自然哲学的数学原理-18

there be given the circle VR, describedfrom the centre C with any intervalCV; and from the same centre describe any other circles ID, KE cutting the trajectory in I and K, andthe right line CV in D and E. Thendraw the right line CNIX cutting the ccircles KE, VR in N and X, and the right line CKY meeting the circleVJi in Y. Let the points I and K be indefinitely near ; and let the bodygo on from V through I and K to k ; and let the point A be the placefrom whence anothe body is to fall, so as in the place D to acquire a velocity equal to the velocity of the first body in I. And things remainingas in Prop. XXXIX, the lineola IK, described in the least given timeTHE MATHEMATICAL PRINCIPLES [BOOK 1trill be as the velocity, and therefore as the right line whose square isequal to the area ABFD, and the triangle ICK proportional to the timewill be given, and therefore KN will be reciprocally as the altitude 1C :that is (if there be given any quantity Q, and the altitude 1C be calledA), as -T-. This quantity call Z, and suppose the magnitude of Q, tooe such that in some case v/ABFD may be to Z as IK to KN, and thenin all cases V ABFD will be to Z as IK to KN, and ABFD to ZZ asIK2 to KN2, and by division ABFD ZZ to ZZ as IN2 to KN2, and thereforeV ABFD ZZ to Z, or as IN to KN; and therefore A x KNQ. x IN。vill be equal to . Therefore since YX X XC is to A X KNZZQ. X IN x CX2as CX2, to AA, the rectangle XY X XC will be equal to-AAv/ABFD ZZ.Therefore in the perpendicular DF let there be taken continually I)//, IVQ ax ex2equal to , =. respectively, and2 v/ ABFD ZZ 2AA V ABFD ZZlet the curve lines ab, ac, the foci of the points b and c, be described : andfrom the point V let the perpendicular Va be erected to the line AC, cutting off the curvilinear areas VD&a, VDra, and let the ordi nates Es:?E#, be erected also. Then because the rectangle D& X IN or DbzR isequal to half the rectangle A X KN, or to the triangle ICK ; and therectangle DC X IN or Dc.rE is equal to half the rectangle YX X XC, orto the triangle XCY; that is, because the nascent particles I)6d3, ICKof the areas VD/>#, VIC are always equal; and the nascent particlesDc^-E, XCY of the areas VDca, VCX are always equal : therefore thegenerated area VD6a will be equal to the generated area VIC, and therefore proportional to the time; and the generated area VDco- is equal tothe generated sector VCX. If, therefore, any time be given during whichthe body has been moving from V, there will be also given the area proportional to it VD/>; and thence will be given the altitude of the bodyCD or CI ; and the area VDca, and the sector VCX equal there o, togetherwith its angle VCL But the angb VCI, and the altitude CI being given,there is also given the place I, in which the body will be found at the endof that time. Q.E.I.COR. 1. Hence the greatest and least altitudes of the bodies, that is, theapsides of the trajectories, may be found very readily. For the apsidesare those points in which a right line 1C drawn through the centre fallsperpendicularly upon the trajectory VTK; which comes to pass when theright lines IK and NK become equal; that is, when the area ABFD igC nl to ZZ.SEC. VI1LJ OF NATURAL PHILOSOPHY. 171COR. 2. So also the angle KIN, in which the trajectory at any placecuts the line 1C. may be readily found by the given altitude 1C of thebody : to wit, by making the sine of that angle to radius as KN to IKthat is, as Z to the square root of the area ABFD.COR. 3. If to the centre C, and theprincipal vertex V, there be described aconic section VRS ; and from any pointthereof, as R, there be drawn the tangent TRT meeting the axis CV indefinitely produced in the point T ; and then joining CCR there be drawn the right line CP, Qequalto the abscissa CT, making an angle VCP proportional to the sectorVCR ; and if a centripetal force, reciprocally proportional to the cubesof the distances of the places from the centre, tends to the centre C ; andfrom the place V there sets out a body with a just velocity in the direction of a line perpendicular to the right, line CV; that body will proceedin a trajectory VPQ,, which the point P will always touch ; and thereforeif the conic section VI。 S be an hyberbola, the body will descend to the centre; but if it be an ellipsis, it will ascend perpetually, and go farther andfarther off in infinilum. And, on the contrary, if a body endued with anyvelocity goes off from the place V, and according as it begins either to descend obliquely to the centre, or ascends obliquely from it, the figure VRSbe either an hyperbola or an ellipsis, the trajectory may be found by increasing or diminishing the angle VCP in a given ratio. And the centripetalforce becoming centrifugal, the body will ascend obliquely in the trajectoryVPQ, which is found by taking the angle VCP proportional to the ellipticsector VRC, and the length CP equal to the length CT, as before. All thesethings follow from the foregoing Proposition, by the quadrature of a certainourve, the invention of which, as being easy enough, for brevity s sake