155. It deserves remark here, that as the momentary change of the simple velocity by any force ƒ depends only on the time of its action, it being=fi (148.) Cor. 1. so the change on the square of the velocity depends on the space, it being=fs. It is the same, whatever is the velocity thus changed, or even though the body. be at rest when the force begins to act on it. Thus, in Indeed all that we know of force is that it is something every second of the falling of a heavy body, the ve which is always proportional to Cor. 2. Uniformly accelerated or retarded motion is the indication of a constant or invariable accelerating force. For, in this case, the areas abfe, acge, &c. increase at the same rate with the times a b, ac, &c. and therefore the ordinates a e, bf, cg, &c. must all be equal; therefore the forces represented by them are the same, or the accelerating force does not change its intensity, or, it is constant. If, therefore, the circumstances mentioned in articles 37 and 38, are observed in any motion, the force is constant. And if the force is known to be constant, those propositions are true respecting the motions. locity is augmented 32 feet per second, and, in every foot of the fall, the square of the velocity increases by 64. 156. The whole area AE ea, expressed by ffs, expresses the whole change made on the square of the velocity which the body had in A, whatever this velocity may have been. We may therefore suppose the body to have been at rest in A. The area then measures the square of the velocity which the body has acquired in the point E of its path. It is plain that the change on v2 is quite independent on the time of action, and therefore a body, in passing through the space AE with any initial velocity whatever, sustains the same. change of the square of that velocity, if under the influence of the same force. Cor. 3. No finite change of velocity is generated in an instant by an accelerating or retarding force. For 157. This proposition is the same with the 39th of the increment or decrement of velocity is always exthe First Book of Newton's Principia, and is perhaps pressed by an area, or by a product ft, one side or fac-mics, in the solution of practical questions. It is to be the most generally useful, of all the theorems in Dynator of which is a portion of time. As no finite space found, without demonstration, in his earliest writings, can be described in an instant, and the moveable must the Optical Lectures, which he delivered in 1669 and pass in succession through every point of the path, so it following years. must acquire all the intermediate degrees of velocity. It must be continually accelerated or retarded. Cor. 4. The change of velocity produced in a body in any time, by a force varying in any manner, is the proper measure of the accumulated or whole action of the force during this time. For, since the momentary change of velocity is expressed by ft, the aggregate of all these momentary changes, that is, the whole change of velocity, must be expressed by the sum of all the quantities ft. This is equivalent to the area of the figure employed in art. 148, and may be expressed by ffi. 154. If the abscissa AE (fig. 8.) of the line a ce be the path along which a body is urged by the action of a force, varying in any manner, and if the ordinates Å a, Bb, Cc, &c. be proportional to the intensities of the force in the different points of the path, the intercepted areas will be proportional to the changes made on the square of the velocity during the motion along the corresponding portions of the path. For, by art. 49. the areas are in this proportion when the ordinates are as the accelerations. But the accelerations are the measures of, and are therefore proportional to, the accelerating forces. Therefore the proposition is manifest. 158. One important use may be made of it at present. It gives a complete solution of all the facts which were observed by Dr Hooke, and adduced by Leibnitz force of moving bodies. All of them are of precisely with such pertinacity in support of his measure of the the same nature with the one mentioned in art. 157, or with the fact, "that a ball projected directly upwards "with a double velocity, will rise to a quadruple height, "and that a body, moving twice as fast, will penetrate "four times as far into a uniformly tenacious mass." The uniform force of gravity, or the uniform tenacity of the penetrated body, makes a uniform opposition to the motion, and may therefore be considered as a uniform retarding force. It will therefore be represented, in fig. 8. by an ordinate always of the same length, and the areas which measure the square of the velocity lost will be portions of a rectangle AEta. If therefore AE be the penetration necessary for extinguishing the velocity 2, the space AB, necessary for extinguishing the velocity I, must be of AE, because the square of I is of the square of 2. 159. What particularly deserves remark here, is, that this proposition is true, only on the supposition that forces are proportional to the velocities generated by them in equal times. For the demonstration of this proposition proceeds entirely on the previously established measure of: Of Moving Forces. Fig. 33. proposition. 