“So far, then, as the results of this incomparable series of observations at Kew and Wansted, considered by themselves alone, can now be stated, the period of the polar rotation at that epoch appears to have been probably somewhat over a year, and certainly shorter by about two months than it is at the present time. The range of the variation was apparently in the neighbourhood of a second of arc, or considerably larger than that shown by the best modern observations.

“Before taking leave of these observations for the present I cannot forbear to speak of the profound impression which a study of them leaves upon the mind, and the satisfaction which all astronomers must feel in recognising that, besides its first fruits of the phenomena of aberration and nutation, we now owe also our first knowledge of the polar motion to this same immortal work of Bradley. Its excellence, highly appreciated as it has been, has still been hitherto obscured by the presence of this unsuspected phenomenon. When divested of its effects, the wonderful accuracy of this work must appear in a finer light, and our admiration must be raised to higher pitch. Going back to it after one hundred and sixty years seems indeed like advancing into an era of practical astronomy more refined than that from which we pass. And this leads to a suggestion worthy of serious practical consideration—whether we can do better in the future study of the polar rotation, than again to avail ourselves of Bradley’s method, without endangering its elegant simplicity and effectiveness by attempts at improvement, other than supplying certain means of instrumental control which would without doubt commend themselves to his sagacious mind.

“In the next article Bradley’s later observations at Greenwich, the results of which are not so distinct, will be discussed; and also those of Brinkley at Dublin, 1808-13 and 1818-22. This will bring again to the surface one of the most interesting episodes in astronomical history,Other puzzles explained. the spirited and almost acrimonious dispute between Brinkley and Pond with regard to stellar parallaxes. I hope to show that the hitherto unsolved enigma of Brinkley’s singular results finds its easy solution in the fact of the polar motion. The period of his epoch appears to have been about a year, and its range more than a second. Afterwards will follow various discussions already more or less advanced towards completion. These include Bessel’s observations at KÖnigsberg, 1820-24, with the Reichenbach circle, and in 1842-44 with the Repsold circle; the latitudes derived from the polar-point determinations of Struve and MÄdler with the Dorpat circle, 1822-38; Struve’s observations for the determination of the aberration; Peters’ observations of Polaris, 1841-43, with the vertical-circle; the results obtained from the reflex zenith-tube at Greenwich, 1837-75, whose singular anomalies can be referred in large part to our present phenomenon, complicated with instrumental error, to which until now they have been exclusively attributed; the Greenwich transit-circle results, 1851-65, in which case, however, a similar complication and the large accidental errors of observation seem to frustrate efforts to get any pertinent results; the Berlin prime-vertical observations of Weyer and BrÜnnow, 1845-46, in which I hope to show that the parallax of Draconis derived from them is simply a record of the change of latitude; the conflicting latitude determinations at Cambridge, England; the Washington observation of Polaris and other close Polars, 1866-87, with the transit-circle; also those at Melbourne, 1863-84, a portion of which have already been drawn upon in the last number of the Journal, and some others. While the list is a considerable one, I shall be able to compress the statement of results for many of the series into a short space.

“In connection with this synopsis of the scope of the investigations, one or two particulars may be of interest, which at the present writing seem to foreshadow the probable outcome. I beg, however, that the statement will be regarded merely as a provisional one. First, while the period is manifestly subject to change, as has already once or twice been intimated, I have hitherto failed in tracing the variations to any regular law, expressible in a numerical formula. Indeed, the general impression produced by a study of these changes in the length of the period is that the cause which produces them operates capriciously to a certain degree, although the average effect for a century has been to diminish the velocity of the revolution of the pole. How far this impression is due to the uncertainty of the observations, and to the complication of the phenomenon with other periodical changes of a purely instrumental kind, I cannot say. Almost all of the series of any extent which have been examined, have the peculiarity that they manifest the periodicity quite uniformly and distinctly for a number of years, then for a while obscurely. In some cases, however, what at first appears to be an objective irregularity proves not to be so by comparison with overlapping series at other observatories.

“Another characteristic which has struck my attention, although somewhat vaguely, is that the variations in the length of the period seem to go hand in hand with simultaneous alterations in the amplitude of the rotation; the shorter periods being apparently associated with the larger coefficients for the latter. The verification of these surmises awaits a closer comparative scrutiny, the opportunity for which will come when the computations are in a more forward state. If confirmed, these observations will afford a valuable touchstone, in seeking for the cause of a phenomenon which now seems to be at variance with the accepted laws of terrestrial rotation.”

