Sir,-Your nameless correspondent, whom I shall call "A. B.," has given us two rules for solving two of the most important cases in spherical trigonometry (see Nos. 1140, 41, and 43). 1st. When the three sides are given to find an angle; and 2nd, when two sides and the included angle are given to find the third side. "A. B." has given us a formula for each case, neither of which are well-adapted for logarithmic calculation, and both of which require, at the same time, no less than three sets of tables, viz.,-1st, a table of the logarithms, sines, and tangents; 2nd, a table of the logarithms of numbers; and 3rd, a table of the natural cosines. Now very few of the common tables contain the natural cosines; and, besides, even with natural cosines at hand, no less than eight steps are required to solve either of the cases. If, however, we use this well-known equation, viz.,—

and A the angle contained between the sides b and c. "A. B.'s" rule embraces two cases. 1st, when the latitude and declination are of the same name; and 2nd, when they are of contrary names. Now

I must tell "A. B." that a case might occur in which, by following his rule, an unscientific calculator would most likely be led into error. For example, suppose the latitude to be 51° 32′ N., the true altitude of the sun's centre, 10° 30', and the

Nat. cos. H. 223891 77° 3' 44". Therefore, the time angle is either 77° 3′ 44′′, or

180°-77° 3′ 44′′-102° 56′ 16′′.

It is, however, the latter angle that must be taken; because the tan. lat. tan. dec. is greater than sin. alt. sec. dec. sec. lat.; consequently, the cos. H is negative; that is, the time angle is obtuse, and 102° 56′ 16′′ - 6 h. 51′ 45′′ from noon; and the apparent time of observation, 5 h. 8′ 15′′ A.M.

=

Solution by the preceding equation. Here a, b, and c are 79° 30′, 38° 28′, and 66° 40'; hence

8 =

a + b + c

2

and s-a=12° 49'.

.. A=102° 56′ 16′′, the same as before. The superiority of this last solution is perfectly apparent.

I shall give another equation for finding the angle A. Let the difference between b and c be called d,

α

2

sec. lat. sec. dec.

of the 2nd case; but this I will make the subject of another communication. KINCLAVEN.

August 12, 1845.

MESSRS. NASMYTH AND MAY'S

The following interesting elucidation of this system from the pen of Mr. Nasmyth, appears in the Mining Journal of last week. We gave the specification of the invention in our Magazine of the 21st June last, and at the same time particularly directed the attention of our readers to the remarkable facts in pneumatics on which it depends.

"In the first place, the object desired to be attained is, to remove the air entirely from the interior of certain large chambers, so that they may, as it were, become vast magazines of vacuum. The ordinary mode of doing this, is to pump out the air by airpumps, which receive their power from a vacuum, created above or below the piston of a steam engine. The principle. I set out upon is simply this-why employ one vacuum to create another? when we could, by the primary process, attain the desired object, without the intervention of any secondary action, or machinery, whatsoever. Now let us examine how this is best to be done. One cubic foot of water, converted into lowpressure steam, will, in round numbers, yield 1700 cubic feet of steam, which will be capable (on being introduced at the upper end of an upright air-tight vessel) of displacing, or forcing out at an aperture below 1700 cubic feet of air; if we now stop the further influx of steam, and close the aperture below, and either permit the steam to condense, per se, or perform that duty by a separate condenser, we shall have for our 1700 feet of steam, 1700 feet of very nearly perfect vacuum (supposing, of course, that our vessel was exactly 1700 cubic feet capacity). Now, if we suppose a communication opened between this magazine of 1700 cubic feet of vacuum, and an atmospheric railway pipe of similar capacity, we shall abstract one-half of its contents of air, and at once reduce it to the state of a vacuum of 7 lbs. to the square inch, or thereabouts. Here, then, we have done some work, so far, with our first 1700 cubic feet of steam. It will be evident that the remaining vacuum in the exhausting chamber, and that in the pipe it has partially exhausted, will be similar in extent namely, each a half perfect vacuum. Now, let us suppose that we have, during the performance of this operation discharged the air from a second chamber of like capacity to the first-viz., 1700 cubic feet, and that that vessel is just filled with steam on a balance with the atmosphere; if before opening the communication between our con

