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BY PROFESSOR DAVIES.

Again, let the parts discriminated in the figure of the last problem be represented similarly in the present one. Then draw the lines cc,, dd, eee, respectively perpendicular to the horizontal traces of the vertical cutting planes (the sides of the railway) and of the surface-plane; and make cc, =uy, dd, uz, ee1 =uv and ee2=xv. Also, draw through c1, d1, z, the parallels to a, A, b, B, AB intersecting in a,, B1.

These parallels, it is obvious from the principles of projection, are the projections on the horizontal plane of the three lines in which another horizontal plane distant from that of the roadway, the height ux, cuts the slopes and surfaceplane respectively. The points a,, ẞ, are therefore the projections of two points, one in each trace, upon the surface-plane by the two slopes.

Also, A,B are two other points in the projections of these traces, and consequently the projections are found by drawing Aa,, ßß,, as in the figure.

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PROBLEM III.

With the same data, to construct the traces of the slopes themselves on the surface-plane.

Construct as in the preceding problems the surface-traces of the vertical planes through Aa,,Bb,, and the horizontal projections Aa,, BB, of the traces of the slopes upon the surface-plane.

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Draw a aa, perpendicular to AB, meeting Aa, in a. Then a, on the horizontal plane is the projection of a, on the surface-plane, and drawing through a, a line a b parallel to the trace AB, this is the line in the surface-plane whose projection is a, a,b,ß,. Draw a,a,, ẞ,62 perpendicular to AB to meet the line ab in a21 B2. Then a, ẞ are the points in the surface-plane whose projections are a,, ß,. Whence, also, Aɑ2, B6, being drawn, they are obviously the traces of the slopes upon the surface plane, which were required in the problem.

PROBLEM IV.

Given the same things as before, and likewise any point whatever in space, to construct the profile ("cross-section") of the cutting which passes through that point.

However the point may be given, its projection on the plane of the roadway can be found by the ordinary processes.

Let p be that projection, through which draw gh perpendicular to the roadway. This will be the base of the section. Produce it to meet the lines Aa,, BB,, (found as before) in g,h,, and draw g, g2 h, h2 perpendicular to AB, meeting Aa, BB, in 92, h. Then gehe is the section of the surface-plane by the profile-plane, and gg, hh are the heights of the points 92, he above the plane of the roadway.

We have now a choice of methods for constructing the figure; but, practically, the following appears to be the simpler:

Make g',h', g, h,, g1g=g191, h'h', =hh, 8182=882, hhhh, the angles g', h' being right angles; join g'g' h'h', and g'h': then the figure gig'2 h'h', is the section required.

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To construct under the same general circumstances the surface traces of the quadrilateral which is formed by the two slopes and two given profile planes.

Let cd, gh be the traces of the profile planes on the plane of the roadway. Produce them to meet Aa,, Bß, in g1, h ̧,k,, 1,, and construct on the surface-plane the points g2, he, l2, k2. Then g2, hq, la, ką will be the trace on the surface-plane required, as is evident.

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Scholium. Since the profile-planes are parallel their sections gl, kal2 with the surface-plane are parallel; and hence it would, in fact, be sufficient to find 92, ha, k2, and draw kl2 parallel to gh. The actual trouble scarcely differs, however, in the two modes of construction.

(To be continued in our next.)

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Now, the bracketed part being an arithmetical series of m-1 terms, we have

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Then to sign (6) adding the term m2 of the equation (a) we get the entire sum=m3. The author of that paper has suc

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ceeded by means of this property in giving = {(n+1): that is, the sum of the

a very elementary investigation of the sum of the cubes of the natural numbers; and not really longer in its work than the method of integration, but much shorter than that by indeterminate co-efficients. It would, however, have been well for him to point out the form of his result, sum

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cubes of the first n natural numbers is equal to the square of the sum of the same numbers themselves.

I am, Sir, your obedient servant, CENTURION.

Home, July 30, 1846.

ON CERTAIN FORMULE RELATING TO THE CIRCLE.

Sir, To carry a little further the first set of approximations which I gave at page 116 of your last number, let v=

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14, r remaining unaltered, then,

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(see page 287 of the last volume of this work.) The correction might of course be continued.

We might, were it advisable, make y &c. expressions involving quadratic

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Since the first introduction of wroughtiron rails for railways, by Birkenshaw, much ingenuity has been exerted to discover a form which, with a minimum of metal, should give a maximum of stability and strength. Considerable difference of opinion exists as to the best form of rail in use at the present day, each particular rail having its advocates. A rail, of which a representation is annexed, has been patented by Mr. Wheeler, which bids fair to dispose of this question, its advantages being of so positive a character as to establish its superiority in all those points so essential to the perfection of railways. The following extract from Mr. Wheeler's specification will at once explain the nature of his invention:

"Fig. 1 is a side elevation of a portion of my improved rail, and fig. 2 a cross section of the same. This rail is trilateral, having three edges or surfaces, A A'A", capable of being used as bearing surfaces for the wheels of carriages to run upon, each of which surfaces may be placed upward in succession as those previously in use become worn or injured. This rail may be secured to the sleepers, as shown in the drawings, where B represents a portion of a sleeper to which a half-round piece of timber, B', is nailed, which exactly fits the underside of the rail, and forms a central bearing for it, while two of its edges, A' A", rest on the body of the sleeper. C is a cast-iron

holding_bracket, bolted down to the sleeper B, a piece of felt or other suitable material being interposed between the sleeper and the rail. Or, the bracket may be formed with a recess on the under side, in which an oak or other wedge may be inserted, as is shown at D."

