This book has been written to provide a discussion of higher dynamics suitable for students of engineering, physics, or astronomy. To a large extent the examples and exercises have been drawn from practical affairs, and have been chosen more for the sake of illustration of physical principles than for their mathematical interest. With hardly an exception, the exercises given at the end of each chapter have been carefully verified, and it is hoped that but few of them are in error. A large number of examples have been worked out in the various chapters, where practical illustration seemed to be required.

A considerable space is devoted to gyrostats and gyrostatic action, and we have used throughout this chapter, and elsewhere, the method set forth in 9 of calculating rates of change of directed quantities for a moving system. This method of proceeding occurred to one of us about fifteen years ago [see Gray's Physics, Vol. I., and we have found it very useful in our teaching, as enabling solutions of difficult problems of rotational motion to be readily built up from first principles. The advantage of the method is most apparent in Chapter IX., which is an expansion of an article on Gyrostats and Gyrostatic Action in Machinery communicated to the Institution of Engineers and Ship-builders in Scotland in 1905. Some elementary accounts of gyrostatic action have appeared during the last two or three years, and it is right to say that we are not indebted to these for our method of treatment.

We have derived assistance from various works, but, as was to be expected, our obligations to Sir George Greenhill's writings are especially great. Besides making additions of the most practical and valuable kind to the science of dynamics, Sir George Greenhill has long advocated the use of units of the sort employed by men to whom a comparison with the force of gravity on a given piece of matter is the most ready means of estimating a force, and protested against. the common dynamical limitation of the word weight. There can be no doubt that the ordinary use of the word in connection with the buying and selling of commodities "weighed" by a balance can never be got over, and that the connotation of the word in that connection is more frequently that of quantity of matter than that of gravity force. And it is better to take advantage of a common connotation than to do something which may tend to confuse it. Hence we have often used the so-called practical units, but without any sacrifice, for none was required, of the real advantages of the absolute system.

Speed and Velocity. We suppose that the direction of motion of a point nowhere undergoes absolutely sudden change. The point, materialised as a particle of matter so small in every dimension that it only serves to mark position in space, therefore moves in a curve in space the direction of the tangent to which is everywhere perfectly definite.

The displacements of the point are in all cases with reference to some system of marks in space which are taken as at rest. Such- a system of marks is called a reference-frame. It may be a curve fixed in space along which the point is constrained to move, or it may be axes of coordinates, for example Ox, Oy, Oz drawn from a point O, in three different directions which are not in one plane. Most frequently they are taken mutually at right angles, and are supposed either to be at rest, or to be in motion in some specified way, with reference to other coordinates which are taken as at rest. The motion of the point along the given curve may be defined by the variation of its distance measured along the curve from a specified fixed point in it, or it may be by the rate of change of the quantities which specify the position of the point with reference to the axes chosen. The relativity of motion will be found discussed in more detail in our elementary treatise.

The curve may in some extreme cases appear to be such as to contradict the condition here stated. For example, a particle ascends under the retarding action of gravity until it is brought to rest and begins to descend. At every instant except that at which it has come to rest, the direction of motion is perfectly definite. The particle does not change the direction of motion suddenly: its speed has been gradually diminished, and it is at rest at the instant of reaching the highest point ; it does not remain at rest for an interval of time however short, but still continues to gain downward velocity, and the instant of rest is the point in time which separates the interval during which the particle ascends from that during which it descends. Again, when a marble is dropped on a stone floor and rebounds, or a cricket-ball is struck by the bat, the direction of motion is changed suddenly, as suddenness is usually understood with respect to ordinary phenomena. But in reality the change of direction of motion occupies an interval of time, that of the duration of collision, though, as reckoned with respect to the time required for ordinary changes which can be followed by the eye, the interval is short.

It must be understood from the outset that in dynamics an instant is not what in ordinary affairs it is often supposed to be, an interval of time of indefinite length: it is not an interval of time at all. It is the final terminus of one interval of time, and the initial terminus of the interval which immediately succeeds. Two planes meet in a line which is not part of either plane, not being a surface in any sense, but is only a dividing mark or common boundary to be crossed by a moving point passing from one plane to the other. A point again is the dividing mark where one part of a line or curve ends and another portion begins: it is not, however it may be indicated by a spot of chalk on a blackboard or a spot of ink on paper, other than merely a mark of position in space. So with an instant in time the distinction between it and an interval of time, must be clearly understood. A pendulum bob at a certain instant is at the extremity of its swing in one direction; but the bob does not remain at rest for any interval of time however short; the swing in the opposite direction begins just when that in the first direction terminates.

The distinction between an instant and an interval of time is the key to the solution of many of the puzzles regarding motion which perplexed the old philosophers. They held that a body could not move from one position to another without occupying in succession a continuous series of intermediate positions, and then came to the conclusion that if that were so motion was impossible, because it was tacitly assumed that occupation of a position implied rest in that position. Each position is occupied at an instant in time, but not during an interval: the true idea in no way negatives the possibility of motion, and the contradiction had no real existence.

At the instant which marks the beginning of an interval of time the moving point or particle is at P v at the instant which marks the end of the interval and the beginning of a succeeding interval it is at P2. P1 , P2 are points on the curve along which the moving point is displaced, and are at a definite distance apart, measured along the curve. We form the ratio s/t, that is the ratio of the numerical value of the distance to that of the time in which it is traversed, and call it the average speed of the moving point during the time t. The unit of speed is thus the speed in which unit of distance is described per unit of time. Speed is thus expressed in feet per second, centimetres per second, miles per hour, or according to any other choice of the units of length and time.

In many cases it would merely cumber our equations to indicate at every symbol or group of symbols the units employed; but when it is necessary, in the statement of results or elsewhere, to specify units we shall do so by adopting for feet per second the symbol f/s, or ft/sec, for centimetres per second cm/s, and so on.

The meaning of the word per is to be observed. The point does not necessarily move for an hour or even for a second. But it traverses the distance s, which may be miles, or only a fraction of an inch in length, at such and such an average rate or speed. For example, the statement that the speed is 60 miles per hour means that if this average speed were maintained constant for an hour the distance traversed would be 60 miles: the actual duration of the motion from P1 to P2 may be only a small fraction of a second. The distance traversed in a given time is equal to the average rate of displacement multiplied by the number of units of time, just as the amount of a workman's earnings for a given time is equal to the product of the rate of wages into the numerical measure of the time. The speed, or rate of displacement, is no more distance traversed than a rate of wages is a sum of money, and must always be expressed as distance per unit of time. The idea of uniform speed presents no difficulty to anyone. Speed, whether the direction of motion remains constant or not, is constant when equal distances along the path are described in equal intervals of time, however small these intervals are taken. The proviso contained in the words italicised is necessary : a train, for example, might run 30 miles in each of successive hours, or 7 1/2 miles in each of successive quarter-hours, even though it stopped at stations: a test by a sufficiently short interval of time would reveal the true variability of the motion.