to

D(x)

And this, by (6.) is equal

D (x)

This formula might also have been deduced by subtracting from the present value of an annuity for the whole life, that of an annuity deferred for n years, since the present value of an annuity during the next n years, and that of an annuity to commence at the end of n years, are evidently together equal to the present value of an annuity for the whole life. Thus, the present value of the life annuity is, (Prob. II.,)

If this be added to 9-7211, which we found by last problem to be the value of an annuity deferred for ten years, on a life of the same age, the sum 17.2948 will be the value of an annuity on the whole life; and it corresponds with the value which we deduced for that age by problem II.

Here we may repeat, once for all, as the remark applies equally to the expressions for all the benefits, that so long as the payments continue the same in number, if they commence a year earlier than we have supposed, the change requisite to be made in the formula, is the D (x + k + 1) D (x + k + 2) D (x) D (x) And the sum of these expressions is, D (x + k + 1) + D (x + k + 2) +

which by (8) is equal to

D(x + k + n). D (x)

;

D (x + k + n),

Example. Required the present value of an annuity of £1 on (20), to be entered upon 10 years hence, and to continue 10 years. Here x = 20, k

=

x=

17159.9452 3818-9594

=

10, and n

= 10.

payment is to be £1; the 2nd, £2; the 3rd, £3, and so on, increasing an

on the Archimedean Screw, or Submarine Propeller," by Elijah Galloway, 1842, which purports to be a history of the screw propeller; and which, from some circumstance, which I will not designate, omits all notice of my propeller, consisting of the segments of a screw, though it is the only one which will accomplish any required speed, and perfect security.

The hope of setting my case before the public, is the occasion of my trespassing on your valuable columns, being assured that your acknowledged liberality, and great regard for equity and fair dealing, will induce you to lend me assistance to assert the justice of my cause over the many pretenders and usurpers of my patent right.

With this view, I have thrown hastily together a few facts in connexion with the history of the screw-propeller.

In the year 1817, there were but four steam-boats on the river Thames, namely, the London Engineer, the Father Thames, the Richmond, and the Hope; the last was the first vessel to which the screw was ever applied. It was fitted with a whole screw, and failed.

In the year 1826, Messrs. Shorter and Lowe, fixed to the bows of the Royal George barge, owned by the Goldsmiths' Company, two parts of a screw, in the form of curved blades. An experiment was made, and this also failed.

In 1834, I produced a model with two blades or segments of a screw, to be placed at the stern of a vessel, which, upon trial, was found to perform well.

In 1836, I became acquainted with a Mr. Joseph J. Oddy Taylor, who was shown the model I had made in 1834, and he entered into a written agreement with me not to take any undue advantage, or deprive me of the benefits of my invention. I made a second model for Mr. Taylor, but for two years, the taking out of a patent was delayed. In 1838, Ï was informed that Mr. Taylor was about to take out a patent for a propeller upon my model, when, to secure my right, I entered a caveat at the proper offices. Soon after, Mr. Taylor did actually make application for a patent, and an appointment having been made for hearing my grounds of opposition to it, I met him at the office of the Attorney-General, Sir John Campbell, who heard the statements of both parties. I produced the agree

ment signed by Mr. Joseph J. Oddy Taylor, and also claimed the model produced by him as being the one made by my own hands. Sir John then asked Mr. Taylor, if he admitted his handwriting to the agreement, when he said, he did; upon which Sir John immediately decided in my favour. A patent was then taken out by me, and the specification was completed, and bears date, 24th March, 1838.

Since that period, my patent has been the sport and plunder of all sorts of pirates. While none have dared to dispute my patent right, some have tried by colourable variations to evade it; others again have boldly applied my segments of a screw, without any variation whatever, and openly set me at defiance, knowing that to a man of my limited means the entanglement of the law would be, as it has been in my case, a denial of justice.

In the autumn of 1838, I fitted my patent propeller, composed of the segments of a screw, to a boat named the Wizard, and made trial of her, having on board Mr. Francis Pettit Smith, the inventor(?) of the Archimedes, or whole screw, and Mr. Wimshurst. We started from Deptford pier to Mr. Wimshurst's yard, and the result of the trip was pronounced to be successful. This was at the time Mr. Wimshurst had the Archimedes on the stocks.

