Preface.

In preparing the present work, which was designed to include in a single volume of moderate compass the elementary portions of Theoretical Mechanics, no formal division of the subject into Kinematics, Statics and Kinetics has been made. The topics often included under the first head it was thought best to introduce separately, each at the point where it is required for immediate application to the treatment of the motions produced by forces. For example, the expressions for radial and transverse accelerations are not introduced until required in the discussion of Central Forces.

The subject of Statics is, to be sure, to a large extent separable from the idea of motion. But, on the one hand, as has been recognized in all recent treatises, the fundamental notions of force are best presented, and the Parallelogram of Forces is best established, on the basis of the Laws of Motion. This requires what may be called a dynamical introduction to Statics. On the other hand, the subject cannot be completed without the Method of Virtual Velocities, an application of the Principle of Work. This principle, which is dynamical as involving forces acting through spaces, advantageously precedes the study of kinetics into which time enters explicitly, and prepares the student for the notion of Kinetic Energy, or work embodied in motion.

Accordingly, in the present volume, Chapter I consists of such a kinetical introduction to the whole subject as is referred to above; Chapters II-VI are purely statical; Chapter VII treats of the dynamical Principle of Work with its application to Statics and to the notion of the Potential Function; and the remaining chapters treat of purely kinetical topics.

The chapters are further subdivided into sections followed by copious lists of graded examples aggregating over 500 in number; many of these were taken from examination papers set at the Naval Academy, and not a few were prepared expressly for this work.

In these examples, as well as in the numerical illustrations introduced in the text, gravitation units of force have for the most part been employed. These units in fact not only have the advantage of being rendered familiar to us by the common usages of every-day life, but they are actually more convenient than absolute units in mechanical problems, since in them the forces arise principally from the weights of bodies. Thus their use is forced upon even those writers who most deprecate the employment of a variable unit of force. The conception of an absolute unit of force, dependent upon mass and motion and not upon weight, is indeed essential to the gaining of correct ideas of the nature of force. Hence the introduction by Prof James Thomson of the poundal, which serves this purpose when the English system of weights and measures is used, has been of very great value. At the same time the employment of the pound as a unit of mass as well as a unit of force has been the cause of confusion, so that a student is sometimes in doubt whether the result of the use of a formula is the number of pounds or of poundals, or, as he may phrase it, whether the formula is expressed in gravitation or in absolute units. To prevent this confusion, care has been taken in the present volume, while using gravitation units, to avoid such expressions as, for example, "a mass of 6 pounds," and to speak instead of "a body whose weight is 6 pounds." There is no doubt that the same body would be intended in either expression, but the former would imply that, in the formula $W = mg$, 6 is the numerical value of $m$, and the latter that 6 is the numerical value of $W$. Inasmuch as the pound, though an absolute "standard" of mass, is properly called and legally styled a "unit of weight," the latter is the more natural course. Accordingly the student is directed on page 13 to follow it and to remember that all the forces are thus expressed in local pounds. If the result is desired in poundals, neither is the formula changed nor is the result found in one unit and then changed to the other, but the number of pounds is taken as the numerical value of $m$.

For the same reason, we should not say that the "weight" of a body varies when it is taken to a place where $g$ has a different value, because the number which legally expresses its weight remains the same. The force of its gravity has indeed changed, but it is (when we use gravitation units) the unit of this force, and not its numerical measure, which has changed.

In the treatment of kinetics, the conception of the forces of inertia has been freely employed, and that without the apologies that some writers have thought necessary. It would seem that the resistance of a body in motion to acceleration in any direction is as much entitled to be regarded as a force as is the resistance of other bodies which, in the case of a body at rest, prevent motion. By including the latter as forces, we obtain the idea of a system of forces in equilibrium; so also, by including the former as forces, we extend this idea to the case of a body in motion, and D'Alembert's Principle presents itself in the form of "kinetic equilibrium," instead of requiring for its statement a set of hypothetical "effective forces."

The study of Mechanics is here supposed to follow an adequate course in the Differential and Integral Calculus, and to form a very important application of its principles. But, when these applications occur, the results are not merely presented in the shape of general formulae in the notation of the Calculus, leaving the student unaided in the process of evaluation. Instead of this, pains has been taken to instruct the student in the methods best adapted in various cases to obtaining numerical results. Particularly in the treatment of statical moments and of momenta of inertia it is hoped that the book will be found a useful supplement to the course of instruction m the processes of integration. Throughout, the practice of relying upon substitution in general formulae is discouraged as far as possible, and the opposite practice inculcated - namely, that of applying general principles directly to the problem in hand.

Special prominence is given to those results which it is the most important to make familiar to the student of Applied Mechanics, and to the readiest ways of recalling them when they have slipped the memory.

Although preference is given to analytical processes, a not inconsiderable use is made of graphical methods. These have, however, been introduced rather as diagrammatic aids to the comprehension of general principles, and to the calculation of numerical results, than as methods of obtaining results by measurement from accurately constructed diagrams - the latter belonging rather to the province of Applied Mechanics.

W. W. J.
April, 1901

Note: Woolsey Johnson published A Treatise on Differential Equations in 1889. A book, published in 1906 under the title Differential Equations, claims to be a 4th edition. However, the Preface in this book was written in December 1905. We give this Preface below.

Preface.

It is customary to divide the Infinitesimal Calculus, or Calculus of Continuous Functions, into three parts, under the heads Differential Calculus, Integral Calculus, and Differential Equations. The first corresponds, in the language of Newton, to the "direct method of tangents," the other two to the "inverse method of tangents"; while the questions that come under this last head he further divided into those involving the two fluxions and one fluent, and those involving the fluxions and two fluents.

On account of the inverse character which thus attached to the present subject, the differential equation must necessarily at first be viewed in connection with a "primitive," from which it might have been obtained by the direct process, and the solution consists in the discovery, by tentative and more or less artificial methods, of such a primitive, when it exists; that is to say, when it is expressible in the elementary functions which constitute the original field with which the Differential Calculus has to do.

It is the nature of an inverse process to enlarge the field of its operations, and the present is no exception; but the adequate handling of the new functions with which the field is thus enlarged requires the introduction of the complex variable, and is beyond the scope of a work of this size.

But the theory of the nature and meaning of a differential equation between real variables possesses a great deal of interest. To this part of the subject I have endeavoured to give a full treatment by means of extensive use of graphic representations in rectangular coordinates. If we ask what it is that satisfies an ordinary differential equation of the first order, the answer must be certain sets of simultaneous values of $x, y$, and $p$. The geometrical representation of such a set is a point in a plane with a direction, so the speak, an infinitesimal stroke, and the "solution" consists of the grouping together of these strokes into curves of which they form elements. The treatment of singular solutions, following Cayley, and a comparison with the methods previously in use, illustrates the great utility of this point of view.

Again, in partial differential equations, the set of simultaneous values of $x, y, z, p$, and $q$ which satisfies an equation of the first order is represented by a point in space associated with the direction of a plane, so to speak by a flake, and the mode in which these coalesce so as to form linear surface elements and continuous surfaces throws light upon the nature of general and complete integrals and of the characteristics.

The expeditious symbolic methods of integration applicable to some forms of linear equations, and the subject of development of integrals in convergent series, have been treated as fully as space would allow.

Examples selected to illustrate the principles developed in each section will be found at its close, and a full index of subjects at the end of the volume.

W. W. J.
Annapolis, Maryland.
December, 1905.