is the weight of a body of the unit of density and of volume equal to the 32.2th part of the unit of volume. The density of distilled water is generally taken as the unit of density, and a cubic foot as the unit of volume. The weight of a cubic foot of distilled water is 1000 oz. avoirdupois, nearly. 45. The equation P=mf must be always understood in accordance with the explanation given in Article (42). As a further illustration, we will apply it to the following problem. A body weighing 24 lbs. is moved by a constant force, which generates in a second a velocity of 3 feet per 1"; find what weight the force would statically support. If we take m to represent the mass of the body and P for the number of lbs. the force would support, g the accelerating force of gravity, we have 24= mg, P=mf, m being the same in both equations. And by the question, ƒ the acceleration produced by Pin the mass m is represented by 3, a foot and a second being the units of space and time,and with the same units, g the accelerating force of gravity is = 32.2. 3 Whence P=24=· -24 = 2·236 lbs. nearly; that is, the 9 32.2 force which acts on the body would support in equilibrium a weight equal to 2.236 lbs. If two bodies A and B are in contact and at rest, we know from statical principles that the pressure which A exerts upon B is equal in magnitule, and opposite in direction to that which B exerts upon A; and again, if two bodies A and B at rest are connected by a fine thread, the strain which the thread exerts upon one of the bodies is equal in magnitude and opposite in direction to that which it exerts upon the other. The question will arise, "Is this the case when the bodies are in motion? or, if the mutual pressures which they exert on each other are not equal, what relation subsists between them?" And to these questions (which must arise in all problems where there is any mutual action between the different parts of a system of bodies), the principles which we have already stated afford no satisfactory answer. It is assumed, however, that when one particle acts on another particle, in motion as well as at rest, the second exerts on the first a force equal in magnitude and opposite in direction to that which the first exerts on the second. If the force which the first exerts on the second be called "action," that which the second exerts on the first may be called "reaction," and the principle just stated may in other words be expressed thus: "Whenever one body A acts on another B, the latter reacts on the former, and this action and reaction are equal in magnitude and opposite in direction." 47. The action here spoken of may be of any kind whatever; as for example, when two bodies in motion or at rest press against each other, their mutual pressures are equal and opposite; or in other words, the action and reaction are equal in magnitude and opposite in direction. Or again, when two particles move in any manner connected by a string, the force which the string exerts on one is equal and opposite to the force it exerts on the other. Or again, if two particles attract or repel each other, the dynamical measure of the force which one of them A exerts upon the other B, is equal to that which B exerts upon A. This principle is frequently embodied in the brief statement that "Action and Reaction are equal and opposite." 48. For some illustration of the third law of motion, we may refer to the observations made with Atwood's machine (Arts. 80—82); but the motions of the heavenly bodies afford the most interesting as well as the most searching test of the truth of the dynamical principles which are employed in investigating them. It has been before remarked, that the laws of motion are enunciated and asserted to be true only with respect to particles, and of course, as we have no practical experience of particles, in the mathematical sense of the word, the student must not expect to find them proved with that degree of strictness which attaches to geometrical demonstration. He is recommended for the present to accept them as conclusions which have been arrived at by philosophers after much painful inquiry and observation, and not to trouble himself much with the particular experiments which may be said to suggest these laws, or with the calculations of more complex phenomena which are based upon them, till he has grasped their meaning, and applied them to a variety of problems. He will then be able more fully to appreciate the bearing of particular experiments on the principles which they are intended to illustrate and confirm. But as no individual experiment will involve one law of motion to the exclusion of the others, the laws of motion must be taken as a whole,-and when we find the observations of numerous and complex phenomena agreeing with calculations based upon these principles and involving them in every variety of combination, we arrive at a moral conviction of their truth. CHAPTER II. OF UNIFORM MOTION AND COLLISION. 49. WHEN a body, regarded as a particle, is subject to uniform velocity in a If then v represent this no extraneous force it moves with straight line (first law of motion). uniform velocity, and s be the length of path described or passed over in any interval of time t, we shall have s = which is the formula for uniform motion. vt; The equation s=vt which connects the three quantities s, v, t will still be true if the path of the body be curvilinear, provided the velocity be uniform: but when the path is not a straight line there must be some force acting on the body which deflects it from a rectilinear path; and if the velocity be uniform, this force must always act perpendicularly to the direction of the body's motion at any time, and the magnitude of this force will depend upon the curvature of the path. This kind of motion however we do not propose to discuss. 50. The position, velocities, and direction of motion, of two particles at any time being given, to find after what interval they will be at an assigned distance from each other, and to determine their position at that time: the motion being in one plane. Let A, B be the position of the particles at first, and AO, BO the directions of their motion. Take 40 to represent the velocity of A, and BT, on the same scale, to represent that of B. Complete the parallelogram AC and join CT. Let a circle with centre O and radius equal to the proposed distance, cut CT in D: join OD and complete the parallelogram PD. Then will P, Q be contemporary positions of the particles originally at A, B, and the time of moving from A to P: that from A to 0:: CD: CT. For by the construction PO: QT= QD : QT=BC : BT= A0: BT; i.e. AP: BQ=A0: BT; that is, AP, BQ are in the proportion of the velocities of the particles, and therefore they are simultaneously at P, Q, and the distance PQ is equal to OD, the distance proposed; and further, time of moving from A to P: time from A to O = AP: A0=BQ: BT = CD: CT by similar triangles. COR. 1. Since the circle with centre O and radius OD will in general cut CT in two points, there will in general be two periods at which the particles are at a distance from each other equal to OD; we leave it as an exercise for the student to form the construction for the other position. COR. 2. Since OD cannot be less than a certain distance, viz. the perpendicular from 0 to CT (unless T and O coincide) we see that the particles will approach each other till their distance is equal to this perpendicular, and is then a minimum, and afterwards they will recede from each other. If P, Q be the centres of two spherical balls, the proposition will enable us to examine the circumstances of their approach to each other &c.;-if the distance PQ = sum of the radii of the balls, we find when and where they will come into con |