Most investigations on the structure of the stars have proceeded on the supposition that their interiors are gaseous throughout. Without accepting this supposition as final truth, we may adopt it for the moment, for the purely opportunist reason that it provides the most convenient line of approach to an excessively difficult problem. A mathematical theorem, generally known as Poincaré’s theorem, proves to be of the utmost service in discussing the internal state of a gaseous star. We have seen how Helmholtz thought that the energy of the sun’s radiation might come from the sun’s contraction, each layer falling in upon the next inner layer as the latter shrunk, and transforming the energy set free by its fall into heat and light. It is easy to estimate how much energy would be set free by a contraction of this kind. For instance, Lord Kelvin calculated that the contraction of the sun, as it shrunk from infinite size to its present diameter of 865,000 miles, would liberate about as much energy as the sun now radiates in 50 million years. In terms of ergs, the sun’s shrinkage would liberate 6 × 10⁴⁸ ergs of energy. Poincaré’s theorem states that the total energy of motion of all the molecules in any gaseous star whatever is equal to precisely half the total energy which the star would have liberated in shrinking down to its present size. The theorem is true quite independently of whether the star ever has so shrunk or not: nothing is involved but the present state of the star. One interesting consequence is that the further a gaseous star shrinks, the hotter it becomes; if a star shrinks to half its present size, the total energy set free by its shrinkage from infinite size is doubled, so that the total energy of motion of its molecules is doubled, and therefore its average temperature is doubled. This is a special case of what is generally known as Lane’s law. Let us go on with our calculation for the special case of the sun. Poincaré’s theorem tells us that, if the sun is gaseous, the total energy of motion of all its molecules is 3 × 10⁴⁸ ergs. The next thing we want to know is how many molecules there are in the sun. The sun’s weight is 2 × 10³³ grammes, but how many molecules are there to a gramme? The answer of course depends on the type of molecule concerned; there are 3 × 10²³ molecules in a gramme of hydrogen, 2 × 10²² in a gramme of air and only 2·5 × 10²¹ in a gramme of uranium. If we suppose the sun to be made of air, it must consist of 4 × 10⁵⁵ molecules, so that the average energy of motion of each molecule must be 7·5 × 10⁻⁸ ergs, and this represents an _average temperature_, for the sun’s interior, of 375 million degrees. In 1907 Emden, by a different calculation, found that if the sun were made of air, the _temperature at its centre_ would be 455 million degrees. Apart from details, it is clear that the interior temperature of a sun made of air would be one of hundreds of millions of degrees.