the formula for the rate of radiation ; and hence (provided k remains constant) a star will radiate the same amount of energy as it passes through all stages of evolution until it becomes too dense to behave as a perfect gas. This agrees with the observed fact that the average magnitudes of the giant stars are approximately the same for all spectral types. The formula shows also that the total radiation (and, subject to calculable corrections, the absolute luminosity) is a function of the mass, and it becomes possible to calculate the mass from the luminosity and vice versa. Thus the great majority of the giant stars of types F and G'are comprised within absolute magnitudes 2 M -o to o M -5; these correspond to masses 0-07 to 1-6 times that of the sun an illustration of the great uniformity of stellar masses. The constant k must be obtained from our general knowledge of the radiation of giant stars of known mass; it is approximately 20 C.G.S. units, that is to say, radiation after passing a column of stellar material of 1/20 gram per sq. cm. would be re- duced to one-third (strictly l/e) of its original intensity. This is a very high opacity, but it is of the same order as that found in labora- tory experiments on X rays, to which the high temperature radia- tion within the stars is closely akin. It is remarkable that k appears to vary very little from one star to another in spite of their consider- able differences of temperature.
Taking account of the deviations from the laws of a perfect gas, the theory can be extended (though with less certainty) to dwarf stars. The most interesting point is to determine the maximum tem- perature attained by stars of different mass. Measuring the mass in terms of the sun, a mass 1 should just attain effective temperature 3,000 (type M), below which it would scarcely appear luminous; a mass $ attains 6,000 (type G) ; mass I attains 9,000" (type AS F); and mass 2-5 attains 14,000 (type_ 65). These results will no doubt be revised when better information is available for check- ing the constants of the theory; but they appear to be reasonably probable.
Radiation Pressure. It is found that the pressure of radiation plays a very important part in the dynamical equilibrium of the giant stars. As already mentioned the stellar material is highly opaque to the radiation ; thus the outflowing radiation exerts a large pressure on the absorbing material, tending to support part of its weight. The fraction of the weight carried by the radiation pressure is the quantity (l 0] in the formula already given. For example (taking molecular weight 3-0) we find:
For star of mass * X sun radiation pressure supports 0-044 f weight ' 5 X " 0-457 "
This gives a clue to the remarkable phenomenon that the masses of the stars are so nearly uniform. Why should the matter of the universe have aggregated into lumps, whose size is almost always between J and 5 X sun? We see that this is just the range for which radiation pressure rises from insignificance to importance, and pre- sumably that fact has determined the size of stellar masses. On feneral grounds it is likely that when radiation pressure counter- alances a considerable part of gravitation, the body would be very liable to disruption; accordingly the chances of survival of stars more than five times as massive as the sun, would be small. The material has_-thus tended to divide and subdivide until the separate masses fell just below the danger limit, fixed by this criterion of radiation pressure; and afterwards there was no cause for further division.
Age of the Stars. In discussions of the evolution and dy- namics of the stars or of systems of stars, the problem arises: What is the time-scale of the process? If astronomers were asked to estimate the length of life of a star from its first luminescence to its final extinction, the answers would probably fall into three groups: (a) The short time-scale, urged by Kelvin, giving a life of about 20 million years; (b) a long time-scale, say io 10 years; (c) an ultra-long time-scale of io 16 years and upwards urged by those who believe the stellar universe to have approximately reached statistical equilibrium. Belief in the short time-scale rests on Helmholtz's theory that the star's heat comes from the gravitational energy converted as it contracts; in recent years, the cumulative evidence against this " contraction theory " of a star's energy has become very considerable, and it now, seems clear that the star must have some much larger store to draw on.
The Cepheid variables may perhaps afford us a measure of the rate at which the evolution proceeds. Like other giant stars, 8 Cephei would need to condense very rapidly if the energy which it radiates came solely from contraction: the increase of density must in fact amount to i % in 40 years. As already explained the period of the light variation is intrinsic, and should there- fore change as the density changes. Calculation shows that the period ought to decrease 40 seconds annually. Now 5 Cephei has been under careful observation since 1785 and the decrease of period is only just detectable. The value given by E. Hertz-
sprung is a decrease of 0-08 second per annum. Thus at the present stage evolution is proceeding at a rate no more than s d of that required by the contraction hypothesis: and some source of energy is being drawn on to prolong the star's life 5oo-fold. Assuming that this store of energy is contained in the star (and not picked up continually from space), it seems clear that it must consist of the sub-atomic energy, releasable when the elements are transmuted or possibly when positive and negative electrons annihilate one another. Since all kinds of energy possess mass, an upper limit to the store can be given; the sun's output of heat could be maintained for is-io 12 years if all the energy contained in it were liberated.
III. INSTRUMENTS
The Hooker telescope, a reflector of too in. aperture, installed at Mount Wilson Observatory, is now (1921) the most powerful telescope in the world. It was brought into regular use in 1919. The mirror is a glass disc of thickness 12-8 in. at the edge and 1 1 -6 in. at the centre, weighing over four tons. It is ground to a focal length of 12-88 metres, but can be used with convex mirrors as a Cassegrain with equivalent focal lengths of 48 and 76 metres. The weight on the polar axis is mainly buoyed by cylinders floating in mercury. A 72-in. reflector has been erected at Vic- toria, B.C. Both these large telescopes are giving excellent performance.
A comparatively small telescope of interesting design was constructed by the late B. Cookson at Cambridge; it is a photo- graphic zenith telescope carried on an annulus which floats in mercury. Rotation about the true vertical is thus secured by flotation instead of by reading spirit levels. After Cookson's death the instrument was removed to Greenwich,' where it has been used with great success for determining latitude variation and the constant of aberration. From seven years' observations the value 2o"-44 I "-013 was obtained for the constant of aber- ration; this is probably the best direct determination of the constant, though scarcely so accurate as the value 2o"-47 ob- tained indirectly from the solar parallax. A- somewhat similar instrument in which rotation about the vertical is obtained by suspension instead of by flotation has been recently installed at Durham Observatory.
An appliance very much used in recent years is a coarse grating consisting of parallel and equidistant metal strips placed in front of the object glass. This grating, with say five "lines" to the inch, seems like a travesty of the diffraction gratings used by physicists; but the action is essentially the same. On either side of the undiffracted image of the star subsidiary images appear, which are in reality spectra of the first, second and higher orders. The distance between the two first order images is pro- portional to the average wave-length of the light ; and hence the grating can be used for determining star colour on a quantita- tive scale. It also provides a convenient means of obtaining images whose intensities are in a definitely known ratio (calculated from the widths of the strips and spaces), which is of great value in determining an absolute scale of photographic magnitudes. These objective gratings appear to have been first used by K. Schwarzschild acting on a suggestion from A. A. Michelson. Another optical device, suggested by A. A. Michelson so long ago as 1890, has recently been used with great success at Mount Wilson. An interferometer consists essentially of two light- collectors of moderate aperture separated by a base-line of con- siderable length (as in a range-finder). The beams of light are then brought together, so that for a point source they produce the usual interference fringes. As the base-line extends or contracts the fringes narrow or widen in proportion. For a double star a length and orientation of the base-line can be found in which the bright fringes of one component fall on the dark intervals of the other component, so that the visibility of the fringes is a mini- mum. In this way the position angle and separation of the com- ponents can be measured with great accuracy, and the method is applicable to double stars too close to be resolved in a telescope; in fact the resolving power of the interferometer is greater than that of a telescope of aperture equal to the base-line. At Mount
