In a previous communication published in the Transactions of the Society (vol. xlv., part ii., p. 407) I attempted to discuss the temperature observations made in Loch Ness during the years 1903 and 1904. Some of the conclusions which I arrived at have not been generally accepted, and in particular limnologists have been slow to acknowledge the existence of the temperature seiche first described by Mr Watson, and which I consider was fully borne out by the observations published in my previous communication. In order to get some ocular demonstration of the possibility of such a phenomenon, I had recourse to laboratory experiments, and it is the description of these experiments which is the main object of the present communication. Besides demonstrating the nature of the temperature seiche, the experiments also throw light on the formation of what has been called by German and Austrian writers the Sprungschicht, and which I now propose to call in English the ‘discontinuity layer,’ which, with the word Sprungschicht, has the merit of being descriptive.

These notes deal mainly—

(1) With the structure of the genital organs of Stylodrilus Gabreteœ (Vejd.), and with its possible identity with Bathynomus Lemani (Grube).

(2) A new species of oligochæt (Stylaria Lomondi).

I owe my opportunity of working on these forms to Sir John Murray, whom I must again thank for his great kindness.

In this paper I wish to refer to some of the more interesting Turbellaria found during the months of July, August, and September 1905 at Ardeonaig on Loch Tay, and during August and the early part of September 1906 at Tarbet on Loch Lomond.

I also wish to describe in more detail a newly discovered Kalyptorhynch, and to put forward some observations on the origin of the nematocysts in some Microstoma lineare.

The opportunity of working on these forms occurred in connection with the work of the Scottish Lake Survey under Sir John Murray, to whom, for his unfailing kindness, I owe a very deep debt of gratitude.

Abnormalities of the digits occur under a great variety of forms, and with such frequency that most medical men interested in the subject are more or less familiar with them.

In studying such cases one cannot fail to be struck with the fact that, although the tendency to abnormality may be inherited, the inheritance is not always “true,” i.e. the precise form is not accurately reproduced in the offspring. Especially, perhaps, is this the case where the abnormality consists in a deficiency of digits. One individual of a family may have four fingers, another three, and another two, or even one, and that one very imperfect. Such cases have often been recorded (see B.M.J., 1886, vol. ii. p. 976).

Bien que les Orcades du Sud aient été découvertes dès 1821, par Powell, sur le Dove, et bien que ces îles aient été visitées à nouveau:

En 1822–23, par Weddell, qui donna le nom au groupe, avec la Jane et le Beaufoy;

En 1838, par Dumont d'Urville, avec l'Astrolabe et la Zélée;

En 1874, par Dallmann, avec le Grönland;

En 1893, par Larsen, avec le Jason;

rien n'était connu de la faune ichthyologique de ces parages au moment où ils furent explorés par l'Expédition Antarctique Nationale Ecossaise (1902–1904).

The body-temperature of the following fishes, crustaceans, and echinoderms has been examined and compared with the temperature of the water in which they live:—Cod-fish (Gadus morrhua), ling (Molva vulgaris), torsk (Brosmius brosme), coal-fish or saithe (Gadus virens), haddock (Gadus œgelfinus), flounder (Pleuronectes flesus), smelt (Osmerus eperlanus), dog-fish (Scyllium catulus), shore crab (Carcinus mœnas), edible crab (Cancer pagurus), lobster (Homarus vulgaris), sea-urchin (Echinus esculentus), and starfish (Asterias rubens). The minimum, maximum, and mean temperature difference for each species are given in the following table:—

The excess of temperature is most evident in the larger specimens. This is well shown in the case of the coal-fish, where in the adult it was 0°·7 C., and in the great majority (11 out of 12) of the young of the first year, 0°·0 C. The body-weight and the conditions under which the fish are captured probably form the most important factors in determining the temperature difference.

In 14 codfish, where the rectal, blood, and muscle temperatures were recorded in the same individual, it was found to be highest in the muscle and lowest in the rectum, the mean temperature difference being 0°·46 C. for the muscle, 0°·41 C for the blood, and 0°·36 C. for the rectum.

