by one radiation, for example the red, we get a system of circular fringes upon both mirrors of the standard; and if these two systems have the same appearance on the upper mirror as on the lower, then we know this fraction is zero; and the number of waves in the second standard is then the nearest whole number to the number determined. If this is not the case, we can by a simple process tell what the fraction is, and can obtain this fractional excess to any required degree of accuracy. As an example, we may multiply the numbers obtained for the first standard by two, and we find 2,424.74 for the number of red waves in standard No. 2. The correct value of this fraction for red light was found to be .93 instead of .74. Thus the same degree of accuracy which was obtained in measuring the first standard can be obtained in all the standards up to the last. We have thus the means of finding accurately the whole number of waves in the last standard. The whole number obtained by this process of "stepping off" for the red radiation of cadmium was found to be 310,678. The fraction was then determined by the circular fringes, as described above, and found to be .48. In the same way the number for the green radiation was determined as 393,307.93; and for the blue radiation as 416,735.86. To give an idea of the order of accuracy, I would state that there were three separate determinations made at different times and by different individuals, as follows:
| Determination | Red | Green | Blue | |||
|
310,678.48 | 393,307.92 | 416,735.86 | |||
|
310,678.65 | 393,308.10 | 416,736.07 | |||
|
310,678.68 | 393,308.09 | 416,736.02 |
The fact that these determinations were made at entirely different times, separated by an interval of whole months, and by different individuals, and that we still were able to get, not only the same whole number of waves, but also so nearly the same fractions, gives us a confidence, which we could not otherwise feel, in the possibilities of the process.
In comparing the standards with one another the temperature made no difference, if only it were uniform throughout the instrument, because two intermediate standards side by side, made of the same substance, would expand in exactly the same way, provided we could be sure that both had the same temperature. But in the determination of the number of waves in standard No. 9 it is extremely important to know the temperature with the very highest degree of accuracy. For this purpose some of the best thermometers obtainable were placed in the instrument, and the thermometers themselves were carefully tested, their errors determined, and other well-known precautions taken. In this way the temperature at which the intermediate standard No. 9 contains the number of waves given above was determined to within one-hundredth of a degree.
The final step in the process is the comparison of the decimeter standard with the standard meter. This is a comparatively simple affair. In fact, it is exactly the same as the comparison of the first intermediate standard with the second, except that the second standard is now ten times as long—which necessitates going through the process ten times instead of twice.
Since in this case also we use the fringes for determining when one end of the standard and the reference plane are in the same plane, the error, as before stated, may be as small as one-twentieth of a wave; so that all the errors added together would be of the order of one-half of a wave, or one quarter of a micron.
The conditions which had to be fulfilled by the instrument which was used for this purpose are, then, these: We have, in the first place, to provide for the displacement of the intermediate standard and of the reference plane in such a way that the parallelism of the mirrors is not disturbed. This necessitates that the ways along which the carriage supporting the mirrors moves be exceedingly true. It took a whole month to perform this part of the work—to get the ways so nearly true that there should be no change in the position of the fringes as the mirrors were moved back and forth. In the second place, we must be able to know the position of the mirrors inside of the box which is placed over the instrument to protect it from temperature changes. To secure this, the carriage which holds the mirrors must be moved by means of a long screw carefully calibrated to within two microns or so. In the third place, since there will be slight displacements, owing to the impossibility of getting the ways absolutely true, it must be possible to correct these displacements. The adjustments for effecting this are shown in Fig. 71. Fourth, we must have a firm support for the longer of the two standards to be compared,
FIG. 73 and a movable support, which moves parallel with itself, for the shorter standard.
The last standard, the auxiliary meter, has to be compared with the standard meter itself, and, therefore, the two must be of similar construction. In other words, in this last comparison we have to resort to the microscope again. For the meter bar which we had in the interferometer itself had two lines upon it as nearly as possible one meter apart, as determined by a rough comparison with the prototype meter. The standard No. 9 had to be compared with this. For this purpose an arm which had a fine mark on it was rigidly fastened to the standard No. 9, and arranged to come in the focus of the microscope. In making this comparison, we must admit, the order of accuracy is not so great. But there are only two of these to make, so that the possible error is the same as that to which we are liable in comparing two meter bars. This error is unavoidable.
The whole instrument had to be placed in a box, which protected it from temperature changes and drafts of air, and had to be placed on a firm pier so as to keep it as free from
FIG. 74 vibration as possible. Finally, the conditions which have been mentioned above for producing a suitable source of light had to be fulfilled. We have thus a fair idea of what conditions had to be met in constructing the complete apparatus for making this comparison.
