Climatic Cycles and Tree-Growth/Chapter 7

by Andrew Ellicott Douglass

Need for such analysis.—During these modern times of rainfall and sunspot records we may compare such records with tree-growth and obtain the interesting correlations exhibited in the last two chapters; but the tree records extend centuries and even thousands of years back of the first systematic weather or sun records of any kind. Without being over-precise or exhaustive, it is interesting to note that California weather records began about 1851. Records on the Atlantic coast began largely in the half-century before that date. London has a rainfall record since 1726, Paris since 1690, and Padua since 1725. Good sunspot records began about 1750, but the number of maxima and minima is known between 1610 and 1750, although the exact dates are uncertain. All this does not carry us very far back, but it serves as an excellent basis for the correct interpretation of the record in the trees.

It would be possible to apply correlation formulas to the Arizona tree records and perhaps to the sequoias and construct a probable rainfall record for long periods of time, but apart from Huntington's study of the "Climatic Factor in History," the chief use of such a record would be in studying the laws which govern rainfall; and this is best done through cycles. We shall find that the sunspot cycle plays an important rôle in rainfall. But we find traces of the solar cycle in nearly all of our tree groups, and evidently the way to read the trees is to study first of all their alphabet of cycles. Hence the best methods of identifying cycles must be used.

Proportional dividers. — If a short series of observations is to be tested for a single period, it can be done by mathematics, but it will take many hours and give a result in terms so precise as often to deceive. This, for example, has been the difficulty with the mathematical solution of the sunspot curve. It seems to the writer that the safer way to solve such a curve is by a graphic process, plotting the curve and applying equal intervals along it. An extremely good instrument for this purpose is the multiple-point proportional dividers. By a system of pivots and bars, 16 or more points are maintained in a straight line and at equal intervals, while the space between two successive points may be drawn out from one-eighth inch to one inch. The remarkable persistence of the half sunspot period in the early Flagstaff trees was detected in this way.

The projection of equal spacing on curves as long as 12 to 15 feet has been done by a 10-foot india-rubber band with small metal clips pinched on at regular intervals. As the band was stretched all the intervals were enlarged by equal amounts, and periodic phenomena were detected. Similar use could be made of the sharp shadows cast by the glowing carbon of an arc-light. The shadow of a transparent scale could easily be cast in all sizes upon a plotted curve. But all these methods of equal spacing on a plotted curve leave far too much to the individual judgment of the investigator.

THE OPTICAL PERIODOGRAPH.

A method of periodic analysis well adapted to the work in hand has been developed by the writer as the need for it became more and more evident. Along with the feeling of need for rapid analysis was the increasing recognition of the desirability of some process which would place mere individual judgment and personal equation as far in the background as possible.

Schuster's periodogram. — In 1898, Schuster suggested the use of the word "periodogram" as analogous to the word spectrogram; that is, a periodogram is a curve or a photograph which indicates the intensity of time periods just as the spectrogram indicates intensity of space-periods or wave-lengths. The spectrogram commonly gives its intensities by varying photographic density along a band of progressive wave-lengths. For the periodogram Schuster made simply a plotted curve, of which the abscissæ represented progressive time-periods and the ordinates represented intensities. He made a mathematical analysis of the sunspot numbers and constructed a periodogram which is reproduced in figure 30. It shows periods at its crest at 4.38, 4.80, 8.36, 11.125, and 13.50 years.

Fig. 30.—Schuster's periodogram of the sunspot numbers.

The optical periodogram. — It is, of course, not necessary that the periodogram should take the form of a plotted curve with intensities represented by ordinates, nor yet need it be exactly like a spectrogram showing intensities by density. The first periodogram produced by the writer is shown in plate 9, a. It is an analysis of the sunspot numbers. from 1755 to 1911. The existence of a rhythm in any specified period is indicated by a beaded or corrugated effect. A line across the corrugations gives in fact the rhythmic vibrations of the cycle. On a moment's examination this periodogram shows much of the information which has been under discussion for many years. The 11-year period is the most pronounced, but it is not so superior to all others as would be expected. It may be of any duration from 11.0 to 11.8 years, but 11.4 is a good average. There is obviously a period somewhere between 9.5 and 10.5 years and one between 8.0 and 8.8, but it is less conspicuous. Faint indications of periods are found near 14 years. The double of 8.4 is seen between 16 and 17 years. The double of the 10-year period shows near the 20 and at 22 the double of the 11 begins.