In my previous paper (see Geological Magazine, March 1906, p. 131) I explained a method of separating igneous rocks into sixteen groups of chemically similar individual rocks. A more minute subdivision appears, however, desirable for the special purpose of identifying and describing particular rocks. I have therefore devised the following further classification with four times as many groups, and I trust that the proximity thus attained between the individual rocks of each group will amply meet all requirements.

Mathematics accepts the Roman numerals as symbols for definite numerical values, without regard to the question of their origin and early history; but this paper is concerned primarily with the form and origin of these symbols, and only incidentally with their numerical values. Yet, though our subject is only indirectly related to mathematics, a survey of some of the hypotheses that have been put forward to account for the forms of these symbols cannot but prove interesting to mathematicians, as the symbols were in general use until the sixteenth century and are still current everywhere, on the dials of clocks, the title-pages of books, and in printed references to authorities.

At the Council Meeting of 5th July 1907, the following communication was received from the Misses Sang, daughters of the late Dr Edward Sang:—

“We, the daughters of the late Dr Edward Sang, LL.D., F.R.S.E., owners under his will of his collection of MS. Calculations in Trigonometry and Astronomy, having by letter of gift of date 12th February 1906 given the above collection to the President and Council of the Royal Society of Edinburgh, and having, by the cancelling on the 24th May 1907 of their acceptance thereof, received back the collection from the President and Council of the Royal Society of Edinburgh, do hereby give the said collection to the British Nation, and do hereby appoint the President and Council of the Royal Society of Edinburgh custodiers of the said collection, in trust for the British Nation, with power to publish such parts as may be judged useful to the scientific world.

Determinants whose elements are themselves determinants made their appearance at a very early stage in the history of the subject, the first foreshadowing of them being contained in Lagrange's “équation identique et très remarquable” of 1773, namely,

where

This, viewed as a result in determinants, is a case of Cauchy's theorem of 1812 regardingthe adjugate, and the adjugate of course is an instance of the special form to which we have now come. Jacobi's theorem regarding any minor of the adjugate has a like history and may be similarly classified. Passing from the case of the adjugate, where each element is a primary minor of the original determinant, Cauchy also considered the determinants of other “systèmes dérivés,” that is to say, the determinants whose elements are the secondary, ternary, … minors of the original, and gave the theorem that the product of the determinants of two “complementary derived systems” is a power of the original determinant, the index of the power being where n is the order of the original determinant and p the order of each element of one of the “derived systems.”

(1) IN a paper by A. Scholtz entitled “Sechs Punkte eines Kegelschnittes” (Archiv d. Math. u. Phys., lxii. pp. 317–324, year 1878) there appears a statement which, after correction of two misprints, runs thus:—

where ξln=ylzn − ynzl. A year or so later there was published a paper by Hunyady with the title “Beitrag zur Theorie des Flächen zweiten Grades” (Crelle's Journ., lxxxix. pp. 47–69), in which it is asserted, again without proof, that

The object of the present note is to formulate and prove a general theorem of which these are cases, and to draw certain deductions therefrom.

When a large earthquake occurs in any part of the earth, its tremors can be recorded on suitable instruments all over the surface of the globe. These instruments are of various types, the most familiar being the horizontal pendulum, the evolution of which we owe to the labours of a small band of enthusiasts who were engaged by the Japanese Government in the late seventies and early eighties to teach the students of Japan the scientific methods of the West. The most conspicuous of these is undoubtedly Professor John Milne, who in his seismological laboratory established in the Isle of Wight continues to study the mysterious movements of the earth. Prompted by him, the British Association has installed some fifty instruments in various parts of the world; and from the accumulating records furnished by these instruments, Milne pursues his seismological studies. In addition to these fifty British Association stations, there have grown up in recent years many seismological laboratories in Europe, Asia, and America; and the data supplied from all these sources place us in a much better position than ever before to draw sure conclusions from the character of the records.