165. Those may be called similar points of space, and similar instants of time, which divide given portions of space or time in the same ratio. Thus, the beginning of the 5th inch, and of the 2d foot, are similar points of a foot, and of a yard. The beginning of the 21st minute, and of the 9th hour, are similar instants of an liour, and of a day. Forces may be said to act similarly when, in similar instants of time, or similar points of the path, their intensities are in a constant ratio. 161. Lemma. If two bodies be similarly accelerated during given times a c and hk (fig. 33.), they are also similarly accelerated along their respective paths AC and HK. Let a, b, c, be instants of the time ac, similar to the instants h, i, k of the time hk. Then by the similar accelerations, we have the force ae: hl=bf: im. This being the case throughout, the area af is to the area hm as the area ag to the area hn. These areas are as the velocities in the two motions (48.) Therefore the velocities in similar instants are in a constant ratio, that is, the velocity in the instant b is to that in the instant i, as the velocity in the instant c to that in the instant k. The figures may now be taken to represent the times of the motion by their abscissæ, and the velocities by their ordinates, as in art. 28. The spaces described are now represented by the areas. These being in a constant ratio, as already shewn, we have A, B, C, and H, I, K, similar points of the paths. And therefore, in similar instants of time, the bodies are in similar points of the paths. But in these instants, they are similarly accelerated, that is, the accelerations and the forces are in a constant ratio. They are therefore in a constant ratio in similar points of the paths, and the bodies are similarly accelerated along their respective paths (155.). For the moveables are similarly urged during the of Moving times of their motion (converse of 156.). Therefore Forces. vft, and vf; but (158.) fs. There. fore fsft and sƒt. COROLLARY. ÷, and ƒ÷. That is, the squares of the times are as the spaces, directly, and as the forces, inversely; and the forces are as the spaces, directly, and as the squares of the times, inversely. 165. The quantity of motion in a body is the sum of the motions of all its particles. Therefore, if all are moving in one direction, and with one velocity v, and if m be the number of particles, or quantity of matter, mv will express the quantity of motion 9, or m v. q 166. In like manner, we may conceive the accelerating forces f, which have produced this velocity v in each particle, as added into one sum, or as combined on one particle. They will thus compose a force, which, for distinction's sake, it is convenient to mark by a particular name. We shall call it the MOTIVE FORCE, and express it by the symbol p. It will then be considered as the aggregate of the number m of equal accelerating forces f, each of which produces the velocity on one particle. It will produce the velocity m v, and the same quantity of motion q. 167. Let there be another body, consisting of n particles, moving with one velocity u. Let the moving force be represented by . It is measured in like manner by nu. Therefore we have, p: ≈≈mv: nu, and p * v: u= : m n ; that is, 162. If two particles of matter are similarly urged by accelerating or retarding forces during given times, And f being = the whole changes of velocity are as the forces and times jointly; or vft. For the abscissæ ac and hk will represent the times, and the ordinates ae and h will represent the forces, and then the areas will represent the changes of velocity, by art. 47. And these areas are as a cxa e to hkx hl. 163. If two particles of matter are similarly impelled or opposed through given spaces, the changes in the squares of velocity are as the forces and spaces jointly; or fs. This follows, by similar reasoning, from art. 49. It is evident that this proposition applies directly to the argument so confidently urged for the propriety of the Leibnitzian measure of forces, namely, that four springs of equal strength, and bent to the same degree, generate, or extinguish only a double velocity. 164. If two particles of matter are similarly impelled through given spaces, the spaces are as the forces and the squares of the times jointly. m 168. In the application of the theorems concerning accelerating or retarding forces, it is necessary to attend carefully to the distinction between an accelerative and a motive force. The caution necessary here has been ge nerally overlooked by the writers of Elements, and this has given occasion to very inadequate and erroneous notions of the action of accelerating powers. Thus, if a leaden ball hangs by a thread, which passes over a pulley, and is attached to an equal ball, moveable along a hori zontal plane, without the smallest obstruction, it is known that, in one second, it will descend 8 feet, dragging the other 8 feet along the plane, with a uniformly accelera ted motion, and will generate in it the velocity 16 feet per second. Let the thread be attached to three such balls. We know that it will descend 4 feet in a second, and generate the velocity 8 feet per second. Most readers are disposed to think that it should generate no greater velocity than 5 feet per second, or of 16, because it is applied to three times as much matter (162.). The error rate in the body by uniformly impelling it along the Of Moving fourth part of the deflective chord of the equicurve Forces. circle. Of Moving error lies in considering the motive force as the same in Forces. both cases, and in not attending to the quantity of matter to which it is applied. Neither of these conjectures is right. The motive force changes as the motion accelerates, and in the first case it moves two balls, and in the second it moves four. The motive force decreases similarly in both motions. When these things are considered, we learn by articles 202 and 207, that the mo tions will be precisely what we observe. Of Deflecting Forces, in General. 