1

ATMOSPHERIC RAILWAY SYSTEM.

denser and this steam-filled vessel, we first open a communication between it and our first vessel, which, as before described, is in the state of half vacuum, it is evident that this first vessel will abstract from the steamfilled vessel a very large portion of steam, until the two are then on a balance; on this simple system of mutual transfer we not only employ the first vessel to act on the second, as a preliminary condenser, but also, as it were, use the steam of the second vessel in great part twice over, inasmuch as this transferred steam will so far act the same as fresh steam from the boiler, in satisfying the wants of the first, or "used up" chamber; this being the case, the second vessel has its vacuum rendered complete, by being brought into communication with the condenser, while the first vessel has its complement of steam made up direct from the boiler, which steam, flowing in at the upper end, performs the air discharging office to perfection. I fear, in my attempts to ren der my system easily understood, I have made what is in reality a most simple and rapid action, appear complex and tedious of performance; but in point of fact, the whole process of displacing the air, condensing the steam which displaced that air, and the transfer of the steam from the one to the other, will be the work of some twenty or thirty seconds of time; and as the vessels which I propose to employ will be upwards of 150 feet in height, and ten feet in diameter, four in number, standing close together inside a tower, we shall have, by their combined action, a magazine of vacuum remarkable perfection, of the vast capacity of 46,800 cubic feet, which would be capable of exhausting seven miles of fifteen inch atmospheric pipe at one masterly stroke, in a few seconds, producing, in that seven miles of pipe, a vacuum equal to 7 lbs. to the inch, which is quite ample for all the purposes of atmospheric railway locomotion. These vacuum chambers, or vessels, will be formed of boiler plate, and lined inside and outside with wood, to prevent any loss of heat, it being important to maintain the chambers near the heat of boiling water, so as not to cause any undue loss of steamthis, however, is a very simple affair.

of

From actual experiments I have made, I find that when low-pressure steam, say lb. to the square inch, is permitted to flow in at the upper end of a tall upright vessel, having an opening below, the included column of air is depressed and forced out with the utmost ease and rapidity, while at the same time, there is no appreciable mix

ture between the steam and air, the two preserving the most remarkable distinctness of separation. This, in fact, forms the grand principle on which I act-namely, the vertical displacement of the column of air by low-pressure steam. At all times it will be most convenient to employ the time between the running of the trains, to prepare the magazine of vacuum all ready for instant action at any moment's notice; in this way a comparatively small boiler will answer. I may mention, that one grand advantage of having a ready-made store of vacuum at hand is, that the closing of the long valve is performed effectually the instant the communication is formed between the vacuum chambers and the atmospheric pipe; this will obviate much source of loss from leakage, which continues so long when the vacuum is produced by the comparatively gradual process of pumping out the air by the ordinary system. I trust the time is not now far distant, when a full-sized, and, therefore, true experiment, will be made, to test, by actual practice, the comparative merits of my direct system, with that of the steam-engine and air-pump system, by erecting a set of my vacuum chambers alongside one of the engines about to be employed on some of the atmospheric railways, obtaining the required steam for my system from the identical boiler, and working the two systems month about, and taking exact account of the coal consumed by each: I should stand or fall by such a true experiment as this, which is beyond all abstract investigations in getting at the real truth."

137

mile and 70 chains in length, with an average inclination of 1 in 48.

The stationary engine is of 25 horses (commercial) power, with a cylinder 20ğ inches diameter and 5 feet stroke, and was erected by Messrs. Stephenson of Newcastle, in the year 1830. It will bring up 35 tons at an average rate of 73 miles per hour, on the incline of 1 in 48, and is, therefore, capable of producing as great a useful mechanical effect as the Dalkey engine of 100 h.p., as it raises more than half the load three times the altitude, on the same length of incline, in nearly the same interval of time.

With a result so important and unexpected, it was necessary, in order to render the deductions perfectly satisfactory, to adopt every practical means of ascertaining the exact power exerted during the experiment; an indicator and counter were therefore attached to the engine.

