748 MECHANICS Rate. 294. The statements in the last two sections are, in f ict, merely particular cases of Newton s two interpreta tions of action in the third law, which have already been discussed ( 165, 167). Analytically, the whole affair is merely this : if s be the space described, v the speed of a particle, dv dv ds dv dt ds dt ds Hence the equation of motion (formed by the second law) .ms mv=f, which gives / as the time-rate of increase of momentum, may be written in the new form dv d , mv ,- = -;- (irat> 2 ) =7 , ds ds^ 2 giving/ as the space-rate of increase of kinetic energy. 295. But a mere rate, be it a space-rate or a time-rate, is not a thing which has objective existence. No one would confound the bank rate of interest with a sum of money, nor the birth or death rate of a country with a group of individual human beings. These rates are, in fact, mere abstract numbers, by the help of which a man may com pute interest per annum from the amount of capital, or the number of infants per annum from the amount of the population. The gradient of temperature, in an irregularly heated body, is a mere vector-rate, by the help of which we can calculate how much energy (in the form of heat) passes in a given time across any assigned surface in the body. To attribute objectivity to a rate is even more ridiculous than it would be to attribute it to a sensation, or to a thought, or to a word or phrase which we find useful in characterizing some material object. 296. On the other hand, all these different kinds of rates have been introduced and continue to be employed, be cause they have been found to be useful. There is no harm done by retaining them, provided those who use them know that they are introduced for convenience of expression, and not because there are objective realities corresponding to them. Even such a term as "centrifugal force" is some times useful ; but always under the proviso that he who employs it shall remember that it is only one side of the stress under which a particle of matter is compelled, in spite of its inertia, to move in a curved line. But the term must be taken, like "algebra," "theodolite," "Abracadabra," or any other combination of letters whose derivation is un certain or unknown, as one and indivisible, to which a certain definite meaning is attached, and as having nothing whatever to do with the meaning or derivation of the word centrifugal, whose embodiment in it is a perennial monument to the memory of an old error. Potential 297. The main characteristics of energy, especially from onergy the experimental point of view, have already been discussed under DYNAMICS (</..) and ENERGY (q.v.). But there is one point of importance connected with it which comes more naturally here than in either of the articles referred to. When two measurable quantities, of any kind, are equi valent to one another, their numerical expressions must involve the same fundamental units, and in the same manner. This is obvious from the fact that an alteration of any unit alters in the inverse ratio the numerical measure of any quantity which is a mere multiple of it. And equivalent quantities must always be expressed by equal numbers when both are measured in terms of the .same system of units. It appears, therefore, from the conservation of energy directly, as well as from the special data in 111, 113, that potential energy must, like kinetic energy, be of dimensions [ML 2 T~ 2 ]. Now it is impossible to conceive of a truly dormant form of energy whose magnitude should depend in any way on the unit of time; and we are therefore forced to the con clusion that potential energy, like kinetic energy, depends kinetic. (in some as yet unexplained, or rather unimagined, way) np.m motion. For the immediate purposes of this article the question is not one of importance. We have been dealing with the more direct consequences of a very compact set of laws, exceedingly simple in themselves, originally based upon observation and experiment, and, most certainly, true. But reason cannot content itself with the mere consequences of a series of observed facts, how ever elegantly and concisely these may be stated by the help of new terms and their definitions. We are forced to inquire into what may underlie these definitions, and the laws which are observed to regulate the things signified by them. And the conclusion which appears inevitable is that, whatever matter may be, the other reality in the physical universe, energy, which is never found unassociated with matter, depends in all its widely varied forms upon motion of matter. In some cases we are sure, in others we can as yet only suspect, that it depends upon motions in a medium which, unlike ordinary matter, has not yet been subjected to the scrutiny of the chemist. But the question, in its generality, is one of the most obscure in the whole range of physics. In the articles ATOM, ATTRACTION, ETHER, will be found nearly all that is yet known on this profoundly difficult subject. But to what is there said must be added the remark that a state of strain of the ether, whether associated with the propagation of light and radiant heat or with a statical distribution of electricity, represents so much " potential " energy, and must in its turn in some way depend on motion. 298. The remarks of Clerk Maxwell on the nature of the evidence for Newton s first law of motion raise a question, in some respects novel, but in all respects well worthy of careful study. He says : "Our conviction of the truth of this law may be greatly strengthened by considering what is involved in a denial of it. Given a body in motion. At a given i:istant let it be left to itself and not acted on by any force. What will happen ? According to Newton s law it will persevere in moving uniformly in a straight line ; that is, its velocity will remain constant both in direction and magnitude. "If the velocity does not remain constant let us suppose it to vary. The change of velocity must have a definite direction and magnitude. By the maxim that the same causes will always pro duce the same effects, this variation must be the same whatever be the time or place of the experiment. The direction of the change of motion must therefore be determined either by the direction of the motion itself, or by some direction fixed in the body. Let us, in the first place, suppose the law to be that the velocity diminishes at a certain rate, which, for the sake of the argument, we may suppose so slow that by no experiments on moving bodies could we have detected the diminution of velocity in hundreds of years. The velocity referred to in this hypothetical law can only be the velocity referred to a point absolutely at rest. For if it is a relative velocity, its direction as well as its magnitude depends on the velocity of the point of reference. If, when referred to a certain point, the body appears to be moving northward with diminishing velocity, we have only to refer it to another point moving northward with a uniform velocity greater than that of the body, and it will appear to be moving southward with increasing velocity. Hence the hypothetical law is without meaning, unless we admit the possibility of defining absolute rest and absolute velocity "It may thus be shown that the denial of Newton s law is iu contradiction to the only system of consistent doctrine about space and time which the human mind has been able to form." This is a good example of a valuable application of a principle which, in its widest scope, is inconsistent with the true foundations of physical science. It is, in fact;, the exceedingly dangerous "principle of sufficient reason "- which requires for its legitimate use the utmost talent and knowledge on the part of the user. 299. But in all methods and systems which involve the idea of force there is the leaven of artificiality. The true laws of motion, based entirely on experiments of the most extensive and most varied kinds, are those of the conserva tion and of the transformation of energy. With the help Max we on inertia Princi] of sufficie reason. True laws oi
motion