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SIZE OF CARRIAGE WHEELS.

that a given wheel may be drawn over a given obstacle with the least possible power, and their solutions therefore, merely determine the radius of a wheel, such that (under the given restrictions) the line of traction will be at right angles to CO (see figure to Mr. Hogan's solution, p. 439, vol. 11.) when the application of the power for that particular wheel will be most advantageous. The question then arises, is it demonstrated that a wheel whose radius (2 feet) answers the conditions of their equation (and in which case the power has been proved to be applied in the best direction possible) would require a less force to draw it over the obstacle, than one of a greater radius? And is it not a question, which their de monstration does not preclude, whe ther on the whole, power would not be gained by increasing CO, because, at the same time that the power is less advantageously applied (the ACO decreasing) the load or resistance acts also at a disadvantage, which may be either more or less than sufficient to compensate for the decrease of power.

To explain myself further, let the load be considered as suspended from the centre of the wheel, and the power and resistance will be applied at C (the extremity of a lever CO, CB

840: 188

being the fulcrum) in the directions CE, CA, and we will also suppose that ECO is not a right anglefrom O draw OX perpendicular to CE*. Then, by mechanics, there will be an equilibrium, when P: W =PO:: XO; now, within certain limits, it will be found that if we augment CA, PO decreases with respect to CO faster than XO does, of course the ratio of XO to PO increases, and less power will be required to sustain the load. Manifestly, therefore, the problem consists in determining CO, such that XO

PO

may be a maximum, and this or any thing equivalent, I do not think your talented correspondents have done.

W⋅:P= {764-176) Hence, if the radius is 7 feet, the ratio of XO to PO is a maximum, and two-thirds of the first-mentioned power is equivalent to the load.

I should not think that any inference of practical utility can be drawn from the solution of the problem, not only because when a wheel is large, the weight and friction is very considerable; but, because the direction in which the power of a horse is applied, in drawing a carriage, should be taken into consideration. A horse draws horizontally, by pushing obliquely upwards; on this account, I should conceive, that if the traces incline downwards from the wheel, such direction would be bad in proportion o the inclination.

I will just remark that 2 feet (or

By way of an example, let the weight of the load be 100lbs., and CO 2 feet, then BO=+='0625— PO 348 and CO (=X0)=1.000, wherefore W:P1000 :: 348= 100lbs. 34.8lbs.-the power required to sustain the load of 100lbs. Now let CO 4 feet, then BO

03125, XO='968, PO="247, then W:P 968: 247=100lbs.::25.5lbs. so that, if the line of traction, be horizontal, a power of nearly 4th less than in the former case is sufficient

In the same manner, when the radi are 7 and 8 feet,

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any number under 4 feet) will not exactly answer the conditions of their equations. The first number that will, is between 6 and 7 feet, in which case the obstacle is to be understood, as placed on the opposite side of the wheel, and the power to be applied by pushing.

To avoid interruption, I assumed that when CO 2 feet, DE would be 4 feet; but, as DE is 4.78 feet, the result 34.8 lbs. is given a little less than it should be; and this rectified would make the power saved still greater.

I am, Sir, your's, &c.
R. C. Jun.

This line may be casily supplied by the reader.

NEW METHOD OF FINDING THE LONGITUDE AT SEA.

DEMONSTRATION OF MR. SINCLAIR'S NEW METHOD OF FINDING THE
LONGITUDE AT SEA.

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103

In the above diagram,

Let Zm, Zs=appt. Zenith distances.
ZM, ZS=true Zenith distances.
hence SM true distance, wanted.

N.V.S.(ZS+ZM)-N.V.S.(ZS-ZM): R2::N.V.S.SM-N.V.S. (ZS-ZM) N. V. S.ZZ. and N. V. S. (Zs+Zm—)N. V. S. (Zs—Zm);R3 ::N.V.S. Sm-N. V. S. (Zs—Zm);N. V. S. ▲Z.

Per Spherics,

And N.V. S. (ZS+ZM)-N.V.S.(ZS-ZM): N.V.S.(Zs+Zm−N.V.S. (Zs—Zm) ::N. V. S. SM—N. V. S. (ZS—ZM):N. V. S. Sm-N. V. S. (Zs—ʼZm.)

Or N.V. S.(ZS+ZM)—N.V.S.(ZS-ZM) × (N. V. S. Sm — N. V. S. (Zs — Zm)

(N.V.S.