“Mr. Chandler’s remarkable discovery, that the apparent variations in terrestrial latitudes may be accounted for by supposing a revolution of the axis of rotation of the earth around that of figure, in a period of 427 days, is in such disaccord with the received theory of the earth’s rotation that at first I was disposed to doubt its possibility. But I am now able to point out a vera causa which affords a complete explanation of this period.Newcomb’s explanation. Up to the present time the treatment of this subject has been this: The ratio of the moment of inertia of the earth around its principal axis to the mean of the other two principal moments, admits of very accurate determination from the amount of precession and nutation. This ratio involves what we might call, in a general way, the solid ellipticity of the earth, or the ellipticity of a homogeneous spheroid having the same moments of inertia as the earth.

“When the differential equations of the earth’s rotation are integrated, there appear two arbitrary constants, representing the position of any assigned epoch of the axis of rotation relative to that of figure. Theory then shows that the axis of rotation will revolve round that of figure, in a period of 306 days, and in a direction from west toward east. The attempts to determine the value of these constants have seemed to show that both are zero, or that the axes of rotation and figure are coincident. Several years since, Sir William Thomson published the result of a brief computation from the Washington Prime-Vertical observations of a Lyrae which I made at his request and which showed a coefficient 0.05. This coefficient did not exceed the possible error of the result; I therefore regarded it as unreal.

“The question now arises whether Mr. Chandler’s result can be reconciled with dynamic theory. I answer that it can, because the theory which assigns 306 days as the time of revolution is based on the hypothesis that the earth is an absolutely rigid body. But, as a matter of fact, the fluidity of the ocean plays an important part in the phenomenon, as does also the elasticity of the earth. The combined effect of this fluidity and elasticity is that if the axis of rotation is displaced by a certain amount, the axis of figure will, by the changed action of the centrifugal force, be moved toward coincidence with the new axis of rotation. The result is, that the motion of the latter will be diminished in a corresponding ratio, and thus the time of revolution will be lengthened. An exact computation of the effect is not possible without a knowledge of the earth’s modulus of elasticity. But I think the result of investigation will be that the rigidity derived from Mr. Chandler’s period is as great as that claimed by Sir William Thomson from the phenomena of the tides.”

“The fact of a periodic variation of terrestrial latitudes, and the general law of that variation, have been established beyond reasonable doubt by the observations collected by Mr. Chandler. But two of his minor conclusions, as enumerated in No. 3 of this volume, do not seem to me well founded. They are—

“1. That the period of the inequality is a variable quantity.

“2. That the amplitude of the inequality has remained constant for the last half century.”

“The question now arises how far we are entitled to assume that the period must be invariable. I reply that, perturbations aside, any variation of the period is in such direct conflict with the laws of dynamics that we are entitled to pronounce it impossible. But we know that there are perturbations, and I do not see how one can doubt that they have so acted as to increase the amplitude of the variation since 1840.”

“The material utilised in the foregoing forty-five series aggregates more than thirty-three thousand observations. Of these more than one-third were made in the southern hemisphere, a fact which we owe principally to Cordoba. It comprises the work of seventeen observatories (four of them in the southern hemisphere) with twenty-one different instruments, and by nine distinct methods of observation. Only three of the series (XXI., XXV., and XXXV.), and these among the least precise intrinsically, give results contradictory of the general law developed in No. 267. This degree of general harmony is indeed surprising when the evanescent character of the phenomenon under investigation is considered.

“The reader has now before him the means for independent scrutiny of the material on which the conclusions already drawn, and those which are to follow, are based. The space taken in the printing may seem unconscionable, but I hope this will be charged to the extent of the evidence collected, and not to diffuseness or the presentation of needless detail; for I have studiously sought to compress the form of statement without omitting anything essential for searching criticism. That it was important to do this is manifest, since the conclusions, if established, overthrow the existing theory of the earth’s rotation, as I have pointed out on p. 21. I am neither surprised nor disconcerted, therefore, that Professor Newcomb should hesitate to accept some of these conclusions on the ground (A. J., No. 271) that they are in such conflict with the laws of dynamics that we are entitled to pronounce them impossible. He has been so considerate and courteous in his treatment of my work thus far, that I am sure he will not deem presumptuous the following argument in rebuttal.

