present value of the perpetuity of 17. per annum, the interest payable m times in a year.

then will

Art. 23.-If in Art. 19, n be infinite, then will

present value of the perpetuity of 17. per annum,

when the annuity is payable u times in a year, and the interest m times.

For a fuller account on this subject, and for a variety of interesting and useful examples, see Mr. Francis Baily's "Doctrine of Interest and Annuities."

From the preceding articles, the state of the population of a country under given circumstances may easily be determined.

If in any place where there is no migration, and the increase of population observe the following law, the amount of the whole population at any given time may be determined as follows: Let P represent the population of a country at any given period; P = B = number of births, and = D = number of deaths in

P

a

a year, then will

population in the first year; from whencea – B-D

a b

=

P

= e.

=A= popu

Now if in Art. 1, for principal, we write population, for r we write e, and for s we put A, we shall have P (I + e)" lation at the end of n years; therefore,

If m P = A, then will (1+e)" = m, from whence n = 1.m · 7. (1 + e) = a period in which the population would be increased m times.

a

a

If the population decrease, we shall have. P = DB decrease of population in the first year, from whence

a b

=

D-B

= = e. By substituting in Art. 5, we get (1—e')".

P

PA decrease of population in n years; therefore,

7

m

1

If = A', then will (1 - e')", from whence n = 1. — ·2. (1 —e') = [l, m + 1. (1 - e1)] = period in which the population would be reduced th part. If the population be

m

1

P

=

increasing, we shall have by substituting in Art. 3, (1+e) population n years since; and if decreasing, we have by Art. 5, P = population n years since. (See Mr. Milne's Annuities, vol. i. p. 103.)

ARTICLE X.

ANALYSES OF BOOKS.

Philosophical Transactions of the Royal Society of London, for 1822. Part II.

WE hasten, by our promptitude in the analysis of this part, which has just been published, to compensate for our tardiness in reviewing the former one. It contains the following papers:

XIX. Experiments and Observations on the Development of Magnetical Properties in Steel and Iron by Percussion. By William Scoresby, Jun. Esq. (Communicated by Sir Humphry Davy, Bart. PRS.)

"Dr. Gilbert, so early as the year 1600, discovered that iron became sensibly magnetic on being hammered and drawn out while lying in a north and south direction; " but Mr. Scoresby cannot discover "that any magnetical effect by hammering has been produced beyond that of occasioning a deviation in the compass needle, or of giving to floating bars or needles the power of conforming their position to that of the magnetic meridian."

Mr. S. having already "succeeded in determining, in a great measure, the principal laws by which the development and destruction of magnetism in iron by percussion, scowering,

filing, bending, &c. are governed," and which have been published in the Edin. Phil. Trans. for 1821, confines himself, in the present communication, "to the application of these laws to practical magnetism; and particularly to the construction of magnets, without the use of any magnetized substance."

"In examining the magnetical effect of percussion on different kinds of iron and steel, two tests were employed; the weight of iron that the body would lift, and the quantity of deviation that it would produce on a magnetic needle when presented to it in a certain position, and at a certain distance. For the first test, common iron nails of different sizes were made use of: they were of the weights of 2, 4, 64, 14, 24, 37, 45, 88, 130, and 188 grs. For the purpose of securing a good and uniform contact with the magnetized bar, the oxide on the ends of the nails wast removed by means of a fine file, and the extremities were then polished by rubbing them on a Turkey stone. The second test I employed consisted of a board two feet in length, with a longitudinal line down the middle divided into inches, and a sensible pocket compass. To guard against the effects of the magnetism of position, the central line of the board was placed exactly in an east and west direction by the compass; and as the board was laid horizontally on a table, this line was known to be in the plane of the magnetic equator, and consequently in a situation in which small bars of iron are not affected by the magnetism of position. In applying this simple apparatus as a measure of magnetism; the bar, whose magnetism was to be examined or compared, was laid along the central line of the board, with its north pole always nearest the compass. The compass was placed with its centre at the commencement of the scale, so that its needle was exactly at right angles to the direction of the bar; and before the deviation took place, its poles were equidistant from the bar. The distance was estimated by the scale on the board, and always represented the space between the north end, or nearest end of the bar, and the centre of the compass. Three hammers were also employed: No. 1, of 22 ounces; No. 2, of 12 ounces; and No. 3, of 24 ounces weight.

With this apparatus, a number of experiments were performed, several tables of which are given: and their general results are stated as follows:

"1. A cylindrical bar of soft steel, 6 inches long, and weighing 592 grains, lifted, after repeated hammering on pewter and stone, grains; but could not be made to lift a nail of 11 grains.

