729 APPENDIX. NOTE A. On 10th May 1831, on the Liverpool line, a malleable iron rail, fifteen feet long, carefully cleaned, and weighing 177 lbs. 104 oz., was laid down. 1833 the same rail was taken up by Mr. J. Locke, then resident engineer on the On 10th February line, and well cleaned as before, and weighed 176 lbs. 8 oz. lost, in twenty-one months, a weight of 18 oz. It had consequently passed on the rail, during that time, was estimated at 600,000. Thus we see that The number of gross tons that had with so considerable a tonnage, and with the velocity of the motion on that railway, the annual loss of the rail was only of its primitive weight; so that it would require more than a hundred years to reduce it to the half of its present strength. NOTE B. Given the area of section of a railway bar below the neutral axis, to find the dimensions of the lower flanch, so that the strength shall be a maximum; the breadth of the middle rib, and the depth of the lower flanch, being also given: let the whole of the sectional area below nn the breadth of rib p q the depth n s the depth of lower flanch r s the tension of the lower fibre =t 1. To find first the expression for the strength of the middle rib: take any variable distance x, then d't::x: tx d' tension of fibre at æ; multiply by the distance, and breadth b, and we have, for the sum of all the resistances, bx2 dx = tb 3 d' this, when x=d', becomes d'2 bt. 2. To find an expression for the strength of the lower flanch: let the breadth =b'; any distance from n n= x; then d' t::x: tx tension of fibre at x; d' multiply by the distance r, and breadth b'; we have, for the sum of all the resistances, d' this taken between the values x=d' and r=d'. therefore the resistance of the rib and lower flanch is, + d12 bt + (d'e- e3 3 d' And the question is, to determine what value must be given to d', that this expression may be a maximum. To effect this it is only necessary to consider d' as variable, to denote it by x, to find the value of the dependent quantity b' in terms of x, to substitute these quantities in the preceding expression, and to make its differential equal to zero. Now since the depth of the middle rib is æ, and breadth b, the area is bæ, and consequently the area of the lower flanch-a-bx, and its depth being e, its breadth a-br = that is, b' a-bx Reducing every term to the denominator 3 r2, and rejecting it, this becomes 2 b x3 + 3 ax2 − 3 b x3 — e2 a + be2 x-3 b x3 + from which may be determined for any given values of a, b, and c. NOTE C. It may be convenient to shew the origin of these formulæ, particularly the third, which is not investigated, except that it has been shewn generally, that if d' denote the depth of the lower fibre below n n, and its tension be made=t, and any variable distance = x, i f = f. xd s from which it follows, that t8' -d' distance of centre of tension; r.dr sum of all the resistances for a unit of breadth, a being taken in fx. its ultimate state. d's, whence the above Now in the rib, when x=d', 'd, and becomes d/2t; but to refer this to the centre of compression c, we have (called the whole depth d) In the same way, calling the tension at x=t', and the breadth (n n-pq) we have, for the resistance of the head, whence the resistance referred to n n is, for the breadth (mm-pq) NOTE D. Extract from "Petition of proprietors of stage coaches employed on the turnpike roads within the county of Lancaster, and travelling in the following line of roads, viz. Liverpool, through Warrington to Manchester. to St. Helens. to Newton and Wigan. to Leigh and Bolton. "The petitioners take leave to exhibit the following facts and statements of calculations, viz. of the taxes paid to government, and tolls paid to the commissioners of the turnpike roads, over which the thirty-three coaches travel between the points before stated, and the cost of working them for the year last past. Presented 3d May 1830." 733 NOTE E. Memoranda relative to the Experiments made at Mr. Laird's works, at North Bicken- The exhausting apparatus consisted of a fan-wheel, with broad radial leaves, revolving within a close box or chamber, placed a little apart from the boiler, but connected with it by a passage leading from the flues traversing the boiler; a short tube above the exhausting chamber passed out to the atmosphere. The furnace was attached to, and placed at the end of, the boiler, opposite to the exhausting apparatus, which latter being put to work, drew through all the turns of the boiler the hot air from the fire, which passed over the throat of the furnace through the bridge flue, and then successively through the other five turns of the flue arranged through the boiler, and finally, was drawn through the exhausting chamber, and passed into the atmosphere. The heat, which in the furnace was extremely intense, was absorbed by the water in the boiler, as the air rushed through the flues, and, when passing up the tube or funnel from the exhausting chamber, was so far cooled, that the hand and arm might be placed, with impunity, down the tube, the temperature not exceeding 180° of Fahrenheit. Not the slightest smoke was perceptible. 6 long. Diameter of exhausting wheel 3 0 0 10 Bridge flue or throat, from the furnace, 2 ft. 6 in. broad, 4 in. wide, 2 ft. deep First turn of the flue, 4 in. wide, 2 ft. deep, 2d, 3d, 4th, and 5th turns, 3 in. wide, 2 ft. deep Whole length of the flues through the boiler Superficial area of the heating surface 5-16 in. iron plate. in. iron plates. 247 square feet. 45 feet. The contents of the water in the boiler, when filled, were from 85 to 90 cubic feet. 33 square feet. The proportion of the heating to the evaporating surface, nearly 7 to 1. 4 ft. 10 in. average depth. Containing about 65 cubic feet. Diameter of the safety valve very nearly five inches, being nineteen square inches area, which was loaded for a pressure of 4 lbs. on the square inch. Giving seventysix for the load. Of this, 66 lbs. of iron were placed in the boiler, and 10 lbs. allowed as the weight of the valve, rod, hook, handle, &c. |