N.V.S.SM—N.V. S. (ZS-ZM))'× (N.V. S. (Zs+Zm) —N. V. S. (Zs--Zm.) Or N. V. S. SM-N. V. S. (ZS-ZM)=N. V. S. (ZS+ZM)-N. V.S.(ZS→ZM) x(N. V. S. Sm-N. V. S. (Zs-Zm.)

N. V. S. (Zs+Zm)—(N. V. S. (Zs-'Zm.), Or N. V. S. SM

=(N.V. S.(ZS+ZM)—N.V.S.(ZS-ZM) × (N.V.S. Sm.—N.V.S. (Zs—Zm.))

N. V. S. (Zs+Zm)-N. V. S. (Zs-Zm)

+N. V. S. ZS-ZM=True distance.

J. S.

1

REMARKS ON MR. SINCLAIR'S METHOD
OP FINDING THE LONGITUDE.

Sir, It gives me considerable pleasure to inform you, for the honour of England as a maritime nation, that notwithstanding the simplicity of the method of clearing the distance in a lunar observation, given by Mr. Sinclair of Dumferline, (No. 315, p. 15,) it is not needed by the teachers of this metropolis, since the common epitomes of Norie and Moore give methods far easier; and even shorter and simpler than these, are given in the linear tables of Thomson, Noric, &c.

In the example, as printed in the Mechanics' Magazine, the subsidiary work for obtaining the natural versed sine to the nearest second, and for finding the log. of the differences for six places of figures are not at all noticed. Each of these operations will occupy nearly as much time as will be sufficient to perform the whole example by Thomson's tables, at least after the formula is written out. In proof of my assertion, the following is Mr. Sinclair's question, as worked by Thomson's tables:

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In the above examples each log. is taken out direct from the tables, requiring no correction whatever.

Another advantage of these tables is, their cheapness, being no more than ten shillings, and they contain every requisite rule and table for finding the longitude by a lunar observation, or by chronometers, likewise directions for knowing the principal stars, with several other useful problems and tables; it is in fact a most complete compendium for the practical navigator.

For the theoretical student I fully agree with the observation of your correspondent, that Robertson's Navigation and Mackey's Longitude are first-rate books on the subject.

Should the necessity of possessing Thomson's tables be urged as an ob jection to his method, in comparison with that of Mr. Sinclair's, I may remark, that tables of natural versed sines, are rarely to be met with in nautical tables, nor do I know of any in the common works on navigation, except in Mackey's Longitude, and this work does not contain the common logs used likewise in Norie's method.

I remain, &c.

ANTHONY PEACOCK, Teacher of Navigation. Calvert-street, St. George's East. Sept. 1, 1829.

HINT TO ASTRONOMICAL TEACHERS.

Sir,—I am sure nothing could be more gratifying to the lovers of science, than to see the proposition of

Log. 1.9041

your able correspondent, Mr. Steele, for a monument to Sir Isaac Newton, carried into effect. My purpose in now addressing you, is not, however, to enforce that proposition, (which is unnecessary) but to mention an idea which the engraving that accompanies Mr. Steele's paper suggested to my mind. It was this, that astronomy might be taught with a great deal more effect, by delineating the hea venly host on the concave or inner side of a sphere, instead of the convex or outer side. Suppose a lectureroom were to be constructed with two hemispherical domes, each representing a portion of the heavens, the principal stars, path of the sun, &c., could any thing be better calculated at once to beautify the room, and to impress on the minds of the students, the relative position of the objects of their study ? Should you think this hint worthy of a place in your interesting periodical, its insertion will oblige,

Your's, &c.
J. W.

MIGRATION OF SWALLOWS. Prince Charles Lucien Bonaparte, when on board the ship Delaware, about five hundred miles from the coast of Portugal, and four hundred from that of Africa, was surprised by the appearance of a few swallows (hirundo rustica et urbica.) An easterly gale of wind prevailed at the time, which had probably blown the swallows out of their course from the main land to Madeira, about two hundred miles distant. These birds were not so much exhausted as might have been expected, from the length of their voyage.

46

PERPENDICULAR ROLLER BLINDS.

IMPROVED SKATE.

