An Investigation of the Principles of Progressive and Rotatory Motion. By the Rev. S. Vince, A. M. of Sidney College, Cambridge. Communicated by George Atwood, A. M. F. R. S.

XXXII. An Investigation of the Principles of progressive and rotatory Motion. By the Rev. S. Vince, A. M. of Sidney College, Cambridge. Communicated by George Atwood, A. M. F. R. S. Read June 15, 1780. The communication of motion from impact is well known to constitute a considerable part of that branch of natural philosophy called mechanics; and as all our enquiries therein are directed, either to assist us in those operations which add to the conveniences of life, or to explain, for the satisfaction of the mind, those changes which we daily see arise from the effects of bodies on each other, it might naturally have been expected that the attention of philosophers would have been engaged, first in the investigation of such cases as most frequently occur from the accidental action of one body on another, before they had proceeded to others less obvious. A little consideration will convince any one how seldom it happens, in the collision of two bodies, that their centers of gravity and point of contact lie in the line of direction of the striking body, yet few writers on mechanics have extended their enquiries any further than this simple case. It must however be acknowledged, that the action of bodies on each other, in directions not passing through their center of gravity, affords a subject at least curious in speculation; for my own part, I have little doubt but that it might be rendered extremely useful to the practical mechanic. I. BERNOULLI was the first who published any thing on this subject. He found the point about which a body at rest would begin to revolve when struck by another body, observing however that D. BERNOULLI had also discovered the same: he has also mentioned the curve described by that point in the progressive motion of the body, and has directed a method of enquiry by which the velocities of the bodies may be found after the stroke, which comprehends all he has done on the subject. Two years afterwards D. BERNOULLI published a paper on progressive and rotatory motion, containing nothing more than what I. BERNOULLI had before given us, and, what is a little extraordinary, says in the introduction, *de tali quidem percussione nihil adhuc, quantum scio, publici juris factum fuit ab iis, qui de motu corporum a percussione egerunt*. EULER has also investigated the velocities of the bodies after impact in a manner somewhat different, but has rendered it much more intricate by a fluxional calculus. To any one, however, who attentively considers the subject, the theory must still appear to be extremely imperfect, as, independent of principles not more self-evident than the propositions they are intended to demonstrate, which both I. and D. Bernoulli have assumed in their investigations, a great variety of other circumstances equally interesting naturally arise in an enquiry into this matter, circumstances absolutely necessary towards understanding the principles of the motion of the bodies after impact. This induced me to consider the subject with some attention, and presuming that I have not been altogether unsuccessful in my endeavours to render the theory more perfect, I determined to lay the result of my enquiries before the Royal Society. I thought it expedient, for the sake of perspicuity, to divide the whole into distinct Propositions; and as the most simple cases are best understood, I have first considered the case of the action of a body on a lever having a corpuscle at each end: and I was the more induced to treat the subject in this manner, as most of the principles can be immediately applied to any number of corpuscles, in consequence of which the general investigations are rendered more easy and satisfactory. PROP PROP. I. Let A and B be two indefinitely small bodies connected by a lever void of gravity, and suppose a force to act at any point D perpendicularly to the lever, to find the point about which the bodies will begin to revolve. From the property of the lever, the effect of the force acting at D (fig. 1.) on the body A is to the effect on B as BD : AD; hence the ratio of the spaces Am, Bn, described by the bodies A and B in the first instant of their motion, will be as $\frac{BD}{A} : \frac{AD}{B}$; join mn, and if necessary produce that line and AB to meet in c, which will manifestly be the point about which the bodies begin to revolve. Hence from similar figures BC : AC :: $\frac{AD}{B}(\propto Bn) : \frac{BD}{A}(\propto Am)$ :: $A \times AD : B \times BD$, or DC = DB : AD + DC :: $A \times AD : B \times BD$, and consequently $DC = \frac{A \times AD^2 + B \times BD^2}{B \times BD - A \times AD}$, and therefore D is the center of percussion or oscillation to the point of suspension c. Cor. 1. Hence, whatever be the magnitude of the stroke at D, the point c will remain the same. Cor. 2. If the force acts at the center of gravity G, the bodies will have no circular motion, for in this case $B \times BD - A \times AD = 0$, and therefore DC becomes infinite. Cor. 3. If the force acts at one of the bodies, the center of rotation c will coincide with the other body. Cor. 4. If the lever had been in motion before the stroke, the point c, at the instant of the stroke, would not have been disturbed. PROP. II. Let a given quantity of motion be communicated to the lever at D, to determine the velocity of the center of gravity G. The space AM, described by the body A in the first instant of motion, is as \( \frac{DB}{A} \): now CG = CD - DG = CD - AG + AD = \( \frac{A \times AD^2 + B \times BD^2}{B \times BD - A \times AD} \) - AG + AD = \( \frac{B \times BD \times BG + A \times AD \times AG}{B \times BD - A \times AD} \); also CA = CD + DA = \( \frac{A \times AD^2 + B \times BD^2}{B \times BD - A \times AD} \) + DA = \( \frac{B \times BD \times AB}{B \times BD - A \times AD} \); hence we have \( \frac{B \times BD \times AB}{B \times BD - A \times AD} \) (AC) : \( \frac{BD}{A} \) (\( \propto mA \)) :: \( \frac{B \times BD \times GB + A \times AD \times AG}{B \times BD - A \times AD} \) (CG) : \( \frac{B \times BD \times GB + A \times AD \times AG}{A \times B \times AB} \) \( \propto GW \) the velocity of the center of gravity; hence if the motion be communicated at G, the velocity becomes as \( \frac{B \times GB^2 + A \times AG^2}{A \times B \times AB} \). Let now the motion, which is supposed to be actually communicated to the rod at D, be equivalent to the motion of a body whose magnitude is G, and moving with a velocity v; then if that motion be communicated at A, the velocity of the center of gravity is well known to be Principles of progressive and rotatory Motion. \[ \frac{G \times v}{A+B} ; \text{ hence } \frac{B \times BG^2 + A \times AG^2}{A \times B \times AB} : \frac{B \times BD \times BG + A \times AD \times AG}{A \times B \times AB} : : \] \[ \frac{G \times v}{A+B} : \frac{G \times v}{A+B} \times \frac{B \times BG \times BD + A \times AD \times AG}{B \times BG^2 + A \times AG} = \text{the velocity of the center of gravity, when the same motion is actually communicated to any point D. Now } BD = BG + GD, \text{ and } AD = AG - GD; \text{ hence } B \times BG \times BD + A \times AD \times AG = B \times BG^2 + A \times AG^2 + GD \times B \times BG - A \times AG = (because B \times BG - A \times AG = 0) B \times BG^2 + A \times AG^2; \text{ consequently the velocity becomes } \frac{G \times v}{A+B}; \text{ and hence the center of gravity moves with the same velocity, wherever the motion is communicated.} PROP. III. Let a given elastic body \( P \), moving with a given velocity, be supposed to strike the lever at the point \( D \) in a direction perpendicular to it; to determine the velocity of the center of gravity \( G \) after the stroke. Suppose first the body to be non-elastic; and let \( v \) be the velocity of the center of gravity after the stroke upon that supposition, and \( V \) the velocity of the striking body: then \( CG : CD :: v : \frac{v \times CD}{CG} = \text{the velocity of the point } D \text{ after the stroke, or of the body } P; \text{ for the same reason } \frac{v \times CA}{CG} \text{ and } \frac{v \times CB}{CG} \text{ equal the velocities of } A \text{ and } B \text{ respectively. Now, because in revolving bodies, the } Mr. Vince's Investigation of the the momenta, arising from the magnitude of the bodies, their distance from the center of rotation and velocity conjointly, remain the same after the stroke as before, we shall have \( P \times V \times DC = \frac{v \times CD^2 \times P}{CG} + \frac{v \times CA^2 \times A}{CG} + \frac{v \times CB^2 \times B}{CG} \), and therefore \( v = \frac{P \times V \times DC \times CG}{P \times DC^2 + A \times AC^2 + B \times BC^2} = \frac{P \times V \times CG}{A + B \times CG + P \times DC} \); hence if \( P \) be supposed an elastic body, we shall have \( \frac{2 \times P \times V \times CG}{A + B \times CG + P \times DC} \) for the velocity of the center of gravity after the stroke, in ipso motus initio. PROP. IV. To determine the motion of the bodies after the first instant, or when they are left to move freely by themselves. The writers on mechanics, from considering the equality of motion on each side the center of gravity, when a body revolves about that point, have inferred, that if a body had a projectile as well as a circular motion communicated to it, the center of gravity would continue to move in a right line, as that point would not be disturbed by the rotatory motion: yet, as, in the case we are now considering, the bodies begin to revolve about a different center, it may be proper to examine more accurately into this matter, and to shew from what what principle it is that the motion of the center of gravity is preserved in a right line. Let a motion perpendicular to the rod be communicated to A (fig. 2.) and then by Cor. 3. Prop. I. B will not be disturbed by such an action; and A will in the first instant have a tendency to revolve about B as a center, and would actually describe the arc AH, if the body B were fixed: let the angle ABH be supposed infinitely small, and let GK be the arc, the center of gravity would have described, and draw the tangents AF, GG to the arcs AH, GG respectively. Now, if A could have moved freely, it would (because AF = AH) have described AF in the same time the arc AH was described, upon supposition that B was fixed; for the radius BA being perpendicular to the circular arc AH, the force of the lever could have no efficacy to accelerate or retard the motion of A in the arc AH, and therefore the velocity in that arc is the same as it would have been if it had moved freely in the tangent: hence HF is that space through which the centrifugal force of A would have carried that body, could it have moved freely; but as A is connected to B by means of the lever, it is manifest that the same force which would have carried A from H to F in the direction of the lever, will, when it has both bodies to move, carry it over a space which is to FH as A : A + B, or as \( bg : bh \), or as \( gk : fh \); hence that space, or the space through which the centrifugal force of \( A \) will draw the lever in the direction \( bh \), is equal to \( kg \); that is, the point \( k \), which is the center of gravity of \( A \) and \( B \), will be found at \( g \), and consequently the center of gravity has preserved its motion uniform in the right line \( gg \), inasmuch as the centrifugal force, acting perpendicularly to the direction of the center of gravity, can neither accelerate or retard its motion. In the same manner it may be proved, that the motion of the center of gravity is continued uniform in the same right line, whatever be the position of the lever. Moreover, as the centrifugal force acts in the direction of the lever, it cannot alter its angular velocity, which will therefore remain as in ipso motus initio. If now we suppose that to the force impressed upon \( A \), two other equal accelerative forces be communicated to \( A \) and \( B \) at the same time, it is evident that no alteration can arise from the actions of the bodies on each other; and the case will then be similar to the motion of the bodies, supposing a single force had been impressed at any point \( D \). The like method of reasoning may be extended to any number of bodies. The same thing may also easily be demonstrated in the following manner. The centrifugal forces of \( A \) and \( B \) (fig. 1.) are respectively \( A \times AC \) and \( B \times BC \); also the centrifugal force of the point \( G \), considering it as having both bodies to move in the direction of the rod, is \( A + B \times GC \), but from mechanics \( A \times AC + B \times BC = A + B \times GC \): hence the centrifugal forces of the bodies \( A \) and \( B \) give the center of gravity a centrifugal force equivalent to its own centrifugal force, which, as the latter would cause that center to move in the tangent \( GG \), the lever not being fixed at \( C \), it is manifest that the former will cause the center of gravity to continue its motion in the same direction. That this motion of the lever, in a direction from the center \( C \), is the only motion which is communicated to it from the effect of the bodies \( A \) and \( B \) is manifest from hence. The bodies begin to revolve freely about the point \( C \), and consequently if the point \( C \) had been fixed, the bodies would have moved on with a uniform angular velocity about \( C \); if therefore we suppose the lever not to be fixed at \( C \), as the efficacy of the centrifugal force which acts in the direction of the lever is now suffered to take place, and no new external force is impressed on either of the bodies, it is manifest, that if in the former case the bodies had no efficacy to disturb the angular velocity of the lever, they cannot have any in the latter, consequently the angular velocity, and from what Mr. Vince's Investigation of the has been before proved, the uniform motion of the center of gravity in a right line, remain unaltered, after the commencement of the motion. PROP. V. In the time the bodies make one revolution, the center of gravity will move over a space equal to the circumference of a circle whose radius is \( CG \) (fig. 1.) From the last Proposition, the angular velocity of the lever is continued uniform; hence the time of a revolution is just the same as if the point \( c \) were fixed, and the bodies were to continue to revolve about that point as a center, in which case the center of gravity \( G \), in the time of a revolution, would evidently describe the circumference of a circle whose radius is \( GC \). This therefore is the space the center of gravity describes in a right line when the bodies move freely, for from the last Proposition that center is carried uniformly forward with the same velocity. Cor. 1. Hence if the magnitude of the force acting at \( D \) vary, the velocity of the center of gravity will vary in the same ratio as the angular velocity. Cor. 2. Hence the point \( D \) may be found, where a force being applied, the bodies shall make one revolution, Principles of progressive and rotatory Motion. tion, whilst the center of gravity moves over any given space \( s \): for let \( p = \) the periphery of a circle whose radius is unity, then \( p : 1 :: s : \frac{s}{p} = \) the radius of a circle whose circumference is the space to be passed over in the time of a revolution, and which must therefore, by the Proposition, be equal to \( GC \); the point \( C \) therefore being determined, \( D \) may be easily found, for from mechanics \( CG \times DG \) is given; and from Cor. 3. Prop. I. when \( D \) comes to \( A \), \( C \) will coincide with \( B \), \[ CG \times GD = AG \times GB, \text{ and consequently } DG = \frac{AG \times GB}{CG}. \] **PROP. VI.** To determine the time of one revolution, supposing everything given as in Prop. III. The point \( D \) being given, we have from Cor. 2. to the last Proposition, \( CG = \frac{AG \times GB}{DG} \); put \( w \) equal the circumference of a circle whose radius is \( CG \), and it appears from the last Proposition, that \( w \) is the space the center of gravity passes over in the time of one revolution; hence, because from Prop. IV. the center of gravity moves uniformly, we have by Prop. III. \[ \frac{2 \times V \times P \times CG}{A + B \times CG + P \times DC} \] Mr. Vince's Investigation of the \[ \frac{1''}{w} : w \times \frac{2 \times V \times P \times CG}{A + B \times CG + P \times DC} = \text{the time of one revolution.