Page:Encyclopædia Britannica, Ninth Edition, v. 15.djvu/710

Rh 678 MECHANICS knowledge of hoiv the tension of the string depends on its length. Thus the tension can be calculated from the relative position of the bullets. Transfer- 12. Scholium to Law HE. On this we will, for the cnce of present, remark only that it furnishes us with the means energy. O f guying directly the transference of energy from one body or system to another. Experiment, however, was required to complete the application of this part of Newton s systematic treatment of the subject. What was wanted, and how it has been obtained, will be treated of later. The first words of the scholium, however, claim for Newton the discovery of the clause we have extracted from it. For they run thus : Hactenus principia tradidi a mathematicis recepta, et experientia multiplici confirmata. Division 13. What has now been said enables us to see the order of the i n which the fundamental ideas should be taken up, so subject. t j ia j. |.j ie necess ij;i e g O f eacn should be provided for before its turn comes. An indispensable preliminary is the study of motion in the abstract, i.e., without any reference to u hat is moving. This is demanded in order that we may be able to apply the Second Law. The science of pure motion, without reference to matter or force, is an extension of geometry by the introduction of the idea of time and the consequent idea of velocity. Ampere suggested for it the term Cinematique, or, as we shall write it, Kinematics. We include under it all changes of form and grouping which can occur in geometrical figures or in groups of points. We shall then be prepared to deal with the action of force on a single particle of matter, or on a body which may be treated as if it were a mere particle. Thus we have the Dynamics of a Particle. This, again, splits into two heads, Statics and Kinetics of a Particle. But all this requires the Second Law only. When we have two or more connected particles, or two particles attracting one another or impinging on one another, the Third Law is required. Next in order of simplicity come the Statics and Kinetics of a Rigid Solid. Then we have to deal with bodies whose form, &amp;lt;fcc., are altered by forces flexible bodies, elastic solids, fluids, &c. Finally, we must briefly consider the general principles, such as &quot; conservation of energy,&quot; &quot; least action,&quot; &c., which are cleducible by proper mathematical methods from Newton s Laws, and of which some at least, if we could more clearly realize their intrinsic nature, would probably be found to express even more simply than do Newton s Laws the true fundamental prin ciples of abstract dynamics. We will not restrict ourselves to one uniform course in the application of mathematical methods. Rather, as con siderations of space require to be attended to, we will vary our methods from one part of the subject to another, so as to exhibit, each at least once, all the more usual processes. And we will endeavour to make the large-type portions of the article, in which only the most elementary mathematics will be introduced, a self-contained treatise which may be read by students of very moderate mathematical knowledge. KINEMATICS. Position and the Means of Assigning it. Position, 14. Motion (or displacement) consists simply in &quot; change of position.&quot; Hence, to describe motion, we must have the means of assigning position. This is, of course, a question of GEOMETRY (q.v.). See also QUATERNIONS. From these articles it appears that the position of one free point with reference to another (all these space relations are relative, as we have already said) depends on three numbers, of which one at least must involve the unit of length. In Cartesian rectangular coordinates, we denote these by x, y, z, which indicate respectively the distance of the point from each of three planes at right angles to each other, and all passing through the origin (or reference point). From another point of view they may be called &quot; degrees regret of freedom.&quot; When the value of one is assigned, say by of free- x = a, dom the point is said to have lost one degree of freedom, or to have had imposed upon it one &quot;degree of constraint.&quot; It of con- must now lie in a plane parallel to the first of the reference straiut planes, and at a distance a from it. When a second degree of constraint is applied, say by y = b, another degree of freedom is lost. The point s position is limited to lie in a second plane in a given position at right angles to the first. It must therefore lie somewhere on the straight line which consists of the series of points common to the two planes. A third degree of constraint z = c takes away its one remaining degree of freedom ; and its position is now definitely assigned as the single point of intersection of three given planes. 15. But constraint may be applied in other ways. Thus if we assign the condition x 2 + y 2 + z~ = a?, we deprive the point of one degree of freedom by com pelling it to remain at a distance a from the origin. It is now limited to the surface of a sphere, but its latitude and longitude on that sphere may be any whatever. Here again the imposition of one degree of constraint has taken away one degree of freedom. 16 In general, one degree of constraint may be ex- Examp! pressed as of con - straiut&amp;lt; This, when has an assigned value, is the equation of a definite surface on which the point must lie. Three such conditions determine the position of the point, and may therefore be looked upon as introducing, rj, , another set of coordinates, which may be used in place of x, y, z. The number of such systems is, of course, unlimited ; but it is often possible to choose one in which the conditions of a problem are much more simply expressed than they were when expressed in x, y, z. The whole question belongs to what is called &quot; change of variables.&quot; To give an elemen tary instance of its use, suppose we take the ordinary simple pendulum, a pellet supported by a fine thread or wire, and oscillating in a vertical plane. If the origin be placed at the point of suspension, and the axis of z be vertical, we have two conditions : where a is the length of the thread ; and ?//x = tana, where a denotes the azimuth of the plans of oscillation. There is but one degree of freedom left, because two degrees of constraint have been imposed. We may choose for this either x, y, or z ; but we should in each case be led to complex expressions. If, however, we consider that all the freedom left to the pendulum is to oscillate in a given plane, we may denote its sole remaining degree of freedom by 6, the angle which the string makes with the vertical ; and form our dynamical equation in terms of this. When 6 is found by dynamical considerations, we have x = a sin 9 cos a, y = a sin 9 sin a , z = a cos 9. Here 6 comes in as what is called a &quot; generalized General coordinate.&quot; If the pendulum be not limited to one plane, the azimuth, or as well as the angle, of the displacement from the vertical may be any whatever. Hence there are two degrees of freedom, which are indicated by the generalized coordinates a and 0.