MR. GURNEY'S STEAM-CARRIAGE. publish the table unless its errors are corrected (which may be done by the column of tangents at p. 81 in the number referred to); but I send it to you as it is, that you may see it as it was formed in the year 1810. I have the honour to be, Sir, A FRIEND TO THE "MECHA- [Our correspondent then presents us with the manuscript copy of his table and rules, which exhibits every mark of being written twenty years ago. The table contains the natural tangents of every degree from 0° to 45°, and the natural cotangents of every degree from 45° to 90°. They are given to three places of decimals; and are usually, though not always, correct in the last figure. They serve, however, to show the accuracy of the construction from which our ingenious correspondent formed his table. The tangents in the first 45° he calls multipliers, that is, multipliers of a base or horizontal line, to obtain a vertical line; the cotangents in the remaining half of the quadrant he calls divisors, and employs for the same purpose. All this, as our readers will perceive, is quite consistent with the directions of P. M. W. in the "Portable Trigonometry;" but it is merely a single case of the utility to practical men of that able production. Some time ago, we received a similar table from Mr. Foord; but it was not so correct as the one of which we are now speaking. Another correspondent informs us, that by means of Professor Littrow's ingenious tables for finding the geocentric places of the planets, and the table of natural sines and tangents in the "Portable Trigonometry," he has computed in much less than one page of letter-paper the right ascension and declination of Mercury for May 5, 1832, the day on which there will be a transit of Mercury over the sun's disc, visible in Britain. Our correspondent finds the right ascension of Mercury on that day to be 2h. 52m., the declination Published in the "Memoirs of the Astronomical Society of London," vol. iii. part 2. 419 16° 44' N., which cannot be far from the truth. These taken together are no mean proofs of the utility of the "Portable Trigonometry."-ED. MEC. MAG.] MR. GURNEY'S STEAM-CARRIAGE. (Extract of a letter from Mr. Herapath to the Editor of " The Times.") "Previous to bringing the carriage 'generally out,' it has been the wish of Mr. Gurney to try it under every possible variety of circumstance, to leave not a shadow of doubt of its practical efficiency. Accordingly, all the opportunities which the severity of the season has afforded have been embraced with anxious sedulity. Sometimes it has been had out in the snow, at others on the ice, up hill and down hill, with a private carriage or an omnibus attached, and sometimes without, but uniformly with the same unqualified success. A very simple contrivance has suggested itself, and been applied, which bids defiance alike to the snow and the ice. On Friday se'nnight, for instance, the carriage with the omnibus went to Holloway and back; and though the road was a continued sheet of ice or snow, it ran thither in 17 minutes. Tuesday, Wednesday, Thursday, Friday, and, I believe, yesterday, it ran for some hours experimentally in Portland-place, Oxford-street, and Regent-street. The carriage is the same which went to Bath and back, and was exhibited to the Duke of Wellington at Hounslow in August last. As from the various statements which have been circulated, the public seems to look with interest to the time when Mr. Gurney's steam-carriages will start for general purposes; in passing, I beg to say, that three or four are already built, and are ready to commence immediately. Some circumstances, however, will occasion them not to start before the beginning or middle of March. At that time, unless any thing should happen to the parties, the public, I believe, may depend on seeing them out for its use. 66 I am, &c. "JOHN HERAPATH. "Cranford, Jan. 31." 420 ON THE VELOCITY ATTAINABLE ON RAILWAYS. Sir, I have been much amused, and I may add instructed, by the ingenuity displayed in the series of papers which have appeared at different times in your pages, during the discussion of the celebrated railway question; I allude particularly to that branch of it which arose respecting the doctrine laid down in The Scotsman, relative to the velocities attainable by carriages, and the power required for that purpose. Not possessing the early parts of the "Mechanics' Magazine," I have not read the beginning of this discussion; but from what passed subsequently, I gather that it was meant to be demonstrated, that with the same expenditure of power per second, a carriage might be moved along a railway with any velocity whatever; and the method of proof was, if I recollect right, by supposing a weight (more than sufficient to overcome the friction) to be attached to the vehicle, by means of a rope passing over a pulley at the end of the plane or railway. Then, since friction is opposed to the descent of the weight by the action of gravity, the difference between this weight and friction will be the motive force at the commencement of motion; but because gravity produces a given acceleration or increase of velocity at the end of each unit of time, and friction causes a given retardation or decrease of velocity at the end of the same units of time (this having been proved by the experiments of Coulomb and Vince): therefore the acceleration arising from the excess of gravity over the friction, must also be a constant quantity, and equal in equal times; and of course a carriage so moved, obeys the laws of uniform acceleration. And hence, because the motive force is constant, it was inferred, that with the same expenditure of power per second, any velocity whatever might be attained. As far as the proof goes, (i. e.) in cases where gravity is the prime mover, there can be no doubt of the correctness of the above assertion; but may not a doubt arise, whether the question in the above case applies to that which they have demonstrated; and may we not reasonably ask, whether there is any mechanical