I omit.PROPOSITION XLII. PROBLEM XXIX.The law of centripetal force being given, it is required to find the motionof a body setting out from a given place, with a given velocity, in thedirection of a given right line.Suppose the same things as inIhe three preceding propositions;and let the body go off fromthe place I in the direction of thelittle line, IK, with the same velocity as another body, by fallingwith an uniform centripetal forcefrom the place P, may acquire inI); and let this uniform force beto the force with which the body1.72 THE MATHEMATICAL PRINCIPLES [BOOK 1.is at first urged in I, as DR to DF. Let the body go on towards k; andabout the centre C, with the interval Ck, describe the circle ke, meetingthe right line PD in e, and let there be erected the lines eg, ev, ew, ordinatelyapplied to the curves BF*, abv} acw. From the given rectanglePDRQ, and the given law of centripetal force, by which the first body isacted on, the curve line BF* is also given, by the construction of Prop.XXVII, and its Cor. 1. Then from the given angle CIK is given theproportion of the nascent lines 1K; KN ; and thence, by the constructionof Prob. XXVIII, there is given the quantity Q,, with the curve lines abv,acw ; and therefore, at the end of any time Dbve, there is given boththe altitude of the body Ce or Ck, and the area Dcwe, with the sectorequal to it XCy, the angle 1CA:, and the place k} in which the body willthen be found. Q.E.I.We suppose in these Propositions the centripetal force to vary in itsrecess from the centre according to some law, which any one may imagineat pleasure; but at equal distances from the centre to be everywhere theBame.I have hitherto considered the motions of bodies in immovable orbits.It remains now to add something concerning their motions in orbits whichrevolve round the centres of force.SECTION IX.Of the motion of bodies in moveable orbits ; and of the motion of theapsides.PROPOSITION XLIII. PROBLEM XXX.Ft is required to make a body move in a trajectory that revolves aboutthe centre offorce in the same manner as another body in the sametrajectory at rest.In. the orbit VPK, given by position, let the bodyP revolve, proceeding from V towards K. Fromthe centre C let there be continually drawn Cp, equalto CP, making the angle VC/? proportional to theangle VCP ; and the area which the line Cp describeswill be to the area VCP, which the line CP describesat the same time, ns the velocity of the describingline Cp to the velocity of the describing line CP ;that is, as the angle VC/? to the angle VCP, therefore in a given ratio,and therefore proportional to the time. Since, then, the area described bythe line Cp in an immovable plane is proportional to the time, it is manifestthat a body, being acted upon by a just quantity of centripetal force maySEC. L。.] OF NATURAL PHILOSOPHY. 173revolve with the point p in the curve line which the same point p, by themethod just now explained, may be made to describe an immovable plane.Make the angle VC^ equal to the angle PC/?, and the line Cu equal toCV, and the figure uCp equal to the figure VCP; and the body being always in the point p} will move in the perimeter of the revolving figurenCp, and will describe its (revolving) arc up in the same time the* theother body P describes the similar and equal arc VP in the quiescov.t figure YPK. Find, then, by Cor. 5, Prop. VI., the centripetal force by whichthe body may be made to revolve in the curve line which the pom* p describes in an immovable plane, and the Problem will be solved. O/E.K.PROPOSITION XLIV. THEOREM XIV.The difference of the forces, by which two bodies may be madi, to KMVGequally, one in a quiescent, the other in the same orbit revolving, i 1 ina triplicate ratio of their common altitudes inversely.Let the parts of the quiescent orbit VP, PK be similar and equal tothe parts of the revolving orbit up,pk ; and let the distance of the pointsP and K be supposed of the utmostsmallness Let fall a perpendicularkr from the point k to the right linepC, and produce it to m, so that mrmay be to kr as the angle VC/? to the /2。-angle VCP. Because the altitudesof the bodies PC and pV, KG andkC} are always equal, it is manifestthat the increments or decrements ofthe lines PC and pC are alwaysequal ; and therefore if each of theseveral motions of the bodies in the places P and p be resolved into two(by Cor. 2 of the Laws of Motion), one of which is directed towards thecentre, or according to the lines PC, pC, and the other, transverse to theformer, hath a direction perpendicular to the lines PC and pC ; the motions towards the centre will be equal, and the transverse motion of thebody p will be to the transverse motion of the body P as the angular motion of the line pC to the angular motion of the line PC ; that is, as theangle VC/? to the angle VCP. Therefore, at the same time that the bodvP, by both its motions, comes to the point K, the body p, having an equalmotion towards the centre, will be equally moved from p towards C ;aridtherefore that time being expired, it will be found