The preliminary part of producing this periodogram is the construction of the "differential pattern" shown in plate 9, b. This pattern is the optical counterpart of a set of columns of numbers arranged for addition, as when one summates a series of annual measures on a 10-year period, for example. The series is arranged in order with the first 10 years in the first line, the second 10 in the second line, and so on. In the case of the pattern the lines are made indefinitely long, so that the optical addition may be done in other directions than merely straight downward, for by making the additions on a slant a different period comes under test.

In order to produce this pattern the sunspot curve was cut out in white paper and pasted in multiple on a black background. The left end of each of the upper lines is the date 1755. Each successive line is moved 10 years to the left, so that passing from above vertically downward each line represents a date 10 years later than its predecessor. This continues from 1755 to 1911, and the lower 10 lines show the latter date at their right ends. It is not necessary that any of the lines should be full length, as we use only a part of each. By passing the eye downward from the top, a period near 10 years will show itself at once by a succession of crests in vertical alinement. If the crests form a line at some angle to the vertical, then the period they indicate is not exactly 10 years. It is more if the slant is to the right and less if to the left. The horizontal lines are spaced the equivalent of 5 years. Hence, if we measure the angle made between a vertical line and a line joining two crests in successive horizontal lines, we may easily calculate by simple formulas the period indicated.

Since the photometric values of all the curves in the diagram are proportional to the plotted ordinates, the photographic summation of the whole pattern in a vertical direction is almost an exact analogue of a numerical summation. This summation is simply done by a positive cylindrical lens with vertical axis. This brings down on the plate a series of vertical lines or stripes. If, now, we cut across these lines with a horizontal slit, the light coming through this slit from one end to the other will be the summation of the diagram in the vertical. But the photographic summation may be done at any slant instead of only in the vertical, and therefore the sensitive plate may be made to summate these curves through a long range of periods. In order to get a long range of periods, the diagram was mounted on an axis with clock-work and slowly rotated in front of a camera with a cylindrical lens for objective, a horizontal slit in the focal plane, and a sensitive plate passing slowly downward across the slit by clock mechanism. In this way a full range of possible periods come under the summing process, and when a real period is vertical the crests of the curves form vertical lines which come down as a series of dots or beads in the slit. When no period is in the vertical the light coming through the slit is uniform. Of course, there is a practical limit to the different angles at which the diagram may be viewed. An angle too far in one direction, making the tested period very small, would require a great number of duplications of the curve, while too great an angle the other way, making the tested period very large, catches the curve in the nonsymmetrical form and introduces errors. In the periodograms actually made of the sunspot curve the minimum period tested was 7 years and the maximum 24. One notes especially that this is a continuous process and that all periods from the minimum to the maximum are tested.

Application to length of sunspot period. — The interest in the sunspot period makes a special consideration of plate 9, c, worth while. This figure is a photograph of plate 9, b, taken out of focus for the purpose of calling attention to certain general features. In b the eye naturally turns to the sharp outlines and notes its minute details. In c the crests of b are changed into large blotches connecting somewhat with their nearest neighbors and varying in intensity. The alinement which they form in a nearly vertical direction is a graphic representation of the period. If the line were exactly vertical the period would be 10 years. The slant to the right shows more than 11. If the line were straight the period would be constant. It is evident that there are several irregularities in it. Having a number of exactly similar lines side by side, the irregularities are repeated in each and thus strike the consciousness with the effect of repeated blows. These irregularities are the discontinuities referred to by Turner in connection with his hypothesis. It is evident at a glance that the sunspot sequence divides itself into three parts, namely, a 9.3-year period, 1750-1790; then an interval of readjustment, 1800-1830, with a 13-year period; and lastly an 11.4-year period lasting to the present time (values approximate).^[1] But the latter is not perfectly constant, for after 1870 there is a change in intensity. The breaks thus shown and Turner's dates of discontinuity are compared in table 6.

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DOUGLASS PLATE 9

A. Periodogram of the sunspot numbers, 1755-1911. Corrugations show periods. The
numbers give length of period in years. The white line is the year 1830 and shows phase.

