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of continuity will be observed in the changes of dip in the needle while it moves over the bar. If the needle is placed so that its hori. zontal axis is at right angles to the axis of the large magnet, then there is a remarkable breach of continuity in the sequences of dip. While moving from one end of the bar to the middle, the needle will be vertical all the time. Suddenly, on passing the middle point, it makes half a revolution, so as to bring the end which before pointed to the zenith towards the nadir. Here, then, in the changes of dip in a needle thus exposed, the law of continuity is not observed. If we substitute for the large magnet the magnetism of the earth, we have the same result. A dipping-needle, placed so that the axis on which it turns is in the magnetic meridian, keeps a vertical position while it is carried from the earth's magnetic poles to the earth's magnetic equator. As it crosses the magnetic equator, its two poles suddenly-exchange positions with each other.
“There is one view to be taken of these facts which does not require us to believe that the law of continuity is disturbed. In both cases, the needle is constrained by its axis; for this axis is put in such a position that the whole force of nature is decomposed into two others, one of which is destroyed by the axis. If we take the action of the free force, the law of continuity prevails both in regard to the direction and the amount of the force. Still, this example will show that in the motions of a machine, or in any case where the forces of nature are artificially modified, it is not always safe to assume, in unqualified terms, the applicability of the law of continuity.
* A case can be supposed in which the force of gravity will be found in the same predicament. We cannot make the experiment, but it is not difficult to imagine what the result would be if the experiment were tried. I suppose a small tunnel to be cut from any point on the earth's surface to the centre, and so on to the opposite hemisphere. A plumb-line, if brought to the extremity of this opening, would point to the earth's centre ; and, if let down into the opening, would still do so, though with diminished force. If it continued to move along the tunnel, at the instant of passing the earth's centre of gravity its direction would suddenly change 180 degrees. In this case, the law of continuity is observed so far as the intensity of the earth's gravity is concerned, but it is broken in regard to the direction of the force. This must necessarily be the case under the influence of central forces, unless there is combined with them another force, like the projectile force, VOL. II.
which, in the solar system, unites with the former to carry the planet in a continuous curve around its centre of motion. If the force of projection be ever so small, the planet will move in a curve, however elongated, and change its direction gradually, though it may be with all the rapidity of the comet shooting through its perihelion. When the projectile force is nothing, the motion is rectilinear, and the direction alters abruptly. Here, also, the case is made easy, and the authority of the law of continuity vindicated. For in this instance, as in all others where motion, and not simply directive power, is considered, the velocity gradually diminishes, and prepares the way for a new motion in the opposite direction.
“ It is well known, that sometimes the law of the forces of nature changes once or more in passing from one condition of nature to another continuously connected with it. Thus the attraction of a solid sphere is as the square of the distance from the centre inversely, so long as the attracted body is on the outside. When the attracted body comes within the surface, the attraction is directly as the distance from the centre. In the case of a hollow shell, the law of its attraction changes more than once. Within the shell, the attraction is constant for all positions. Outside, it obeys the same law as in a solid sphere. In the thickness itself, the attraction is subject to a third law. The centre of gravity of the attracted body will pass abruptly from one to another of these three conditions; but it is not always safe to represent the whole body by its centre of gravity. As the small body is passing through the surfaces of the large one, neither of the three laws stated above is applicable. Probably no single law will follow the body through the various positions involved in the entering of one of the bodies into the other. The law itself probably changes every instant, and thus the three partial laws, which are so discontinuous, and which are derived from a consideration of only the centre of gravity, will appear to be continuously connected when those links which are neglected when we study the motions wholly through the centre of gravity are restored. The mathematical function itself, therefore, if made so general as to include all the conditions of the experiment, might possibly be continuous from first to last. At any rate, if we give full weight to this apparent breach of continuity in the present mathematical expression of the law of attraction, it by no, means follows that the body which is attracted and passes into these various exposures will change its velocity abruptly, as it comes under the sovereignty of one or the other of these laws. The laws may be different, widely different, in themselves, and yet in positions near together each may give a velocity not very different from what the others would have done. Therefore the abrupt change of the law will produce only a gradual change in the velocity of the moving body. This consideration is sufficient to show that the law of continuity is observed, to the exclusion of violent changes in matter. Nevertheless, a mental shock will be occasioned if the law itself shall not appear upon deeper investigation to retain, unbroken and unimpaired, its simplicity and integrity.”
Three hundred and fifteenth meeting. January 31, 1849. — QUARTERLY MEETING. The PRESIDENT in the chair.