After this time, a vessel with my patent segments was worked in the Thames, above and below bridge, about October, 1838, with a party of gentlemen on board, and found completely to answer every expectation. Among these gentlemen I may mention the names of Mr. Anderson, Mr. Cox, and Mr. Soaper, of the Central Coast of America Company.

Since my patent was granted there have been several others taken out, which are all, more or less, infringements of mine.

The first of these was Mr. J. J. O. Taylor, who has never, however, acted upon it. Then came a boat by Capt. Carpenter, R.N., and worked by handlabour, but that did not succeed.* After this, a patent was taken out by Capt. George Smith, R.N., who has admitted

A great mistake, as Mr. Lowe will see, by referring to Mech. Mag., No. 976, p. 329; and No, 978, p. 364.-Ed. M. M.

to me that his invention was similar to mine; and I must acknowledge, to his honour as a gentleman, that he has never attempted to carry out his patent. Another screw propeller was brought out by a Mr. Hunt, and patented in that name, which came so palpably from my patent segments, that none but experiments have ever been attempted by it.* The next patent granted was to Mr. Blaxland, whose specification is expressed in terms so closely resembling mine, that the description reads as if it had had mine for the groundwork.

I have thus briefly alluded to the history of the screw propeller, as applicable to my segments. There are other propellers invented, as Fyfes', Rennie's, Napier's; but it is needless to refer to all.

I have already trespassed too far on your courtesy, to enter at length into the proceedings of the "Ship Propeller Company," under the management of Mr. F. Pettit Smith; but there are one or two statements put forth in their prospectus which I cannot forbear briefly noticing. The prospectus alleges that "the Archimedes steam-boat has proved the superiority of Mr. Smith's Patent Screw Propeller" whereas, any success attending the Archimedes was gained when my patent segments were applied, instead of Mr. Smith's whole screw, The voyage across the Bay of Biscay, from Plymouth to Oporto, so boastingly referred to, was performed by a propeller of my segments. The Archimedes left port with Mr. Smith's patent, which was afterwards unshipped, and my patent segments fitted on, with which she performed the voyage, as the testimony of the gentlemen who freighted her" will prove. On her return she got foul of the "Goodwins," and was obliged to come back to the Thames with my propeller still fixed to her stern. Another statement made by the Company is, that the whole of the patents have been purchased by them, which is not true. My patent remains in my own right and possession; and what the Company could not purchase, they have been unscrupulous enough to appropriate without leave or license.

I desire no favour beyond my right;

Here, again, we must say, Mr. Lowe is greatly in error. Mr. Hunt's patent is for the combination of a steering and propelling apparatus in one; and he expressly states, that he lays no claim to any particular sort of propelling blade.-Ed. M. M.

and if my submarine propeller be deficient in producing the two most essential requisites for a propeller to possess, viz., security and speed, then I should expect it to fall into deserved neglect.

There may be some merit in producing a valuable appendage to the power of steam, (which has cost me many years of labour and expense,) and my ambition may be gratified at the success of my ingenuity; but there should be some more solid advantages accruing to me, in the shape of a fair recompense.

Your giving publicity to this account may form the first step towards vindicating my right; and, trusting the day of retribution will not now be long delayed, I remain, Sir,

With great respect,
Your most obedient humble servant,
JAMES LOWE.

October 24, 1842.

THE GREAT BRITAIN-IRON PLANKING.

Sir, Having resided near Bristol, during the time that the principal portion of the outward sheeting of the hull of the then Mammoth, now Great Britain, was being laid on, it struck me very forcibly, that the method adopted, that of riveting together oblong sheets of iron, of short lengths, was a most clumsy mode of proceeding, and utterly unworthy of the advanced state to which the mechanical arts have arrived; and I should have communicated to you my views on the subject long ago, had I not expected that the remedy, which seemed so obvious to me, must, ere this, have occurred to others more interested in its adoption, and provided with the machinery necessary to carry it out. As that, however, appears not to have been the case, I shall at once state my plan, which is to substitute for the iron sheets now used, what may be termed "Iron Plank." I would have the iron intended for ship-building, (and for divers other purposes as well) rolled out into slips of a length and breadth suitable to the purposes to which they will have to be applied, and for large ships I would say the longer the better. There can be no difficulty in the thing, as the requisite machinery attached to a powerful rolling mill will readily present itself to the mind of any intelligent mechanic.