B. Differential pattern used in making the periodogram, consisting of the sunspot curve
mounted in multiple.

C. Same pattern photographed out of focus to show discontinuities in the vertical lines.

D. Sweep of sunspot numbers, 1755-1911.

E. Differential pattern of sunspot numbers made by the periodograph process.

Table 6.—Discontinuities in the sunspot cycle.

Periodogram.	Turner.
	1766
Between 1788 and 1804.	1796
Between 1830 and 1837.	1838
Between 1870 and 1884.	1868
	1895

By means of this diagram one can discover at a glance the origin of many of the periods which Michelson thought were illusory and in which opinion he was largely right. We can plainly see a 9.3-year period in the early part of the curve. Let us call this part of the sequence A_n and its broken continuation near the center S_n, and the lower and later part giving the 11.4-year period C_n. Thus we get at once three periods, 9.3, 11.4, and something over 13 years. If, now, we bring the average A_n into line with the average C_n as the periodograph does, we get 11.4 years. If we bring the average A_n-1 into line with the C_n-1, we get close to 10 years. If we bring into line A_n and the heavier parts of C_n-2, we get 8.4 years or thereabouts. And at 5.6 years we find a period which is just half of C_n and at 4.7 the half of A_n, and so on. It is like a checker-board of trees in an orchard; they line up in many directions with attractive intensity. But plate 9, c, helps remove some of the complexity of the sunspot problem. It shows us that while these various periods are apparent, they are improbable and needless complications. The diagram supplies a basis for profitable judgment in the matter. Hence to avoid just such awkward cases as the sunspot curve, a differential pattern is considered to be a necessary accompaniment of the periodogram in doubtful cases.

Production of differential pattern.—The work described above, consisting particularly in the production of a periodogram from the differential pattern, was done at Harvard College Observatory in 1913. The next fundamental improvement in the apparatus was in 1914, and consisted in a method of producing the differential pattern without all the labor of cutting out the curves. It was simply the combination of a certain kind of focal image called a "sweep" and an analyzing plate. A single white or transparent curve on a black background is all that is now needed as a source of light. An image of this is formed by a positive cylindrical lens with vertical axis. In the focal plane image so produced each crest of the curve is represented by a vertical line or stripe and the whole collection of vertical lines looks as if it has been swept with a brush unevenly filled with paint and producing heavy and faint parallel lines. Each of these lines represents in its brightness the ordinate of the corresponding crest. The sweep of the sunspot numbers is shown in plate 9, d. Any straight line whatever in any direction across this sweep truly represents the original curve, not as a rising and falling line but in varying light-intensity. A plate with equally spaced parallel opaque lines, called the analyzer or analyzing plate, is placed in the plane of this sweep. These lines may be seen in plate 9, e. When the analyzer is turned at a small angle to the lines of the sweep, each transparent line shows the full curve or a substantial part of it in its varying light intensities. 'These numerous reproductions are all parallel to each other, separated by equal dark lines, and each one is displaced longitudinally with reference to its neighbors, thus presenting the characteristics of the differential pattern. By twisting the analyzer with reference to the sweep while the two remain in parallel planes, different periods may be tested; for as the analyzer twists, each reproduction varies in respect to its length and its displacement from its adjoining neighbors above and below. When a period is formed it shows itself, just as in the original differential pattern, by rows of dark and light spots in alinement more or less perpendicular to the analyzing lines, as in plate 9, e. These light and dark rows are analogous to interference fringes and are identical with the elaborate but provokingly useless designs on a wire screen in front of its reflection in a window, or with the parallel fringes when two sets of parallel lines are held at a slight inclination to each other.^[2] Alinements are always best recognized by holding the paper edgewise and looking at the diagram at a low angle rather than in a perpendicular direction.

