10. The mathematical difficulties of the subject commence when we attempt to discuss the law according to which the velocities of the molecules are grouped about their mean value. We are of course at liberty to consider an imaginary gas in which the velocities are grouped at the outset according to any law we please, but in general every collision which occurs will tend to change this law. The problem before us is to investigate whether there is any law which remains, on the whole, unchanged by collisions; and if so whether the velocities of the molecules of a gas, starting from some arbitrarily chosen law, will tend after a sufficient time to obey some definite law which is independent of the particular law from which the gas started.

There are two totally distinct methods of attacking these problems, and these are given in this chapter and the next, the relation between them being discussed in Chapter IV. The present chapter contains the classical method of which the development is due mainly to Clerk Maxwell and Boltzmann (see § 60 below).

The definition of Density.

11. There is no difficulty in defining the density of a continuous substance. If we take a small volume v, enclosing a given point P, and denote by m the mass of matter contained within this volume, then the assumption of continuity ensures that as the volume v shrinks until it is of infinitesimal size, while still enclosing the point P, then the ratio m/v will approach a definite limit ρ, and we define the density at the point P as being the value of the limit ρ.

I give here, at the outset of her married life, the best picture I have been able to draw of my mother's character. Her letters no doubt reveal it by degrees thoroughly and truly, but I think what I have to say may fill up the gaps. It must, however, be kept in mind that I am speaking of a much later time, as my memories of her are naturally more vivid in her later-middle and old age.

These old letters speak of her as gay and merry, and I have been told by old friends of hers that she had the charm of abounding life and high spirits. In my childish reminiscences, I think of her as serene but somewhat grave. The merriment, the jokes, the fun, would all be from my father. One can realise how heavy was the burden of anxiety borne by her so calmly, from seeing what deep effect it produced on her character.

Her charm is difficult to express, but all who knew her well felt its power. Acquaintances at first sometimes strangely misunderstood what she was, and felt awed by a certain reserved gravity of expression before she spoke. One thing is certain that I never knew anyone who was so naturally good.

I mean that I have known those who impress one as having conquered their evil, but with her there seemed no evil to conquer. It was impossible to imagine a selfish or vain thought, an unjust action or untruthful word, an unkind or harsh judgment. Everything about her was wholesome and natural.

My husband gave up his post in the Ecclesiastical Commission in the early part of 1892. In anticipation she writes:

The Grove,Jan. 13th, 1892.

The following was written whilst Bessy was away from home, so that my mother was without her usual companion.

The Grove,Jan. 15th, 1892.

525. IN preceding chapters we saw how the classical system of mechanics failed to account for certain phenomena of physics, and it became clear that these could only be explained in terms of a new system of dynamics which must be supposed to supersede the old classical system when we have to deal with phenomena conditioned by the fine-scale structure of matter.

The first instance of the failure of the classical mechanics was provided by the phenomenon of black-body radiation. On the assumption that black-body radiation represents a state of thermodynamical equilibrium, we found that the observed distribution of radiant energy in the spectrum of an ideal black body could be explained on the supposition that the energy of each part of the radiation-producing mechanism fell into “quanta.” Corresponding to a simple harmonic vibration of frequency v, there was supposed to be an energyquantum of amount hv, where h is Planck's universal constant of which the value has already been given. Calling this energy ∈, it was found that the energy of vibration could be 0, ∈, 2∈, 3∈,… or any integral multiple of ∈, but could not contain fractional parts of ∈. The energy, after retaining one of these values for a certain time, had to be supposed to jump abruptly to some other value, which in turn was retained until another jump occurred.