For this Edition the work has been thoroughly revised, and a considerable portion of it, with the view of rendering the explanation of principles more copious, re-written. The general plan, which will be sufficiently understood from the Table of Contents, does not differ from that of the two former Editions. The Diagrams, instead of being placed as before at the end of the book, are here inserted in the text; a change which it is hoped the reader will find advantageous. The Tenth Chapter contains a Collection of Problems. These the Author has solved as much as possible by general methods for the purpose of illustrating the general principles previously laid down: for which reason he has not in any case been anxious to solve a problem by the shortest possible method, or by elegant devices adapted only to that particular case. In this Edition, several new Articles have been introduced, and amongst them a new method of considering the Perturbation of a Pendulum (Art. 157), which, as it appears more simple than any before published, is substituted in the room of the method before given. The investigations in the former editions, respecting centrifugal force, being incomplete in consequence of not determining its direction as well as its magnitude, have in this edition been replaced by other methods, which, though not so brief, are free from the objection alluded to.

The author has not uniformly adhered to one system of Differential notation, believing that each has the advantage in particular cases; but in general the subscript notation has been preferred, partly because without being less expressive it occupies less space in type than its rival, and is more quickly written; but chiefly because it ensures that the unpracticed student in using it cannot lose sight of the independent variable, an event which is not of very uncommon occurrence when the notation is employed. When several variables occur in an expression, it may happen to be necessary to perform the operation of differentiation upon it on several distinct hypotheses; and hence arises the necessity as well as the difficulty of inventing a system of differential notation, which shall simply but sufficiently mark the various hypotheses on which the operation is to be effected.

When two bodies moving directly towards each other meet, or if one overtake the other, upon their surfaces first coming in contact, the motion of each receives a check, which to our senses appears to take place instantaneously. That a finite, though very small, time is really occupied in effecting the changes in their motions, is rendered certain by a closer examination of the experiment. For, if two ivory balls, one of which is stained with ink, be made to touch each other, at the point of contact, upon the clean ball, there will be seen an extremely minute spot of ink. But if they be made to strike against each other with considerable velocities, the spot of ink will be very much enlarged; and the greater the striking velocities, the larger will be the spot. Now, as two spheres can only touch each other in a single point, the appearance of a large round spot transferred from the stained ball to the clean ball, shews that their figures during the impact have been compressed at and near the point of first contact; hence, during the impact the centres of the balls have approached nearer to each other than the sum of their radii, which is their distance when merely in contact. This approach through a finite space being accomplished with finite velocities, must necessarily have occupied a finite time. We can now see also how the change in the velocities is affected. At first meeting the balls begin to press against each other; - this pressure goes on increasing as the balls become more compressed, until the compression has reached its maximum, at which moment the pressure has also obtained its greatest value, and the approach of their centres having ceased, the velocities of the balls are equal. At this moment the natural effort of the substance of the balls to restore their former figure (that of perfect spheres) causes them to thrust one another apart with considerable violence, (yet this operation also occupies a finite time,) and produces the phenomenon of rebounding. That property of matter by which the balls make this effort to regain their former figure is called elasticity. If the balls regain their figure with the same force as was required to compress them, the elasticity of the substance of which they are composed is said to be perfect. No substance of this kind has hitherto been found in nature. All bodies are therefore considered as imperfectly elastic.

27. During the whole process of compression and separation detailed above, there exists a mutual pressure, which, though continually varying, is always the same for both, and which is employed in retarding (or accelerating as the case may be) the motions of the balls until they separate.

The product of mass into velocity is called momentum and hence the result of this article may be thus stated in words,

In the direct impact of two bodies the momentum lost by one of them is equal to that lost or gained by the other.

There is no part of the investigation in the last article which requires the bodies to be in actual contact: if by any other means they are enabled to exert in the line which joins them such an action upon each other, as is equivalent to an equal pressure applied to each in opposite directions, the same result will follow, viz. that

It is easily shewn that the attraction which two bodies exert upon each other is of this nature; for if the attracting bodies be placed at the opposite ends of a rigid rod no motion will ensue, which proves that they exert equal pressures upon the rod in opposite directions. The rod, as long as it remains between the bodies, prevents motion, but the instant it is removed, the same forces which caused them to exert equal pressures, now cause them to rush towards each other. Hence the force of attraction is equivalent to an equal pressure applied to each body in opposite directions. Consequently when two bodies A, B move towards each other by mutual attraction,

In the direct mutual action of two bodies the momentum lost or gained by one is equal to that lost or gained by the other.

30. This was formerly the third Law of Motion, and was stated thus, "Action and Reaction are equal and opposite." We may regard it as a deduction from our second and third Laws, and as it is a result which admits of many simple and varied experimental means of examination, it furnishes us with easy tests of the truth of those laws from which it has been derived. Thus, let A, B be two balls of different substances, magnitudes, and weights, suspended by two strings CA, DB from two such points C, Z) that when at rest the balls just touch each other and have their centres in the same horizontal line. This arrangement is necessary in order to insure direct impact when the balls are elevated and suffered to fall through such circular arcs aA, bB, (whose centres are C, D) that they may strike together at the moment when they reach their lowest positions. By having the strings of different lengths, if necessary, and letting the bodies start at proper times, we may insure their meeting together at the lowest points of their respective arcs, and with any velocities we please. After striking together they will rebound, and by measuring the respective altitudes through which they ascend we shall be able to calculate the velocities with which they rebounded, that is, their velocities after impact. The velocities before impact are also known, and thus we can ascertain whether the momenta lost by the balls are equal. This experiment has been repeatedly tried, and been attended with the most satisfactory results, and hence affords us strong evidence of the truth of these Laws of Motion. If it be thought necessary, the effect of compression may be examined apart from that of elasticity by fixing a steel point in one of the bodies at the point of impact, which will cause them to adhere together after striking.

31. But, again, it is evident from Art. 27, that if two bodies A, B are always subject to equal pressures, by whatsoever means those pressures are produced, if only they act in the directions of motion of the bodies to which they are respectively applied, the same result follows; and consequently, a A, B act upon each other through any machine, so as to satisfy these conditions, viz. that equal pressures are always exerted on them in their respective directions of motion, it is true in this case also, as in the case of direct action, that the momentum lost or gained by one of them is equal to that lost or gained hy the other.

33. Now if force be the cause of motion it seems natural to expect, since cause and effect are necessarily proportional, that the quantity of force employed ought to be exactly proportional to the quantity of motion produced or destroyed. We shall shew that this follows from the measures of force and motion which we have given.