ART. X. Philofophical Tranfactions, Vol. LXXV. for 1785. Part II. concluded. See Review for May laft, Art. I. Art. 16. Of the rotatory Motion of a Body, of any Form whatever, revolving, without Restraint, about any Axis paffing through its Centre of Gravity. By Mr. John Landen, F. R. S. T THE refearches of M. D'Alembert concerning the preceffion of the equinoctial points, and the nutation of the earth's axis, publifhed in 1749, feems to have given the firft occafion, both to him and M. L. Euler, of 'confidering in what manner to determine the motion of a body of any form whatever, and acted on by any forces whatever. And in order to this, Euler found it neceffary to fhew what axes a body might revolve around, without nutation, or, fo that the centrifugal force of the oppofite particles being every where equal, fhould cause no vacillation of the axis, but that it fhould remain at reft whilft the body revolved round it. The refult of his inquiries, and also of those of M. de Segner, in his Specimen Theoria Turbinum, was, that every body had at least three fuch axes paffing through its centre of gravity, and perpendicular to one another: and these axes are therefore called fixed or permanent axes of rotation. If a body, by means of fome external impulfe impreffed obliquely to one of these axes, be made to revolve round fome other axis or line paffing through the centre of gravity of the body, fuch axis will be no longer permanent, but its poles will have a motion; or, rather, according to Mr. Landen, it will have a different axis, and new poles every inftant; the determination of which is a matter of fome difficulty; and Mr. Landen's folution, given in the Paper before us, is confeffedly different from thofe of the two celebrated mathematicians above mentioned. It is manifeft, that every axis paffing through the centre of gravity, about which a globe can revolve, is a permanent axis; and Mr. Landen has fhewn, in the Philof. Tranfact. for 1777, that every fuch axis in a cylinder of uniform matter, whofe length is to its radius as 3 to 1, will alfo be permanent: and in his Mathematical Memoirs he has fhewn how to determine the ratio of the dimenfions of a cone, conoid, prifm, pyramid, &c. that shall have the like property. When the axis,' fays Mr. L. about which a body may be made to revolve, is not a permanent one, the centrifugal force of its particles will disturb its rotatory motion, fo as to caufe it to change its axis of rotation (and confequently its poles) every inftant, and endeavour to revolve about a new one: and I cannot think it will be deemed an uninterefting propofition, to determine in what track, and at what rate, the poles of fuch momentary axis will be varied in any body whatever: as, without the knowledge to be obtained from the folution of fuch problem, we cannot be certain whether the Earth, or any other planet, may not, from the inertia of its own particles, fo change its momentary axis, that the poles thereof shall approach nearer and nearer to the prefent equator; or whether the evagation of the momentary poles, arifing from that caufe, will not be limited by fome known leffer circle. Which, certainly, is an important confideration in aftronomy; efpecially now that branch of science is carried to great perfection, and the acute aftronomer endeavours to determine the motions of the heavenly bodies with the greatest exactnefs pofiible.' In the Philof. Tranfa&. for 1777, I gave a fpecimen of this theory, as far as it relates to the motion of a spheroid, and a cylinder. The improvements I have fince made in it, enable me now to extend it to the motion of any body whatever. M. L. Euler, and M. D'Alembert reprefent the angular velocity, and the momentum of rotation of the revolving body, as always variable, when the axis about which it has a tendency to revolve is a momentary one, except in a particular cafe. By my investigation it appears, that the angular velocity and the momentum of rotation will always be invariable in any revolving body, though the axis about which it endeavours to revolve be continually varied; and the tracks of the varying poles upon the furface of the body are thereby determined with great facility. It is not only obfervable, that the track which the varying poles take in the furface, are fuch that its momentum of rotation may continue the fame whilft its angular velocity continues the fame; but in a given body, there is only one fuch track which a momentary pole can purfue from a given point. If the angular velocity and the momentum of rotation of a revolving body were to vary according to the computations adverted to above, it would follow, that a body might acquire an increase of force from its own motion, without being any way affected by any other body whatever, as the fame