The analyzing plate resembles a coarse grating with equally spaced parallel lines. Much difficulty was experienced in making it. It is most satisfactory if made on glass with strong contrast between the opaque and transparent parts. The grating now in use was produced by photographing a 10-foot sheet of coordinate paper upon which 165 lines of black gummed paper had been carefully fastened. The coordinate lines permitted the spacing to be done with exactness. The width of the transparent space throughout was three-tenths of the distance from center to center. This was carefully photographed by a good lens at different distances. Glass prints were made from each negative and are still in use.^[3]

<poem>Theory. — The formula for the period is very simple: Let y = length of curve in years or other time-unit employed. l = length of curve image across sweep lines in centimeter or other unit of length. s = spacing center to center of analyzing lines in unit of length. Then${l \over s}$ = number of analyzing lines in curve when lines are parallel to sweep. ${ys \over l}$ = number of years in 1 line when lines are parallel to sweep.

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Now, taking analyzing lines aa¹ and bb¹ in figure 31 as horizontal, and letting the sweep be inclined as a small angle δ with the analyzing lines, the number of lines required to cross the sweep in the direction ab perpendicular to analyzing lines will be increased and hence the value in years between two analyzing lines will be decreased; hence

$\scriptstyle {ys \over l}cos\delta$ = years per line from a to b.

If the fringe is perpendicular to the analyzing lines, its period is the distance ab in years and we have for this special case:

$\scriptstyle p_{1}={ys \over l}cos\delta$.

Fig. 31.—Diagram of theory of differential pattern in periodograph analysis.

If, however, the fringe takes some other slant, as the direction ac, making the angle θ with the analyzing lines, then the period desired is the time in years between a and c. That equals the time between a and b less the time from b to c. Now bc in years would equal $\scriptstyle {\overline {ab}}cot\theta$ except for the fact that the horizontal scale along bc is greater than the vertical scale along ab in the ratio $\scriptstyle {cos\delta } \over {sin\delta }$ and therefore a definite space interval along it means fewer years in the ratio of $\scriptstyle {sin\delta } \over {cos\delta }$. Hence we have:

bc (in years) = ab (in years) tan δ cot θ

or

$\scriptstyle P=p_{1}\left(1-\tan \delta \cot \theta \right)$

which is the period required.

The separation of the fringes needs to be known at times in order to find whether one or more actual cycles are appearing in the period under test. In figure 31

$\scriptstyle {\overline {ab}}=s$

$\scriptstyle {\overline {ad}}={s \over {sin\delta }}$

$\scriptstyle {\overline {ac}}={{ssin(\theta -\delta )} \over {sin\delta }}$

which is the width required.

THE AUTOMATIC OPTICAL PERIODOGRAPH.

The present apparatus combines the two processes whose development has been described above. The second process developed is really the first one in the present instrument.

The curve. — The curve is prepared by cutting it out in a thick coordinate paper. The space between the curved line and the base is entirely removed and the curve becomes represented by area. In order to make the density still greater, the paper is painted with an opaque paint so that the brilliant light passing through will come through only the curve itself and not the paper. A special window-shutter is made to occupy the lower 2 feet of the window, whose width is some 50 inches. The curtain can be drawn down to the top of this, excluding the light around the edges. This window-shutter has a door in the upper part to give access to the interior. Within this box is a sloping platform upon which a mirror 8 by 46 inches is placed. This mirror is about 35° from the horizontal position and when looked at from a horizontal direction it reflects the sky from near the zenith. On the side of this box toward the room is a slit 45 by 3 inches in size. This extends horizontally and is on a level with the mirror. Below this slit is a narrow groove for taking the lower edge of the curve paper and above this slit is a strip of wood on hinges, so that when the lower edge of the curve is placed in the narrow groove below, this hinged strip closes down on the top and holds the curve in place directly in front of the mirror. Looked at from a horizontal direction within the room, the curve is seen brightly illuminated by light from the sky not far from overhead.

Track and moving mechanism. — About 7 feet from the curve the track begins and extends back 45 feet in a perpendicular direction. The track consists of 3 rails. The center rail is of uniform height and takes the single rear wheel, whose motion controls the movement of the film at the back of the camera. The right-hand rail is also uniform in height and supports one of the front wheels. The left rail is variable in height and supports the driving-cone, which serves as the other front wheel. The cone is 6 inches long and 3 inches in greatest diameter. It rests on a side rail whose elevation and distance from the center can be altered. The purpose of this particular mechanism is to vary the speed with which the camera travels along the track, for the time of exposure is approximately proportional to the square of the distance from the curve, and therefore when the camera travels from the near position to the far position it must slow down in rate as it goes along. The left rail, therefore, at the near position is close to the center and low down; in the middle and outer parts of the track it gets farther away and higher up, since the parts of the cone near the vertex travel on it. The axis of the cone carries a bevel gear meshing with another bevel attached to a vertical axis with a worm gear at the top, which the electric motor drives with a belt connection. In order to aid the motion of the camera, a cord passes from its back to the outer end of the track and by a system of pulleys and weights exerts a slight constant force. The motor is so connected that the camera travels away from the curve. The details here described may be seen in plate 10.

