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Consequently, 8 =

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R" 1

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is the present worth of an annuity of one

pound certain, for n 1 years, and is therefore given in the Tables. Let it be re

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nd) Hence, P =

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(ad - d wand) = 1 − ( d (w + 1) + a + nd),

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Q. E. D.

Example 1.-Required the present worth of an annuity of 17., depending on the life of a person aged 20, for 7 years, interest 5 per cent-probabilities from the Carlisle Tables, and the first payment in advance.

Here d = 42, n = 6, w + ] 5.3294;
R = 1.340095, n d = 5836 and a =

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⚫.4.9572 + 1 5.9572 is the required value, which, by another method, we find to be true in every figure.

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lisle Table, for the same age and rate per cent, is 15.856.

De Moivre supposes 86 years as the maximum limit of life; but the value of life annuities computed from this hypothesis gives results even lower than those deduced from the Northampton Table,

with the exception of the values on very young or old ages. If, instead of 86, we take 96 as the maximum age, the results will be more in accordance with the Carlisle Table, which, with some slight modifications, is now used in many of the London Assurance Offices.

Table showing the Values of an Annuity of 11. on a single life, according to De Moivre's Hypothesis, and De Moivre's Hypothesis Modified; also, the Northampton and Carlise Tables: interest 3 per cent.

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Sir,-With regard to the spontaneous heating of cast iron, several instances are reported; for instance, Professor Daniel, in examining a cube of grey cast iron, by muriatic acid obtained a porous spongy substance; untouched by the menstruum it was easily cut off by a knife, had a dark grey colour like plumbago, and when placed in considerable quantity on blotting paper to dry, it spontaneously heated, ignited, and scorched the paper: its properties were not impaired by being left for weeks in the solution of iron or in water.

Mr. D. Williamson, in his work on "Engines of War," states than an iron gun of the Florida, of the invincible Armada, sunk off one of the western islands of Scotland (Mull) when raised by Sir Archibald Grant and Captain Roc in 1740, became red hot in less than a quarter of an hour, and was cool in three hours; also that a gun raised off Carlscrona, having been in the sea 50 years, became very hot

on exposure to the atmosphere: one third was converted into a mass very like plumbago. Mr. Dean raised some cast-iron guns and balls on the June 16th, 1836, from the Mary Rose, sunk off Spithead, July 18th, 1545, which became very hot and fell to pieces. Balls of 30 lbs. were reduced to 19.8 and 70 lbs. balls to about 45 lbs.

Perhaps some person would inform us, through the medium of your Journal, whether any similar phenomenon has been observed with regard to the cast iron raised from the wreck of the Royal George, and if so, what is supposed to be the cause, and the chemical action that takes place. Is it that the heat becomes susceptible by reason of any sudden contraction? Or does the cast iron become minutely divided, and therefore its affinity for oxygen so much increased as to take fire in the atmosphere?

I remain, Sir, your obedient servant,

October 20, 1842.

I. K. F.

RAILWAY REFORM.

Sir,-There is a standing jest among those who love to laugh at philosophers, about Sir Isaac Newton's having caused a hole to be cut in his study door for his cat to come in by, and by its side another for her kitten; a story which, as the Italians say, 66 se non e vero, e ben trovato." Now, however ridiculous this may seem, something like a parallel to it may be seen on all our railroads, which, to the great cost, and almost ruin, of their shareholders, have been made double, when a single one might have sufficed; for in order to work a single railroad with as much efficacy as a double one, we have only to construct, at about half its length, a single sinuosity, either to the right or to the left, to enable the train coming up, to pass the train going down, and take its place, and, vice versa, the opposite train. And this passage might be safely used by making a short halt at each end of the sinuosity. To accomplish this great saving, it is only necessary that the two trains should always start at the same hour, and stop at each end of the sinuous curve, until by a signal one train enters the curve, and the other proceeds in a straight line. I need not, I believe, take up more room in your useful Magazine by detailing the immense saving that would accrue by this simple and not expensive process, as it is evident it would nearly reduce the whole expenditure by one half; and if constructed on a plan, the model of which I have just sent to France and Germany (for in England we never

ent is so simple as hardly to require any further explanation; at the same time, we shall be glad to receive from him any additional elucidations which he may think likely to be of interest to our readers.-ED. M. M.]