power or agent within the reach of human ingenuity that acts in a manner similar to gravity, (i. e.) producing equal accelerations in equal times: for this is the test to which they have subjected themselves, by assuming a weight acting by the force of gravity as their motive power, in lieu of some machine? The question, then, I imagine, will be this-Does the accele ration arising from a succession of impulses by one given body in motion upon another in motion, when the velocity of the impelling body (and consequently its force) is constant, increase equally in equal times? Let us take the case of two bodies whose weights are A and B, and velocities a and b, moving in the same direction on the line passing through their centres of gravity, and in an unresisting medium. We will suppose them imperfectly elastic-that the force of compression is to the force of elasticity, as 1:m, and that A is the impelling body. By Wood's Mechanics, pr. 47, the velocity gained by B in the direction of A.'s motion, after the collision or im(1+m) A pact, will be represented by A+B x(a-b); and the velocity lost by A, (1+m) B A+B in the same direction, is (a-b): then b+ will be B.'s actual velocity after the Aa+Bb+Amx(a-b) A+B first impact = Now, suppose that some time after the first impact, A.'s velocity and energy should be renewed as at first assumed, and again overtake B moving with its acquired velocity, then, by substitution in the original equation or expression, we shall have (1+m) Ax (a-Aa+Bb+Am (a-b) A+B A+B for the velocity gained by B after (1 + m) A A+B (1+m) A the second impulse = Ba-Bb-Am (a —b) A+B ⋅ b + (a−b) = a is the ultimate velocity attainable by B; that is, just equal to that A. It has hitherto been assumed, that each impulse happened after some interval of time, and in the mean time, that the body B moved uniformly with its acquired velocity; but as we are at liberty to assume those times or intervals as short as we please, suppose them to be exceedingly small or evanescent, and then we have the effect of continued uniform pressure, acting mechanically upon another body in motion; and these effects must be in accordance with the above law. Indeed, Dr. Wood says "The effects of pressure and impact are manifestly of the same kind, and produced in the same way. Excess of pressure on the one side produces momentum, and equal and opposite momenta support each other by opposite 421 numerator, the series will decrease ad infinitum. Cor. 1.-If mo, or the bodies, are perfectly hard and inelastic, the exA (a-b) pression for the nth term is X B n- 1 A+B A+B Cor. 2.-If they are perfectly elastic, or m=1, then it becomes (a-b) x B-Am-1 A+B 2 A A+B In a similar way, when A and B move in opposite directions (in which case the relative. velocity is (a+b), the velocity lost by A after the nth im(1+m) A. pact= ・Bm n~1 a- -b⋅ A+B A+B In general, therefore, the increments will be expressed by a series, such as the following: a, ar, ar2, ar3, where is a fractional quantity: hence, to determine the greatest possible velocity which B can attain after an infinite number of impulses, we must sum the series, which is = (1+m) A (1+m) A × (a—b) = a−b pressure. Thus also pressures may be compared either by the weights they would sustain, or the momenta they would generate, under the same circumstances." P. 30. In estimating the effects of pressure as above, the intervals of time being evanescent, we cannot appreciate the successive increments of velocity; but by adding together any given number of the terms of the series we obtain (as is well known) another geometrical progression, in which the increments for any given space of time, as one second, can be successively measured: thus, a+ar, a + ar × r2, a+arxr, &c., which is the original series under a new form-a geometrical progression, having a different first term and ratio. The result of all this is, that if a body be impelled (or drawn, for the effect is manifestly the same) by 422 ON THE VELOCITY ATTAINABLE ON RAILWAYS. another in an unresisting medium, let the latter be ever so small, the former will ultimately attain the velocity of that by which it is moved, and no more. The only difference in applying large and small bodies moving at the same rate, as a motive force, being this, that the larger they are, the quicker will the series converge, and the sooner will B attain the same speed as A, as a reference to the se ries will show. If, therefore, we wish to obtain mechanically a double velocity in B, we must also give A a double velocity; but then double power is expended contrary to what has been asserted in The Scotsman. At the same time it will be remarked that we can, if we make use of different bodies as our prime mover or mechanical agent, obtain ultimately even a greater velocity in the body moved, at the same or less expense of power per second; for if A be decreased, and a increased in the same or a greater proportion, the different values of Aa, which represent the force or momentum of A, may be either equal to, greater, or less than its former value; but a necessarily increases, and therefore the velocity communicated will be increased ac cordingly: but this is not (generally speaking) the sense in which the comparative expenditure of power is meant to be taken-it is to be confined to the same body, whatever it may be. The last observation will, I think, be found (subject to the effects of friction and the resistance of air, &c.) to be in accordance with the fact. The wind may act with the same force as water, but the former necessarily moves faster than the latter, and will, cæteris paribus, produce a greater velocity than the latter. stances the cause of its being made use of is the misapprehension of the term "moving force," as applicable to constantly accelerating forces, which are only known to us by the effects produced; and, therefore, can only be compared by such effects under similar circumstances, Hence Wood's "Mechanics"-" The accelerating force is measured by the velocity uniformly generated in equal