We shall now show how these conditions were actually fulfilled in the apparatus that was used for the experiment.
Fig. 73 gives a plan of the entire arrangement. It is easy to recognize the vacuum tube which serves as a source of light and the arrangement of the plates in the interferometer. This arrangement is the same as that shown in Fig. 72. In order to have but one radiation at a time in the instrument, the light from the tube is passed through an ordinary spectroscope. Thus the light from the tube Z is brought to a focus on the slit t1. It is then made parallel by means of the lens x2 and passes through the prism W, which is filled with bisulphide of carbon. The
FIG. 75 lens x3 forms the spectral images of the slit t1 in the plane of the slit t2. The arm ZW of the spectroscope can be moved so as to bring either the red, the green, or the blue spectral image upon this slit, from which it passes into the instrument.
Fig. 74 is a view of the plan of part of the instrument. The arrangement of surfaces shown diagramatically in Fig. 72 is readily recognized. All of the plates, I may state, instead of being rectangular, have a circular border, because in this form they can be worked true more readily.
Fig. 75 represents a vertical cross-section of the same instrument. It will be noted that the reference plane is divided into sections. This is done in order to enable us to determine very accurately the position of the interference fringes. The two intermediate standards will be recognized at the right.
Fig. 76 represents the actual instrument in perspective. In this the two microscopes, with their arrangement for producing an illumination on the meter bar by means of reflected, light, are shown. On the left are the handles which turn the two screws. One of these moves the intermediate standard
FIG. 76 and the other moves the reference plane. The complete instrument in the case which protects it against temperature changes is shown in Fig. 77.
This investigation was reported in the spring of 1892 to Dr. Gould, who at that time represented the United States in the International Committee of Weights and Measures. It was principally through his goodness that I was asked to carry out the actual experiments at the International Bureau of Weights and Measures at Sèvres. Many of the accessories that were required for the instrument which has just been described had to be made in this country, and were taken over and installed in one of the laboratories of the Bureau.
The standard meter itself is kept in a vault underground and under double lock and key, and is inspected only once in ten years, and even then it is not handled any more than
FIG. 77 is absolutely necessary. It took the better part of an entire year to accomplish the work as it has been described. The final result of the investigation was that the number of light waves in a standard meter was found to be, for the red radiation of cadmium 1,553,163.5, for the green 1,966,249.7, for the blue 2,083,372.1—all in air at 15° C. and at normal atmospheric pressure.
It is also worth noting that the fractions of a wave are important, because, while the absolute accuracy of this measurement may be roughly stated as about one part in two million, the relative accuracy is much greater, and is probably about one part in twenty million.
The question may be asked: What is the object of making such determinations, when we know that the standard itself would not change by any amount which would vitiate any ordinary measurements? The reply would be that, while the care taken of the standards is pretty sure to secure them from any serious accident, yet we have no means of knowing that any of these standards are not going through some slow process of change, on account of a gradual rearrangement of the molecules. Now that we have compared the meter with an invariable standard, we have the means of detecting any slow change and of correcting the standard which has been vitiated by such process. Thus it is now possible to control, by reference to the standard light waves, the standard of length. The standard light waves are not alterable; they depend on the properties of the atoms and upon the universal ether; and these are unalterable. It may be suggested that the whole solar system is moving through space, and that the properties of ether may differ in different portions of space. I would say that such a change, if it occurs, would not produce any material effect in a period of less than twenty millions of years, and by that time we shall probably have less interest in the problem.
SUMMARY
1. We find that three propositions for expressing our standard of length in terms of some invariable length in nature have been made, namely:
- a) Measurement of the seconds pendulum.
- b) Measurement of the earth's circumference.
- c) Measurement of light waves.
The first two, as well as the first plan proposed for carrying out the third, i. e., the method of the diffraction grating, have been found deficient in accuracy.
2. The second or interference method of utilizing light waves, while ideally simple in theory, necessitates in practice an elaborate and complicated piece of apparatus for its realization. But, notwithstanding the delicacy of the operation, it is capable of giving results of such extraordinary accuracy that, were the fundamental standard lost or destroyed, it could be replaced by this method with duplicates which could not be distinguished from the originals.
- ↑ Better, the image of d in a and b, which in the figure would coincide with the front surfaces of m and n.