169. It was observed, in art. 71, that a curvilineal Therefore no finite change of direction can be pro- 170. The most general and useful proposition on this subject is the following, founded on art. 75. The forces by which bodies are deflected from the tangents in the different points of their curvilineal paths are proportional to the squares of the velocities in those points directly, and inversely to the deflective chords of the equicurve circles in the same points. We may still express the proposition by the same symbol, f= where ƒ means the intensity of the deflecting force. 171. We may also retain the meaning of the proposition expressed in article 76, where it is shewn that the actual linear deflection from the tangent is the third proportional to the deflective chord and the arch described in a very small moment. For it was demonstrated in that article (see fig. 18.) that BZ: BC BC: BO. We see also that B b, the donble of BO, is the measure of the velocity, generated by the uniform action of the deflecting force, during the motion in the arch BC of the curve. 172. The art. 77. also furnishes a proposition of frequent and important use, viz. The velocity in any point of a curvilinear motion is that which the deflecting force in that point would gene REMARK. 137. The propositions now given proceed on the sup position that, when the points A and C of fig. 18. after continually approaching to B, at last coalesce with it, the last circle which is described through these three points has the same curvature which the path has in B. It is proper to render this mode of solving these questions more plain and palpable. If ABCD (fig. 34.) be a material curve or mould, Fig. 34. and a thread be made fast to it at D, this thread may be lapped on the convexity of this curve, till its extremity meets it in A. Let the thread be now unlapped or EVOLVED from the curve, keeping it always tight. It is plain that its extremity A will describe another curve line A b c. All curves, in which the curvature is neither infinitely great nor infinitely small, may thus described by a thread evolved from a proper curve. The properties of the curve A b c being known, Mr Huyghens (the author of this way of generating curve lines) has shewn how to construct the evolved curve ABC which will produce it. From this genesis of curves we may infer, 1st, that the detached portion of the thread is always a tangent to the curve ABC; 2dly, that when this is in any situation Bb, it is perpendicular to the tangent of the curve Abc in the point b, and that it is, at the same time, describing an element of that curve, and an element of a circle abx, whose momentary centre is B, and which has Bb for its radius. 3dly, That the part 6A of the curve, being described with radii growing continually shorter, is more incurvated than the circle ba, which has B b for its constant radius. For similar reasons the arch bc of the curve A bc is less incurvated than the circle a bx. 4thly, That the circle a b x has the same curvature that the curve has in b, or is an equicurve circle. Bb is the radius, and B the centre of curvature in the point b. ABC is the CURVA EVOLUTA or the EVOLUTE. A b c is sometimes called the INVOLUTE of ABC, and sometimes its EVOLUTRIX. 174. By this way of describing curve lines, we see clearly that a body, when passing through the point b of the curve A b c may be considered as in the same state, in that instant, as in passing through the same point b of the circle abx; and the ultimate ratio of the deflections in both is that of equality, and they may be used indiscriminately. The chief difficulty in the application of the preceding theorems to the curvilineal motions which are observed in the spontaneous phenomena of nature, is in ascertaining the direction of the deflection in every point of a curvilineal motion. Fortunately, however, the most important cases, namely those motions, where the deflecting forces are always directed to a fixed point, afford a very accurate method. Such forces are called by the general name of Of Moving the circles, and are proportional to the square of the veForces. locities, directly, and to their distances from the centre, inversely. For, since their motion in the circumference is uniform, the areas formed by lines drawn from the centre are as the times, and therefore (72.) the deflections, and the deflecting forces (164.) are directed to the cen Of Moving curvilineal path, are inversely as the cubes of its distances tre. Therefore, the deflective chord is, in this case, a "÷, therefore da2÷d × 1,÷, therefore the diameter of the circle, or twice the distance of the body from the centre. Therefore, if we call the distance from the centre