Of the other individuals that may be designated as Trustees by the said Peter Cooper, in the written act by which he may convey or devise the land and edi fice as aforesaid. If the said Peter Cooper shall have so elected, in and by said act, he shall also be a Trustee of said Board during his natural life. Of the Trustees so to be designated by the said Peter Cooper, the two first named by him, and designated by the numbers one and two, shall hold their offices until the first day of January of the first political year succeeding that in which such conveyance or devise shall take effect: the two next, named and designated by the numbers three and four, until the first day of January, in the second political year; the two next, named and designated by the numbers five and six, until the first day of January, in the third political year; the two next, named and designated by the numbers seven and eight, until the first day of January, in the fourth political year; the two next, named and designated by the numbers nine and ten, and all others hereinafter named, until the first day of January, in the fifth political year, succeeding that in which such conveyance or devise shall take effect. At the expiration of said respective terms of office, or in case of any vacancy in either, by resignation, death or otherwise, during the term, such office shall be thereafter respectively filled in the following manner: The vacancy in the office of the Trustee number one, by such person as the Governor of this State may appoint: The vacancy in the office of the Trustee number two, by such person as the General Society of Mechanics and Tradesmen in the city of New York may appoint, and in such mode as their by-laws may prescribe: The vacancy in the office of the Trustee number three, by such person as the Judge of the District Court of the United States for the Southern District of New York may appoint: The vacancy in the office of the Trustee number four, by such person as a majority of the Governors of the Society of the "New York Hospital" may appoint: The vacancy in the office of the Trustee number five, by such person as a majority of the Judges of the Superior Court of the city of New York may appoint: The vacancy in the office of the Trustee number six, by such person as the Corporation of the Chamber of Commerce in said city may appoint, and in such mode as their by-laws may prescribe: The vacancy in the office of the Trustee number seven, by such person as a majority of the Trustees of the Astor Library may appoint: The vacancy in the office of the Trustee number eight, by such person as a majority of the Directors of the Mercantile Library may appoint.: The vacancy in the office of the Trustee number nine, by such person as a majority of the New York Society Library may appoint: The vacancy in the office of the Trustee number ten, by such person as a majority of the Trustees of the New York Historical Society may appoint: Every Trustee, to be appointed in the manner specified in this section, shall hold his office for the five years next succeeding the appointment, ending with 31st day of December, in the fifth year; and all vacancies occurring subsequently to the first appointment, shall thereafter be filled respectively by the authorities indicated in this section: Every such act of appointment of a Trustee shall be in writing, and shall be duly filed in the office of the Clerk of the County of New York, and a duplicate thereof shall be delivered to the Board before the Trustee takes his seat. In case any of the courts, corporations, or public officers specified in the preceding section shall cease to exist, or shall omit to exercise the authority therein committed to them, the said Board of Control shall apply to the Legislature to substitute such other court, corporation or public officer as the Legislature may deem expedient. The seat of any member of the Board of Control, who may absent himself without its permission, unless prevented by sickness, for five regular monthly meetings in a single year, may be vacated by a majority of the remaining members. SEC. 5. The body corporate, hereby created, may take and hold the property above mentioned, and may lease such portions of the building as they may deem most conducive to the interests of the institution, and may receive its rents and revenues, and any other donations or endowments which may be made in aid of the objects herein above expressed, and apply the same, or the income thereof, to the enlargement or improvement of their means of instruction; and they may confer such degrees and diplomas for proficiency in science, art, philosophy or letters as may be appropriate, subject to the conditions contained in the act of conveyance or devises aforesaid. SEC. 6. The Board of Control shall appoint all professors, teachers, and other officers necessary for the conduct of the institution, and regulate their salaries, emoluments and tenure of office; and shall apply all the rents and revenues of said property, as well before as after the conveyance thereof, to the body cor. porate, hereby created, to the necessary expenses of the institution, including the preservation, renovation, and repair of the edifice, and the proper maintenance and increase of the apparatus and