} \] Cor. Hence the angular velocity being inversely as the time of a revolution, will vary as \( \frac{A + B \times CG + P \times DC}{V \times P \times CG \times W} \). PROP. VII. The point c, as the center of gravity moves forward, will describe the common cycloid. From the description of the common cycloid it appears, that the center of the generating circle passes over a space equal to the circumference of that circle whilst it makes one revolution. With the center G (fig. 3.) and radius GC, describe the circle CXY, and draw CR, CW perpendicular to ABC, and let the circle CXY be supposed to revolve on the line CR; then will the center G move over a space equal to the circumference of the circle CXY whilst it makes one revolution, and the point c will describe the common cycloid: but from Prop. v. the point G will move over a space equal to the circumference of a circle whose radius is GC, whilst the bodies, and consequently GC, make one revolution; and hence the point c will describe the same curve as before, that is, the common cycloid. PROP. PROP. VIII. Let a motion be communicated to the lever obliquely, to determine the point about which the bodies begin to revolve. Let \(FD\) (fig. 4.) represent the force communicating the motion at the point \(D\), which resolve into two others \(FH\), \(HD\), the former \(FH\) parallel to the lever, and the latter \(HD\) perpendicular to it. Let \(c\) be the point about which the bodies would have begun to revolve, had the force \(HD\) only acted, and which may be found by Prop. I: and suppose in this case \(mgn\) to have been the next position of the lever after the commencement of the motion, or that the bodies \(A, B,\) and center of gravity \(G\), had been carried to \(m, g\) and \(n\) respectively. But as the force \(FH\) acts at the point \(D\) at the same time in the direction of the rod, if we take \(Gq : Gg\) as \(FH : HD\), then whilst the center of gravity would have moved from \(G\) to \(g\) in consequence of the force \(HD\), it will by means of the force \(FH\) be carried in the direction of the lever from \(G\) to \(q\), and also every other point of the lever will be carried in the same direction with the same velocity; take therefore \(Ap\) and \(Br\) each equal to \(Gq\), and complete the parallelograms \(AA, GW\) and \(BB\), and the bodies \(A, B,\) and center of gravity \(G\) will, at the end of that time, be found found at \(a\), \(b\) and \(w\) respectively, and \(awb\) will be the position of the lever. Now it is evident, that \(c\) is not the point about which the bodies begin to revolve, for (considering the lever to be produced to \(c\)) that point must have moved over a space \(cc\) equal to \(gq\), when the lever is come into the position \(awb\): draw \(co\) perpendicular to \(cb\), and \(go\) perpendicular to \(gw\), and \(o\) will be the center of rotation at the commencement of the motion. For conceive \(co\) to be a lever, then the lever \(abc\) has a circular motion about \(c\), whilst that point is moving from \(c\) to \(e\), and consequently the point \(o\) is carried forward in a direction parallel to \(cc\) by this motion; but as the lever \(co\) is carried by a circular motion about \(c\) in a contrary direction, it is evident that that point of the lever \(co\) must be at rest where these two motions are equal, as they are in contrary directions. Now the velocity of \(c\) in the direction \(cc\): velocity of \(g\) about \(c\) :: \(gq\): \(gg\) :: (by sim. triang.) \(co\): \(cg\), and the velocity of the point \(g\) about \(c\): velocity of the point \(o\) about \(c\) :: \(cg\): \(co\); hence ex aequo the velocity of \(c\) in the direction of \(cc\), or of \(o\) in the direction \(op\) parallel to \(cc\), is equal to the velocity of the same point \(o\) in a contrary direction arising from its rotation about \(c\), and consequently \(o\) being a point at rest, must be the center of rotation in ipso motu initio. Also, because \(ma\) is equal and and parallel to \( nb \), \( ab \) must be equal and parallel to \( mn \), therefore the angular velocity is just the same as if the force \( fh \) had not acted. The center \( o \) of rotation at the beginning of the motion being thus determined, every thing relative to the motion of the bodies, after they are at liberty to move freely, may be determined as in the preceding Propositions. Cor. 1. Hence it appears, that whatever be the magnitude or direction of the force communicating the motion, or the point at which it acts, the center of gravity will move in a line parallel to the direction of the force, for the triangles \( fhd \), \( gqw \) being similar, \( cw \) must be parallel to \( fd \). Cor. 2. The same is manifestly true for any number of bodies; for let (fig. 5.) \( e \) be a third body, and conceive it to be connected with the other two bodies \( a \) and \( b \) in their center of gravity \( g \); then if \( fd \) represents the force acting at the point \( d \), it is evident from the last Corol. and the second Prop. that the center of gravity moves with the same velocity and in the same direction, as if the same motion had been communicated at \( g \) in a line \( rg \) parallel to \( fd \), and that the center of gravity has the same velocity communicated to it, as if the two bodies had been placed at \( g \); conceive therefore the bodies \( a \) and \( b \) to be placed at \( g \), and let the force act at \( d \), and then then from the last Corol. the center of gravity $g$, of the three bodies, will move in a line parallel to the direction of the force communicated. In the same manner it may be proved for any number of bodies. **SCHOLIUM.** The method here made use of to determine the point of rotation *in ipso motu initio*, when a single force acts at any point $D$, may be applied, when any number of forces act at different points at the same time. For let (fig. 1.) $\alpha, \beta, \gamma, \&c.$ represent the forces acting on the lever at the points $D, E, F, \&c.