The following substantial advantages are claimed for this form of rail over those heretofore employed:

1. Great strength, and impossibility of the rail springing or curling at the ends. 2. Greater steadiness from the character and position of its bearing upon the continuous sleeper. 3. Increased safety, inasmuch as wheels with deeper flanges can advantageously be used on this rail. 4. Greater durability, arising firstly from the form of the rail itself, secondly from its having three bearing surfaces available in succession.

While the alarmists are so loudly predicting a probable deficiency and greatly enhanced cost of iron, it is of no small importance to manipulate that material into the most advantageous form. Railway companies are also well aware that everything tending to increased durability of their permanent way is of the utmost importance to their interests, and may, in the long run, prove a remedy for the depreciative consequences of rivalry and competition.

B.

SUGGESTIONS FOR PROMOTING GREATER SECURITY ON RAILROADS.

Sir,-At a time when so many accidents are occurring on our railroads, owing to the ease with which carriages can escape from the form of the present rails, you will, I am sure, allow me to suggest an improvement tending to greater safety, although it might augment the expense,

which ought to be a matter of inferior consideration. I would propose that the rail should form one side of the Roman letter V, at the base of which there should be a horizontal plane, on which the wheels should run that carried the engine, so that unless it could surmount

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an angle of 75 degrees, it could never escape. Both rails being bevelled outwards, it seems nearly impossible that the carriages should get off the rails on either side, and the wheels being lenticular, would be favourable to their overcoming oscillation, being constrained by the acute angle of the sides to preserve its line.

Such a wheel should be formed by the intersection of two large circles, the base of the lens thus produced being truncated with a slightly curved edge, and working on an axle or pole, of equal dimensions throughout. The reason for giving the wheels a convex form, is because it gives the broadest nave, which contralizes and ensures an equal pressure on its truncated rim, whose edges are rounded to prevent friction against the sides of the sloping wing. Perhaps it may not be necessary that the diagonal sides on this construction should be of iron; hard oak might be substituted for it, being never exposed to pressure from the impossibility of the wheel ever getting a bite on them.

A truck road on this principle I proposed some fifty years ago, to convey heavy goods from Staines to London, (that road being then in the winter in many places dangerous to pass, and frequently repaired with fascines or fagots.) The plan was very coolly received, except by Dr. Anderson, author of the "Bee," and Mr. Wilson, a barrister, and absolutely ridiculed by the wiseacres of Windsor-so little was then thought of a mode of conveyance of this kind.'

I remain, Sir, yours, &c.,
GEO. CUMBERLAND.

July 24, 1846.

RECENT AMERICAN PATENTS. [Selected from Mr. Keller's Reports in the Franklin Journal.] IMPROVEMENT IN SHIPS' ANCHORS. Nicholas P. Isaacs, and James Raisbeck.The object or this improvement is to prevent the fouling of the cable upon the stock and arms of the anchor, which in the present mode of making anchors is attended with much inconvenience.

Claim." These additions and modifications to and about the common ship anchor, and their application, for the purposes herein minutely described, are improvements which we claim to be of our own invention, which are briefly enumerated as follows:

"1st, The method by which the cable is

prevented from fouling upon the stock of the anchor, by attaching protectors to the extremities thereof, which extend down and are fastened to the shank at a point below the trend of the shank. These protectors, from their oblique position, clear the cable whenever it touches it. They also obviate another difficulty which is frequently presented, when the anchor is lashed to the cathead, which prevents the entangling of the ropes, and the obstructing and tearing of the sails upon the portion of the naked stock which stands above the gunwale; this is prevented by warding off whatever comes in contact with it.

"2d, The mode of protecting the arms of the anchor against the fouling of the cable, by means of moveable plates of metal attached to the flukes by means of rings, or hinges, and covering the entire arm-the same being combined and operating as herein set forth. The inner ends of these plates move longitudinally upon ridges or projections made upon each side of the shank by means of a connecting chain or rope, by which, when the one plate is forced down upon the arm of the anchor, the other is drawn up to a position at right-angles, or nearly so, with the shank; so that when the arm lies buried in the soil, the opposite one stands defended by the raised plate against the fouling of the cable, which plate assumes a position nearly perpendicular to the soil.