This is not perhaps the most ap propriate season for submitting the fol lowing improvement to your readers, but frosts, in this country, are quickly come and gone," and it may be better to anticipate, than delay till they arrive, any subject connected with them. No doubt, many of the readers of the "Mechanics' Maga zine," and those, at least who are skaters, have felt the inconvenience occasioned by the shifting of the toe from its right position on the skate, after the strap has been slightly stretched, and not to mention the fracture of a handsomely shaped toe, (of the skate of course, I mean) in the attempt to knock it into its right position, may have suffered as much torture from the buckle of the cross strap from the toe to the instep, as some of our modern beaux and belles from the infliction of a tight shoe in a ball room. To alleviate the sufferings of our fellow-creatures, I take it, is one of the noblest attributes of humanity, and shall therefore think myself entitled to no small share of their gratitude in proposing the above improvement. Let a small toe-piece of strong leather, A, similar to that upon the French clogs, be well secured to the front part; the necessity for a toe-strap will be done away with, and one from the middle of the skate, B, and another, rather backwarder than usual, from the heel at C, will not only keep it steady to the foot, but be found to give much less inconvenience than the present, often painful method of securing them-the heels must have spikes, not screws. Norwich,

July 22nd, 1829.

J. O. B.

105

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PERPENDICULAR ROLLER BLINDS.

Sir, The blinds of which I now take the liberty to send you a de scription, are intended either for horizontal or perpendicular rollers, and they can be drawn without using the ordinary endless cord. They are not

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106

DR. LARDNER'S LECTURES ON MECHANICS.

In this diagram, which only shews the upper part, in a bird's eye view, it will be observed that the roller for winding the blind upon is to have a grooved wheel at both ends, and that if the blind B is wound up from right to left as at E, the cord F must be the reverse, that is, from left to right. The cord F G H is fastened at F to the roller wheel, and passing over a small wheel H, which can be screwed into the casement, is made secure to the rod B. The other end of the roller, &c., is to be prepared after the same manner. If the blind is placed horizontally, the cord may be taken hold of at G on either side of the window, to assist in raising or lowering.

Very neat blinds for the lower row of panes might be made of squares of silk in an oblong frame, each piece being made to draw upon this plan, by using catgut instead of cord; and blinds made in this style would likewise be very suitable for carriage windows. I am, Sir, Your's most respectfully, Henry Ď

EFFECTS OF REFRACTION.

Captain Scoresby, on his return from his first landing on the east coast of Greenland, at Cape Lister, in lat. 70° 30 N. gave the following interesting account in his journal of an extraor dinary instance of the optical phenomena produced by refraction. "It was about 11 P. M., the night was beautifully fine, and the air quite mild. The atmosphere, in consequence of the warmth, being in a highly refractive state, a great many curious appearances were presented by the land and icebergs. The most extraordinary effect of this state of the atmosphere, however, was the distinct inverted image of a ship in the clear sky, over the middle of the large bay or inlet before-mentioned the ship itself being entirely beyond the horizon. Appearances of this kind I have before noticed, but the peculiarities of this were-the perfection of the image, and the great distance of the vessel that it represented. It was so extremely well defined, that, when examined with a telescope, made by Dollond, I could distinguish every sail, the general rig of the ship,' and its

particular character; insomuch that I confidently pronounced it to be my father's ship, the Fame, which it afterwards proved to be though, on comparing notes with my father, I found that our relative position, at the time, gave the distance from one another nearly thirty miles, being about seventeen miles beyond the horizon, and some leagues beyond the limit of direct vision. I was so struck by the peculiarity of the circumstance, that I mentioned it to the officer of the watch, stating my full conviction that the Fame was then cruising in the neighbouring inlet."

SINKING OF MOUNTAINS.

Geologists have recorded many instances of mountains siuking into the earth. The following are remarkable :

:

In the south of France, on the 23d of June, 1827, a mountain, belonging to the chain of the Cevennes, sunk with an awful crash into the valley of Pradines, overwhelming a small village in its course, and spreading devastation to a considerable distance.

In 1806, a beautiful valley, interspersed with pleasant villages, in Schweitz, a canton in Switzerland, became a scene of awful calamity. During the peaceful serenity of a summer's evening, the inhabitants of five villages, amounting to two thousand souls, were involved in sudden destruction by the falling of the north-east projection of the Rosenberg mountain, which covered more than three square miles of fertile country.

The ignorant attribute phenomena such as these, to the malicious agency of fiends and evil spirits; and foolishly resort to spells and exorcisms to save themselves. The student of Nature views them merely as the effect of plain natural causes-the sudden expansion of frozen particles of water-the ignition of combustible bodies-or the mere action of the law of gravity.

Z.

NOTES OF DR. LARDNER'S LECTURES ON MECHANICS.

Continued from page 63.

Ir a non-elastic body strike another non-elastic body in a slanting direc tion, it will not remain stationary, by reason of the resolution of the forces.

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