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394. The only effects of the intermolecular forces would be exhibited in molecular alterations of the form or volume of the masses involved. But as these (practically) remain almost unchanged, the forces which produce, or tend to produce, changes in them may be left out of consideration. Thus we are enabled to investigate the action of machinery by supposing it to consist of separate portions whose forms and dimensions are unalterable.
395. If we go a little farther into the question, we find that the lever bends, some parts of it are extended and others compressed. This would lead us into a very serious and difficult inquiry if we had to take account of the whole circumstances. But (by experience) we find that a sufficiently accurate solution of this more formidable case of the problem may be obtained by supposing (what can never be realized in practice) the mass to be homogeneous, and the forces consequent on a dilatation, compression, or distortion, to be proportional in magnitude, and opposed in direction, to these deformations respectively. By this farther assumption, close approximations may be made to the vibrations of rods, plates, etc., as well as to the statical effects of springs, etc.
396. We may pursue the process farther. Compression, in general, develops heat, and extension, cold. These alter sensibly the elasticity of a body. By introducing such considerations, we reach, without great difficulty, what may be called a third approximation to the solution of the physical problem considered.
397. We might next introduce the conduction of the heat, so produced, from point to point of the solid, with its accompanying modifications of elasticity, and so on; and we might then consider the production of thermo-electric currents, which (as we shall see) are always developed by unequal heating in a mass if it be not perfectly homogeneous. Enough, however, has been said to show, first, our utter ignorance as to the true and complete solution of any physical question by the only perfect method, that of the consideration of the circumstances which affect the motion of every portion, separately, of each body concerned; and, second, the practically sufficient manner in which practical questions may be attacked by limiting their generality, the limitations introduced being themselves deduced from experience, and being therefore Nature's own solution (to a less or greater degree of accuracy) of the infinite additional number of equations by which we should otherwise have been encumbered.
398. To take another case: in the consideration of the propagation of waves on the surface of a fluid, it is impossible, not only on account of mathematical difficulties, but on account of our ignorance of what matter is, and what forces its particles exert on each other, to form the equations which would give us the separate motion of each. Our first approximation to a solution, and one sufficient for most practical purposes, is derived from the consideration of the motion of a homogeneous, incompressible, and perfectly plastic mass; a hypothetical substance which, of course, nowhere exists in nature.
399. Looking a little more closely, we find that the actual motion differs considerably from that given by the analytical solution of the restricted problem, and we introduce farther considerations, such as the compressibility of fluids, their internal friction, the heat generated by the latter, and its effects in dilating the mass, etc. etc. By such successive corrections we attain, at length, to a mathematical result which (at all events in the present state of experimental science) agrees, within the limits of experimental error, with observation.
400. It would be easy to give many more instances substantiating what has just been advanced, but it seems scarcely necessary to do SO. We may therefore at once say that there is no question in physical science which can be completely and accurately investigated by mathematical reasoning in which, be it carefully remembered, it is not necessary that symbols should be introduced), but that there are different degrees of approximation, involving assumptions more and more nearly coincident with observation, which may be arrived at in the solution of any particular question.
401. The object of the present division of this work is to deal with the first and second of these approximations. In it we shall suppose all solids either RIGID, i.e. unchangeable in form and volume, or ELASTIC; but in the latter case, we shall assume the law, connecting a compression or a distortion with the force which causes it, to have a particular form deduced from experiment. And we shall also leave out of consideration the thermal or electric effects which compression or distortion generally produce. We shall also suppose fluids, whether liquids or gases, to be either INCOMPRESSIBLE or compressible according to certain known laws; and we shall omit considerations of fluid friction, although we admit the consideration of friction between solids. Fluids will therefore be supposed perfect, i.e. such that any particle may be moved amongst the others by the slightest force.
402. When we come to Properties of Matter and the Physical Forces, we shall give in detail, as far as they are yet known, the modifications which farther approximations have introduced into the previous results.
403. The laws of friction between solids were very ably investigated by Coulomb; and, as we shall require them in the succeeding chapters, we give a brief summary of them here; reserving the more careful scrutiny of experimental results to our chapter on Properties of Matter.
404. To produce sliding of one solid body on another, the surfaces in contact being plane, requires a tangential force which depends -(1) upon the nature of the bodies; (2) upon their polish, or the species and quantity of lubricant which may have been applied; (3) upon the normal pressure between them, to which it is in general directly proportional ; (4) upon the length of time during which they have been suffered to remain in contact.