Many attempts have been made to interpret the systematic proper motions of the stars as the result of a movement of the solar system in space. In recent years several elaborate and extensive investigations have been made, but the increase of the available material in amount and accuracy has not led to a corresponding increase in the accuracy of the so-called “Solar Apex.” Instead, determinations from different groups of stars have shown larger differences than can be attributed to accidental error, and in particular a determination by Kobold, using Bessel's method, has given results at variance with those obtained by Airy's method.

1. Specimens of steel cooled from high temperatures are, as regards their magnetic properties, in a “sensitive state.”

2. After magnetisation, reversal of the field reduces the susceptibility.

3. Cyclic reversals of the field between definite limits bring the magnetic quality to a definite lower value for fields lying between these limits.

4. So long as the field is not changed in sign, the “sensitive state” persists.

5. Cyclic reversals of the field between narrow limits reduce the susceptibility for fields beyond these limits, but not to its lowest possible value.

6. The rate of cooling from the high temperature—if not very rapid—has little influence on the magnitude of the effect.

7. The “sensitive state” does not wear off with time.

8. Mechanical vibration of the specimens considerably reduces the phenomenon.

9. The “sensitive state” is most pronounced in hard steels.

10. The effect is induced to a slight degree in some varieties of steel by temperatures as low as 100° C. It increases with increasing temperatures to about 700° C. Further increase in temperature has little influence.

11. Similar effects are noticeable in cast iron and in cobalt. The “sensitive state” is absent, or is present to only a very slight extent, in soft iron.

12. The effect has a maximum value for fields which give large values of μ, and tends to zero as the saturation point is approached.

§ 1. THE usual method of proving that a function defined as the limit of a sequence of continuous functions is continuous is by proving that the convergence is uniform. This method may fail owing to the presence of points at which the convergence is non-uniform although the limiting function is continuous. In such a case it would be necessary to apply a further test, e.g. that of Arzelà (“uniform convergence by segments”).

In some cases the continuity may be proved directly by means of a totally different principle, without reference to modes of convergence at all. It is, in fact, a necessary and sufficient condition for the continuity of a function that it should be possible to express it at the same time as the limit of a monotone ascending and of a monotone descending sequence of continuous functions.

THIS paper was begun about the close of 1906, in order to fulfil a promise given at the end of the paper “On the convective equilibrium of a gas under its own gravitation only,” published in the Philosophical Magazine, 1887; and part of it was communicated by Lord Kelvin to the Royal Society of Edinburgh at its meeting on 21st January 1907. Since then, however, important additions have been made to it, and the subject has been dealt with more fully than was originally intended. Unfortunately the manuscript was left incomplete at Lord Kelvin's death. It ended with § 35.

MY last communication in reference to the history of skew determinants dealt with the period 1827–1857 (Proc. Roy. Soc. Edin., xxiii, pp. 181–217). The present paper continues the history up to the year 1865, but in addition contains an account of two writings belonging to the previous period, namely, by Brioschi (1855) and Bellavitis (1857).

§ 1.THE objects of the present paper are, first, to describe certain curves which approximate to the graphs of the time of sunset (or sunrise) and end of twilight (or daybreak) all the year round for various latitudes; second, to tabulate the yearly phenomena of light and darkness for different latitudes under various conditions.

The time of sunset depends upon the latitude and the sun's declination,

and the extent of its variations depends upon the obliquity of the ecliptic. The extent of the variations of twilight depends also upon a certain angle a, which is the maximum depression of the sun's centre below the horizon for which the light reflected by the upper strata of the atmosphere is sensible.

FOR the past two years I have been engaged in the investigation of the iron-water of Scotland, England, and Wales, and altogether have examined over a hundred samples. This publication is a preliminary notice of some new species of iron-bacteria which I have found in the course of this investigation. At present the number of known forms belonging to this class is six. In this paper I wish to outline the main characteristics of five new species, reserving for later papers the detailed accounts of their life-histories.