percuffive force, applied at the fame diftance from the momentary axis, would not always destroy the rotatory motion of the body, which furely cannot poffibly be true. From the common principles of mechanics, I conclude that a revolving body, not affected by any external impulse, can no more acquire an increase in its momentum of rotation, than any other body, moving freely, can acquire an increafe in its momentum, in a given direction, without being impelled by gravity or fome other And the truth of this conclufion, which is hereafter proved by other reasoning, may be eafily inferred from the property of the lever; feeing that the joint centrifugal force of the particles of the revolving body, which is the only disturbing force, has no tendency to accelerate or retard their motion about the momentary axis, but only to alter the pofition of fuch axis, the direction in which that force acts being always in a plane wherein that axis will be found. By the theory explained in this Paper, it appears that a paral lelopipedon may always be conceived of fuch dimenfions, that being, by fome force or forces, made to revolve about an axis, paffing through its centre of gravity, with a certain angular velocity, it fhall move exactly in the fame manner as any other body will move, if made to revolve, by the fame force or forces, about an axis paffing through its centre of gravity; the quantity of matter (as well as the 7 initial initial angular velocity) being fuppofed the fame in both bodies; and due regard being had, in the application of the moving force or forces, to the correfponding planes in the bodies. Therefore, as we may from thence always affign the dimenfions of a parallelopipedon that fhall be affected exactly in the fame manner as any other body will be affected, as well with regard to the centrifugal force of the respective particles of the bodies, as to the action of equal percuffive forces, or ofcillation; it will, after fhewing how the dimenfions of fuch parallelopipedon may be computed, be only neceffary, in inveftigating the propofition under confideration, to determine the tracks and velocities of the poles of the momentary axis, about which any parallelopipedon may be made to revolve. This evagation of the pole of a revolving body, does not arife from gravity, the attraction of any other body, or any external impulse whatever; but is only the confequence of the inertia of matter, and must neceffarily enfue in every body in the univerfe revolving without restraint about any line paffing through its centre of gravity, that is not a permanent axis of rotation.' And, fuppofing the earth's rotatory motion to be disturbed only by the centrifugal force arifing from the inertia of its own particles, the track of polar evagation will be nearly circular and very small, or the pofition of the axes but very little altered.' However, after all that Mr. L. has done upon this fubject, it does not appear clear to us, that a revolving body can alter its axis of rotation, without altering its angular velocity, and confequently its momentum. If it move about the fame axis while only the poles of this axis evagate in the spherical furface, the angular velocity may continue uniform: but this does not feem to be Mr. L.'s meaning. In any one pofition, every particle in the body will have its own proper circulating velocity about the then axis; and will endeavour, as it were, to preferve it. And altering the axis will not immediately deftroy this tendency, without at the fame time altering the angular velocity. We may very poffibly be mistaken, but we think this will make it necef fary to confider fome fuch forces as thofe introduced by Meffrs. Euler and D'Alembert.-It is difficult to exprefs ourselves, fo as to be clearly understood, on this intricate fubject without schemes; we will therefore endeavour, as the matter is very important, to place it in a fomewhat different light: premifing, that Mr. Landen, as well as the other two gentlemen above-mentioned, fuppofes the centre of gravity of the revolving body to be always at reft.-Let then a body, of fuch a form as to have only three permanent axes of rotation, be at reft in free space, till it is put in motion by fome force, or impulfe, acting in a direction perpendicular to one, and in the fame plane with, and parallel to another of its permanent axes of rotation, while the centre of gravity is kept at reft, by an equal force applied to that centre in an oppofite direction, but parallel to that by which the motion is produced: we fuppofe, no one will deny, that in confequence of fuch impulfe, but after it has ceafed to act, the body will con P 4 tinue tinue to move with an uniform angular velocity about the permanent axis which is perpendicular to both thofe before mentioned. But, fuppofe at the fame time that each of the other permanent axes is impelled by a fimilar force, and the centre of gravity in like manner kept at reft by equal and oppofite ones acting against it, we think it undeniable, that, on their ceafing to act, the body will alfo move with equable angular velocities about the other two permanent axes of rotation, and confequently about all three at the fame time: for the motions, being refpectively perpendicular, cannot difturb one another. So here then, we have an inftance of a body revolving without restraint, not in the manner determined by Mr. Landen, but in all refpects agreeable to the folution of M. L. Euler.