It does not (except in extreme cases where scratching or abrasion
takes place) depend sensibly upon the area of the surfaces in contact. This, which is called Statical Friction, is thus capable of opposing a tangential resistance to motion which may be of any requisite amount up to u R; where R is the whole normal pressure between the bodies; and j (which depends mainly upon the nature of the surfaces in contact) is the co-efficient of Statical Friction. This co-efficient varies greatly with the circumstances, being in some cases as low as 0.03, in others as high as 0.80. Later we shall give a table of its values. Where the applied forces are insufficient to produce motion, the whole amount of statical friction is not called into play; its amount then just reaches what is sufficient to equilibrate the other forces, and its direction is the opposite of that in which their resultant tends to produce motion. When the statical friction has been overcome, and sliding is produced, experiment shows that a force of friction continues to act, opposing the motion, sensibly proportional to the normal pressure, and independent of the velocity. But for the same two bodies the co-efficient of Kinetic Friction is less than that of Statical Friction, and is approximately, the same whatever be the rate of motion.
405. When among the forces acting in any case of equilibrium, there are frictions of solids on solids, the circumstances would not be altered by doing away with all friction, and replacing its forces by forces of mutual action supposed to remain unchanged by any infinitely small relative motions of the parts between which they act. By this artifice all such cases may be brought under the general principle of Lagrange ($ 254).
406. In the following chapters on Abstract Dynamics we will confine ourselves strictly to such portions of this extensive subject as are likely to be useful to us in the rest of the work, or are of sufficient importance of themselves to warrant their introduction-except in special cases where results, more curious than useful, are given to show the nature of former applications of the methods, or to exhibit special methods of investigation adapted to the difficulties of peculiar problems. For a general view of the subject as a purely analytical problem, the reader is referred to special mathematical treatises, such as those of Poisson, Delaunay, Duhamel, Todhunter, Tait and Steele, Griffin, etc. From these little is to be learned save dexterity in the solution of problems which are in general of no great physical interest—the objects of these treatises being professedly the mathematical analysis of the subject; while in the present work we are engaged specially with those questions which best illustrate physical principles.
STATICS OF A PARTICLE-ATTRACTION.
407. We naturally divide Statics into two parts—the equilibrium of a Particle, and that of a rigid or elastic Body or System of Particles whether solid or fluid. The second law of motion suffices for one part-for the other, the third, and its consequences pointed out by Newton, are necessary. In the succeeding sections we shall dispose of the first of these parts, and the rest of this chapter will be devoted to a digression on the important subject of Attraction.
408. By $ 221, forces acting at the same point, or on the same material particle, are to be compounded by the same laws as velocities. Therefore the sum of their resolved parts in any direction must vanish if there is equilibrium; whence the necessary and sufficient conditions.
They follow also directly from Newton's statement with regard to work, if we suppose the particle to have any velocity, constant in direction and magnitude (and § 211, this is the most general supposition we can make, since absolute rest has for us no meaning). For the work done in any time is the product of the displacement during that time into the algebraic sum of the effective components of the applied forces, and there is no change of kinetic energy. Hence this sum must vanish for every direction. Practically, as any displacement may be resolved into three, in any three directions not coplanar, the vanishing of the work for any one such set of three suffices for the criterion. But, in general, it is convenient to assume them in directions at right angles to each other.
Hence, for the equilibrium of a material particle, it is necessary, and sufficient, that the (algebraic) sums of the applied forces, resolved in any one set of three rectangular directions, should vanish.
409. We proceed to give a detailed exposition of the results which follow from the first clause of § 408. For three forces only we have the following statement.
The resultant of two forces, acting on a material point, is repre
sented in direction and magnitude by the diagonal, through that point, of the parallelogram described upon lines representing the forces.
410. Parallelogram of forces stated symmetrically as to the three forces concerned, usually called the Triangle of Forces. If the lines representing three forces acting on a material point be equal and parallel to the sides of a triangle, and in directions similar to those of the three sides when taken in order round the triangle, the three forces are in equilibrium. Let GEF be a triangle, and
B let MA,MB,MC, be respectively equal and parallel to the three sides EF, FG, GE of this triangle, and in directions similar to the consecutive directions of these sides in order. The point M is in equilibrium.
F 411. [True Triangle of Forces. Let three forces act in consecutive directions round a triangle, DEF, and be represented respectively by its sides: they are H not in equilibrium, but are equivalent to a couple. To prove this, through D draw DH, equal and parallel to EF, and in it introduce a pair of balancing forces, each equal to EF. Of the five forces, three, DE, DH and FD, U
E are in equilibrium, and may be removed, and there are then left two forces, EF and HD, equal, parallel, and in dissimilar directions, which constitute a couple.]
412. To find the resultant of any number of forces in lines through one point, not necessarily in one planeLet MA, MA,, MA, MA, repre
D" sent four forces acting on M, in one plane; required their resultant. Find by the parallelogram of forces,
D'" the resultant of two of the forces, MA,
D and MA. It will be represented by MD. Then similarly, find MD”, the resultant of MD (the first subsidiary resultant), and MĂ,, the third force. Lastly, find MD'”, the resultant of MD and MA, MD" represents the
AI resultant of the given forces.
Thus, by successive applications of the fundamental proposition, the resultant of any number of forces in lines through one point can be found.
413. In executing this construction, it is not necessary to describe