SERRET. Propriétés Géométriques relatives à la Théorie des Fonctions Elliptiques.-L. viii. 495. Note à l'occasion du Mémoire de M. William Roberts, &c.-L. ix. 160. Mémoire sur la Représentation Géométrique des Fonctions Elliptiques et Ultra-elliptiques. Addition au mémoire précédent.-L. x. 257 and 286. It was on this memoir that M. Liouville made so favourable a report to the Institute. V. R. p. 72. Développemens sur une Classe d'Equations relatives a la Représentation des Fonctions Elliptiques.-L. x. 351. Note sur les Courbes Elliptiques de la Première Classe.-L. x. 421. Sur la Représentation des Fonctions Elliptiques de Première Espèce. -Camb. and Dublin Math. Journ. i. p. 187. SOHNCKE. Æquationes Modulares pro Transformatione Functionum Ellipticarum et undecimi et decimi tertii et decimi septimi ordinis.—-C. xii. 178. M. Sohncke here gives the results which he investigates by a general method in the following paper. Equationes Modulares, &c.-C. xvi. 97. V. R. p. 68. TALBOT. Researches in the Integral Calculus.-Phil. Trans. 1836, p. 177; 1837, p. 1. V. R. p. 41. On Comparative Analytical Researches on Sea Water. IN a paper read today in the Chemical Section, I have tried to show that in the ocean between Europe and America, the greatest quantity of saline matter is found in the tropical region far from any land; in such places 1000 parts of sea water contain 366 parts of salt. This quantity diminishes in approaching the coast, on account of the masses of fresh water which the rivers throw into the sea; it diminishes likewise in the westernmost part of the Gulf-stream, where I only found it to be 35.9 in 1000 parts of the water. By the evaporation of the water of this warm current, its quantity of saline matter increases towards the east, and reaches in N. lat. 39° 39′ and W. long. 55° 16', its former height of 36.5. From thence it decreases slowly towards the north-east, and sea water, at a distance of sixty to eighty miles from the western shores of England, contains only 35'7 parts of solid substances; and the same quantity of salt is found all over the north-eastern part of the Atlantic as far to the north as Iceland, always at such a distance from the land that the influence of fresh water from the land is avoided. From numerous observations made on the shores of Iceland and the Faröe islands, it is evident that the water of the Gulf-stream spreads over this part of the Atlantic Ocean, and thus we see that water of tropical currents will keep its character even in high northern latitudes. Besides the southerly direction which any current flowing from the northern polar regions must take, it will, according to well-known physical laws depending upon the rotation of the earth, always take a direction towards the west, and thus be driven towards the eastern shores of the continents, while any tropical current flowing towards the north will, according to the same law of rotation, take a direction towards the western shores of the continents. This is at present the case in the Atlantic Ocean, and its effects upon the shores of Europe, which by a branch of a tropical current are surrounded by warm water, produce a mild and moist climate. The water of the different seas is much more uniform in its composition than is generally believed. In that respect my analyses agree with the newer analyses of atmospheric air, in showing that the differences are very slight indeed. Sea water may contain more or less salt, from a very small quantity, as in the interior part of the Baltic, to an amount of 37.1 parts in 1000 parts, which I found in water from Malta, and which is the greatest quantity I ever observed; but the relative proportion of its constituent saline parts changes very little. In order to get rid of those differences which might arise from the different quantity of saline matter in sea water, I have compared sulphuric acid and lime with chlorine, and the following results are the mean of many analyses : In the Atlantic, the proportion between chlorine and sulphuric acid is 10,000 1188; this is the mean of twenty analyses, which differ very little from each other. In the sea between the Faröe islands, Iceland and Greenland, the same proportion, according to the mean of seventeen analyses, is 10,000: 1193. In the German Ocean, according to ten analyses, it is 10,000: 1191. In Davis's Straits, according to the mean of five analyses, it is 10,000: 1220. In the Kattegat, according to the mean of four analyses, it is 10,000: 1240. Thus it appears that the proportion of sulphuric acid increases near the shores, a fact which evidently depends upon the rivers carrying sulphate of lime into the sea. The proportion between chlorine and lime in the Atlantic Ocean, according to the mean result of seventeen analyses, is 10,000: 297; and in the sea between Faröe and Greenland, according to the mean of eighteen analyses, it is 10,000: 300. In the longitude of Greenland, and more than 100 miles to the south of the southernmost point of that large tract of land, sea water contains only 350 in 1000 parts. In going from this point towards the north-west it decreases constantly, and in Davis's Straits, at a distance of about forty miles from the land, it only contains 32.5 parts of salt in 1000 parts of sea water. This character seems to remain in the current which runs parallel to the shores of North America; and at N. lat. 4310 and W. long. 46 the sea water contained only 33-8 parts of salt. Thus tropical and polar currents seem not only