d, we have ƒ/7. v2 d 176. These forces are also as the distances, directly, and as the square of the time of a revolution, inversely, For the time of a revolution (which may be called the PERIODIC TIME) is as the circumference, and therefore as the distance, directly, and as the velocity, inversely. Therefore t, and, and v÷ d d d 177. These forces are also as the distances, and the square of the angular velocity, jointly. For, in every uniform circular motion, the angular velocity is inversely as the periodic time. Therefore, calling the angular velocity, a, a, and da, —⇒ and therefore fd a'. I d 178. The periodic time is to the time of falling along half the radius by the uniform action of the centripetal force in the circumference, as the circumference of a circle is to the radius. For, in the time of falling through half the radius, the body would describe an arch equal to the radius (37.-6.) because the velocity acquired by this fall is equal to the velocity in the circumference (167.). The periodic time is to the time of describing that arch as the circumference to the arch, that is, as the circumference is to the radius. 179. When a body describes a curve which is all in one plane, and a point is so situated in that plane, that a line drawn from it to the body describes round that point areas proportional to the times, the deflecting force is always directed to that point (72.) 180. Conversely. If a body is deflected by a force always directed to a fixed point, it will describe a curve line lying in one plane which passes through that point, and the line joining it with the centre of forces will describe areas proportional to the times (73.) The line joining the body with the centre is called the RADIUS VECTOR. The deflecting force is called CENTRIPETAL, or ATTRACTIVE, if its direction be always toward that centre. It is called REPULSIVE, or CENTRIFUGAL, if it be directed outwards from the centre. In the first case, the curve will have its concavity toward the centre, but, in the second case, it will be convex toward the centre. The force which urges a piece of iron towards a magnet is centripetal, and that which causes two electrical bodies to separate is centrifugal. 181. The force by which a body may be made to describe circles round the centre of forces, with the angu lar velocities which it has in the different points of its f d4 I d3 This force is often called centrifugal, the centrifugal force of circular motion; and it is conceived as always acting in every case of curvilineal motion, and to act in opposition to the centripetal force which produces that motion. But this is inaccurate. We suppose this force, merely because we must employ a centripetal force, just as we suppose a resisting vis inertiæ, because we must employ force to move a body. 182. If a body describe a curve line ABC by means of a centripetal (fig. 35.) force directed to S, and vary- Fig. 35according to some proportion of the distances from it, ing and if another body be impelled toward S in the straight line ab S by the same force, and if the two bodies have the same velocity in any points A and a which are equi distant from S, they will have equal velocities in any other two points C and c, which are also equidistant from S. Describe round S, with the distance SA, the circular arch A a, which will pass through the equidistant point a. Describe another arch Bb, cutting off a small arc AB of the curve, and also cutting AS in D. Draw DE perpendicular to the curve. The distances AS and a S being equal, the centripetal forces are also equal, and may be represented by the equal line AD and ab. The velocities at A and a being equal, the times of describing AB and ab will be as the spaces (14.). The force ab is wholly employed in accelerating the rectilineal motion along a S. But the force AD, being transverse or oblique to the motion along AB, is not wholly employed in thus accelerating the motion. It is equivalent to the two forces AE and ED, of which ED, being perpendicu lar to AB, neither promotes nor opposes it, but incur vates the motion. The accelerating force in A there. fore is AE. It was shewn, in art. 48, that the change of velocity is as the force and as the time jointly, and therefore it is as AEX AB. For the same reason, the change of the velocity at a is as ab × a b, or ab3. But, as the angle ADB is a right angle, as also AED, we have AE: AD=AD: AB, and AE X AB=AD', ab. Therefore, the increments of velocity acquir ed along AB and a b are equal. But the velocities at A and a were equal. Therefore the velocities at B and b are also equal. The same thing may be said of every subsequent increase of velocity, while moving a long BC and b c; and therefore the velocities at C and c are equal. The same thing holds when the deflecting force is directed in lines parallel to a S, as if to a point S' infi nitely distant, the one body describing the curve line VA'B', while the other describes the straight line VS. 183. The propositions in art. 73. and 74. are also true in curvilineal motions by means of central forces. When Of Moving Forces. Fig. 36. Fig. 36. When the path of the motion is a line returning into itself, like a circle or oval, it is called an ORBIT; otherwise it is called a TRAJECTORY. The time of a complete revolution round an orbit is called the PERIODIC TIME. or 2 arc1 c c arc2 C 222 C expresses the velocity generated by this force, during the description of the arc, or the velocity which may be compared directly with the velocity of the motion in the arc. The last is the most accurate, because the velocity generated is the real change of condition. 