horizontal axis; but as the theory of such waves was foreign to the subject of the book, he deferred until now the publication of the investigation on which that statement was founded. Having communicated some of the leading principles of that investigation to Mr. William Froude in April 1862, the author was informed by that gentleman that he had arrived independently at similar results by a similar process, although he had not published them. The introduction of Proposition II. between Propositions L. and III. is due to a suggestion by Mr. Froude. The following is a summary of the leading results demonstrated in the paper:— Proposition I.-In a mass of gravitating liquid whose particles revolve uniformly in vertical circles, a wavy surface of trochoidal profile fulfils the conditions of uniformity of pressure, such trochoidal profile being generated by rolling, on the under side of a horizontal straight line, a circle whose radius is equal to the height of a conical pendulum that revolves in the same period with the particles of liquid. Proposition II.-Let another surface of uniform pressure be conceived to exist indefinitely near to the first surface: then if the first surface is a surface of continuity (that is, a surface always traversing identical particles), so also is the second surface. (Those surfaces contain between them a continuous layer of liquid.) Corollary. The surfaces of uniform pressure are identical with surfaces of continuity throughout the whole mass of liquid. Proposition III.-The profile of the lower surface of the layer referred to in Proposition II. is a trochoid generated by a rolling circle of the same radius with that which generates the upper surface; and the tracing-arm of the second trochoid is shorter than that of the first trochoid by a quantity bearing the same proportion to the depth of the centre of the second rolling circle below the centre of the first rolling circle, which the tracing-arm of the first rolling circle bears to the radius of that circle. Corollaries. The profiles of the surfaces of uniform pressure and of continuity form an indefinite series of trochoids, described by equal rolling circles, rolling with equal speed below an indefinite series of horizontal straight lines. The tracing-arms of those circles (each of which arms is the radius of the circular orbits of the particles contained in the trochoidal surface which it traces) diminish in geometrical progression with a uniform increase of the vertical depth at which the centre of the rolling circle is situated. The preceding propositions agree with the existing theory, except that they are more comprehensive, being applicable to large as well as to small displacements. The following is new as an exact proposition, although partly anticipated by the approximative researches of Mr. Stokes: Proposition IV.-The centres of the orbits of the particles in a given surface of equal pressure stand at a higher level than the same particles do when the liquid is still, by a height which is a third proportional to the diameter of the rolling circle and the length of the tracing-arm (or radius of the orbits of the particles), and which is equal to the height due to the velocity of revolution of the particles. Corollaries.-The mechanical energy of a wave is half actual and half potentialhalf being due to motion, and half to elevation. The crests of the waves rise higher above the level of still water than their hollows fall below it; and the difference between the elevation of the crest and the depression of the hollow is double of the quantity mentioned in Proposition II. The hydrostatic pressure at each individual particle during the wave-motion is the same as if the liquid were still. In an Appendix to the paper is given the investigation of the problem, to find approximately the amount of the pressure required to overcome the friction between a trochoidal wave-surface and a wave-shaped solid in contact with it. The application of the result of this investigation to the resistance of ships was explained in a paper read to the British Association in 1861, and published in various engineering journals in October of that year. The following is the most convenient of the formulæ arrived at:-Let w be the heaviness of the liquid; f the coefficient of friction; g gravity; v the velocity of advance of the solid; L its length, being that of a wave; z the breadth of the surface of contact of the solid and liquid; the greatest angle of obliquity of that surface to the direction of advance of the solid; P the force required to overcome the friction; then P=fwLz (1+4 sin2 B+sin1 B). 