Thus, the Sun has existed for one-third of the total history of the universe. Some stars are older than the Sun; some are younger. In Chapter 6, our examination of main sequence models neglected the question of how stars form.

Adoption is the formal act by which the form and content of a treaty are settled. The classic principle was that, unless otherwise agreed, adoption needed the consent of all the states participating in drawing up the text. But often, in the case of a multilateral treaty drawn up at an international conference, a text is adopted by a specified majority, although consensus will normally be sought. Adoption of a bilateral treaty text is normally achieved by initialling. The process of establishing the final text is known as authentication and consists of a formal indication that a document contains the authentic and definitive text. Treaties adopted within an international organisation will follow that organisation’s rules. Sometimes, a diplomatic conference will conclude with a summary of proceedings known as a ‘final act’, to which a treaty adopted by that conference will be annexed. ‘Conclusion’ of a treaty is not defined in the Vienna Convention on the Law of Treaties but normally refers, in the case of a bilateral treaty, to signature of that treaty, and, in the case of a multilateral treaty, either to signature of the final act or the date on which the treaty is opened for signature.

The Fourier series is introduced as a very useful way to represent any periodic signal using a sum of sinusoidal (“pure”) signals. A display of the amplitudes of each sinusoid as a function of the frequency of that sinusoid is a spectrum and allows analysis in the frequency domain. Each sinusoidal signal of such a complex signal is referred to as a partial, and all those except for the lowest-frequency term are referred to as overtones. For periodic signals, the frequencies of the sinusoids will be integer multiples of the lowest frequency; that is, they are harmonics. Pitch is a perceived quantity related to frequency, and it may have a complicated relationship to the actual frequencies present in terms of the series. For periodic signals, changes in the relative phase of the partials do not change the perception of sounds that are not too loud.

However, although stars are generally in hydrostatic equilibrium, all stars are also variable. We can distinguish broad families of variation. The most spectacular variation involves catastrophic changes such as supernova explosions. Some stellar variability arises from external influences, such as accretion of mass from a disk or a companion.

When a new state comes into being, it is important to know which treaties, bilateral or multilateral, applied to it when it was part of another state, and which still apply to it. The Vienna Convention on Succession of States in Respect of Treaties 1978 seeks to regulate the issue, but only twenty-three states were parties by July 2022. Certain customary law principles, which the chapter outlines, can, however, be stated with reasonable confidence. The chapter also examines the case of former colonies and other overseas territories, and the approaches known as universal succession and the clean slate doctrine. It takes a detailed look at the two German states, the former Soviet Union, the former Yugoslav Republics, the former Czechoslovakia, and Hong Kong and Macau. It also discusses the role of the depositary in cases of treaty succession.

The Vienna Convention on the Law of Treaties 1969 contains the body of rules governing the law of treaties. Its scope covers treaties between states. The rules largely reflect customary international law. So even where states are not parties to the Convention, its rules will often apply to their treaty relations by virtue of customary international law. Treaties between a state and an international organisation, or between international organisations, are not covered by the 1969 Convention. But international organisations may be party to the separate Vienna Convention of 1986, which adapts the rules of the 1969 Convention to international organisations. If they are not parties to the 1986 Convention, customary international law will apply to their treaty relations. The 1969 Convention does not cover oral agreements, which are in any case extremely rare. The 1969 Convention does not have retrospective effect. The implications of this are discussed.

Full powers is a document produced as evidence that the person named in it is authorised to represent their state in performing certain acts in relation to the conclusion of a treaty, in particular its signature. The production of full powers is a fundamental safeguard for other states that they are dealing with a person with the necessary authority. It should be distinguished from credentials which authorise a person to represent a state at an international conference, in an organisation, or in ordinary bilateral relations with another state. Full powers are not required for heads of state, heads of government and foreign ministers. Full powers may also be dispensed with by the states concerned if it appears from their practice or other circumstances that it was their intention to consider the person as representing the state for such purposes. A state may issue its permanent representative to an international organisation with general full powers, which gives authority for that person to sign treaties generally adopted within that organisation. The content and procedure for the issue of full powers is discussed.

One-dimensional traveling wave pulses are defined and generalized to sinusoidal traveling waves. The mathematical description of sinusoidal waves is considered in depth. It is shown that any traveling wave can be considered the sum of standing waves, and vice versa. Longitudinal polarization is introduced and contrasted with the previous examples involving transverse polarization. The effects of dispersion, where the wave speed may depend on frequency, on wave propagation are briefly discussed.

The distinction is made between linear and nonlinear physics problems. Whereas the linear problems can be solved as a sum of simpler problems, nonlinear situations cannot be treated this way. That has implications for the solutions for the bowed and blown (wind) instruments, where the driving force is nonlinear, so the resonant modes cannot be treated individually. The stick-slip mechanism for bowed instruments is used as an example where friction provides the nonlinear force. The driving terms for reed instruments are also shown to be nonlinear and can be, in part, understood by thinking about negative resistance. The nonlinear coupling between the modes has implications for the overall tuning and for the frequencies of the overtones.