collections; and, while so used, neither the land, nor the building, nor its contents, nor any funds or donations in aid of its legitimate objects, shall be subject to taxation. It shall not be lawful for said Board to sell or mortgage the said land or edifice, or any parts thereof, nor to contract any pecuniary engagement exceeding the revenues of the current year. No member of said Board shall receive any pecuniary compensation for his services. SEC. 7. The Supreme Court shall possess and exercise a supervisory power over said institution, and may, at any time, on the written application either of three members of said Board of Control, or of twenty graduates of the institution, of at least five years' standing, require from its Trustees, collectively or individually, a full account of the execution of their trust. Every Trustee may freely publish, at any time in his discretion, any matter within his knowledge relating to such institution, or its management in any respect, including any discussions in the Board of Control in relation to any matter whatever; and shall be bound fully to disclose the same, whenever required, either by said Superior Court, or a committee of either branch of the Legislature. Full minutes shall be kept by the said Board of all their proceedings, and the yeas and nays shall be recorded on any vote on the request of any member. Neither the said Board of Control, nor any member thereof, shall, in any way, take into account any religious tenet or opinion of any professor or teacher, or of any candidate for any office in said institution, on any appointment to or removal from such office; nor of any student applying for admission into said institution, or competing for any of its honors or advantages: nor shall they permit any professor or teacher in said institution, to make any discrimination among its students on account of their religious tenets or opinions. If any Trustee of said institution, after due inquiry by the Supreme Court, and sufficient notice, shall be found unfaithful or culpably negligent in the discharge of his duty, the said Court shall remove him from office. SEC. 8. The Board of Control shall annually, in the month of January, present to the Common Council of said city a full report of all their receipts and disbursements during the year, and of the progress and condition of the institu tion, and shall also transmit a duplicate copy of the said report to the Legisla ture, and shall, at all times, furnish any further information in respect to their funds, revenues, and course of instruction, which the Legislature or Regents of the University may require. SEC. 9. The Legislature, at any time, may alter, amend, or repeal this act. SEC. 10. This act shall take effect immediately. REESE This sheet (page 529 to page 542,) should have been bound up with No. XI. XVI. EDUCATIONAL MISCELLANY. ON THE MOTION OF THE GYROSCOPE AS MODIFIED BY THE RETARDING BY MAJOR J. G. BARNARD, A. M. Corps of Engineers U. S. A. In a previous paper (see article in this Journal for June, 1857, to which this paper is intended to be supplementary,) I have investigated the "Self-sustaining power of the Gyroscope" in the light of analysis. From the general equations of "Rotary motion" I have deduced the laws of motion for the particular case of a solid of revolution moving about a fixed point in its axis of figure, (or the prolongation thereof). I have shown that such a body, having its axis placed in any degree of inclination to the vertical, and having a high rotary motion about that axis, will not, under the influence of gravity, sensibly fall; but that any point in the axis will describe "an undulating curve whose superior culminations are cusps lying in the same horizontal plane;" that this curve approaches more and more nearly to the cycloid, as the velocity of axial rotation is greater; that when this velocity is very great the undulations become very minute and "the axis of figure performing undulations too rapid and too minute to be perceived, moves slowly about its point of support." I have shown how the direction and velocity of this gyration are determined by the direction and velocity of axial rotation and the distance of the center of gravity of the figure from the point of support, and that the remarkable phenomenon exhibited by the gyroscope is but a particular case due to a very high velocity of axial rotation, of the general laws of motion of such a body as described, which embrace the motion of the pendulum in one extreme and that of the gyroscope in the other, and that intermediate between these two extreme cases (for moderate rotary velocities) the undulations of the axis, will be large and sensible. I have likewise shown that whenever, to the axis of a rotating solid, an angular velocity is imparted, a force which I have called "the deflecting force" acting perpendicular to the plane of motion of that axis, is developed, whose intensity is proportional to this angular velocity, and likewise to the rotary velocity of the body; and that it is this deflecting force which is the immediate sustaining agent, in the gyroscope. In the above deductions of analysis is found the full and complete solution of the "self-sustaining power of the gyroscope." To make the character of the motion indicated by analysis, No. 11.