“It should be said, first, that in beginning these investigations last year, I deliberately put aside all teachings of theory, because it seemed to me high time that the facts should be examined by a purely inductive process; that the nugatory results of all attempts to detect the existence of the Eulerian period probably arose from a defect of the theory itself; and that the entangled condition of the whole subject required that it should be examined afresh by processes unfettered by any preconceived notions whatever. The problem which I therefore proposed to myself was to see whether it would not be possible to lay the numerous ghosts—in the shape of numerous discordant residual phenomena pertaining to determinations of aberration, parallaxes, latitudes, and the like—which had heretofore flitted elusively about the astronomy of precision during the century; or to reduce them to tangible form by some simple consistent hypothesis. It was thought that if this could be done, a study of the nature of the forces, as thus indicated, by which the earth’s rotation is influenced, might lead to a physical explanation of them.

“Naturally, then, I am not much dismayed by the argument of conflict with dynamic laws, since all that such a phrase means must refer merely to the existent state of the theory at any given time. When the 427-day period was propounded, it was as inconsistent with known dynamic law as the variation of it now appears to be. Professor Newcomb’s own happy explanation has already set aside the first difficulty, as it would appear, and advanced the theory by an important step. Are we so sure yet of a complete knowledge of all the forces at work as to exclude the chance of a vera causa for the second?”

“The philosopher should be a man willing to listen to every suggestion, but determined to judge for himself. He should not be biassed by appearances; have no favourite hypothesis; be of no school; and in doctrine have no master. He should not be a respecter of persons, but of things. Truth should be his primary object. If to these qualities be added industry, he may indeed hope to walk within the veil of the temple of Nature.”

“We now come upon a new line of investigation. Heretofore, as has been seen, the method has been to condense the results of each series of observations into the interval comprised by a single period, then to determine the mean epoch of minimum and the mean range for each series, and, finally, by a discussion of these quantities, to establish the general character of the law of the rotation of the pole. It is now requisite to analyse the observations in a different way, and discover whether the deviations from the general provisional law, in the last column of Table II., are real, and also in what manner the variation of the period is brought about. The outcome of this discussion, which is to be presented in the present paper, is extremely satisfactory. The real nature of the phenomenon is most distinctly revealed, and may be described as follows:—

“1. The observed variation of the latitude is the resultant curve arising from two periodic fluctuations superposed upon each other. The first of these, and in general the more considerable, has a period of about 427 days, and a semi-amplitude of about 0.12. The second has an annual period with a range variable between 0.04 and 0.20 during the last half-century. During the middle portion of this interval, roughly characterised as between 1860 and 1880, the value represented by the lower limit has prevailed, but before and after those dates, the higher one. The minimum and maximum of this annual component of the variation occur at the meridian of Greenwich, about ten days before the vernal and autumnal equinoxes respectively, and it becomes zero just before the solstices.

“2. As the resultant of these two motions, the effective variation of the latitude is subject to a systematic alternation in a cycle of seven years’ duration, resulting from the commensurability of the two terms. According as they conspire or interfere, the total range varies between two-thirds of a second as a maximum, to but a few hundredths of a second, generally speaking, as a minimum.

“3. In consequence of the variability of the coefficient of the annual term above mentioned, the apparent average period between 1840 and 1855 approximated to 380 or 390 days; widely fluctuated from 1855 to 1865; from 1865 to about 1885 was very nearly 427 days, with minor fluctuations; afterwards increased to near 440 days, and very recently fell to somewhat below 400 days. The general course of these fluctuations is quite faithfully represented by the law of eq. (3), (No. 267), and accurately, even down to the minor oscillations of individual periods, by the law of eq. (15), hereafter given, and verbally interpreted above. This law also gives a similarly accurate account of the corresponding oscillations in the amplitude. The closeness of the accordance between observation and the numerical theory, in both particulars, places the reality of the law beyond reasonable doubt.”

“A very important conclusion necessarily follows from the agreement of the values of the 427-day term, deduced from the intervals between the consecutive values of T in Table XII., namely, that there has been no discontinuity in the revolution, such as Professor Newcomb regarded as so probable that he doubted the possibility of drawing any conclusions from the comparison of observations before and after 1860 (A. J., 271, p. 50).

“The present investigation demonstrates that the way out of the apparently irreconcilable contradiction of theory and observation in this matter does not lie in the direction of discrediting the observations, as he is inclined to do. On the contrary, the result is a beautiful vindication of the trustworthiness of the latter, and, at the same time, of the theory that demands an invariable rate of motion; providing a perfectly fitting key to the riddle by showing that another cause has intervened to produce the variability of the period. I feel confident that Professor Newcomb will agree with the reality of the explanation here set forth, and will reconsider his view that the perturbations in the position of the Pole must be of the nature of chance accumulations of motion, a view which he then considered necessary to the maintenance of the constancy in the period of latitude-variation.”