"2. The same bar hammered vertically upon a parlour poker, also held erect, after 22 blows, lifted with the lower end, which was a north pole, 88 grains; and on using a larger hammer, received a considerable increase of power, producing a deviation of the compass, three inches distant, of 34 degrees: further hammering, it was found, rather diminished than increas

ed the effect. On the bar being inverted, so that the north pole was upward, the magnetism was very nearly destroyed by a single blow; while two blows changed the poles. Hammering the end of the bar in the plane of the magnetic equator also destroyed the polarity; but the effect was not fully produced until many blows had been struck.

"When the poker had been previously hammered in a vertical position, an increase of magnetic effect on the bar was obtained, a single blow being now sufficient to enable the bar to lift about 20 grains; and when the end was hammered into a kind of cup, so as to be easily bruised, the bar was by one blow rendered capable of lifting between 30 and 40 grains. After 10 blows, the highest effect obtained in all the experiments was produced, the same bar readily lifting a nail of 188 grains, being nearly one-third of its own weight!

"The magnetism by percussion was found by subsequent experiments to be augmented when the length of the bars was increased; thus a quarter-inch cylindrical bar of steel five inches long, after receiving 20 smart blows, produced a deflection of the needle, at the distance of three inches, of 13°, and lifted 64 grains. Another piece of the same bar 7 inches long, after similar treatment, produced a deviation of 24°, and lifted 45 grains; and a third bar of the same kind 12 inches long, after 20 similar blows, occasioned a deviation of the compass of 33°, and easily lifted 88 grains. The shortest bar, it was observed, received the full effect by the first two blows; but the others continued to increase in energy as the percussion was continued. These bars did not receive a power equal to that first used; the cause was probably their greater hardness.

"3. A strong magnet properly tempered was injured in whatever position it was hammered, but most rapidly when the north pole was upward. After no further diminution of its magnetism could be produced with the south end upward, a quick loss of power was effected by hammering it with the north pole upward. But after the magnetism had been reduced to a certain extent by hammering in both positions, the power became nearly stationary; so that on striking it in any position with the same hammer, very little change of intensity occurred."

Besides these results, the author mentions the effect of percussion on soft steel magnets, on soft iron not magnetized, and on cast iron. One of the first capable of lifting upwards of 1000 grains, when placed vertically upon the poker with its north pole upward, had its magnetism destroyed by five blows. A bar of soft iron of the same size and form as the steel bar first described, and weighing about 600 drains, was hammered for a considera. ble time while held vertically upon the poker. The greatest effect which he could produce with the large hammer was a deflection of the compass needle, at the distance of three inches, of 13 degrees. In this state it lifted a nail of 64 grains, but

refused one of 11 grains weight. A similar bar of cast-iron became capable of lifting 37 grains; and after it had acquired this power, its magnetism was nearly destroyed by five blows with the north pole upward.

The strong magnetising effect of percussion on soft steel induced Mr. Scoresby to apply the property to the formation of magnets. In accomplishing this object he took particular care that no magnetic substance should be used in the process, which he describes in the following terms:

"I procured two bars of soft steel, 30 inches long, and an inch broad; also six other flat bars of soft steel 8 inches long, and half an inch broad, and a large bar of soft iron. The large steel and iron bars were not, however, absolutely necessary, as common pokers answer the purpose very well; but I was desirous to accelerate the process by the use of substances capable of aiding the developement of the magnetical properties in steel. The large iron bar was first hammered in a vertical position. It was then laid on the ground with its acquired south pole towards the south; and upon this end of it, the large steel bars were rested while they were hammered; they were also hammered upon each other. On the summit of one of the large steel bars, each of the small bars held also vertically was hammered in succession, and in a few minutes they had all acquired considerable lifting powers. Two of the smaller bars connected by two short pieces of soft iron in the form of a parallelogram, were now rubbed with the other four bars in the manner of Canton. These were then changed for two others; and these again for the last two. After treating each pair of bars in this way for a number of times, and changing them whenever the manipulations had been continued for about a minute, the whole of the bars were at length found to be magnetized to saturation, each pair readily lifting above eight ounces!"

XX. On the Alloys of Steel. By J. Stodart, Esq. FRS. and Mr. M. Faraday, Chemical Assistant in the Royal Institution. (Communicated by J. Stodart, Esq. FRS.)

We purpose to give this most important paper entire in an early number of the Annals.

XXI. Some Observations on the Buffy Coat of the Blood, &c. By John Davy, MD. FRS.

This communication consists of observations on the three fol lowing subjects; which, though important to the medical philo sopher, are devoid of general interest; viz. the cause of the buffy coat which appears on blood drawn from persons labouring under inflammatory disease; the fallacy of a prevalent opinion that the age of those morbid adhesions connecting serous membranes, which are so often met with in dissection, may be guessed at by their strength; and the effusions of serum found after death in the cavities of serous membranes. The author thinks, in con