-[IV., No. 2.]—34. sensible to the eye, it is only necessary to attach to the ordinary gyroscope, in the prolongation of the axis, an arm of five or six inches in length, and having an universal joint at its extremity, and to swing the instrument as a pendulum; or, the extremity of an arm of such a length may be rested in the usual way, upon the point of the standard, when, with the centre of gyration removed at so great a distance from the point of support, the undulatory motion becomes very evident. But it cannot fail to be observed that the motion preserves this peculiar feature but for a very short period. The undulations speedily disappear; instead of periodical moments of rest (which the theory requires at each cusp) the gyratory velocity becomes continuous, and nearly uniform and horizontal; and it increases as the axis (owing to the retarding influences of friction and the resistance of the air) slowly falls. In short, the axis soon seems to move upon a descending spiral described about a vertical through the point of support. The experimental gyroscope, in its simplest form consists of two distinct masses, the rotating disk, and the mounting (or ring in which the disk turns). The point of support in the latter, though it gives free motion about a vertical axis, constrains more or less, the motion of the combined mass about any other. The rotating disk turns at the extremities of its axle, upon points or surfaces in the mass of the mounting, with friction; it is rare, too, that the point of support, of the mounting, is adjusted in the exact prolongation of the axis of the disk. Without attempting to subject to analysis causes so difficult to grasp as these, I shall first attempt to show, by general considerations, what would be the immediate influence of the retarding forces of friction and the resistance of the air upon our theoretical solid; and then point out the further effect due to the discrepancies of figure, above indicated. Leaving out of consideration the minute effect of friction at the point of support, these forces exert their influence, mainly in retarding the rotary velocity of the disk. Friction-at the extremities of the axle of the disk, and the resistance of the air, at its surface, are powerful enough to destroy entirely in a very few minutes, the high velocity originally given to it. It is in this way, mainly, that they modify the motion indicated by analysis. If the rotary velocity remained constant while the axis made one of the little cycloidal curves aba', (fig. 1) the deflecting force would be just sufficient, as I have shown (p. 556 of the article cited) to lift the axis back to its original elevation a', and to destroy, entirely, the velocity it had acquired through its fall cb. If, at a', the rotary velocity n underwent an instantaneous diminution, and remained constant through another undulation, a curve, of larger amplitude and sagitta a'b' a" would be described, and the axis would again rise to its original elevation a", and again be brought to rest. We might then, on casual considera tion of the subject, expect to see the undulations become more and more sensible as the rotary velocity decreased. The reverse, however, is the case, as I have already stated. In fact, the above supposition would require the rotary velocity n to be a discontinuous decreasing function of the time; whereas it is, really a continuous decreasing function. It is undergoing a gradual diminution between a and a'. The deflecting force, which is constantly proportional to it, is therefore insufficient to keep the axis up to the theoretical curve aba', but a lower curve ab, a, is described; and when the culmination a, is reached, it is below the original elevation a'. But the 2d of our general equations for the gyroscope (4), [afterwards put under the sim 2 = ple form (eq. ())v, 29] which is inde γ pendent of n, shows that the angular velocity of the axis will always be that due to its actual fall h below the initial elevation. On reaching the culmination a, therefore, the axis will not come to rest, but will have a horizontal velocity due to the fall a'a, and the curve will not form a cusp but an inflexion at a,. The axis will commence its second descent, therefore, with an initial horizontal velocity. It will not descend as much as it would have done had it started from rest with its diminished value of n; and, for the same reason as before, will not be able as again, to rise high as its starting point a, but to a somewhat lower point a, and with an increased horizontal velocity. These increments of horizontal velocity will constantly ensue as the culminations become lower and lower, while on the other hand, the undulations become less and less marked, as indicated by the figure. I have stated in my former paper (p. 559) that a certain initial horizontal angular velocity such as would "make its corresponding deflecting force equal to the component of gravity, g a2 1. sin 9, would cause a horizontal motion without undulation." This horizontal velocity is rapidly attained through the agencies just described: or, at least, nearly approximated to, and the axis, as observation shows, soon acquires a continuous and uniform horizontal motion. On the other hand, this sustaining power being directly pro |