“It is therefore proved beyond reasonable doubt that the directions of the rotations is from West to East in both elements; whence the general form of the equation for the variation of latitude adopted in A. J., 284, p. 154, eq. (19). It may be thought that too much pains have been here bestowed upon a point which might be trusted to theory to decide. I cannot think so. One of the most salient results of these articles has been the proof of the fact that theory has been a blind guide with regard to the velocity of the Polar rotation, obscuring truth and misleading investigators for a half a century. And even if we were certain, which we are not, that the fourteen months’ term is the Eulerian period in a modified form. It would still be necessary to settle by observation the direction of the annual motion, with regard to which theory is powerless to inform us. To save repetition of argument, I must refer to the statement in A. J., 273, pp. 68, 70, of the principles adopted in beginning these inquiries in 1891.”

“The ground is now cleared for examination of the only topic remaining to be covered, to establish, upon the foundation of fact, every point in the present theory of these remarkable movements of the earth’s axis. This is the question of the possibility that these movements are not real, but merely misinterpretations of the observed phenomena; being in whole or in part an illusory effect of instrumental error due to the influence of temperature. Such a possibility has been a nightmare in practical astronomy from the first, frightening us in every series of unexplained residuals, brought to light continually in nearly all attempts at delicate instrumental research. A source of danger so subtile could not fail to be ever present in the mind of every astronomer and physicist who has given even a superficial attention to the question of the latitude variations, and there is no doubt that some are even now thus deterred from accepting these variations as proved facts. Perhaps the most explicit and forcible statement of the doubts that may arise on this subject has been given very recently by Mr. Cornu. The views of so distinguished a physicist, and of others who are inclined to agree with him, call for careful attention, and cannot be neglected in the present closing argument upon the theory presented in these articles. It is unnecessary, for the purpose of disposing of objections of the sort raised by Cornu, to insist that it is not sufficient to show that the observed variations, attributed to the unsteadiness of the Earth’s Pole, are near the limit of precision attainable in linear differential measures, and in the indication of the direction of gravity by means of the air bubble of the level; or to show that there are known variations in divided circles and in levels, dependent on temperature and seasons. Nor need we require of objectors the difficult, although essential, task—which they have not distinctly attempted—of showing that these errors are not eliminated, as they appear to be, by the modes in which astronomers use their instruments. Neither need we even urge the fact that a large portion of the data which have been utilised in the present researches on the latitude were derived by methods which dispense with levels, or with circles, a part of them indeed with both, and yet that the results of all are harmonious. On the contrary, let us admit, although merely for argument’s sake, that all the known means of determining the direction of gravity—including the plumb-line, the level, and a fluid at rest, whether used for a reflecting surface or as a support for a floating instrument—are subject to a common law of periodical error which vitiates the result of astronomical observation, obtained by whatever methods, and in precisely the same manner. Now, the observed law of latitude variation includes two terms, with periods of fourteen and twelve months respectively. Since the phases of the first term are repeated at intervals of two months in successive years, and hence in a series of years come into all possible relations to conditions of temperature dependent on season, the argument against the reality of this term, on this ground, absolutely fails, and needs no further notice. As to the second, or annual term, while the phases, as observed in any given longitude, are indeed synchronical with the seasons, they are not so as regards different longitudes. If, therefore, the times of any given phase, as observed in the same latitude, but in successively increasing longitudes, occurred at the same date in all of them, there would be a fatal presumption against the existence of an annual period in the polar motion. If, on the contrary, they occur at times successively corresponding to the differences of longitude, the presumption is equally fatal to the hypothesis that they can possibly be due to temperature variation as affecting instrumental measurement. But the facts given in the foregoing section correspond most distinctly to the latter condition. Therefore, unless additional facts can be brought to disprove successively these observed results, we may dismiss for ever the bugbear which has undoubtedly led many to distrust the reality of the annual component of the latitude-variation, while they admit the existence of the 427-day term.”

“It would seem as if the mind of the great discoverer must combine contradictory attributes. He must be fertile in theories and hypotheses, and yet full of facts and precise results of experience. He must entertain the feeblest analogies, and the merest guesses at truth, and yet he must hold them as worthless till they are verified in experiment. When there are any grounds of probability he must hold tenaciously to an old opinion, and yet he must be prepared at any moment to relinquish it when a clearly contradictory fact is encountered.”