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A model for colliding objects in the asteroidal belt is formulated. An integro-differential equation describing the evolution of a system of particles undergoing inelastic collisions and fragmentation is derived and solved for steady-state conditions. It is found that the number density of particles per unit volume in the mass range m to m + dm is Am−a dm, where A and α are constants (provided that certain conditions are satisfied). The population index α can then be derived theoretically; for asteroids and their debris, α = 1.837, in agreement with an empirical fit to the observed distribution. Various statistical properties of the distribution can be derived from the model. It is found that, for asteroidal objects, catastrophic collisions constitute the most important physical process determining particle lifetimes and the form of the particle distribution for particles sufficiently large that radiation effects are unimportant. The lifetime of the largest asteroids is found to be of the same order of magnitude as the probable lifetime of the solar system; therefore, some of the largest asteroids may have survived since the time of creation, whereas most smaller ones have not and are collisional fragments, according to the present model.

Several attempts have been made lately to simplify the finding of bearings of stars with a view to their more general use for marching at night during military operations. Some of these consist of pocket-books of tables giving the bearings for different times and latitudes, whilst others are movable diagrams or mechanical contrivances of a more or less useful nature. The most recent of the latter is " The Mechanical Star Bearing-Finder," devised by Mr. E. T. Goldsmith, R.N., Senior Naval Instructor, Royal Naval College, Keyham, which has just been published by Messrs. G. Philip & Son. Upon a strip of linen, measuring about 3J by 13J inches, are shown, in their relative positions, the most suitable stars for marching purposes in the South of England and the North of France, and on a line running along the bottom of this star-map are the months and days. This strip-like linen star-map is made to slide along in a mounted cardboard cover, upon the lower edge of which are given the hours of the day and night. Next there is a transparent celluloid protractor upon which curves are drawn giving the bearings of any of the stars seen on the chart through the protractor, when the latter is set for any time by means of a suitable pointer. In using this contrivance for finding the bearings of stars, first the star chart is moved along until the date is in a line with one ofthe edges ofthe time scale at the bottom of the outer cover, and then the celluloid bearing protractor on the top of the star chart is placed so that an arrow in its centre is opposite the time for which the bearing is required. When this is done the bearing of any star on the star-map that is visible through the celluloid protractor can with care be read off to within a degree or two. Allowance has been made for " variation," so that the bearings read off the protractor are magnetic and not true bearings, as is usually the case with devices of this sort. It seems doubtful if this is an advantage, especially as the variation is not the same for the south of England as for northern France, and is changing every year. By a simple arrangement for giving the proper declination of the sun, the same protractor can be used for finding the bearing of the sun for any date and time. Additional protractors are in preparation for use in other areas, of which those for the south of Scotland and the north of England are now ready. There is no doubt that this device is ingenious, but how far it will prove of practical utility for the purpose of star marching remains to be seen. Since the time must be known, it is also a question whether a set of pocket tables prepared for the same stars and limits of latitudes, giving the bearings for every degree at definite times would not have been of more service, handier to use, and have given greater accuracy. In the explanatory text there are one or two mistakes in the spelling of star names that might be corrected in another edition ; for instance Fomalhaut is spelt Formalhaut, and a Ophiuchi is given as a Opiuchi. E. A. R.