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And the sum of the first n terms, commencing with the present age,

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And the sum of the first n terms, commencing with the (x + k)th

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(8.)

(9.)

R (x + k + n)

If in the expressions (7) we suppose n = 1, they become, respectively,

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to, as well as others which may be derived by transposition from those we have given, we do not think it necessary to set down, since their mode of derivation is so extremely simple; and we shall accordingly refer, as we have occasion for them, to the expressions which we have shown to contain them.

If, in the expressions (10), we substitute +1 for x, we obtain the same expressions as we should obtain by making n in (6)=1. And in like manner, by substituting in (10) x + k, and x+k+1 for x, we obtain the same expressions as we should obtain from (9) and (8) respectively, by making n=1 in these expressions. The expressions just referred Again, since N (x) = D (x + 1) + D (x + 2) N (x + 1)

N (x+2):

=

=

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D (x + 2)

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S(x), the resulting equation will be,

and so on; if we add these equations together, observing that

N (x) + N(x + 1) + N (x + 2) + &c.

=

+ 2) + 4 C (x + 3) + &c.

.

(11.)

S (x)
D (x + 1) + 2 D (x + 2) + 3 D (x + 3) + 4 D (x + 4) + &c. . (11.)
And in the same way we should obtain
R (x) = C (x) + 2 C (x + 1) + 3 C (x
Sometimes, in the Commutation Tables,
the assurance columns are not exhibited.
We therefore proceed to show how their

place may be supplied by means of the annuity columns,

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and so on,

Again, since C (x) = v D (x)

(x + 1) v
D (x + 1)

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1)

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C (x + 1) = v D (x + 1) D (x+2)

C (x + 2) = v D (x +

C (x+3) = v D (x +

2) — D (x + 3)

3) – D (x + 4)

to the end of life; if we add these equations, observing that
C (x) + C (x + 1) + C (x + 2) + &c.

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=

M (x);

= v [D (x) + D (x + 1) + &c.] = v N (x − 1 ) ; N(); the resulting equation will be

+ &c.

1)

=3

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R, in terms of S, and which is obtained in precisely

R (x) v S (x 1) S (x) The expressions just deduced may be exhibited in a somewhat different form, which is, in certain circumstances, more convenient than the other.

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(13.)

Since by (10), N (x) = N (x − 1 ) − D (x) The first of the foregoing expressions, becomes by substitution,

N (x-1)+D(x) = D(x) – N (x − 1 ) + v N (x − 1) - D (x) − (1 − v) N (x − 1) The corresponding expression for R (x), obtained in the same way, is, R(x)=N(x-1)-(1-v) S(x-1) (14.)

Of the foregoing expressions, those numbered (1) to (11), are extensively useful, in deducing and simplifying the formulæ in the practical application of the tables; and the remaining expressions numbered (12) to (14), besides their use in supplying the place of the assurance columns when required, serve also to verify the tables.

We have not space here to give examples illustrative of any of the foregoing formulæ. But these arc so very simple, that this does not seem at all necessary.

For the convenience of those who may wish to test the table prefixed to the present paper, by means of the formula of verification given above, we subjoin the following elements.

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and its logarithm is 98296666. Also 1-v= 03846154; and its logarithm is 58502665. We make a single remark before closing the present paper. The formula we have demonstrated may appear to some at first view, both complex and confused. They are nevertheless eminently remarkable for their simplicity and symmetry. This, a farther and practical acquaintance with them will make abundantly manifest.

In our next paper we propose to show how the present values of the simple benefits may be deduced from the Commutation Tables.

Hermes-street, Pentonville.

G.

MR. ZANDER'S TABLE OF THE PERFORMANCES OF VESSELS ABOVE BRIDGELAW OF THE VELOCITY OF STEAM-VESSELS.

Sir, Since my letter to you of the 27th inst. I have been further studying the table given in your 1001st Number, page 359, of the comparative performances of seven steam-vessels; and combined therewith, that law in hydrostatics quoted in the excellent paper of "O. B. F." on

Mr. Booth's system of propelling," (No. 992, Aug. 13, page 149.) "If any body move through a fluid at rest, the force or resistance of the fluid against the body will be as the square of the velocity, and density of the fluid."