$ respectively; then from the same principles the effect of all the forces on $A$: the effect on $B$ :: $\frac{\alpha}{AD} + \frac{\beta}{AE} + \frac{\gamma}{AF} + \&c.$ :: $\frac{\alpha}{BD} + \frac{\beta}{BE} + \frac{\gamma}{BF} + \&c.$ which quantities put equal to $P$ and $Q$ respectively, and then $\frac{P}{A} : \frac{Q}{B} :: AM : BN :: AC : BC$, from whence it appears, that (putting $GC + GA = AC$ and $GC - GB = BC$) the distance $GC = \frac{A \times Q \times AG + B \times P \times BG}{B \times P - A \times Q}$. The same conclusion might have been deduced from this consideration; that if any number of forces act on a lever, the effect on any point of that lever is just the same as if a force, equivalent to the sum of these forces, had had acted at their common center of gravity, find therefore their common center of gravity, and conceive a force equivalent to them all to be communicated to that point, and the Problem is reduced to the case of the first Proposition. If any of the forces had acted on the opposite side of the lever, such forces must have been considered as negative. If there be any number of bodies placed on the lever, and a single force acts at D, it will appear from the same principles that the point C, about which they begin to revolve, will be the point of suspension to the center of percussion D; and the same conclusion will be obtained, if the bodies be not situated in a straight line. As a direct investigation, however, is always to be preferred to conclusions drawn from induction, it may be thought proper, before we apply any of the foregoing principles to the case of the action of bodies upon each other by impact, to shew how such a direct investigation to determine the point about which a body, having a motion communicated to it, begins to revolve, may be obtained; previous to which, however, some further considerations are necessary. PROP. IX. If a force acts upon a body in any given direction not passing through the center of gravity; to determine the plane of rotation, the direction in which the center of gravity begins to move, and its motion after. Conceive a plane $AYBZ$ (fig. 6.) to be supported upon a line $AB$ passing through its center of gravity $G$, and suppose a force to act at any point $D$ in that line, and in a direction perpendicular to the plane; then it is manifest, that such a force can give the plane no rotatory motion about $AB$. Imagine now the support to be taken away whilst the force is acting at $D$, then it is evident, that as the plane had no tendency to move about $AB$ as an axis, and the taking away of the support can give it no such motion, it will, by Cor. 2. Prop. VIII. begin its progressive motion in the direction in which the force acts; and as the force is supposed not to act at the center of gravity, it must at the same time have a rotatory motion about some axis, which, as it has no motion about $AB$, must lie somewhere in the plane, and perpendicular to $AB$; and consequently in ipso motus initio the plane of rotation must be perpendicular to the plane $AYBZ$. Let \( \text{LCM} \), perpendicular to \( \text{AB} \), be the axis about which the plane begins to revolve, and \( p, q \) be two equal particles of the plane similarly situated in respect to \( \text{AB} \), also \( qb, pa \) perpendicular to \( \text{LCM} \). Now the centrifugal force of \( p \), or its force in the direction \( ap \) is \( p \times ap \), and that of \( q \) in the direction \( bq \) is \( q \times bq \); to determine how these forces will affect the motion of the plane, we may observe in the first place, that the force \( p \times ap \), acting at \( a \) in the plane, must tend to give it a motion about an axis perpendicular to the plane; but as an equal force \( q \times qb \) acts at \( q \) to give it a motion in a contrary direction, it is evident that the two forces will destroy each other, so far as they tend to generate any motion in the plane about an axis perpendicular to it; and hence it is manifest, that if the parts of the plane \( \text{AYB}, \text{AZB} \), be similar, and similarly situated in respect to \( \text{AB} \), the plane, after the commencement of the motion, will have no tendency to revolve about an axis perpendicular to it. Also, as the centrifugal force of each particle acts in a direction parallel to \( \text{AB} \), it can give the plane no tendency to revolve about that line as an axis, and consequently the plane of rotation will be preserved as in ipso motus initio. Conceiving therefore the plane on each side the line \( \text{AB} \) to be similar, and similarly situated, suppose another plane to be fixed upon this, whose parts on each side \(AB\) are similar, and similarly situated, and the force to act as before, then it is manifest, that as each plane endeavours to preserve the same plane of ro- tation, the two planes connected will also continue to move in the same plane of rotation, for the action of one plane on another, on each side the plane of rota- tion, being equal, cannot tend to disturb the motion in that plane; and as this must be true for any number of planes thus similar and similarly situated, it is evident, that if a force should act upon a body, and each sec- tion, perpendicular to the direction of the force, should be similar on each side the plane passing through the di- rection of the force, and the center of gravity of the body, that that plane would be the plane of rotation in which the body would both begin and continue its mo- tion. It appears also from what has been proved, that if every section on each side that plane had not been si- milar, the plane of rotation would not