582 DYNAMICS although, it may be argued that the displacement of each part of the system takes place in the same interval of time, and that the geometrical conditions enable us to compare the displacements experienced by different parts during the same time, and thus lead us to a comparison of velocities, yet it must be observed that this is only a comparison amounting simply to a relation between corresponding displacements, and does not involve time explicitly, since the whole displacement may take place in a time as long or as short as we please, for we do not consider it. More over, the actual motion of any part may be made uniform or varying in any arbitrary manner without any account being taken of it. In fact it is simply two or more con figurations of the material system which are compared together, and, though for the sake of distinction we call one the initial and another the final configuration, we might as well distinguish them in any other manner and without stating which follows the other. Indeed we contemplate them as co-existent during the act of comparison. Hence we may complete the science of displacement or pure mechanism without ever considering force, or being able to measure time or even to define equal intervals. Kinematics. If to our conceptions of space and of dis placement we couple that of time as a measurable quantity, we are led to compare the rates of non -simultaneous as well as of simultaneous displacements, and are consequently obliged to measure the rate at which displacement occurs by the change of position experienced in a definite interval of time by the body, figure, or point we are regarding. Rate of change of position measured thus we call velocity. The next step in the same direction is the consideration of the rate at which velocity changes, or acceleration, and thus the association of our conception of space with that of time as a measurable quantity opens up to us that branch of dynamics which we call kinematics. Matter. Having considered displacement in connection with the time during which it occurs, the next step leads us to take account of the thing displaced, and here we are obliged to contemplate matter directly. Matter, like time and space, we do not attempt to define, but treat it as a primary conception, its more obvious properties making themselves known to all through daily experience. Force. The change of the motion of material bodies brings us at once, through the introduction furnished by the first law of motion, to the conception of force, which may be defined in terms of three primary quantities, viz., space, time, and matter. The second law of motion expresses the manner in which matter is affected by force, and teaches us how to measure force by the observation of its effects. The science of dynamics in its restricted sense is that which treats of the consequences arising from the relations of matter to force, and before we can proceed in this science beyond the first step we must become acquainted with the second law of motion, while kinematics requires for its complete development only the first law of motion, its range being thereby sharply defined and separated from that of the rest of dynamics. The laws of motion, like other natural laws, must be understood to express merely the properties of natural bodies as we find them, and within the degree of accuracy to which our experiments can be relied on. We might, of course, have started with any hypotheses we liked respecting the relations of force to matter, and upon these hypotheses arid our conceptions of time and space have constructed a purely theoretical system of dynamics which would have been perfectly self-con sistent ; but our conclusions might, or might not, have agreed with observations of natural phenomena. If we found an agreement between the results of the application of our theory to special problems and the solutions of the corresponding problems as worked out objectively in nature, we should have reason to believe that our hypotheses agreed with the facts, or, in other words, that they were true, and we should then raise them to the dignity of natural laws. It is on evidence of this kind that our acceptation of all natural laws is based. If our conclusions were inconsistent with natural phenomena our system of dynamics would be an abstract, instead of a natural, science if, indeed, it might be called a science at all and would be valuable merely as an intellectual exercise. In the case of such an abstract science we are not even bound to adopt the axioms respecting the properties of space which are usually accepted, but may confer upon our " space " any number of dimensions and any properties we please. Stress. Though the conception of a single force is con venient, it nevertheless results from a mere process of mental abstraction. We never meet with a single isolated force in nature, but each is accompanied by an equal and opposite force acting in the same straight line, and when we speak of one without the other we do so merely for the sake of brevity. The third law of motion implies this statement, though it has also a wider signification. The action and reac tion which are thus always inseparably linked together may be conveniently called a stress, of which the two forces are opposite aspects. Thus it appears that there is nothing in nature corresponding to what we are accustomed to call a single force ; stresses, indeed, abound, and may be produced whenever we please, but we are completely ignorant of their existence except when they change the relative velocities of different portions of matter. Then, and then only, do they appeal to our senses. Statics. The investigation of the conditions under which a system of stresses produces no displacement of the bodies between which they act constitutes the science of statics, and will be discussed under the head of MECHANICS. Measurement of Force. Since force can be defined in terms of epace, time, and matter, it follow that the measurement of a force ought to involve measurements of these three quantities and of them only. Now it is plain that any force whatever may be chosen as the unit in terms of which other forces should be expressed, provided it is capable of being reproduced at all times and in all places with precision. We all now believe that the quantity of matter in a body is unchanged by changing its position or by the simple lapse of time, and we also believe that the region of space which we inhabit is sufficiently homoloidal to allow us to compare distances in different directions, at different places, and at different times. Moreover, the first law of motion, as has been stated above, provides, when proper precautions are taken, a method of measuring time which satisfies the requirements of the mind, while the rotation of the earth affords a practical measure of time sufficiently exact for the most refined experiments we can execute. Therefore a unit of force which depends only on the units of length, mass, and time will be the same at all places, and, so far as our experience allows us to judge, at all times. Such a unit is termed an absolute unit. Not only force but every other quantity dealt with in dynamical science, as well as every physical quantity whose relations to space, mass, and time are known, can be measured in terms of a unit of its own kind which depends only on the fundamental units of length, mass, and time, and is then said to be expressed in absolute measure. The three primary units must be chosen in an arbitrary manner, and their permanence must be considered a matter of definition ; but when these have been once fixed, all the absolute units derived from them are perfectly determinate and invariable. If a foot, a pound, and a second be chosen as units, the corresponding absolute unit of force is called a poundal ; while if the primary units be a centimetre, a
gramme, and a second, the unit of force is called a dyne.