Now, I have assumed that the square

the velocity) by the square of the velocity of the paddles, and the result appears to confirm what I have asserted in my letter of the 27th, that the area of efficient paddle surface does not exceed the length of the paddle-board of both wheels, multipled by its greatest depth in the water. But more minute statistical tables are required of the correct dimensions of a great number of steam-vessels, their draught of water, loaded and unloaded, their exact speed per hour under similar circumstances, and the correct number of revolutions of the paddles per hour instead of per minute before any very satisfactory conclusion can be pronounced. The following will be the area of acting paddle-board surface, according to the foregoing calculation, if correct.

feet of dynamical resisting surface as-
signed to each of the seven steamers
mentioned in the above table is cor-
rect; and therefore that those numbers
multiplied by the square of the velocity
of the vessel is the amount of resistance
which the paddle-wheels must necessarily
overcome by acting against the water in
the contrary direction; which (when
reduced to the direct action of the pad-
dles) will be equal to the number of
square feet of acting surface, multiplied
by the square of the velocity, at which
the floats revolve. As there appears to
be some doubt as to what is the area of
resisting surface of the paddles, I have
endeavoured to arrive at it, by dividing
the resistance to the vessel (being the
area of resisting surface, by the square of
Era. resisting surface 26.485 sq. feet × velocity 912
velocity of the paddles

Thunder and Lightning.

= 12.172

19.081 x 102

14.99

10.877

13.2752

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These seven vessels, on account of their depth in the water, not much exceeding the depth of the paddles, give a nearer result than probably would be the case with vessels of greater draught of water, where some correction might be found necessary for unequal immersion.

66

I am quite convinced, with your correspondent "OB F," that founded upon the law of quadruple resistance octupal power is the necessary consequence;" but the velocity and density being both doubled, appear to me to be the cause of the quadruple resistance only. The power of the steam engine, as applied in the vessel, is divided into two parts, one half acting directly against the water by the paddles, and the other half, or counter action, is that by which the vessel is propelled; and in order to double the velocity of the vessel, the force of the paddles against the water

must be quadrupled, to counterbalance the quadruple resistance which the vessel will experience; thereby requiring an eightfold increase of the engine's power. If a stationary engine were fixed upon the shore, and a vessel attached thereto by a rope, in that case, quadruple the power in the engine would produce double the velocity in the vessel; but an engine upon the vessel, compared with this upon the shore, must have eightfold the power to overcome the same resistance in the vessel. Supposing the resistance of the vessel at one mile per hour to be repre

By the help of my wheel for measuring the velocity of fluids, (the correctness of which I have stal further ascertained by measuring a distance of half a mile upon a canal bank, and leaving a mark at each furlong, when each wheel passed parallel ther to through the canal, recorded exactly the same distance,) a correct result might be obtained of the speed of any steamer upon the Thames, by noticing their velocity with and against the stream, and the velocity of the tide at the same time.

sented by 10, the following table will show the resistance to 10 miles per hour, and the power which an engine on board ought to have to overcome that resistance. But a vessel being provided with a sufficient power to propel her at the rate of 5 miles per hour, it would appear to require one of quadruple the power only to propel her at 10 miles per hour. Is this view of the "octuple power" correct? Or are we to understand thereby, that the power required increases as the cube of the velocity, which was my first impression? The latter ought indeed to convince proprietors of steamvessels, how extravagant would be the attempt greatly to exceed the velocity already acquired in steam navigation, when the difference of power at a speed of 8 or 10 miles per hour would be as 512 to 1000.

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ON LIGHTNING CONDUCTORS, ETC.-BY J. R. HILL, ESQ., C.E. Sir,-In your 998th Number, Mr. J. Roberts claims the merit of having proposed wire rope conductors for shipping some years ago. If wire ropes were at that time known, Mr. R. was fairly entitled to credit for suggesting the application; but by far greater credit is due to any individual who will encounter all the difficulties, disappointments, expense, and the numerous discouraging circumstances attendant on introducing valuable improvements. Had Mr. Roberts done this, he would have deservedly obtained that consideration which he now claims for a written suggestion. I do not imagine that Mr. Smith claims the invention of wire conductors, though, having been the first maker, and perhaps I may be allowed to say, inventor of the wire rope and of its application for ships' conductors,

it cannot be disputed that he deserves the credit of its useful application. Ships having wire standing rigging require no other provision as conductors than a perfect metallic communication from one set of shrouds to another, and from the lower ones to the water. This communication cannot be so easily effected by metallic substances in any other state as that of wire rope.

Mr. Roberts justly appreciates the value of conducting the electric fluid over the side, instead of through the hold of the ship. Mr. R. says, I am wrong in supposing that the electric fluid is conducted over the surface of metallic bodies, but that it is in the ratio of the mass; on this I shall feel most happy to be informed, for I can easily believe it is not a settled question. It is not enough for me to be corrected by saying "you are wrong"-I wish to know what is right.