The Corresponding Secretary read a letter from James Hall, Esq., of Albany, acknowledging the notification of his election as a Fellow of the Academy, and presenting the first volume of his work on the Paleontology of New York.
The Corresponding Secretary also presented from Dr. Bache a copy of his report on the progress of the United States Coast Survey, accompanied by the request that the Academy would submit it to a careful examination, and make such suggestions as might be called for upon the scientific character and value of the survey as now carried on, or which might tend to give greater efficiency to the work. The subject was referred to a committee, consisting of Professor Peirce, Professor Lovering, Mr. Treadwell, and Mr. J. I. Bowditch.
A note from Dr. John Ware, resigning his place on the Rumford Committee, having been read by the President, Mr. Treadwell was appointed to fill this vacancy.
Professor Arnold Guyot, late of Neuchatel, was elected a Fellow of the Academy.
At the request of the committee on the establishment of permanent marks to indicate the water-level on our coast, Lieutenant Davis and Mr. E. C. Cabot were added to the committee.
Mr. Desor exhibited a minute crustacean animal, found in Cochituate water, with a drawing of the same, which he pronounced to be a species of Calanus.
Three hundred and sixteenth meeting.
February 6, 1849. — Monthly Meeting. The Vice-President, Mr. Everett, in the chair.
The Corresponding Secretary read a letter from Professor Arnold Guyot, signifying his acceptance of the Fellowship of the Academy. He also read a communication from Mr. James D. Dana, of New Haven, upon the importance of having å larger edition of the scientific works embodying the results of the United States Exploring Expedition under Captain Wilkes. Whereupon the President was requested to address the Joint Library Committee of Congress, and to call attention to the memorial presented by the Academy upon that subject, on a former occasion.
Professor Peirce, from the committee appointed on that subject, read the following report upon the results of the United States Coast Survey, and it was ordered that a copy therof be forwarded to Dr. Bache, the Superintendent of the Survey, and another to the Hon. Robert C. Winthrop, Speaker of the House of Representatives.
“ The committee of the American Academy of Arts and Sciences, to which were referred the report of the Superintendent of the United States Coast Survey, and the letter of the Superintendent requesting the opinion of the Academy thereupon, respectfully submit the following report.
“ The present Superintendent of the Coast Survey of the United States was designated, in 1843, as the proper successor of Mr. Hassler, with extraordinary unanimity, by the science of the whole country. It was believed that his great scientific capacities, nurtured at West Point, and grown to maturity under the most favorable opportunities for development in his own country and during his European travels, united with his admirable judgment and enlarged administrative ability, were precisely adapted to the conduct of this national work, so important to commerce and navigation, and so interesting to science. After the lapse of five years from the date of his appointment, it is deemed reasonable to comply with Professor Bache's request to make a careful examination of his labors, and inquire if the high expectations which had been formed have been realized ; if there has been so rich a harvest of valuable results as might have been anticipated; if the best methods of observation have been uniformly adopted; and if the survey has been conducted throughout with proper economy and despatch. Your committee have made this examination to the best of their ability, and have thought it advisable to present their conclusions in as simple and condensed a form as possible. The tone of the report is necessarily laudatory, for the committee are persuaded that the minutest and most conscientious scrutiny will find every thing to approve and nothing to condemn.
" 1. The methods and instruments of observation appear to be, in all cases, the most convenient and accurate which are known; while, in some striking instances, they are such as were not known or tried in geodetical operations before they were introduced upon this survey. It is, moreover, grateful to record, that some of the most important of these improvements are of American origin. The committee beg leave to refer to some examples.
“ The apparatus for measuring the base-lines is so portable, that six and three fourths miles are measured in ten working-days, and so accurate, that the whole amount of possible error in this distance would not exceed half an inch. This beautiful apparatus, which is incomparably superior to any which has ever before been adopted, is in principle and combination the invention of the Superintendent himself. It is a compeAsating system, and is in this respect closely allied to the elegant arrangement invented and used by Mr. Borden in the survey of the State of Massachusetts, but the method of compensation is novel and original in an essential and characteristic feature.
“ The method which has been finally adopted for the measurement, astronomically, of differences of latitude is that which was invented by Captain Talcott, late of the Corps of Engineers of our army, and which had not before been used for geodetical purposes. A full description of this method has been recently published in an unusually handsome form by the Topographical Bureau of the War Department, in a memoir written by Captain T. J. Lee, which contains some exam