The slide valves of the engine were very inaccurately set during the first seven experiments; the eighth was made the next day, after the slides had been adjusted; this experiment, therefore, shows a better result. Further alterations of the slides, had it been possible to have made them, within the time allotted for these experiments, would have ensured still more satisfactory results.

Much misapprehension has prevailed from comparing the atmospheric with the locomotive system, in consequence of using the same arguments as were brought forward in favour of the stationary power, and it will be apparent, from reading the Report of the Atmospheric Committee,* that they have been misled by these arguments, and that they would have arrived at the same conclusions if they had been considering the system of stationary engine and rope traction, and for precisely the same reasons. A comparison of the atmospheric system with that of rope traction is therefore important, in order to ascertain the practicability, or the applicability of the atmospheric system. These experiments may, it is contended, be received as strong arguments against the commercial practicability of the atmospheric system.†

The exact inclination of each portion of the plane was accurately ascertained, posts were placed at intervals of 5 chains, and the time of passing each post was carefully registered. The tables of experiments give the weights of the carriages and load of each train, the number of revolutions of the engine, the speed of the piston per minute, the average pressure upon the piston, the

For this Report see Mech. Mag. vol. xlii. p. 293. The experiments were entrusted to Mr. Gastineau, who followed with great accuracy the author's instructions.

pressure in the boiler, and the estimated power of the engine.

From these comparative tables the following results are obtained.

1st. That an engine of 25 commercial h.p. on the Whitstable line, working at an average of 50 h.p. of 33,000 lbs., gives a useful mechanical result, nearly equal to the Dalkey engine of 100 commercial h.p., working at 160 h.p.; or in other words, it raises trains of 30 tons on an incline of 1 in 48, at the same speed that the Dalkey en-/ gine raises trains of 66 tons on an incline of I in 138, which is evidently nearly an equal mechanical effect.

2ndly. That the lost power on the Whitstable Incline averages four-tenths of the whole power, while on the Dalkey line it averages eight-tenths of the whole power.

3rdly. That the loss in transmitting the power to the train, as shown in Mr. Samuda's experiments, exceeds eight-tenths of the whole power, while in his statement sent to the Committee it was estimated at oneseventh, which explains the difference in the calculations of the power required on the atmospheric lines.

4thly. The amount of lost power appears, in Mr. Stephenson's experiments, to be nearly uniform, throughout all the degrees of vacuum; this at first sight appears extraordinary, but it is evidently caused by the circumstance of the useful mechanical effect

being computed by multiplying the resistance due to friction and gravity into the mean velocity in feet, without regarding the resistance of the air, which was greater with the low degrees of vacuum, because the velocities were greater.

It is, however, difficult to estimate this, as the velocities were variable; but assuming 20 lbs. as adopted by Mr. Stephenson, for 30 miles per hour, it will affect the result, to the amount of about 25 per cent., which may be taken as an approximation to the truth; but whatever it may be, it is evident, from the uniformity of the results, that the effect of the increased leakage, at the higher degrees of vacuum, does not exceed the increased resistance of the air, and thereby proves that the leakage is not the main source of lost power.

The experiments by Mr. Samuda and M. Mallet show a less mechanical effect than those by Mr. Stephenson; which arises evidently from their holding the train by the breaks until the vacuum was made, in order to obtain as great a maximum speed as possible.

With a result showing so large an amount of lost power, it becomes interesting to en

A

quire to what it is to be attributed, and how the total amount is made up; but this would require experiments more especially devoted to that question. It is, however, evident that a large portion is of a description which (although not lost in a theoretical point of view) is not under mechanical control, either with heavy or light loads.