somewhere in theline mkr, which, passing through the point k, is perpendicular to the linepC ; and by its transverse motion will acquire a distance from the line174 THE MATHEMATICAL PRINCIPLES [BOOK J.C, that will be to the distance which the other body P acquires from theline PC as the transverse motion of the body p to the transverse motion ofthe other body P. Therefore since kr is equal to the distance which thebody P acquires from the line PC, and mr is to kr as the angle VC/? tothe angle VCP, that is, as the transverse motion of the body p to thetransverse motion of the body P, it is manifest that the body p, at the expiration of that time, will be found in the place m. These things will beso, if the bodies jo and P are equally moved in the directions of the linespC and PC, and are therefore urged with equal forces in those directions.I: ut if we take an angle pCn that is to the angle pCk as the angle VGj0to the angle VCP, and nC be equal to kG, in that case the body p at theexpiration of the time will really be in n ; and is therefore urged with agreater force than the body P, if the angle nCp is greater than the anglekCp, that is, if the orbit npk, move either in cmiseqnentia, or in antecedenticijwith a celerity greater than the double of that with which the lineCP moves in conseqnentia ; and with a less force if the orbit moves slowerin antecedent-la. And ihj difference of the forces will be as the intervalmn of the places through which the body would be carried by the action ofthat difference in that given space of time. About the centre C with theinterval Cn or Ck suppose a circle described cutting the lines mr, tun produced in s and , and the rectangle mn X nit will be equal to the rectan-*//? n ^* */?? ^"le mk X ins, and therefore mn will be equal to . But sincemtthe triangles pCk, pCn, in a given time, are of a given magnitude, kr andmr. a id their difference mk, and their sum ms, are reciprocally as the altitude pC, and therefore the rectangle mk X ms is reciprocally as thesquare of the altitude pC. But, moreover, mt is directly as |//z/, that is, asthe altitude pC. These are the first ratios of the nascent lines ; and hencer - that is, the nascent lineola mn. and the difference of the forcesmtproportional thereto, are reciprocally as the cube of the altitude pC.Q.E.D.COR. I. Hence the difference of the forces in the places P and p, or K and/.*, is to the force with which a body may revolve with a circular motionfrom R to K, in the same time that the body P in an immovable orb describes the arc PK, as the nascent line m,n to the versed sine of the nascentmk X ms rk2arc RK, that is, as to ^g, or as mk X ms to the square ofrk ; that is. if we take given quantities F and G in the same ratio to oneanother as the angle VCP bears to the angle VQ?, as GG FF to FF.And, therefore, if from the centre C, with any distance CP or Cp, there bedescribed a circular sector equal to the whole area VPC, which the bodyOEC. IX.l OF NATURAL PHILOSOPHY. 175revolving in an immovable orbit has by a radius drawn to the centre debribedin any certain time, the difference of the forces, with which thebody P revolves in an immovable orbit, and the body p in a movable orbit, will be to the centripetal force, with which another body by a radiusdrawn to the centre can uniformly describe that sector in the same timeas the area VPC is described, as GG FF to FF. For that sector andthe area pCk are to one another as the times in which they are described.COR. 2. If the orbit YPK be anellipsis, having its focus C, and itshighest apsis Y, and we suppose thethe ellipsis upk similar and equal to ..it, so that pC may be always equal /to PC, and the angle YC/? be to the ;angle YCP in the given ratio of G 。to F ; and for the altitude PC or pC 。we put A, and 2R for the latus rec- /t。turn of the ellipsis, the force with *which a body may be made to revolve in a movable ellipsis will be asFF RGG RFF- + --rg , and vice versa./Y A. A.Let the force with which a body mayrevolve in an immovable ellipsis be expressed by the quantity , and the-. 7force in V will beFFBut the force with which a body may revolve ina circle at the distance CY, with the same velocity as a body revolving inan ellipsis has in Y, is to the force with which a body revolving in an ellipsis is acted upon in the apsis Y, as half the latus rectum of the ellipsis to theRFFsemi-diameter CY of the circle, and therefore is as , =- : and tluRFFwhich is to this, as GG FF to FF, is as -~py^~~~: and this force(by Cor. 1 cf this Prop.) is the difference of the forces in Y, with which thebody P revolves in the immovable ellipsis YPK, and the body p in themovable ellipsis upk. Therefore since by this Prop, that difference atany other altitude A is to itself at the altitude CY as -r-, to ^TF- the same AJ CYJR C^ ("* R P^ T*1difference in every altitude A will be as -3:. Therefore to theFFforce -T-:, by which the body may revolve in an immovable ellipsis VPK176 THE MATHEMATICAL PRINCIPLES [BOOK I.idd the excess -:-=A , and the sum will be the whole force A-rA-r -。