to be different in respect to their temperature, but also in the quantity of salt which they contain; from which it appears, that while the quantity of water carried away from the tropical sea by evaporation is greater than that which rain and the rivers give back to that sea, the reverse takes place in the polar seas, where evaporation is very small and the condensation of vapour very great. The circulation must on that account be such, that a part of the vapour which rises in tropical zones will be condensed in polar regions, and in the form of polar currents flow back again to warmer climates. Although my analyses are only made on water from the ocean between Europe and America, yet little doubt can be entertained that that part of the ocean which separates America from Asia is constituted in a similar manner, and that currents flowing from the poles are the rule, and currents flowing towards the poles the exception. Lime is rather rare in the sea around the West Indian islands, where millions of coralline animals constantly absorb it, the proportion according to five analyses being 10,000: 247; and it is rather copious in the Kattegat, where the numerous rivers of the Baltic carry a great quantity of it into the ocean. The proportion is there, according to four analyses, 10,000: 371. On the Calculation of the Gaussian Constants for 1829. By A. ERMAN. As purely theoretical speculations on natural phænomena remain highly unsatisfactory until they can be founded on a sufficient number of observations, in the same manner collections of the most careful observations must be almost useless before they are thoroughly elaborated according to a given theory. Nay, the accumulation of observed numbers, notwithstanding the value they possess when viewed by themselves, may even become injurious to science, by retarding its progress. Indeed the aspect of progressively increasing, but not duly elaborated, materials, must at last give rise to the apprehension, both on the part of those engaged in furnishing them, and of every one interested in the results to be gathered from them, that the means may be wanting to bring such a stock of matter to bear for their proper purpose. The loss of the whole, that is to say, of data which have not been acquired without the exertion of considerable scientific labour, and which seemed pregnant with beautiful germs, would then be a most discouraging consequence. The British Association for the Advancement of Science has many times proved itself convinced of the truth of this principle. A resolution adopted by the Association in 1833, during its first meeting at Cambridge, warded off the peril just mentioned, even from a department of science whose long-established rate of progress had not been able to protect it sufficiently against such a risk. The reduction of the Greenwich observations of planets, undertaken in consequence of this resolution, and now published by order of the Lords Commissioners of the Admiralty, has been fully appreciated by all astronomers, and particularly by the late M. Bessel, who in the last moments of his life welcomed it as the beginning of a new period in astronomy. Moreover, the condition that a uniform progress of observation and calculation is equally indispensable in less-developed or only nascent branches of physical science, has been expressed by the British Association at its second meeting at Cambridge in 1845; first, by several of the members being inclined to raise the question, whether the continuation of magnetic and meteorological observations were desirable, as long as a great part of the materials collected by them are still waiting their first employment; and, secondly, by including the calculation of the Gaussian constants of terrestrial magnetism for 1829 within the sphere of their own operations, being pleased at the same time to entrust me with the superintendence of the same, and to place at my disposal the sum of £50, granted for this purpose for the year 1845 to 1846. I shall endeavour to point out in a few words the fruits this arrangement seems to promise, and the results it has already obtained. I think we are authorised to suppose that all those phænomena which we have learned to express by numbers, with the help of remarkably accurate instruments, will at length lead us to a theory of the forces which produce them; and that, in consequence, the intrinsic value of observations on such phænomena a value which hitherto could not be demonstrated-will then at once become most evident. It was this expectation alone which often encouraged observers to persevere in labours apparently rather tedious, and the zeal with which the meteorological and part of the magnetic variations are pursued by your members in British and colonial observatories, is, I think, attributable to the same cause. In the branches of physics which they cultivate, these observers, it is true, have still to look to futurity for both kinds of progress, viz. the discovery of an abstract theory, and the true establishment of the same by means of observed numbers. As to the first and most import ant of these steps, they have a consolation in the fate of their predecessors in most similar labours: I allude to the long series of philosophers who devoted themselves during the first thirty years of this century to ascertaining the mean values of magnetic elements for as many points on the surface of the globe as possible, and whose undertakings are so carefully recorded by one of them-I mean