"3rd, We also claim, for clearing the cable from off the arms of the anchor whenever it comes in contact with it, the combination of the hinged sliding plates with the two guards of the same curvature of the arms and crown, consisting of narrow strips of iron extending from fluke to fluke, and welded upon the sides of each. These guards are also attached to the crown of the anchor by means of a short shoulder, leaving a space between them and the arms of the anchor.

"These guards, so situated, create a great width across the crown of the anchor, presenting an enlarged and curved form, by means of which width, and the curvature of the guards, it is impossible to foul the cable effectually upon the arms, and it requires but the smallest force to cause the anchor to cant and bite the ground. The space left between the guards and arms presents a means by which the fish-hook can be fastened about the guards or arms in such a manner as to fish the anchor."

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RECENT AMERICAN PATENTS.

caoutchouc applied and cemented to its exterior surface or surfaces, or of two or more layers of such stocking-knit cloth, having one or more sheets or layers of caoutchouc interposed between, and connected or made to adhere to them."

Claim.-"I claim the above new manufacture, or the combination of the two elastic materials, stocking-knit fabric and caoutchouc, whether the latter be applied to the exterior surface or surfaces of said fabric, or between two or more pieces or layers of said fabric or cloth, substantially as set forth."

IMPROVEMENTS

IN THE COMPOUND

HYDRO-OXYGEN OR HYDROSTATIC BLOW

PIPE.

Robert Hare, M.D.

Claim.-"What I claim as my improvement on the hydro-oxygen blowpipe, and in the processes for supplying this instrument with gas is as follows:

"In the first place, I claim the employment of more than one jet pipe, or several jet pipes, so associated as to produce a large flame by the combination of several smaller flames or jets; or of one jet pipe, so crammed with coarse wire as to cause the gaseous mixture to escape into the air at the point of efflux in several jets, as already described, in lieu of one, as in using the instrument originally invented by me, and in all subsequent modifications prior to the present; for, although in the concentric pipes used first by me, and afterwards by Daniells and Maugham, the hydrogen escaped from the pipe so as to mingle with the oxygen passing out through another, still more than one pipe or one aperture yielding one jet was not used for the efflux of the mixture of the two gases employed. Moreover, although tubes, crammed with fine wire, have been employed as safetytubes to prevent the retrocession of the flame, coarse wires have never been heretofore used to cram the tube at the point of efflux, so as to divide the the gas at the place of inflammation.

"I also claim the employment of a jet, which may be made from between two flat surfaces, whether channelled or not, or, in other words, from a fissure or keef, or row of holes, as set forth.

"I likewise claim the refrigeration of the jet pipe or pipes, by passing it or them through a case or box filled with water or other refrigerating substances, as heretofore described.

"I claim, also, the employment of a vessel provided with a tray and rod like that last described, in combination with another vessel of the same capacity, of which the bottom is above the middle of that first mentioned, both vessels communicating by

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means of adequate pipes and cocks, so that the gas may pervade both simultaneously, or, by shifting a cock, may be prevented from reaching the additional vessel; after which, the acid being driven into the second vessel, its contact with the zinc is prevented or diminished, as more fully made intelligible by the description.

"I further claim the employment, for the support of the platinum, of a brick of kaolin, or other infusible earth, resting on a platform so balanced on a point, or a ball, or in any other way, as to be kept horizontal by attaching to it a weight, and supported by a lever turning on a universal joint, and as heretofore described.

"I claim, also, the employment of any variety of carburetted hydrogen for the fusion of platina, in lieu of hydrogen, in the apparatus above described."

OR

IMPROVED METHOD OF ADJUSTING THE BED OR PLATEN OF TOGGLE-JOINT PROGRESSIVE LEVER PRESSES. S. W. Bullock.-The bed or platen of the press has inclined ends that slide on blocks, which are made to slide horizontally in the frame of a press, so that when they are drawn together by a double screw, they force the platen out, and when moved apart they permit the platen to be moved in.

Claim.-"What I claim herein as new, and desire to secure by letters patent,' is the discovery of the peculiar principle or feature exhibited in the accompanying drawings and specification, to wit:-Regulating the bed or platen of toggle-joint and other presses (to suit various sized bales or packages) by its connection with some moving part of, or connected with, the press, and the holding said platen at any and all points of its range, with equal power or force of resistance, as is indicated by the position and power of the press, and in accordance with the size of the bale, using for that purpose any known mechanical arrangement that will produce the intended effect, as herein set forth."

PROPELLER FOR BOATS. Horatio Hubbell.-Claim.-"I am aware that conical propellers have been heretofore proposed to be used, but these differed materially from that herein described, they having been furnished with propelling vanes, buckets, or threads, set spirally on the outside of the cone; and to such propellers I do not therefore make any claim; but what I do claim as new, and desire to secure by letters patent, is the within described manner of forming a propeller for steamboats and other vessels, by placing vanes buckets within a hollow cone, which is made to revolve with its open mouth, or base, in the direction of the vessel's wake, and its

or

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