With heavy loads and a high vacuum, the power of the engine required to form the vacuum will amount to the greater proportion of the whole power exerted, and which will be practically lost; with the other extreme, of light loads and a small vacuum, the friction of the air in the pipe becomes an equally large proportion of the power expended, because the whole bulk of air must be exhausted, however small the load may be. In fact, the atmospheric medium of traction may be practically represented by a rope without weight, but so extremely elastic, that with a high vacuum the larger proportion of the power of the engine is required to give it sufficient tension to sustain the required load. With small loads, the friction of the air on the sides of the tube will also form a large proportion of the power exerted by the engine, so that there will be one particular vacuum and load, with any given engine and pipe, in which the sum of these losses will become the least possible; but whatever this may be, is evident the amount must be fatal to the atmospheric system as a mode of applying stationary power.

There is also a loss of power necessarily arising from the construction of the airpump, and, in addition to that from leakage, there will be a certain loss from the atmosphere not having time to act with full effect on the piston.

It must be apparent, that the loss from leakage does not form a large proportion of the lost power, and there is, therefore, but little field for mechanical improvements, which has been urged as an argument in favour of the system.

This loss of time and of power, from the interval necessary to get up the required vacuum, will have the effect of reducing the average rate of travelling on a single line of atmospheric pipe, considerably below that of a locomotive line; because such loss of time must occur at the meeting of every train, and with trains running at intervals of half an hour, this delay must necessarily occur every quarter of an hour.

Let A, B, C, D, in the annexed diagram, represent the position of the engines on an atmospheric line, the first section, A to B, being similar to the Dalkey line.

The best result obtained from the experiments on the Dalkey line was the transport of a train of 30 tons in five minutes. Now, assuming three trains per hour to be required, if there was only one engine, the train, when it arrived at B, would have to wait five minutes, which would make the total time of traversing 14 miles ten minutes, being an average speed of 10 miles per hour, although the extreme speed may have reached 35 miles per hour. If the engines at B, C, D, were double engines, the average speed would be doubled with double the number of trains.

139

From this it appears, that on a railway formed of sections similar to the Dalkey line, with single engines, or 100 h. p. (of 33,000 lbs.) per mile, a train of 30 tons would make three trips each way per hour, at an average speed of 10 miles; and if the engines were double (i. e. 200 h. p. per mile), six trips each way per hour might be effected, at an average speed of 21 miles per hour, which is the simplest case of a long

Therefore, in a line with meeting points every five miles, with similar gradients and power to the Dalkey line, i. e. 105 h. p. per mile (without spare engines), the greatest effect will be to transport trains of 30 tons every half-hour, at an average speed of 30 miles; which is evidently an insufficient power to insure punctuality with the ordinary traffic of a railway.

[Here follow in a tabulated form the experiments with the Tyler Hill Engine, illustrated by indicator diagrams; after which comes a Table, which we shall give next week, exhibiting the general results as contrasted with those furnished by the Dalkey Engine.]

G Dble... Eng.

14 15 miles.

evidently without a knowledge of how the apparent difficulties in the application were proposed to be overcome, or were actually avoided. He would instance only a few points, and leave to others, better qualified than he was, the task of refuting the charge of impossibility. It was not proposed to use any other than the engines of the main line, for working the sidings, which could be laid in, without at all interfering with the continuity of the main line. Level crossings were quite as practicable as on locomotive lines. There was even an additional security, as by a simple contrivance, consisting of a cylinder and piston connected, with the main pipe, the platform which, when down, formed the protection of the valve, under the crossing, could be raised when the vacuum was being formed, and thus, not only became a signal that a train was about to pass, but also formed a barrier, for preventing anything from traversing at an inopportune moment. He could not understand the necessity for bringing two trains together, as had been assumed; but if that did occur, a little extra power might be used in that particular instance, in the same way as in an emergency, another locomotive would be added to the ordinary train engine. As respected the liability to be thrown off the rails by impediments, he must contend, that the position assumed was not supported by facts. On the Dalkey line, there were curves of 130 yards radius, which were constantly traversed at a speed of 35 miles per hour! yet no accident had occurred. It was well known that locomotive engines were not in the habit of traversing curves of that radius, at such a speed. He could not agree with the statement of the comparative cost of the two systems. He thought that the author had underrated the actual cost of locomotive haulage; while he had overrated, not only the cost of that by the atmospheric system but also the amount of power employed