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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 1802, the author was informed by that gentleman that he hnd 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 bv Mr. Froude.

The following is a summary of the leading results demonstrated in the payer:—

Proposition I.—In a mass of gravitating liquid whoso particles revolve urutonnly 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.

1'roposition 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 laver 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 nrst rolling circle bears to the radius of that circle.

Corollaries.—The profiles of the surfaces of uniform pressure and of continuitr form an indefinite series of trochoids, described by equal rolling circles, rolling whit 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 tho approximative researches of Mr. Stokes :—

Proposition IV.—The centres of the orbits of the particles in a given surface of equal

{iressure stand at a higher level than the same particles do when the liquid is still, >y a height which is a third proportional to tho diameter of the rolling circle and the length of tho tracing-arm (or radius of the orbits of the particles), and which is equal to the height duo 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 tho 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 tho result of this investigation to the resistance of ships was explained in a paper read to the British Association in 1801, and published in various engineering journals in October of that year. The following is the most convenient of the formula) arrived at:—Let w be the heaviness of the liquid; f the coefficient of friction; g gravity; v tho velocity of advance of the solid; L it* length, being that of a wave; s the breadth of the surface of contact of the solid and liquid; p the greatest angle of obliquity of that surface to the direction of advancecients of the general term of the hinomial theorem, as explained in the first memoir.

In this tho expansion was effected in terms of p and Jt, hut we may suppose the expnnsion effected in terms of (p) alone. In that case the coefficient of the general term would he symbolical, and a function of (V). 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 (ir'+fl (p) Air)" in terms of (ir). The general cases of division yet remained to be worked. This has been effected by Mr. Spottiswoode in a voir able and beautiful paper published in the 'Philosophical Transactions' for 1862. He has there given in full the division of

4* (p) «"+<pn-1 GO +<j>n(p) **-*+ &c + <*>„ 0)

internally and externally by ^r, (p) n-+^0 (p); secondly, tho division of

<Pn GO * +^-i0>)t»"i+*»_s GO .*"-•+ +<*><, 00

internally and externally by

GO Tm++m-i GO T"-1+r-m-» GO. T-~3+. ...++. GO;

thirdly, the division of

PH *. <*>„_, «+/>"-'<!>„-» (*)+••■ •+*» W

internally and externally by

Ho has fully investigated the conditions that the divisor in each case may bo 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 hero 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 nuthor considered that tho 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 vet 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, tho 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 represented by the German Fclenographers. The sketch accompanying this communication, taken nt 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 enst and west sides, rising- into a considerable mountain mass at the north angle marked B by Beer and Miidler, 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 Ilerschel II."

[graphic]

Beer and Madler thus describe the table-land:—

"South-easterly of Horrebow is a large plateau, fourteen German miles broad, and from twenty to twenty-rive German miles long, appearing less from foreshortening The western border stretches from the western corner of Horrebow to that of Pvthagoras, and is rather steep. An offshoot from the same stretches to Anaxininniler. The southern boundary is denoted bv the crater Horrebow B (+58° 0' Lat., and —42° 0' Long.), the northern boundary bv two craters e and/Pythagoras. It rises on the enst, 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 019 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 —4~>° 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, contain'!;-; 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, 18G2, Mnrch 12, 6" to 10h 30m G. M. T., moon's age 12**18, morning illumination. Instrument employed, the Royal Astronomical Society's Sheepshanks telescope No. 6, aperture 2 75 inch.

"South of the crater or depression Ilerschel II. is another, well defined, but net 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 Madler; 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 Madler [Horrebow A"; the three form a triangle: the two remaining craters are near together, and nearly enst of A; the largest is marked d by Beer and Miidler, the other e. All the craters are shown on the map. [Note.—The crater d is referred to in the foregoing translation ns/Pythagoras; Beer and Madler 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 Ilerschel 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 poinw 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 Ilerschel II. and " South," and this is the inert conspicuous portion of it; but on the night of the 31st of January, 1803, under the morning illumination, it was seen to extend to the north of a crater then coining into sunlight eastward of "South," which it is proposed to designate "Babbage.'' A

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