2 g In ordinary cases, the value of f for water sliding over painted iron is 0036. The quantity L (1+4 sin B+sin') is what has been called the "augmented surface." In practice, sin' ẞ may in general be neglected, being so small as to be unimportant. Some Account of Recent Discoveries made in the Calculus of Symbols. Before the publication of Professor Boole's memoir on a "General Method in Analysis," which appeared in the Philosophical Transactions' for 1844, those mathematicians who adopted the symbolical methods suggested by the researches of Lagrange and Laplace, confined themselves to the use of commutative symbols, and the science was consequently very limited in its applications. It received a fresh impulse from the very remarkable memoir of Professor Boole mentioned above, in which an algebra of non-commutative symbols was invented and applied to the integration of a large class of linear differential equations. It occurred to the author that the proper method of extending the calculus was to construct systems of multiplication and division for functions of non-commutative symbols. This he accordingly effected in his memoir published in the Philosophical Transactions' for 1861. As the symbols are non-commutative, two distinct systems of multiplication and division, internal and external, arise for each class of symbols employed. Let p and be two symbols combining according to the law ƒ (π).pm=pm ƒ (ñ+m), where f() is any function of (T), then he gave, in the memoir alluded to, equations to determine the conditions that a symbolical function such as n-2 p2 þn (#)+p”—1 Þn−1 (#)+p”—2 $n−2(π)+ &c. +$。 (#) may be divisible internally and externally without a remainder by the symbolical function py, ()+1⁄4。 (π), where Pn (T), Pn-1 (T), Pn-2 (π)... Þ。 (T), V, (T) and y。 (T) 0 are all rational functions of (#), or, in other words, that pу, (Ã)+↓。 (π) may be an internal or external factor of p" (T)+pa¬1 În−1 (π)+ &c., and also an equation to determine the condition that ↓, (p). π+。 (p) may be an internal factor of P3 (p).3 +2 (p) π2 + 1 (p). π +。 (p). 1 0 He then gave some theorems for the transformation of certain functions of these symbols, which lead to some very curious theorems in successive differentiation: he has treated this part of the subject more fully in the Philosophical Magazine for April 1862. In a subsequent part of his paper in the 'Philosophical Transactions,' he established binomial and multinomial theorems for these symbols, by showing how to expand (p2+p✪ (π))” and (p2+p¤~1 0, (π)+p2¬2 02 (π)+ 0,(T)+....)" in terms of (p) and (T). At the end of the paper he gave some methods for solving differential equations by a process analogous to the "Method of Divisors" in the theory of algebraical equations. In his second memoir "On the Calculus of Symbols," published in the Philosophical Transactions' for 1862, he has shown how we may find the highest common internal divisor of functions of non-commutative symbols, and also how we may resolve them in all possible cases into two equal factors, a process analogous to that for extracting the square root in common algebra. He then investigated the theory of multiplication in this calculus more generally. He gave a rule to find the symbolical coefficient of pm in a continued product of the form (p+02 (π)) (p+02 (π)) (p+l3 (π)) . . . . . . . . . (p+02 (T)). After this he resumed the consideration of the binomial and multinomial theorems explained in the former memoir. He gave the numerical calculation of the coeffi cients of the general term of the binomial theorem, as explained in the first memoir. In this the expansion was effected in terms of p and, but we may suppose the expansion effected in terms of (p) alone. In that case the coefficient of the general term would be symbolical, and a function of (#). He had calculated its value in the memoir, and also the value of the corresponding general symbolical_coefficient in the multinomial theorem supposed expanded in powers of p alone. He concluded the paper by giving a method to expand the reciprocal binomial (π2+0 (p) dπ)” in terms of (). The general cases of division yet remained to be worked. This has been effected by Mr. Spottiswoode in a very able and beautiful paper published in the 'Philosophical Transactions' for 1862. He has there given in full the division of Pn (p) "+n-1 (p) "1+$n-2 (p) π"-2+ &c. .....+$。 (p) π internally and externally by , (p) + (p); secondly, the division of Þn (P) ☎ +Pn_1(p) π”− 1 +Pn-2 (P). π" -2 + ·· ··· +&。 (p) internally and externally by ¥m (p) πTM+¥m~1 (P) πTM−1+¥m−2 (p). π−3+. . . . +¥o (P) ; thirdly, the division of p2 Þ2 (π)+p”-1 Þn-1 (#)+p2¬2 Pn−2 (π)+........+$。 (π) internally and externally by .... He has fully investigated the conditions that the divisor in each case may be an internal or external factor of the dividend, and his results, which are expressed by means of determinants, will be found extremely interesting. The author in conclusion states that he believes the form in which the calculus now stands will be permanent, and that subsequent improvements will be very much based on extending systems of multiplication and division to other symbolical expressions, in which the laws of symbolical combination are different from those here assumed. On some Models of Sections of Cubes. By C. M. WILLICH. These were carefully-executed models, designed to illustrate certain simple propositions in solid geometry relative to the volumes, &c. of solids formed by the section of a cube by planes. The author wishes, at the same time, to place on record the simple fraction, which gives an extremely close approximation to the side of a square equal in area to a circle of which the