186. A body may describe, by the action of a centripetal force, the direction of which passes through C (fig. 36.), a figure VPS, which figure revolves (in its own plane) round the centre of forces C, in the same manner as it describes the quiescent figure, provided that the angular motion of the body in the orbit be to that of the orbit itself in any constant ratio, such as that of m to n. For, if the direction of the orbit's motion be the same with that of the body moving in it, the angular motion of the body in every point of its motion is increased in the ratio of m to n+m, and it will be in the same ratio in the different parts of the orbit as before, that is, it will be inversely as the square of the distance from S (75.). Moreover, as the distances from the centre in the simultaneous positions of the body, in the quiescent and in the revolving orbit, are the same, the momentary increments of the area are as the momentary increments of the angle at the centre; and therefore in both motions, the areas increase in the constant ratio of m to m+n (75.). Therefore the areas of the absolute path, produced by the composition of the two motions, will still be proportional to the times; and therefore (73.) the deflecting force must be directed to the centre S; or, a force so directed will produce this compound motion. 187. The differences between the forces by which a body may be made to move in the quiescent and in the moveable orbit are in the inverse triplicate ratio of the distances from the centre of forces. Let VKSBV (fig. 36.) be the fixed orbit, and upk bu the same orbit moved into another position; and let Forces. Vpn NoNtQV be the orbit described by the body in Of Moving absolute space by the composition of its motion in the orbit with the motion of the orbit itself. If the body be supposed to describe the arch VP of the fixed orbit while the axis VC moves into the situation u C, and if the arch up be made equal to VP, then p will be the place of the body in the moveable orbit, and in the compound path V compound path Vp. If the angular motion in the fixed orbit be to the motion of the moving orbit as m to n, it is plain that the angle VCP is to VC p as m to m+n. Let PK and pk be two equal and very small arches of the fixed and moving orbits. PC and pc are equal, as are also KC and k C, and a circle described round C with the radius CK will pass through k. If we now make VCK to VCn as m to m+n: the point n of the circle K kn will be the point of the compound path, at which the body in the moving orbit arrives when the body in the fixed orbit arrives at K, and pn is the arch of the absolute path described, while PK is described in the fixed path. In order to judge of the difference between the force which produces the motion PK in the fixed orbit and that which produces pn in the absolute path, it must be observed that, in both cases, the body is made to approach the centre by the difference between CP and CK. This happens, because the centripetal forces, in both cases, are greater than what would enable the body to describe circles round C, at the distance CP, and with the same angular velocities that obtain in the two paths, viz. the fixed orbit and the absolute path. We shall call the one pair of forces the circular forces, and the other the orbital. Let C and c represent the forces which would produce circles, with the angular velocities which obtain in the fixed and moving orbits, and let O and o be the forces which produce the orbital motions in these two paths. These things being premised, it is plain that o-c is equal to O-C, because the bodies are equally brought towards the centre by the difference between O and C and by that between o and c. Therefore o-O is equal to c-C (A). The difference, therefore, of the forces which produce the motions in the fixed and moving orbits is always equal to the difference of the forces which would produce a circular motion at the same distances, and with the same angular velocity. But the forces which produce circular motions, with the angular motion that obtains in an orbit at different distances from the centre of forces, are as the cubes of the distances inversely (175.). And the two angular motions at the same distance are in the constant ratio of m to m+n. Therefore the forces are in a constant ratio to each other, and their differences are in a constant ratio to either of the forces. But the circular force at different distances is inversely as the cube of the distance (121.). Therefore the difference of them in the fixed and moveable orbits is in the same proportion. But the difference of the orbital forces is equal to that of the circular. Therefore, finally, the difference of the centripetal (A) For let A o, AO, A c, AC represent the four forces o, O, c, and C. By what has been said, we find that c=OC. To each of these add Oc, and then it is plain that o O=c C, that is, that the difference of the circular forces c and C is equal to that of the orbital forces o and O, VOL. VII. Part II. ᅥ 3 R |