The Arithmetical, or, as we should say, Algebraic, Problems of Diophantus involve a single unknown quantity, symbolised by ς′ or ς°′. The number for which this symbol stands is described as πλῆθος μονάδων ἄλογον, and is spoken of as ὁ ἀόριστος ἀριθμός, the undefined number, or simply as ὁ ἀριθμός, the number, par excellence, of the problem in question.

By all commentators, with but one exception so far as I know, the ς′ of Diophantus is taken as an abbreviation of ἀριθμός, and I find the same statement made unquestioningly in the standard Palæography of Gardthausen. The mathematicians, while expressing no doubt as to the fact, show, however, in some cases, a clear enough perception of the difficulty of accounting for such an abbreviation, and are not all agreed as to its more precise origin. Nesselmann, Cantor, and most others, treat the ς as the final letter of ἀριθμός: Heath, who devotes many pages to a discussion of the subject, puts forward, as an alternative and preferable hypothesis, a suggestion that the ς may be a corrupt cursive abbreviation of the two first letters αρ of the same word.

Clerk Maxwell, in his Electricity and Magnetism, vol. i. § 304, makes the following remark about the rotatory coefficient:—“It should be found, if anywhere, in magnets which have a polarisation in one direction, probably due to a rotational phenomenon in the substance.”

The current which should arise from such a coefficient was first observed by Hall. He passed a current through a strip of metal; he then found two points on opposite sides of the strip, which, while the current was flowing, were at the same potential, and which therefore indicated no current when joined to a galvanometer. The plate was next brought into a uniform magnetic field, and when everything was steady the two points previously at the same potential were no longer so, and a current flowed through the galvanometer. This effect is observable in all conductors.

Kundt has shown that the transverse effect in iron, cobalt, and nickel is proportional to the magnetisation. Such an effect, where the magnetisation appears in the first power, we shall call a Hall effect. In applying the same method to bismuth, he found that no transverse effect was given by the thin plates of the electrolitically deposited metal used by him. That this absence of transverse effect is not characteristic of all bismuth so prepared has been shown experimentally. The question is, What relation, if any, exists between it and the magnetisation? To settle this it is necessary to compare the transverse effect in any given plate with some other effect in the same plate whose relation to the magnetisation is known. Such an effect is the variation of resistance. Goldhammer has shown that this latter is proportional to the square of the magnetisation.

About two years ago I communicated to this Society a paper on the above subject, which was printed in the Philosophical Magazine (1894). The object of the investigations therein discussed was the determination of the law of decrease of torsional oscillations when the range of oscillation was large in comparison with the palpable limits of elasticity. An equation of the form

where y represents the range of oscillation, and x represents the number of oscillations which have taken place since the commencement of the observations in any one experiment, was found to give an exceedingly close representation of the results. The values of the quantities n, a, and b depend on the magnitude of the initial oscillation, and on the previous treatment of the wire. It was also found that, when the oscillations were allowed to die away to a sufficient extent, the value of n tended to diminish. The oscillations were practically isochronous.

This investigation first commenced in the examination of a few special points which had arisen in connection with Professor Ewart's investigations on the Cranial Nerves and Lateral Sense Organs of Elasmobranchs. As this examination revealed facts of more interest than was expected, it occurred to me to investigate very thoroughly the whole of the cranial nerves of this animal, and to publish the results as a continuation of the work already alluded to. The research turned out to be much easier and more interesting than I had anticipated, and, as regards its cranial nerves at any rate, Chimæra is unrivalled among vertebrates, first, for the ease with which its nerves may be dissected, and second, for the almost ideal results that are to be obtained. The endless perplexity and mystification produced by the study of the trigeminal and facial nerves of the cartilaginous fishes,—dispelled to a very great extent, it is true, by the researches of Marshall and Spencer,—would probably have never arisen if Stannius had made a careful examination of the roots of the 5th and 7th nerves of his Chimæra. This, however, he omitted to do in the very form—in fact, as far as we know, the only form—which would have repaid the investigation ; and hence the embryologist stepped in and made a discovery that the anatomist had just, and only just, failed to grasp. The more important facts, therefore, relating to the cranial nerves of the Holocephali are, it is claimed, here recorded for the first time.

Since the foundation of the Scottish Meteorological Society in 1856, a mass of information regarding the climate of Edinburgh has been reduced and published in the Society's Journal from time to time, the data being furnished to that body by its local observers. In addition, numerous unreduced manuscript books of meteorological observations which extend back to the year 1764 are further available for discussion. In 1861 the late Principal Forbes communicated to this Society a paper on the “Climate of Edinburgh,” two elements of climate being alone discussed, viz., temperature and rainfall, and for a period of but forty years. The reduction of the observations taken virtually without a break during the past 132 years was therefore desirable for the discussion of cyclical and other weather changes, for which records covering a long period are absolutely necessary. Through the kindness of the Councils of the Scottish Meteorological Society, the Royal Society of Edinburgh, and the Royal Meteorological Society, the loan of the manuscript registers kept in Edinburgh and its immediate vicinity was obtained.

By films of iron, nickel, and cobalt are here meant electrolytic deposits of these metals on platinised glass. The magnetic properties of such films are interesting in various ways. For example, when they are longitudinally magnetised, we have the case of a magnet where the ratio of length to breadth is very great, and where, consequently, the demagnetising force is very small, e.g., with a film of a 5000th part of a millimetre in thickness, and 5 mms. in length, we have a ratio in length to thickness of 25,000: 1; at the same time the film is so small that it can be tested in very high uniform fields.