diameter is unity. ASTRONOMY. Some Cosmogonical Speculations. By ISAAC ASHE, M.B. The author considered that the present planiform condition of the system disproved the common view that it had formerly been a gaseous sphere, and proved that it had originally been a liquid plane, as Saturn's rings are at present; nor yet in a heated condition, since he thought that, though capable of transformation, heat could no more be absolutely lost than its equivalent, motion. The planets had, doubtless, been originally molten; but this heat the author ascribed to the collision of particles, during their formation, from the liquid plane described. This formation he ascribed to the development of a centre of attraction in the liquid plane, and showed how, in a revolving plane, a diurnal rotation from west to east might hence be originated, the particles so attracted acting as a mechanical "couple" of forces on the planet during its formation. From the distance between the interior and exterior planets, he inferred the former existence of two rings, as in the system of Saturn, the asteroids being probably formed from small independent portions of matter between these rings. He considered that the planets also first existed individually as planes, basing this view on the uniformity of plane observed in the orbits of the satellites. The satellites themselves he considered to have been formed from portions of matter left behind during the contraction into a globe of such a plane, which had at first occupied the whole space included within the present orbits of the satellites. This view of the formation of the satellites he based on the fact that the period of diurnal rotation in each of them corresponded with the period of its revolution round its primary, which he showed would be the case with any body whatever, if so left behind or lifted off a planet. The author then discussed the chemical changes that would ensue on the surface of the earth after it had assumed the globular form. Oxidization of its metallic constituents would absorb a vast proportion of its gaseous matter, and the formation of water would remove a great deal in addition. Hence the absence of atmosphere or water on the moon's surface might be accounted for, as she would carry off with her only th portion of the gaseous elements of the planet, and her surface exposed to the chemical action of those elements would be much more than th that of the earth. Water also might be quite absorbed on her surface in the formation of hydrates of the alkaline and earthy bases. This On the earth, sodium would unite with chlorine, and common salt would result; and to the large amount of salt so formed the author ascribed the saltness of the ocean; rivers could only carry to the sea salt obtained from soil originally deposited by the ocean, and which must therefore have derived its salt from the sea. process must be still going on, and hence Dr. Ashe inferred that the sea could never have become salt, or be now increasing in saltness, from that cause; hence he dissented from that view, which was the one universally put forward by geologists. On a Group of Lunar Craters imperfectly represented in Lunar Maps. One of the objects of lunar maps should undoubtedly be such a representation of the forms of the irregularities of the moon's surface, that a student may readily, at the suitable epochs, ascertain the general outlines and configurations of the parts which he is studying, so as to be certain that he has not misapprehended either the position or form of any particular portion of the lunar surface. A map constructed for a given epoch, at the full for instance, that shall give those features by which every crater, mountain-chain, and plain may be instantly recognized, is at the present moment a desideratum. Indeed, on such a map some craters would not find place. A certain angle of illumination is necessary to bring out saliently the distinguishing features of a crater or mountain-chain; and a series of maps that would exhibit each to the best advantage, must include as many distinct epochs of illumination in their construction as there are meridians encircling the lunar globe. One of the greatest monuments of the skill and industry characterizing astronomical science is undoubtedly Beer and Mädler's large map of the Moon. To the student of selenography it is invaluable; his progress would be slow without it. The writer of this paper cannot, however, agree with Crampton "that every mountain and every valley, every promontory and every defile on the moon's surface, finds its representative on that map." On the contrary, in his examination of the lunar surface, he has met with several instances of features not recorded thereon, a recent instance of which forms the subject of the present paper. In the neighbourhood of a fine chain of craters that come into sunlight from ten to thirteen days of the moon's age, and are well seen under the evening illumination from twenty-one to twenty-four days of the moon's age, lying in the northern regions of the moon from 57° to 74° N. Lat., and from 25° to 50° E. Long., and designated Philolaus, Anaximenes, and Anaximander, with