In my juvenile days I was fond of electric experiments, and on using a prime conductor, made of tin plates, which was covered with a coat of black varnish, I invariably found that every discharge from the conductor brought off a speck of the varnish, laying bare the surface of the tin. I have frequently conducted powerful discharges over clean paper, with nothing more than a line made by a black lead pencil, in which case there was a measurable metallic surface, though but a mass too insignificant to support the solid theory. Dr. Priestley, wishing to satisfy his mind on the question, dipped a perfectly clean chain into melted resin, till it had a coating of a considerable thickness, and on sending a discharge through it, found that all the resin had been dispersed, leaving the chain perfectly bare. Other experiments in favour of the surface theory might be adduced, but I do not know of one supposing the contrary; though, no doubt, much may be said on both sides. fusion of metallic leaves, wires, &c., only appears to prove that the quantity of surface is inadequate to the quantity of elec tricity, passing over, and therefore does not affect the question. Dr. Franklin supports the theory of the mass, but without any further authority than hypothetical reasoning. In the excellent philosophical work of Mrs. Somerville, it is asserted that electricity is conducted over, and in proportion to the surface. If Mr. Roberts, who appears an able writer, and is, I doubt not, equally able as

The

an experimenter, will be kind enough to give the form of an experiment, which shall satisfactorily prove the matter, I, for one, shall feel exceedingly obliged.

The experiments described in your No. 1002, which took place at Portsmouth, in presence of Sir Edward Codrington, appeared to have been a brilliant display of electricity, rather than leading to any useful result; as they proved nothing that was not before known. They failed to prove that conducting lightning into the hold of a ship, where by carelessness and inattention the conductors may be broken, lateral discharges take place, and serious consequences ensue by combustible substances taking fire, is safer than conducting it over the side, and from thence to the water.

I am, Sir,

Your obedient servant,
J. R. HILL.

98, Chancery-lane, November 3, 1842.

BLASTING ROCK-SECURITY FROM ACCI

DENTS-SAND TAMPING.

Sir, I would not have troubled you with any remarks on the letter addressed to you by "W. C." (No. 1001), who seems to entertain different opinions from myself on some practical operations in rock blasting, but for the sake of humanity.

"W. C." states that the workmen unanimously prefer the old plan of firing the holes with straw, to the employment of Bickford's fuse; by which he manifestly implies that its use is attended with inconvenience.

Now I can state on the contrary, that having watched the introduction of this fuse to blasting-works, in very many places and during several years, I have never in a single instance met with master, overseer, or workman, who did not express the highest approbation of it; so much so, that I have known great impatience manifested for a renewed supply, when it had run short.

I am indeed inclined to believe that "W. C." speaks from report, and that he has not himself any actual knowledge of the application of the fuse, otherwise he would not have overlooked the very striking security it affords against the distressing accidents that frequently occur to workmen in the act of ramming down tamping, and sometimes in the act of firing.

If a comparison be made between the number of accidents that occur to men who use the fuse, and those who employ straws on the old plan, during the execution of the same amount of work, where tamping is rammed down in the usual manner, the difference will be found to be very great.

As regards the question of tamping with loose sand, there is no doubt but that its application is so easy, and free from almost a possibility of accident, that if it can be made to afford sufficient resistance, its use would be most advantageous.

"W. C." states that by leaving a space between the powder and the sand, the desired resistance will be obtained, and that he has never placed more than 18 or 22 inches of sand over any charge of powder.

I should be glad to know by what easy and cheap contrivance he preserves the vacant space between the powder and the sand; and I am strongly inclined to doubt the effect being so great as he thinks. The following experiment would, however, make this more clear.

In a hole in solid rock of 1 or 2 inches diameter and from 2 to 3 feet deep, insert from 18 to 22 inches of sand over a small charge of powder, say 1, 2, or 3 oz., with any space you please between the two, and my opinion is that even the smallest charge would blow out the sand, provided there were no peculiar resistance obtained from the contrivance used for preserving the space between the powder. and the sand.

If such small charges on trial should have the effect I anticipate, it would prove that no essential advantage is gained by the space left as described.

Your obedient servant,

J. F. B.

THE ECONOMY OF THE ATMOSPHERIC
RAILWAY SYSTEM.

[From a Lecture delivered by Professor Vignoles, before the Royal Cornwall Polytechnic Society, October 6, 1842.]

It was well known to have been a matter of great discussion whether, on the Liverpool and Manchester line, stationary power should not be used instead of locomotiveand it was not without great hesitation that the latter was adopted. The objection to the stationary power was, the vast amount of force requisite to overcome the friction of the ropes, sheaves, &c., by which much power

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