The differential pattern mechanism. — The camera is divided into three separate compartments, to each of which access is obtained by a sliding door moving in grooves on the side. The front compartment produces the differential pattern. It is about 7 inches long by 5 inches wide in the clear and 4 inches high. It is nearly divided into two parts by a partition which comes down from the top at about 2 inches from the front end. This partition does not go down to the floor of the compartment, but leaves a space of about an inch. A hole 1.5 inches in diameter is cut through the front of this compartment a little above its center, and another hole of the same size to match is cut through this partition, while at the back of this compartment a large opening is made a little over 2.5 inches wide and about 2 inches high. The lens is carried on a special carriage consisting of a horizontal and a vertical part. The vertical piece has a hole 1.5 inches in diameter cut in it, and the lens is mounted over the hole. The lens now in use consists of a spherical lens concavo-convex 2 inches in diameter and 12 inches in focus placed on the inside, and a positive cylindrical lens of the same size and focus placed on the outside with axis vertical. The convex side of each lens is placed outward. The lens carriage is placed partly under the partial partition and the lens in its holder comes directly between the two holes mentioned. When the sliding door of the compartment is down, the compartment is sufficiently light-tight to fulfill all the requirements of a camera. The movable carriage of the lens is mounted on two small glass tubes and runs between guides. A spring at its back end pulls it toward the position of focus for distant objects, where its motion is stopped by a pin. A long screw is passed through a, hole in the bottom of the camera box and enters the bottom of this lens carriage, so that an automatic arrangement outside and underneath the camera can regulate the focus. This consists of a vertical axis with two lever arms. The upper lever arm is a short one connected to the screw which comes from the lens board. The lower lever arm is some 4 inches below the upper and goes off in a direction nearly at right angles; it carries on its end a wheel in a horizontal position. This wheel is so placed that it runs on an especially arranged track attached to the side of the center rail of the main track. By varying the elevation of this special focussing track in different parts of the main track, the focus of the lens can be automatically controlled.

At the back of this first compartment is the analyzing plate, the same plate used in previous work. The spacing of its lines is 0.5 mm. from center to center. The proportionate transparent part is about three-tenths of the center-to-center measurement. The area covered by these lies is 1 by 3 inches, making about 156 lines. The photograph is transparent with dense black lines in it. The glass has been cut down to a convenient size, and this plate is mounted at the back of the first compartment with the film side of the plate toward the back. This plate is over the large opening at the back of the first compartment. The differential pattern is formed automatically by the lens on this plate. The plate is held in a fixed position with its lines nearly vertical but inclined about 12° to the lines of the sweep formed by the lens. This produces fringes more or less horizontal in direction. Varying periods are tested by changing the distance from the curve which alters the scale of the sweep while the analyzing lines are unchanged. As the scale of the sweep changes, the fringes appear to rotate about the center of the differential pattern. Immediately behind the analyzing plate are two condensing lenses described in the next topic. They bring the general beam of light to a focus about 6 inches back of the plate. For visual work a movable mirror, just back of the plate, reflects the beam outside the camera box, through an eyepiece to the eye. For photographic work a small total-reflection prism and simple lens are inserted about 5 inches back of the analyzing plate. These throw the beam outside into a special camera attachment in which ordinary films or plates may be used.