necessarily have continued the same after the commencement of the mo- tion. Hence all bodies, formed by the revolution of any plane figure, will have the axis about which they were generated, a fixt axis of rotation; to determine, however, every other axis of a body about which it would continue to revolve, would be foreign to the subject of this paper. Supposing therefore the plane of rotation to continue the same same (for in this paper I mean to confine my enquiries to such cases) imagine all the particles of the body to be referred to that plane orthographically, which supposition not affecting the angular motion of the body, the centrifugal force of all the particles, to cause the body to revolve about an axis perpendicular to that plane, will remain unaltered. Let LMNO (fig. 7.) be that plane, and suppose a force to act at A in the direction PA lying in the same plane, which produce until it meets LN, passing through the center of gravity G, perpendicularly in D; then by Cor. 2. Prop. viii. the center of gravity G will begin its motion in a line parallel to PA, or perpendicular to LN; and consequently the center C, about which the body begins to revolve, must lie somewhere in the line LN. Now the centrifugal force of any particle p is \( p \times pc \); let fall pa perpendicular to LN, then the effect of that force at c, in a direction perpendicular to LN, will be \( p \times pa \), and in the direction CL it will be \( p \times ca \); but as the sum of all the quantities \( p \times pa = 0 \), and the sum of all the quantities \( p \times ca = \) the body multiplied into CG, it follows from the same reasoning as in Prop. III. that the point G will continue to move in a direction perpendicular to LN; and also, as the forces \( p \times ca \) act in a direction perpendicular to that in which the center of gravity moves, its motion must be continued. nued uniform. In the following Propositions, therefore, we suppose the axis of the body, after the commencement of the motion, to continue perpendicular to the plane passing through the direction of the force, and the center of gravity of the body, and that the body itself is orthographically projected upon that plane; also in the case of the action of two bodies on each other, the plane passing through the direction of the striking body and point of percussion is supposed to pass through the centers of gravity of each body; that the axis of each body after it is struck continues perpendicular to that plane, and that each body is reduced to it in the manner above described. PROP. X. To determine the point about which a body, when struck, begins to revolve. Let LMNO (fig. 7.) represent the body, G the center of gravity, and PA the direction of the force acting at A, which produce till it meets LN, passing through G, perpendicularly in the point D; draw PB perpendicular to PC, on which (produced if necessary) let fall the perpendicular DW; C being supposed the point about which the body begins to revolve, and which, from the last Proposition, is somewhere in the line LN. Because the body, in consequence sequence of the force acting at D, begins to revolve about c, and consequently if immediately after the beginning of the motion a force were applied at D equal to it, and in a contrary direction, the motion of the body would be destroyed, it is evident, that the efficacy of the body revolving about c, to turn the body about D, should any obstacle be opposed to its motion at that point, must be equal to nothing; for were it not, the body, when stopped at D, would still have a rotatory motion about that point, and consequently two equal and opposite forces applied at D would not destroy each others effects, which would be absurd. Now the force of a particle p, in the direction pw, being \( p \times pc \), its efficacy to turn the body about the point D is \( p \times pc \times dw \); but by sim. triang. \( dw : db :: ac : pc \), \( \therefore dw = \frac{db \times ac}{pc} \), and consequently the efficacy to turn the body about D \( = p \times db \times ac = p \times ca \times dc - cb = p \times ca \times dc - p \times pc^2 \); hence the sum of all the \( p \times ca \times dc \) — the sum of all the \( p \times pc^2 = 0 \), and consequently CD \( = \frac{\text{sum of all the } p \times PC^2}{\text{sum of all the } p \times Ca} \), therefore D is the center of percussion, the point of suspension being at c. Cor. From this and the preceding Proposition it appears, that every thing which was proved in Prop. v. vi. vii. holds here also in the case of the action of one body on another. PROP. PROP. XI. Let a body \( P \) (fig. 8.) moving with the velocity \( v \), strike the body \( Q \) at rest in the point \( A \), and in a direction \( AD \) passing through the center of gravity of the striking body; to determine the velocity of each body after the stroke, supposing them to be elastic. The solution of this Proposition depending on the same principles as that of Prop. III. we shall have, putting \( v \) equal the velocity of the center of gravity \( G \) after the stroke, on supposition that the bodies were non-elastic (\( DGC \) being supposed perpendicular to \( AD \), and \( C \) the point about which the body \( Q \) begins to revolve) \[ V \times P \times CD = \frac{v \times P \times CD^2}{CG} + \frac{v \times \text{sum of all } p \times C^2}{CG}, \quad \text{and consequently} \] \[ v = \frac{V \times P \times CD \times CG}{\text{sum of all } p \times pC^2 + P \times CD^2}; \quad \text{but it is well known,} \] that the sum of all the \( p \times pC^2 = CG \times CD \times Q \), and hence \[ v = \frac{V \times P \times CG}{Q \times CG + P \times DC}, \quad \text{and therefore if the bodies be sup-posed elastic, we have } \frac{2P \times V \times CG}{Q \times CG + P \times DC} \text{ for the velocity of} \] the center of gravity \( G \) after the stroke. Now to de-termine the velocity of \( P \), we have \( \frac{P \times V \times CD}{Q \times CG + P \times DC} \) equal its velocity after the stroke from single impact, and con-sequently \( v - \frac{P \times V \times CD}{Q \times CG + P \times DC} = \frac{Q \times V \times CG}{Q \times GC + P \times DC} \) is the velocity. lost by \( P \) from simple impact; hence if the bodies be elastic, \( \frac{2 \times Q \times V \times CG}{Q \times GC + P \times DC} \) will be the velocity lost by \( P \) if elastic, and consequently the velocity of \( P \) after the stroke \[ = V - \frac{2 \times Q \times V \times CG}{Q \times GC + P \times DC} = \frac{P \times DC - Q \times GC}{Q \times GC + P \times DC} \times V. \] Cor. 1. If the direction \( AD \) passes through \( G \), then \( OA \) being equal to \( OD \), we have \( \frac{2PV}{Q + P} = Q \)'s velocity, and \( \frac{P - Q}{P + Q} \times v = P \)'s velocity, which is well known from the common principles of elastic bodies. Cor. 2. If \( P \times DC = Q \times GC \), or \( P : Q :: GC : DC \), then will the body \( P \) be at rest after the stroke. Cor. 3. If \( Q \) were infinitely great, the velocity of \( P \) after the stroke would be \( = -v \) as it ought, for \( P \) would then strike against an immovable obstacle. Cor. 4. Whatever motion \( Q \) gains from the action of \( P \), it would lose, if, instead of supposing \( P \) to strike \( Q \), \( Q \) were to move in an opposite direction, and strike \( P \) at rest with the same velocity with which \( P \) struck \( Q \); in such case, therefore, the velocity of \( Q \) after the stroke would be \( V - \frac{2P \times GC \times V}{Q \times GC + P \times DC} = \frac{Q - 2P \times GC + P \times DC}{Q \times GC + P \times DC} \times v. \) Cor. 5. Hence if \( P \) be infinitely great, or \( Q \) be supposed to strike an immovable object, its velocity after the stroke will be \( = \frac{DC - 2CG}{DC} \times v \): hence when \( DC = 2GC \), the body \( Q \) will have no progressive motion after the stroke, stroke, but would in such case, if \( p \) were immediately taken away, continue to revolve about a fixed axis. It may also be observed, that when \( DC \) is greater than \( 2GC \), or the velocity of \( Q \) is positive, that, because it is impossible for \( Q \) to continue its progressive motion, it is only to be understood, that if immediately after the impact the body \( p \) were removed, the body \( Q \) would then proceed with such a velocity. Cor. 6. Suppose the bodies to be non-elastic, and let \( M \) be the magnitude of a body placed at \( D \), which, being acted upon by \( p \), shall have the same velocity generated as was before generated in the point \( D \) of the body \( Q \); then by the common rule for non-elastic bodies, the velocity of \( M \) after the stroke will be \( \frac{P \times V}{P + M} \), and hence \( \frac{P \times V}{P + M} = \frac{P \times V \times DC}{Q \times CG + P \times DC} \), consequently \( M = Q \times \frac{GC}{DC} \). Cor. 7. If a given quantity of motion were communicated to any point of the body \( Q \), the progressive motion of that body after the stroke would be the same. For suppose the magnitude of the body \( p \) to be diminished fine limite, and its velocity to be increased in the same ratio, then, because \( \frac{P \times V \times CD}{Q \times CG + P \times DC} \) (which is the velocity of \( p \) after the stroke, if the bodies be non-elastic) = (because \( p \) is infinitely small) \( \frac{P \times V \times CD}{Q \times CG} \), the velocity of \( p \) after the stroke Principles of progressive and rotatory Motion. Stroke from simple impact is finite, consequently its motion must be infinitely small, and therefore \( P \) must have communicated all its motion to \( Q \): now in this case the velocity of \( Q \left( = \frac{P \times V \times CG}{Q \times CG + P \times CD} \right) = \frac{P \times V}{Q} \), which quantity is independent of the place where the force acts; in the same manner it would appear if we had supposed the bodies elastic. PROP. XII. Supposing every thing given as in the last Proposition, except that the direction \( AD \) does not pass through the center of gravity \( g \) of the striking body; to determine the velocity of each body after the stroke. Let \( AD \) (fig. 9.) be produced to meet \( FGO \) passing through \( g \), the center of gravity of the striking body, perpendicularly in \( F \), and suppose \( O \) to be the point of the body \( P \) which is not disturbed by the action of \( P \) on \( Q \): now it appears from Cor. 6. Prop. XI. that if both bodies were non-elastic, and a body equal to \( Q \times \frac{CG}{CD} \) were placed at \( D \), the velocity of that body, from the action of \( r \), would be equal to the velocity of the point \( D \) of the body \( Q \); for the same reason, therefore, it appears, that if, instead of supposing Mr. Vince's Investigation of the P to strike Q in the direction FA, a body equal to \( P \times \frac{GO}{FO} \) were to strike Q at the same point and in the same direction (which direction is supposed to pass through the center of gravity of that body) the effect on Q would be the same; hence, if in the quantity \( \frac{V \times P \times CD}{Q \times GC + P \times DC} \), which from the last Prop. expresses the velocity of the point D after the stroke, on supposition that the bodies are non-elastic, we substitute for P a body equal