Physics

Applications of Differential Equations.

It is worth while illustrating the bearing of a differential equation, and the way in which it embodies a whole series of phenomena which can be dissected from it by the arts of pure mathematics in a purely mechanical though recondite manner, and then handed back to the physicist for interpretation. Naturally the physicist and the mathematician may be combined in the same individual in ex ceptional cases, and as has recently been said (Nature, May 26, 1928) : "It is interesting to speculate whether the intellectual gulf separating the great physical scientists from the rest of their fellows is not greater now than at any previous period in the world's history—whether they have not usurped the place of the metaphysicians." That was true in Newton's day, and is true now ; though no one claims that any single individual possesses the exceptional power displayed by Newton.

i A differential equation which may serve as an illustration is the following : H is any quantity, no matter what, representing some salient feature in the kind of subject dealt with ; it is allowed to have a slope of variable gradient in any of the three directions in space, and to have a rate of variation and indeed an accelerated or variable rate of variation in time. The equation expresses the relation between these extremely abstract possibilities as affecting the (perhaps unknown) quantity H, while A, B, C are physical constants characteristic of the special conditions appropriate to some particular problem, each being capable of a physical inter pretation in any definite case.

When the conditions are such that A is zero, the equation can be made to represent the whole theory of the conduction of heat; and it was thus worked out by the great mathematician Fourier into a complete analysis of diffusion by a series of har monics. It gives primarily the flow of heat through a conductor of any shape, and it was applied by Lord Kelvin, after Fourier, to the cooling of the earth, and to diffusion in gases and liquids, and to many other phenomena. In fact the same differential equation represents his theory of signalling through an Atlantic cable. The working out of the consequences of any equation may be treated as a mathematical exercise ; the interpretation of those consequences in any particular case belongs to the physicist. When the conditions are such that only B is zero, the equation represents the phenomenon of waves, or at least of such waves as travel with a definite velocity independent of wave-length, like air waves and ether waves.

When C is zero the equation represents the settling down of a disturbance, whether electrical or material, after the forces which originated the disturbance have ceased to act. The neutralization of the C term avoids reference to space-considerations and makes it an equation in time alone, like a law of cooling or of leakage, which last is so simple a case as not even to involve A; while if the right-hand side be changed to C H the equation represents all the laws of electrical oscillations as worked out by Lord Kelvin in 1853 and now applied in wireless wave meters. The A term

then represents inertia, or the magnetic property called self-induc tion; the B term signifies resistance, especially the kind of electri cal resistance involved in Ohm's law. When the conditions are such that A and B are both zero, things have become stationary, and the equation reduces to the one so extensively applied by Laplace to solve static gravitational problems. It embodies the conditions which gravitational fields must satisfy, and develops into part of the theory of astronomy.

When none of the coefficients is zero, and all three terms have to be taken into account (especially if a term involving H is added in some cases), the equation represents waves or diffusions travelling through a more complicated medium, at speeds varying with or at least dependent on the wave-length. Waves such as these can occur in the ether when it is interfered with or loaded by electrified particles, or when a medium is a polarizable insulator or a conductor. It can be made to represent the reflection of light by metals, as well as the transmission of waves through the Heaviside layer of ionized air, now becoming familiar to wireless amateurs. It also gives the Heaviside theory of cable signalling, which is more complete than Lord Kelvin's was, and which, ap plied in practice in America and elsewhere, has much improved long-distance telephony. Needless to say the equation in full form can do much more than that. It, or rather an interlacing set of such equations, represents the most general kind of wave ; and the kingdom dominated by waves is now rapidly expanding.

The solution of such equations taxes the powers of all but high mathematicians and may lead to new functions of vast interest and importance. Skilled manipulation of the symbols by a pure mathematician is neither aided nor hindered by information about the particular things they may be intended to represent. The laws are just as valid and self-consistent even if the symbols represent imaginary quantities, as they sometimes do. In the interpretation of the integrated solution however there must be a return to reality.

The vast multitude of facts and phenomena which can thus be treated by a differential equation, so that previously unsuspected consequences can be deduced, is surely of high interest even to those whose education does not enable them to follow the details. The reason why they are so powerful in enabling latent facts to be brought to light is plain enough. It is because the actual proc esses at work are compactly, precisely and completely formu lated, in all their interactions, by these equations, so that they embody the essence of the phenomenon ; their solution and con sequences can subsequently be worked out by the few who possess sufficient skill and insight. This method of procedure—from for mulation, through solution, to interpretation—is indeed the high est example of the advantage possessed, and the power gained, by those who have successfully attained the aim of all science, "rerum co gnoscere causes."