The periodogram mechanism. — The remainder of the camera is especially for the purpose of producing the periodogram from the differential pattern. Almost in contact with the analyzing plate is a condensing lens consisting of two cylindrical lenses about 2 inches in diameter and 6 inches focus; these are mounted with vertical axes and with their convex sides toward each other. The aperture of the condenser is about 0.75 inch in vertical height and 1.75 inches in length. The purpose of these condensers is to coverge the light which comes through the analyzing plate on the slit at the back. The second compartment is nearly the same size as the first, namely, about 6.5 inches long. At its front end is the analyzing plate with the condensers and at its back in the same optical axis is a vertical slit about 1 inch long and 1 mm. wide. The sides of this slit are beveled so that the slit itself is at the back. In the middle of this compartment is a powerful cylindrical lens or combination of lenses with horizontal axis. This lens is made up of 4 separate positive cylindrical lenses, each 2 inches in diameter and 6 inches focus. These all have their convex sides toward the common center. They are mounted on a movable carriage of wood which slips in place or may be removed entirely. The aperture of this lens system is about 1.5 inches long by 0.75 inch high. The effect of the condensing lens and of this cylindrical lens is to cast in the plane of the slit an area of light whose size is essentially a repro

DOUGLASS PLATE 10

A. The automatic optical periodograph.

B. Differential patterns of Sequoia record, 3200 years at 11.4.

duction of the aperture of the objective, namely, 1 inch high by 0.25 inch wide, but the detail in this area of light is brought in focus by the cylindrical lens and integrates the horizontal lines of the differential pattern. When, therefore, the differential pattern shows a series of horizontal fringes, they become reproduced by a series of horizontal lines crossing the slit, while in the slit itself they appear as a series of dots. When a period is disclosed by proper position of the camera, it will produce horizontal lines on the analyzing plate. A series of black and white dots, therefore, go through the slit into the final compartment; but when the distance is such that the lines on the differential pattern are at some slant, then, the integration carried into the slit being still horizontal, the illumination in the slit is uniform. In this way the beaded or corrugated effect in the slit indicates a period at that particular distance from the curve.

In order to read off periods directly in the final result without the necessity of making exact measures, an automatic signal or period indicator is introduced in this second compartment. Above the upper and lower ends of the slit are placed small pieces of mirror at 45°, and corresponding to these there are two small holes 0.25 inch in diameter in the side of the box. Outside of these holes again is a mirror at 45° reflecting light from the curve in the window. So long as the holes are open, direct light from the curve is reflected by the two sets of mirrors through the slit on to the film beyond, as will be described. A shutter is placed over the outer holes in the box with a lever carried down to the vicinity of the central rail. On the end of the lever arm is a wheel. At proper intervals small pieces of wood are placed in the side of the track, so that as the wheel passes over them the shutter is opened and light passes to the mirrors and makes a dot or a line on each side of the film in the third compartment. In this way marks can be placed on the film independent of the periodogram, and yet they can be spaced exactly to represent the different periods tested. Special periods, for example 5 or 10 years, etc., are indicated by the extra length and density of the marks produced. These appear on the margins of the periodograms in plate 11.

The final compartment at the rear contains a drum on a vertical axis which is slowly rotated as the whole mechanism moves along the track. The rear wheel resting on the center rail is connected by gearing to the drum, so that 1 mm. on the drum represents 42.7 mm. or 1.7 inches on the track. This makes a convenient length for the final periodogram. The drum can be detached, carried to a dark room to have a film pinned to its periphery, returned in a special light-tight box, and mounted on its axis for an exposure. The times of exposure depend on characteristics of the curve under test, but it is necessary to allow about 35 minutes for the range from 4 to 15 years, and several times that for the range from 15 to 25 years. Plates 10, 11, and 12 illustrate the apparatus and the periodic analysis produced. Periodograms. — Plate 11, which has been arranged to illustrate the work of the periodograph, shows several of the early periodograms which are comparatively free from obvious instrumental defects. In each the range of periods is marked on the left margin. Periods are indicated by the vertical band or ribbon breaking up into a series of horizontal dots or beads. For example, plate 11, a, is a periodogram of the 5-year standard period made for the purpose of calibrating the work of the periodograph. The 5-year period is very prominent near the top of the diagram in the plate. At 10 years its first harmonic appears with double crest, showing still that it is a 5-year period. At 15 years the second harmonic shows with a triple crest, and at 7.5 years the 3/2 overtone is evident with 3/2 crests. These overtones are always readily distinguished from the fundamental on the differential pattern. The differential pattern of this 5-year standard is shown in plate 12, q. The instrument is set for analysis at 5.0 years. In this position the integrating lens sums up the rows of light crests as a series of dots on the periodogram.