to \( P \times \frac{GO}{FO} \), we shall have \( \frac{V \times P \times DC \times gO}{Q \times GC \times FO + P \times gO \times DC} \) for the velocity of the point D from the action of P; and consequently \[ \frac{2 \times V \times P \times GC \times gO}{Q \times GC \times FO + P \times gO \times DC} = \text{the velocity of the center of gravity } G \text{ of the body } Q, \text{ after the stroke, if the bodies be perfectly elastic. To determine now the velocity of the striking body, let } OF, \text{ perpendicular to } OG, \text{ be the space described by the point } O \text{ in the first instant of time after the stroke, which, as that point is not disturbed by the action of the bodies on each other, may represent the velocity of } P \text{ before the stroke, and let } FB \text{ represent the velocity of the point } F \text{ after the stroke; join } FB, \text{ and draw } GD \text{ perpendicular to } OG, \text{ and then will } GD \text{ represent the velocity of the center of gravity } G \text{ of the striking body after the stroke. Draw } FC \text{ perpendicular to } FA, \text{ and produce } GD \text{ to meet } FC \text{ in } E; \text{ now the velocity } lost by P at the point F by simple impact being equal to \[ V - \frac{V \times P \times DC \times gO}{Q \times GC \times FO + P \times gO \times DC} = \frac{V \times Q \times GC \times FO}{Q \times GC \times FO + P \times gO \times DC}, \] we shall have be the velocity lost by the point F, on supposition that the bodies are perfectly elastic (supposing of to represent the value of v) equal to \[ \frac{2 \times V \times Q \times GC \times FO}{Q \times GC \times FO + P \times gO \times DC}, \] and therefore by sim. triang. fc (FO) : cb :: fe (og) : ed = \[ \frac{2 \times V \times Q \times GC \times gO}{Q \times GC \times FO + P \times gO \times DC} = \text{the velocity lost by the center of gravity } g, \] and hence \( V - \frac{2 \times V \times Q \times GC \times gO}{Q \times GC \times FO + P \times gO \times DC} = \) \[ \frac{V \times Q \times GC \times FO + V \times P \times gO \times DC - 2 \times V \times Q \times GC \times gO}{Q \times GC \times FO + P \times gO \times DC} = \text{the velocity of } P \text{ after the stroke.} \] Now, as it appears from Prop. IX. that the progressive motion of a body, when left to move freely, continues uniform and in the same direction, it follows, that the expressions for the velocities of each body in the first instant after the stroke, both in this and the preceding Proposition, will represent the uniform progressive velocities with which the bodies will continue to move, and consequently the place of each body, at the end of any given time after impact, may easily be determined. Cor. I. If the direction FA passes through g, then fo and go becoming infinite, we shall have \[ \frac{2 \times V \times P \times GC}{Q \times GC + P \times DC} \] for the velocity of Q, and \[ \frac{V \times P \times DC - V \times Q \times GC}{Q \times GC + P \times DC} \] for the velocity Mr. Vince's Investigation of the velocity of \( p \), agreeable to what was proved in the last Proposition. Cor. 2. Hence the point about which \( p \) begins its rotatory motion may easily be found; for produce (if necessary) \( fb \) and \( of \) to meet in \( a \), and \( a \) will be the point required; and by sim. triang. \( bc' (= \frac{2 \times v \times q \times gc \times fo}{q \times gc \times fo + p \times g \times dc}) \) \( cf : : fo (= v) : oa = \frac{q \times gc \times fo + p \times g \times dc}{2 \times q \times gc} \), and hence \( ra = \frac{p \times g \times dc - q \times gc \times of}{2 \times q \times gc} \). Cor. 3. If, instead of supposing \( q \) to have been at rest, it had been moving forward in a direction parallel to that of the body \( p \), with the velocity \( v \), the motion of each body after the stroke may easily be determined: for considering \( p \) as acting upon \( q \) with the velocity \( v-v \), we shall have by this Proposition (putting \( 2m = \frac{2 \times p \times gc \times go}{q \times gc \times fo + p \times g \times dc} \)) \( v-v \times 2m = \) the velocity communicated to \( g \), therefore \( v+v-v \times 2m = \) the velocity of \( q \) after the stroke: also \( v-v \times m \times \frac{cd}{cg} = \) the velocity gained by the point \( d \) from simple impact, and consequently the velocity of that point after \( = v+v-v \times m \times \frac{cd}{cg} \), hence \( v-v-v-v \times m \times \frac{cd}{cg} = \) the velocity lost by \( p \) at the point \( f \) from simple impact, therefore \( p \)'s velocity after the stroke \( = v-v-v-v-v \times m \times \frac{cd}{cg} \times \frac{g \times o}{fo} \). In the same manner manner it might have been determined, had Q moved in an opposite direction. Cor. 4. Hence also we may easily determine the motion of each body after the stroke, supposing Q had not been moving in a direction parallel to the motion of P, by resolving Q's motion into two parts, one parallel to the motion of P, and the other perpendicular; and finding by the preceding what would be the effect of the parallel motions, and then compounding Q's motion, after the stroke from that consideration, with the motion it had in a direction perpendicular thereto before the stroke. Cor. 5. The point a of the body P will describe (when that body after the stroke has any progressive motion) the common cycloid. Cor. 6. Hence, therefore, the times of the revolutions of each body may be determined as in Prop. vi. Cor. 7. If the bodies had any rotatory motion before impact, every thing relative to the motion of the bodies after the stroke might have been determined from the same principles.