Let Y1, Y., Ys, Y4, &c. represent the successive values of y; then we shall have by Taylor's theorem

and as x may be any function whatever, if we substitute for it suc

If now we transfer the exponents of the powers of dy to the characteristic d, the second members of the preceding equations

will become

provided in the developements of the first members we transfer the exponents of the powers of dy to the characteristic d.

Hence we may

obtain

And since the second members of these equations are identical with the corresponding members of the equations represented by (A), we shall obtain

And substituting the last found values of y,, Y., Y 3 Yn r

....

we shall obtain

We understand that the fine collection of minerals which belonged to the late Rev. Richard Hennah, of St. Austell, in Cornwall, will be sold by Mr. King early in June next.

XII. New Arrangement of Primitive Rocks.

Raumer, a German geologist of considerable eminence, has just published a small tract on the Granite of the Riesengebirge. The following is the succession of rocks which he describes as occurring in that country :—

1. Central Granite. It is not intermixed with the adjacent rocks, and does not send out veins from its mass into the rocks that rest upon it.

2. Gneiss and Granite Formation.-This formation rests immediately on the central granite, and the gneiss and granite alternate with and pass into each other.

3. Green-slate.-This formation is principally composed of hornblende, and rests sometimes on No. 2, sometimes on No. 1.

4. Gneiss. It rests upon the gneiss-and-granite formation, or the green-slate, and subordinate to it is a bed of mica-slate. 5. Mica-slate.-It contains great beds of lime-stone.

6. Clay-slate.

It deserves particular attention that all these five different formations are wrapped round the central granite in what is called a mantle-form position.

XIII. Ordnance Maps of British Counties.

The circumstances which were thought to render expedient the suspension of the publication of the Ordnance Maps being now removed, the publication of them is resumed, and they may be obtained, as formerly, at the Drawing-room in the Tower, or of Mr. Faden, Charing Cross. As the suspension was only intended to be temporary, not merely the operations of the trigonometrical survey, but those of the mapping and engraving, have been regularly carried on during that period, under the superintendance of

Col. Mudge; so that several county maps will be ready for delivery almost immediately. The maps of Cornwall, Devonshire, Dorsetshire, Hampshire (including the Isle of Wight), Sussex, and that part of Kent which squares in on the Sussex side, with the general work, will be published in a very few weeks: and a separate map of the Isle of Wight is now on sale. The maps of all the contiguous counties north of these are in the hands of the engravers : and that of the whole county of Kent is re-engraving, and in a state of forwardness. When the several plots and portions now planning by the surveyors are finished, at least three-fifths of England and Wales will be ready to be placed successively in the hands of the engravers; and the work will be carried on with all possible expedition consistent with accuracy. These maps are on a scale of an inch to a mile, a scale that admits of an attention to minutiæ which must, of necessity, be disregarded in maps of smaller size. Hence it may not only be expected that the general outline, and the prominent physical circumstances, shall be correctly delineated, but that the minuter points and peculiarities, which are interesting to the topographer and the antiquarian, shall be permanently marked, and readily traced, in these maps.

XIV. New Hygrometer; and on the Mode of cutting Glass. By Mr. Robert Burrhard.

(To Dr. Thomson.)

MY DEAR SIR, Waddon House, March 15, 1816. As you were kind enough to notice an article I sent you a short time since, I am induced to trouble you again, by sending the sketch of a hygrometer on a new construction. As I have but just made the above, I cannot as yet state the results; but I think it will be very sensible and permanent; at least, it will answer as a comparative instrument to make others. Should you consider it worthy a place in your Annals, I shall feel myself honoured by your notice, and remain, dear Sir, very respectfully,

ROB. W. Burrhard.

The above-mentioned instrument (Plate L., Fig. a,) consists of the segment, A, B, C, made by joining together three very light pieces of flat brass, of which the pin, e, is the centre, on which hangs the long thin wire, D, to the lower end of which is attached a large piece of sponge, F. The arc, C, is to be divided decimally from 1 to 100. At g is a male thread, with a heavy nut, for the purpose of adjusting the pin, D, to 1; that is, to the beginning of the scale, or quite dry. At E is a common scale, beam, pin, and check, f, made with a fine bearing. If we suppose the sponge attached, and the instrument hung up, it must be adjusted by turning the heavy nut, g, either nearer or further from the point of bearing, f, until the line, D, with the sponge perfectly dry, comes exactly to the mark 1. The sponge must then be saturated with

moisture, which, from its increased weight, will make the arc, C, recede from 1 to 7 (but which should have been divided decimally); of course the intermediate numbers on the scale denote the degrees of dryness or moisture. From the greater part of this hygrometer being metal, it will not be liable to err, like the catgut, whipcord, &c.; and it may be adjusted at any time by merely drying the sponge, and noting whether it hangs directly at 1, which it must be made to do by turning the nut, g. The points of suspension, e and f, should be further apart than I have drawn them, as the instrument will be more sensible of change.

A more direct way of cutting a glass flask is, first to scratch it in the direction intended to be cut with a glazier's diamond, or common flint will answer; and then, by applying a red-hot poker to the end of the scratch, it may be carried in any direction with perfect ease.

ARTICLE X.

New Patents.

SAMUEL JEAN PAULY, of Knightsbridge, Middlesex, engineer; for an article or substance for making without seams, coats, great coats, waistcoats, habits, cloaks, pantaloons, mantles, stockings, socks, and any other kind of clothing; covers for umbrellas, and hats, and mattrasses; seats and cushions filled with atmospheric air. March 23, 1816.

SAMUEL BROWN, of Westgate, Norfolk, ironfounder; for improvements upon the swing and wheeled plough-carriage and plough-shares. March 23, 1816.

ROBERT CAMERON, Jun., of Edinburgh, paper-maker; for a machine for manufacturing paper on a principle entirely new. March 23, 1816.

EMERSON DOWSON, of Welbeck-street, ironmonger, and JOHN ISAAC HAWKINS, of Tichfield-street, engineer; for improvements or additions to grates and stoves, and an instrument, machine, or apparatus for supplying grates and stoves with fuel. March 23,

1816.

URIAH HADOCK, of Holloway, chemist; for a new species of paint, colour, and cement, for painting and colouring and preserving the interior and exterior of houses, ships, and other things. March 23, 1816.

WILLIAM MACNAMARA, of East Smithfield, plate-glass manufacturer; for a method or methods of manufacturing glass. March 23, 1816.

JOHN SORBY, the younger, of Sheffield, edge-tool maker; for a method of making an augur, for the use of shipwrights, millwrights, carpenters, and other artificers, upon a new and improved construction. March 23, 1816.