together, or at least approximately so, and the joint result may present a very complicated subject of investigation. The tides of the ocean consist of a series of superimposed undulations. Not only are there the ordinary semi-diurnal tides caused by sun and moon, but a series of minor tides, such as the lunar diurnal, the solar diurnal, the lunar monthly, the lunar fortnightly, the solar annual and solar semi-annual are gradually being disentangled by the labours of Sir W. Thomson, Professor Haughton and others.
Variable stars present interesting periodic phenomena; while some stars, δ Cephei for instance, are subject to very regular variations, others, like Mira Ceti, are less constant in the degrees of brilliancy which they attain or the rapidity of the changes, possibly on account of some longer periodic variation.15 The star β Lyræ presents a double maximum and minimum in each of its periods of nearly 13 days, and since the discovery of this variation the period in a period has probably been on the increase. “At first the variability was more rapid, then it became gradually slower; and this decrease in the length of time reached its limit between the years 1840 and 1844. During that time its period was nearly invariable; at present it is again decidedly on the decrease.”16 The tracing out of such complicated variations presents an unlimited field for interesting investigation. The number of such variable stars already known is considerable, and there is no reason to suppose that any appreciable fraction of the whole number has yet been detected.
Principle of Forced Vibrations.
Investigations of the connection of periodic causes and effects rest upon a principle, which has been demonstrated by Sir John Herschel for some special cases, and clearly explained by him in several of his works.17 The principle may be formally stated in the following manner: “If one part of any system connected together either by material ties, or by the mutual attractions of its members, be continually maintained by any cause, whether inherent in the constitution of the system or external to it, in a state of regular periodic motion, that motion will be propagated throughout the whole system, and will give rise, in every member of it, and in every part of each member, to periodic movements executed in equal