-Again, if the centre of gravity be kept at reft as before, while, instead of three impulfes against the permanent axes, the fingle, or momentary, axis is obliged, by a force acting against it, to move in the fame direction and with the fame velocity as before, we do not fee what should hinder it from continuing fuch motion, whereby the whole body muft ftill move in the fame manner as it did after the three before-mentioned impulfes. But moreover, if, while the centre of gravity is ftill kept at reft in the fame manner, the body be impelled by a force acting in any other direction obliquely to its permanent axes, it is not clear to us that the motion about those axes will not still be uniform, and confequently the compound motion of the fame nature as before. We think keeping the centre of gravity at rejt will oblige the body, on account of the equilibrium, to revolve uniformly about the permanent axes. For no one fingle force acting at the centre of gravity, though both equal, parallel, and contrary in direction to the fingle impelling one, will be able to keep that centre at reft; it muft, to do that, require the three above-mentioned forces, or others equivalent to them. And the making use of a contrary hypothefis, or fuppofing that the centre of gravity is at reft, without taking into the account how it is kept fo, may very poffibly occafion errors. Ait. 18. A Defcription of a new Syftem of Wires in the Focus of a Telescope, for onferving the comparative right Afcenfions and Ďeclinations of celestial Objects; together with a Method of investigating the fame when obferved by the Rhombus, though it happen not to be truly in an equatorial Pofition. By the Rev. Francis Wollafton, LL. B. F. R. S. This confifts in placing a fquare angularly to M. Caffini's wires at 45°, which Mr. Wollafton thinks will answer better than the rhomboid or fyftem of wires invented by Dr. Bradley. Art. 22. Sketches and Defcriptions of three fimple Inflruments for drawing Architecture and Machinery in Perspective. By Mr. James Peacock. Not to be understood without the plates, Art, Art. 24. An Account of the Measurement of a Bafe on Hounflow This long and laborious bufinefs appears to have been undertaken in confequence of a paper tranfmitted in October 1783, by Comte d'Adhemar the French Ambaffador, to Mr. Fox, then Secretary of State, being a Memoir of M. Caffini de Thury, in which he fets forth the great advantage that would accrue to aftronomy by measuring a feries of triangles from London to Dover, there to be connected with thofe already executed in France, by which combined operations the relative fituations of the two most famous Obfervatories in Europe, Greenwich and Paris, would be more accurately afcertained than they are at prefent; for, according to him, the uncertainty hitherto, all circumftances confidered, has been amazingly great indeed. In order to this, it was neceffary to have an extensive baseline measured with all poffible accuracy, as a beginning to fuch feries of triangles, and we have here a very full account of the performance of this on Hounflow-Heath, from a place called King's Arbour, at the north-weft extremity, beween Cranford Bridge and Longford, to Hampton Poor-Houfe, near Bushy Park, being upwards of five miles. This was done, first with a chain of 100 feet long, but of a new and very proper conftruction for the purpofe; then with deal reds, and 3dly with glass rods. Having first measured 274 chains, and marked it with a picket, and afterwards done the fame with deal rods, they found that the interfection on the tripod terminating the 27400 fect, only overfhot the picket anfwering to the 274th chain by two inches and nine tenths. Yet, for reafons that they have given at large, but for which we must refer to the Paper itself, they preferred the measurement with glafs rods. Were our opinion of any weight, we should prefer the use of fuch a chain, with plenty of help, as they had, to all other methods, both for fimplicity and accuracy. Wi. ART. XI. Airopaidia: Containing the Narrative of a Balloon Ex- HIS gentleman appears to have made a more fuccessful THE far |