Plate 11, b, is the analysis of a mixed standard used for calibrating the instrument. The curve contains sharp triangular crests at intervals representing periods of 7, 9, 11, 13, and 17 years, all mixed together and no two starting intentionally from the same point. These are all separated in the periodogram and the overtones of some may be seen. Such overtones can be distinguished from the fundamental on the differential pattern.

Plate 11, c, gives a periodogram of the sunspot numbers from 1610 to 1910, using before 1750 the probable times of maxima suggested by Wolfer. The best period is at 11.1 as usually quoted. If the variation from 1750 only is taken, the best period comes at 11.4. This periodogram shows a period at about 8.6. The degree of accuracy with which one can pick out the periodic point is a real criterion of the accuracy of the result selected. The differential pattern of this same series of sunspot numbers will be found in plate 12, a, in which the vertical rows of crests are readily distinguished. The sudden change in direction of the lines a little below the center of this and the two following periodograms is an instrumental defect due to slight unevenness in the track and therefore is without significance.

Plate 11, d and e, give an analysis of the Arizona 500-year record. The chief points of interest are the well-defined double-crested 11.6-year period and the 19-year and 22-year periods. Other weaker periods may be seen from place to place.

Resolving power of the periodograph. — The accuracy with which a period can be determined by the periodograph may be readily observed in the differential pattern and the periodogram. The pattern indicates a period by showing a row of light spots or crests in line. The accuracy

DOUGLASS PLATE 11

A. Periodogram of standard 5-year period.

B. Periodogram of mixed periods.

C. Periodogram of sunspot numbers, 1610-1910.

D. Periodogram of Flagstaff 500-year record, to show cycles between 4 and 15 years of length.

E. Periodogram of same continued to 25 years.

of the period is the accuracy with which the direction of this line can be ascertained. This depends on the length of the row of crests, on the shortness of each crest, and on their individual regularity or alinement. These characteristics may be noted in the plates and especially in plate 12, q. Expressed in other terms, these resolving features are respectively as follows: (1) Number of cycles covered by the given curve. (2) Shortness of maxima in relation to length of cycle; if the maximum is sudden and sharp, as in rainfall, the accuracy may be very great; if the maximum is long, as in a sine curve, the accuracy is less. (3) Regularity in the maxima and freedom from interference. These features all appear in the differential pattern and hence the accuracy of any period is its most evident feature and all observers can judge it equally well. It is exactly analogous to the accuracy of a straight line passed through a series of plotted points which theoretically ought to form a straight line but which do not do so exactly.

The most important part of the constructed instrument which may alter the accuracy of analysis is the analyzing plate. The accurate spacing and parallelism of the lines is a mechanical feature and can be produced with care and attention to details, but the relation of width of transparent line to center-to-center spacing of the lines is a matter of judgment and the necessities of photography. As this relative width increases, the length of each crest in the pattern becomes longer and the row of crests becomes wider and less definite in direction. If the maxima in the curve under test are of the sine-curve type this relation is less important, for the light crests in the pattern will be long in any case, but for sharp, isolated maxima resolution is lost if the width of the transparent line is too great. In the instrument now constructed the ratio of transparent line to center-to-center spacing is 3:10, but a smaller ratio such as 1:10 could advantageously be used in certain cases if there is sufficient light to make photography easy.

The accuracy in reading a periodogram is at once apparent on its face. When the number of cycles is great as in plate 11, a, the rhythmic or beaded effect is short and very limited in extent, as in the 5-year period there indicated, and the period is accurately told. But if the number of cycles is reduced (as in plate 11, b or c) the periodic effect in the photograph extends over a greater range and its center can not be told with the same precision. The accuracy of estimation in the periodogram is therefore the actual accuracy of the result.

1. In discussing the periodicities of sunspots (1906², pp. 75-78) Schuster divided his 150 years, from 1750 to 1900, into two nearly equal parts. He found in the first part two periods of 9.25 and 13.75 years acting successively, and in the second part, a period of 11.1 years.
2. Roever (1914), has used somewhat similar interference patterns to illustrate very beautifully certain lines of force.
3. A very superior analyzing plate has recently been made from a ruled screen such as is commonly employed in half-tone engraving.