an unnamed crater between Anaximenes and Anaximander, are three crater-form depressions, of which there are numerous examples. on the moon's surface, the usual characteristics being, 1st, an extensive floor, exhibiting a variety of surface in different specimens, often pierced with small craters and diversified with hills; 2nd, a more or less perfect rampart, here and there pierced with craters, and rising into elevated peaks, so that the entire depression is readily recognized as a distinct formation, completely separated from its surrounding neighbours. Two such depressions, lying nearly in the same meridian, and connected by a table-land or plateau, are very imperfectly, if at all, represented by the German selenographers. The sketch accompanying this communication, taken at Hartwell, on Sept. 18, 1862, under the evening illumination, exhibits the general characters of the northern depression, viz. a floor pierced by a line of eruption (a common feature in several lunar forms), a nearly continuous rampart on the east and west sides, rising into a considerable mountain mass at the north angle marked B by Beer and Madler, pierced by the crater Horrebow, and connected by the steep rocks that form the north_boundary of the plateau. It is proposed, in accordance with a suggestion by Dr. Lee, to designate this depression "Herschel II." Beer and Mädler thus describe the table-land : "South-easterly of Horrebow is a large plateau, fourteen German miles broad, and from twenty to twenty-five German miles long, appearing less from foreshortening. The western border stretches from the western corner of Horrebow to that of Pythagoras, and is rather steep. An offshoot from the same stretches to Anaximander. The southern boundary is denoted by the crater Horrebow B (+58° 9′ Lat., and -42° 0′ Long.), the northern boundary by two craters e and ƒ Pythagoras. It rises on the east, in three great steep mountains of a very dark colour, straight up to the plateau, and only faint traces extend from thence still further towards the east. The most southerly of these three mountains is 919 toises high, while all three of the mountains appear to be exactly similar to each other in height, form, and colour. "The surface of the plateau itself has, besides several craters, among which Horrebow A(+58° 40′ Lat., and -45° 30′ Long.), 2·67 German miles in diameter, is the largest, deepest, and brightest,-only a few scarcely perceptible ridges, and may accordingly be considered as an actual level. But whether this landscape, containing nearly 200 square German miles, is to be distinctly recognized as one connected whole, depends very much upon illumination and libration.” It is proposed to designate this table-land" Robinson," in honour of the Astronomer of Armagh. The following description of the same table-land is taken from the author's observations, dated London, 1862, March 12, 6 to 10h 30m G. M. T., moon's age 124-13, morning illumination. Instrument employed, the Royal Astronomical Society's Sheepshanks telescope No. 5, aperture 2.75 inch. "South of the crater or depression Herschel II. is another, well defined, but not so large. Between the two is a table-land, in which at least five craters have been opened up. Two are in a line with Horrebow; both are given by Beer and Mädler; the northern one is marked B [Horrebow B], the southern is undesignated. The principal crater in this table-land is marked A by Beer and Mädler [Horrebow A]; the three form a triangle: the two remaining craters are near together, and nearly cast of A; the largest is marked d by Beer and Mädler, the other e. All the craters are shown on the map. [Note.-The crater d is referred to in the foregoing translation as ƒ Pythagoras; Beer and Mädler thus speak of it :-"Through an oversight, the lettering Pythagoras d occurs twice on our map; once for a slightly depressed crater on the edge of the previously-described plateau."] "The table-land lies nearly in the direction of the meridian: the mountains on the north slope, or rather their rugged and precipitous slopes, dip towards the large crater Herschel II.; while those on the south [the three dark mountains before mentioned] dip towards the other and smaller crater, which it is proposed to designate South.' On the west the table-land abuts on the border of the Mare Frigoris, while on the east it extends to some mountain-ranges beyond Anaximander." [The reader will notice a discrepancy in the descriptions as regards the points of the lunar horizon. It was thought better to leave each description as given by the writers, rather than attempt a conversion of them; especially as future observers can decide upon which they will adopt, consistent with the principles of lunar topography.] The form of the table-land before described is irregular. In the sketch it appears to be confined to the area between Herschel II. and "South," and this is the most conspicuous portion of it; but on the night of the 31st of January, 1863, under the morning illumination, it was seen to extend to the north of a crater then coming into sunlight eastward of "South," which it is proposed to designate "Babbage." A |