want of an elementary work on dynamics, in which Newton's laws of motion are taken as the basis of the science, and in which as much space is devoted to unit questions as their importance demands, has for some time necessitated a MS. on this subject for the use of pupils; this MS. forms the first Chapter of the present work. I am glad to avail myself of this opportunity of expressing my sense of the gratitude I owe to W. H. Besant, Esq., M.A., F.R.S., late Fellow and Lecturer of St John's College, Cambridge, not only for many valuable suggestions during the preparation of this work, but for a sound instruction in the principles of dynamical science, which form the only foundation on which a superstructure of Natural Philosophy can be based, and for much kindness shewn in many ways. I have also to express my indebtedness to Professor J. Clerk Maxwell, LL.D., F.R.S., &c., for several ideas in different portions of the work, and my sincere thanks to John Cox, Esq., B.A., Scholar of Trinity College, Cambridge, for his kindness in perusing the proof-sheets. I have also to thank my friends Messrs W. M. Hicks, B.A., Scholar of St John's College, and R. F. Scott, B.A., Scholar of St John's College, for their kindness in working many of the examples. The Note on the Third law of Motion at the end of the book may be read by the student after reading the laws of motion in the first chapter, but he will be in a better position to understand it when he has read more of the subject. WM. GARNETT. ST JOHN'S COLLEGE, ELEMENTARY DYNAMICS. CHAPTER I. ON THE GEOMETRY OF A MOVING POINT, AND THE FUNDAMENTAL LAWS AND PRINCIPLES OF DYNAMICS. 1. A POINT is said to be in motion when it changes its position relative to surrounding objects. From this definition it will be seen that all cases of motion which will come under our consideration are essentially relative; in fact we have no means of measuring absolute motion or of determining whether any given point is absolutely at rest in space, or not, and it may even be doubted whether the human mind is capable of forming a distinct conception of absolute motion. We shall not attempt to give a definition of space: our idea of it must be considered a primary conception. Time is defined by the metaphysician as "the succession of ideas;" the physicist treats time, like space, as a primary conception. Equal times are those intervals during which the earth turns through equal angles relative to the fixed stars, and any duration of time may be measured by the angle turned through by the earth during the interval. The most obvious unit of time is therefore the sidereal day, or the period during which the earth makes a complete rotation on its axis relative to the fixed stars. The unit generally adopted is the second of mean solar time. 2. The velocity of a point is the rate at which it is changing its position relative to surrounding objects; in other words, the degree of speed with which it is moving. G. D. 1 A point is moving with uniform velocity when it passes over equal distances in equal intervals of time: under other circumstances its velocity is said to be variable. Velocity is measured, when uniform, by the space passed over in a unit of time; when variable, it is measured at any instant by the space which would be passed over in the unit of time, if the velocity remained constant during that unit and the same as at the proposed instant. The unit of velocity is taken as that velocity with which a point will pass over the unit of length in the unit of time. The numerical measure of the velocity of a moving point is therefore the number of units of length passed over by the point in the unit of time, if the velocity be uniform; but, if variable, its numerical measure at any instant is the number of units of length which would be passed over by the point in the unit of time, if its velocity remained constant during that unit and the same as at the proposed instant. Now if the unit of length be increased or decreased, the unit of time remaining the same, the space passed over in the unit of time by a point moving with unit velocity is increased or decreased, and therefore the unit of velocity is changed, in that same ratio. If, on the other hand, the unit of time be increased or decreased, the unit of length remaining the same, the time required by a point moving with unit velocity to pass over the unit of length is increased or decreased accordingly. Now the longer the time occupied by a point in moving over the same distance, the less must be its velocity, and the shorter the time the greater the velocity. Hence the unit of velocity must vary inversely as the unit of time, if the unit of length remain constant. Also we have just shewn that the unit of velocity varies directly as the unit of length when the unit of time is kept constant. Therefore, when all are allowed to vary together, the unit of velocity will vary directly as the unit of length, and inversely as the unit of time. Todhunter's Algebra, Art. 425.) (See 3. The mathematical expression for any physical quantity always consists of two factors, one being the unit of the same kind as the thing considered, the other representing |