times; no regard being had to the quantity of matter moved.” And "the moving force" is measured by the momentum uniformly generated in a given time, or M and BV, where M moving force, B mass, and V, equals the velocity uniformly generated in a given time, i. e. the accele rating force. Now, these measures are merely comparative; they indicate the intensities of any two or more given forces, which are to be compared by their effects; but they do not inform us of the nature, power, or energy of such uniform forces themselves. On the other hand, the general acceptation of moving force, as applied to machines, &c., is the momentum, i. e. the product of the moving body into its velocity. It will be observed, that by supposing A.'s velocity to be continually resumed after impact or pressure (A losing as much momentum as B gains), nothing more is meant than that its energy or force against B remains unchanged and invariable; although of necessity A can move with no more velocity than B, yet the force or momentum of A should not be measured by the product of its mass and actual velocity, but such a velocity as it would move with, supposing the opposing body instantaneously withdrawn. The tension of the string, on the supposition that B is drawn by A, must vary according to the difference between the velocity with which A would move and that with which it does move, that is, as a−b; but by our preceding series it will be seen, that the differences of velocity between A and B are in geometrical progression, as well as the increments; B-Amn-1 for the part (a-b) x2 A+B is, this difference after the nth imI think, moreover, that in some in- pact. The tension, therefore, varies It appears to me that the very worst of all methods that could have been devised for settling the question, is that of calling in aid the effects of gravity. I think I have shown that the mechanical accelerating force is a continually decreasing quantity; on the other hand, the acceleration of gravity is uniform and constant. On this ground alone it appears to me, the latter cannot lead to correct results. CYLINDRICAL RAILWAY CARRIAGE. .87 477 1*AR PO 1 ไ in this proportion, and ultimately becomes o, when the two bodies move at the same rate. If the effects of friction be traced, even the foregoing velocity cannot be quite attained; and the limit will be continually narrowing as the friction is greater. Yours, &c. R. C. Jun. Sir, I would inquire through the medium of your useful Magazine, why the use of linen thread, especially in sewing, has of late years given place to cotton; an article so much inferior in strength and durability, which, however, the manufacturers have brought to very great perfection, but of which I know not why flax should not be equally susceptible. Does cotton possess any peculiar quality to render it more generally useful than flax, or is it that the raw material is obtained at less expense? The bad quality of much of the paper of the present day, owing apparently to the use of cotton instead of linen rags, chiefly suggested this inquiry. I am, yours, &c. Dec. 9, 1829. E. W. G. CYLINDRICAL RAILWAY-CARRIAGE. FL. The following account of a very novel and ingenious description of railway-carriage, invented by a Mr. P. Fleming, engineer, of New York, is given by Dr. Jones, the Superintendent of the Patent-office at Washington, in a recent Number of the "Journal of the Franklin Institute." V "The carriage is a cylindrical body, which may have an axis passeing through it, or gudgeons affixed to, and projecting from, its ends, for the af purpose of drawing it. The wheels are iron rims placed round the cylinder so as to encompass it like hoops; these stand at a proper distance from each other, to run upon the rail; they are provided with flanches, or have their faces finished in any form suitable to the rail upon which they are to run. In the inside of the cylinder may be stowed boxes, barrels, bales, -or other goods to be transported. 423 When bars of iron, lumber, or other articles of considerable length have to be carried, the traction is performed in a different way; the carriage is then a hollow cylinder, not furnished with ends; the iron bars, boards, or plank, are passed entirely through it, and, of course, do not admit the employment of an axle, or gudgeons. In this case an endless rope is passed round the middle of the cylinder, which is furnished with double rows of pegs to form a groove, or checks, to retain the rope, or band, in its proper place. This rope also passes over a pulley, which is attached to the horse, or other drawing power, so as to work like the large and small wheels of a lathe with their bands. Two, three, or more cylindrical carriages may be made to follow each other, when connected by bands in the same way. "Under this arrangement it is evident that whatever is carried must roll with the carriage, but in transporting some kinds of goods, and particularly in carrying persons, this would, to say the least of it, be very inconvenient. To obviate this objection, a second cylindrical body is placed inside of the first, and is made sufficiently small to revolve within it. This is suspended upon the axis, or gudgeons, and is weighted on one side, so that whilst the outer cylinder rolls upon the road, the inner one will not revolve with it. It is proposed sometimes to make this suspension by the agency of friction-wheels, so as to leave but little more friction than that which results from the rolling of the carriage. The patentee says "What I claim is the use of a cylinder, or other volume of revolution, on a railway, as a carriage, or vehicle for transportation. 6 "I also claim as my invention the use of the endless rope in the manner above described, for progressive motion. By means of this use of the cylinder and traction-rope, friction is saved, or avoided, to a greater degree than by any machine now known. The traction-rope may be employed separately from the cylindrical railway-carriage, in any other machine where similar progressive motion is required.' ' ܐ܂ |