-RGG RFF,.-5 by which a body may revolve in the same time in the mot-A.able ellipsis upk.COR. 3. In the same manner it will be found, that, if the immovable orbit VPK be an ellipsis having its centre in the centre of the forces C} andthere be supposed a movable ellipsis -upk, similar, equal, and concentricalto it; and 2R be the principal latus rectum of that ellipsis, and 2T thelatus transversum, or greater axis ; and the angle VCjo be continually to theangle TCP as G to F ; the forces with which bodies may revolve in the im-FFA FFAmovable and movable ellipsis, in equal times, will be as ^ and -p~RGG RFF+ A.-3 respectively.COR. 4. And universally, if the greatest altitude CV of the body be calledT, and the radius of the curvature which the orbit VPK has in Y, that is,the radius of a circle equally curve, be called R, and the centripetal forcewith which a body may revolve in any immovable trajectory VPK at the placeVFFV be called -f-=Trri , and in other places P be indefinitely styled X ; and thealtitude CP be called A, and G be taken to F in the given ratio of theangle VCjD to the angle VCP ; the centripetal force with which the samebody will perform the same motions in the same time, in the same trajectoryupk revolving with a circular motion, will be as the sum of the forces X -f-VRGG VRFF~A*COR. 5. Therefore the motion of a body in an immovable orbit beinggiven, its angular motion round the centre of the forces may be increasedor diminished in a given ratio; and thence new immovable orbits may befound in which bodies may revolve with new centripetal forces.COR. 6. Therefore if there be erected the line VP of an indeterminate-p length, perpendicular to the line CV given by position, and CP be drawn, and Cp equal to it, making the angle VC/? having a given ratio to the angle VCP, the force with which a body may revolvein the curve line Vjo/r, which the point p is continually describing, will be reciprocally as the cubeCof the altitude Cp. For the body P, by its vis inertia alone, no other force impelling it, will proceed uniformly in the rightline VP. Add, then, a force tending to the centre C reciprocally as thecube of the altitude CP or Cp, and (by what was just demonstrated) theSEC. IX..J OF NATURAL PHILOSOPHY. 177body will deflect from the rectilinear motion into the curve line Ypk. Butthis curve ~Vpk is the same with the curve VPQ found in Cor. 3, PropXLI, in which, I said, hodies attracted with such forces would ascendobliquely.PROPOSITION XLV. PROBLEM XXXLTo find the motion of the apsides in orbits approaching very near tocircles.This problem is solved arithmetically by reducing the orbit, which abody revolving in a movable ellipsis (as in Cor. 2 and 3 of the aboveProp.) describes in an immovable plane, to the figure of the orbit whoseapsides are required ; and then seeking the apsides of the orbit which thatbody describes in an immovable plane. But orbits acquire the same figure,if the centripetal forces with which they are described, compared betweenthemselves, are made proportional at equal altitudes. Let the point V bethe highest apsis, and write T for the greatest altitude CV, A for any otheraltitude CP or C/?, and X for the difference of the altitudes CV CP :and the force writh which a body moves in an ellipsis revolving about itsp-p T? C* f^T? F*Ffocus C (as in Cor. 2), and which in Cor. 2 was as -r-r -。 -.-3 ,FFA + RGG RFF ,that is as, -^ , by substituting T X for A, will be- ARGG RFF + TFF FFXcome as-p . In like manner any other centripetal force is to be reduced to a fraction whose denominator is A3, andthe numerators are to be made analogous by collating together the homologous terms. This will be made plainer by Examples.EXAMPLE 1. Let us suppose the centripetal force to be uniform,A3and therefore as 3 or, writing T X for A in the numerator, asT3 3TTX + 3TXX X3=-. Ihen collating together the correspon- A3dent terms of the numerators, that is, those that consist of given quantities,with those of given quantities, and those of quantities not given with thoseof quantities not given, it will become RGG RFF -f- TFF to T3 asFFX to 3TTX -f 3TXX X3, or as FF to 3TT + 3TX XX.Now since the orbit is supposed extremely near to a circle, let it coincidewith a circle; and because in that case R and T become equal, and X isinfinitely diminished, the last ratios will be, as RGG to T2, so FF to3TT, or as GG to TT, so FF to 3TT; and again, as GG to FF, so TTto 3TT, that is, as 1 to 3 ; and therefore G is to F, that is, the angle VC/?to the angle VCP, as 1 to v/3. Therefore since the body, in an immovable178 THE MATHEMATICAL PRINCIPLES [BOOK Iellipsis, in descending from the upper to the lower apsis, describes an angle,if I may so speak, of ISO deg., the other body in a movable ellipsis, and therefore in the immovable orbit we are treating of, will in its descent from180the upper to the lower apsis, describe an angle VCjt? of ^ deg. And this。/ocomes to pass by reason of the likeness of this orbit which a body actedupon by an uniform centripetal force describes, and of that orbit which a