Lord Kelvin: An account of his scientific life and work

by Andrew Gray

In the second edition many topics are more fully discussed, and the contents include a very valuable account of cycloidal motion (or oscillatory motion, as it is more usually called), and of a revised version of the chapter on Statics which forms the concluding portion of the book, and which discusses some of the great problems of terrestrial and cosmical physics.

Various speculations have been indulged in, from time to time, as to the respective parts contributed to the work by the two authors, but these are generally very wide of the mark. The mode of composition of the sections on cycloidal (oscillatory) motion gives some idea of Thomson's method of working. His proofs (of "T and T-dash" as the authors called the book) were carried with him by rail and steamer, and he worked incessantly (without, however, altogether withdrawing his attention from what was going on around him!) at corrections and additions. He corrected heavily on the proofs, and then overflowed into additional manuscript. Thus, when he came to the short original § 343, he greatly extended that in the first instance, and proceeded from section to section until additions numbered from § 343a to § 343p, amounting in all to some ten pages of small print, had been interpolated. Similarly § 345 was extended by the addition of §§ 345 (i) to 345 (xxviii), mainly on gyrostatic domination. The method had the disadvantage of interrupting the printers and keeping type long standing, but the matter was often all the more inspiring through having been produced under pressure from the printing office. Indeed, much was no doubt written in this way which, to the great loss of dynamical science, would otherwise never have been written at all.

The kinematical discussion begins with the consideration of motion along a continuous line, curved or straight. This naturally suggests the ideas of curvature and tortuosity, which are fully dealt with mathematically, before the notion of velocity is introduced. When that is done, the directional quality of velocity is not so much insisted on as is now the case: for example, a point is spoken of as moving in a curve with a uniform velocity; and of course in the language of the present time, which has been rendered more precise by vector ideas, if not by vector-analysis, the velocity of a point which is continually changing the direction of its motion, cannot be uniform. The same remark may be made regarding the treatment of acceleration: in both cases the reference of the quantity to three Cartesian axes is immediate, and the changes of the components, thus fixed in direction, are alone considered.

There can be no doubt that greater clearness is obtained by the process afterwards insisted on by Tait, of considering by a hodographic diagram the changes of velocity in successive intervals of time, and from these discovering the direction and magnitude of the rate of change at each instant. This method is indeed indicated at § 37, but no diagram is given, and the properties of the hodograph are investigated by means of Cartesians. The subject is, however, treated in the Elements by the method here indicated.

Remarkable features of this chapter are the very complete discussion of simple harmonic or vibratory motion, the sections on rotation, and the geometry of rolling and precessional motion, and on the curvature of surfaces as investigated by kinematical methods. A remark made in § 96 should be borne in mind by all who essay to solve gyrostatic problems. It is that just as acceleration, which is always at right angles to the motion of a point, produces a change in the direction of the motion but none in the speed of the point (it does influence the velocity), so an action, tending always to produce rotation about an axis at right angles to that about which a rigid body is already rotating, will change the direction of the axis about which the body revolves, but will produce no change in the rate of turning.[20]

A very full and clear account of the analysis of strains is given in this chapter, in preparation for the treatment of elasticity which comes later in the book; and a long appendix is added on Spherical Harmonics, which are defined as homogeneous functions of the coordinates which satisfy the differential equation of the distribution of temperature in a medium in which there is steady flow of heat, or of distribution of potential in an electrical field. This appendix is within its scope one of the most masterly discussions of this subject ever written, though, from the point of view of rigidity of proof, required by modern function-theory, it may be open to objection.

In the next chapter, which is entitled "Dynamical Laws and Principles," the authors at the outset declare their intention of following the Principia closely in the discussion of the general foundations of the subject. Accordingly, after some definitions the laws of motion are stated, and the opportunity is taken to adopt and enforce the Gaussian system of absolute units for dynamical quantities. As has been indicated above, the various difficulties more or less metaphysical which must occur to every thoughtful student in considering Newton's laws of motion are not discussed, and probably such a discussion was beyond the scheme which the authors had in view. But metaphysics is not altogether excluded. It is stated that "matter has an innate power of resisting external influences, so that every body, as far as it can, remains at rest, or moves uniformly in a straight line," and it is stated that this property—inertia—is proportional to the quantity of matter in the body. This statement is criticised by Maxwell in his review of the Natural Philosophy in Nature in 1879 (one of the last papers that Maxwell wrote). He asks, "Is it a fact that 'matter has any power, either innate or acquired, of resisting external influences'? Does not every force which acts on a body always produce that change in the motion of the body by which its value, as a force, is reckoned? Is a cup of tea to be accused of resisting the sweetening influence of sugar, because it persistently refuses to turn sweet unless the sugar is put into it?"

This innate power of resisting is merely the materiæ vis insita of Newton's "Definitio III," given in the Principia, and the statement to which Maxwell objects is only a free translation of that definition. Moreover, when a body is drawn or pushed by other bodies, it reacts on those bodies with an equal force, and this reaction is just as real as the action: its existence is due to the inertia of the body. The definition, from one point of view, is only a statement of the fact that the acceleration produced in a body in certain circumstances depends upon the body itself, as well as on the other bodies concerned, but from another it may be regarded as accounting for the reaction. The mass or inertia of the body is only such a number that, for different bodies in the same circumstances as to the action of other bodies in giving them acceleration, the product of the mass and the acceleration is the same for all. It is, however, a very important property of the body, for it is one factor of the quantum of kinetic energy which the body contributes to the energy of the system, in consequence of its motion relatively to the chosen axes of reference, which are taken as at rest.

The relativity of motion is not emphasised so greatly in the Natural Philosophy as in some more modern treatises, but it is not overlooked; and whatever may be the view taken as to the importance of dwelling on such considerations in a treatise on dynamics, there can be no doubt that the return to Newton was on the whole a salutary change of the manner of teaching the subject.

The treatment of force in the first and second laws of motion is frankly causal. Force is there the cause of rate of change of momentum; and this view Professor Tait in his own writings has always combated, it must be admitted, in a very cogent manner. According to him, force is merely rate of change of momentum. Hence the forces in equations of motion are only expressions, the values of which as rates of change of momentum, are to be made explicit by the solution of such equations in terms of known quantities. And there does not seem to be any logical escape from this conclusion, though, except as a way of speaking, the reference to cause disappears.