MR. CLEMENT'S NAUTICAL INVENTIONS. The Sillometer, Marine Thermometer, and Steam Thermometer.

Since we gave a description of these ingenious instruments, (see Mech. Mag., No. 989, p. 84.,) a very complete series of trials have been made of them, by command of the Lords of the Admiralty, on board of H. M. S. V. Lightning, during several voyages from Woolwich to Portsmouth and back, and also up and down the river Medway. We have been favoured with a sight of the Official Reports of these trials, and with permission to make the extracts from them which we subjoin. The results, it will be seen, are in the highest degree satisfactory, and in perfect accordance with those previously obtained by the officers of the French Royal Marine. We may hope, therefore, to see these instruments now generally adopted, both in our government and merchant services; the Marine Thermometer, more especially in all vessels which are exposed to the risk of coming in contact with floating icebergs, or have to navigate unknown, or imperfectly known seas.

1. Trials with the Sillometer.

adopt new plans in great works), (During Passage from Woolwich to Ports

which includes a great saving in the transverse sleepers, as well as an additional security in the construction of the wheels which support the engine, and would also reduce the expense considerably, whilst it would render the railway more lasting, and easier to be laid down, than any that has as yet been adopted to my knowledge. You will oblige me by inserting this in your next Magazine, and if any of your correspondents are desirous of seeing my new plan, and you can make room for it, I shall have great pleasure in sending it for your pages. I am, yours, &c.,

GEORGE CUMBERLAND, SEN.

Oct. 20, 1842. Culver-street, Bristol.

[The plan of our esteemed correspond

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Thermometer would indicate the approach to rocks and icebergs, from the influence these bodies are known to have on the temperature of the sea, for a considerable distance."

It will be observed, that, when off the Nab Head, the temperature indicated by the Marine Thermometer was 14.25° centig.; but, on the return voyage, the

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temperature which it indicated off the
"aris-
same point was 16.20 and 16'5,
ing," says the official report, from
the circumstance of our course being
more distant from the light vessel:"
another striking illustration of the cor-
rectness with which the instrument de-
notes the recession from, or approach to,
shoal water.

III. Trials with Steam Thermometer.
(During passage from Woolwich to Portsmouth.)

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"The dial of the marine thermometer is on deck, and shows by inspection merely, the approach to shoal and deep water, also, (as it is inferred,) the approach to land, rocks, and ice-bergs.

"The steam thermometer is so placed, that the officer on deck observes the degree of temperature (and pressure) of the steam in the boilers, so that, if from any cause, the temperature of the steam in the boilers becomes higher or lower than it should be, it immediately becomes known; and, in high-pressure engines more especially, the approach to such a degree of temperature and pressure as may be dangerous is easily observed.

"The sillometer has also a dial upon deck, which constantly shows the number of miles per hour that the vessel is going; consequently, it is easy to discover, under all circumstances, what is the best trim of the

Portsmouth Harbour.

vessel, and the most advantageous quantity and distribution of the sails, for obtaining the greatest speed. As the sillometer shows immediately, the effect which every alteration in the sails or trim of the ship has on its velocity, it follows also, that ships fitted with the sillometer can constantly maintain the speed that may have been agreed upon, and so keep company together, and maintain the same relative position, though, from the darkness of the night, or thickness of the weather, they cannot see each other.

"To ascertain the distance run after any number of hours, it is simply to take the number of minutes one of the watches of the sillometer has gained over the other, and to multiply that number by 6, there results the distance run in miles.

"The instruments of Monsr. Clement, fitted on board the Lightning, appear to be well and securely placed; nor does there appear to be any danger of their being deranged."

We subjoin the account of the additional trials made with the thermometer in the Medway.

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