Col. Sabine, in his admirable report on magnetic intensity. They too were long enough under the necessity of restricting the immediate application of their operations to refuting some evidently superficial or erroneous theoretical views, and then, after detaching from their results every accidental influence, to register them in the annals of science, as contributions to a theory which they only hoped might be attained. But M. Gauss's admirable theorem, that any terrestro-magnetical element, that is to say, any observable part of the intensity of magnetic force at a given point of the earth, or any angle formed by this force with a given plane or line, can be represented by combining with given functions of the latitude and longitude of this point a limited number (probably twenty-four) of constant quantities, and the way pointed out by him for deriving these constants from a sufficient number of observed mean values of magnetic elements, have in a short time so completely realized these hopes, that a great encouragement was held out, both to former observers of mean magnetic elements, and to those who were then, and are still employed in less-advanced branches of physics: nevertheless this encouragement was but an imperfect one. Το complete it, the possibility of applying those former observations had to be changed to a reality. On this account I am inclined to think that the committee appointed to conduct the cooperation of the British Association in the system of combined magnetic and meteorological observations, have particularly contributed to the satisfaction of their own observers, by encouraging the calculation of the Gaussian constants for 1829; for, by so doing, they in the first place have confirmed their adhesion to the general principle, that no set of observations whatever must remain longer than is indispensably necessary without reduction to theory; and secondly, they have made the immediate application of the mean magnetic values for 1845, that may be furnished by the combined British and Russian observatories, the more probable, as it will then be already preceded by a similar application of the analogous values for 1829. Besides this, to prove the influence of your resolution on the department of science most directly connected with it, I may remark that a more and more exact determination of magnetic constants (the Gaussian) is equally indispensable at the present moment (and for the same reason), as the obtaining of the constants for planetary orbits was formerly, from the moment in which Kepler's and Newton's discoveries opened a possibility of arriving at them. Whatever may be the analogies once to be found in magnetism for the secular variations and other perturbations of planetary orbits, the entrance into these untouched fields of science cannot fail to be effected by fixing the actual values of Gaussian constants. It was under these circumstances that I long ago felt it to be a debt I had contracted towards science, that the magnetic elements which I observed from 1828 to 1830, at about 650 equidistant stations, on a line encircling the globe, between latitudes 67° north and 60° south, conjointly, perhaps, with the magnetic elements that had been observed in Europe during the same years, should be fully applied to the development of the now existing theory. For the undertaking of such a work, however, it was evidently necessary to have more time at my disposal than I have ever enjoyed. Henry Petersen, too, a most industrious and talented young mathematician, who in 1842 had undertaken and performed at my request a small part of M. this comprehensive task, found his leisure hours unequal to its completion. Now, on the contrary, the support of the British Association has enabled and induced this gentleman to suspend his other official duties for the year just expired, and to devote himself entirely to the prosecution of the work in question, in which his success will, I think, be appreciated by the Association, from the results which I have the pleasure of laying before you, accompanied by some remarks on the means employed to obtain them. What proportions these one year's results bear to the final term of the whole labour, and how far they deserve to be continued, is a subject which I shall take the liberty of touching upon at the conclusion of this paper. The object of the calculations committed to my superintendence may be stated to consist in finding, by a sufficiently large series of observations, twenty-four corrections,— to be singly applied to the twenty-four preliminary values, which M. Gauss assigned to the constants of terrestrial magnetism, and in calculating at the same time the probable errors of the so-corrected constants. To this effect (preserving the literal denominations used in M. Gauss's theory of terrestrial magnetism, and marking by AX, AY, AZ, Aw, wad, yai, the differences (theoretical value-actual (or observed) value)), the following expressions have been derived : 0=AX+Ag1, sinu-cos u(Ag11.cosλ+Ah11.sinλ)+2 cosu.sinu.Ag2,0 ~ 11 3 cos3 u -1 cosu). 5 (Ag3,1. cos λ +Ah3‚1. sin λ)— sin u. (3 cos2 u−1)(Ag3,2. cos 2 x 8 u)sinu.(Ag (Ag11.cos &+Ah1‚1.sin λ) 7 +Ah+2.sin 2λ)-(4 cos u-1)sin u(Agt,3.cos 3 A+ A+,.sin 3x) 0=AY+Ag11.sinλ —Ah1‚1. cos λ+cos u (Ag2,1. sin λ—Ah2,1. cos λ) 3 +3 sin2 u (Ag3,3. sin 3 λ—Ah3,3. cos 3 λ)+(cos3 u—; 1 (Ag1‚1‚ sin λ—Ah11‚.cos 2) + (2 cos2 u——)sin u (Ag1‚2. sin 2 › (Ag4,2. —Ah112 cos 2 λ)+3 cos u. sin2 u (Ag4,3 . sin 3 λ — Ah1‚3 cos.3 λ) (1.) (2.) |