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under different circumstances, and perhaps there can be no general comparison between the two.

The toothed-rail system used on the Mt. Washington and the Rigi Railways is so familiar that mere mention of those celebrated roads will stand in lieu of a full description. The name "third grip rail used on the Mt. Cenis Railroad, explains itself.

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be transported from a higher to a lower evel.

If to the train at the summit of the incline a locomotive is attached whose weight is W, and if, for the moment, friction is neglected, it is evident that the entire train will be capable of elevating a counter weight weighing P+W+ W', W' being an additional weight, due to the traction of the locomotive, and Many other ingenious methods of depending upon the inclination of the overcoming elevation have been pro- plane. The counter weight will, in turn, posed, but explanations of them, al- be capable of elevating a dead weight though perhaps interesting, would not P+W+W' or a train weighing P+W+ be compatible with the limits of this paper. Let us, therefore, proceed to explain the device proposed in this

paper.

The adhesive power of a locomotive is evidently the limit of its tractive power, and may be substituted for it in determining the maximum load. It varies greatly according to the condition of the track, but may be safely taken at onefifth the insistent weight on drivers multiplied by the cos. angle of inclination.

2W' provided it is also drawn by a locomotive. By properly choosing the inclination of plane and weight of counter weight, the system will be under the In order to make plain the problem full control of the locomotive, and enunder consideration, it may be repeated tirely independent of the preponderence in a more specific form. Let us suppose of the traffic down the plane. But bethat a trunk line wishes to extend a fore discussing this case let us determine branch into some mining district where the value of W'. the elevation to be overcome is great, but the character of the country such that most of it may be mastered in one or two short inclines, if safe and economical means of climbing them can be devised. In overcoming ordinary grades, and in all attempts to climb steep gradients, the real process is a change of kinetic energy, or energy of motion into potential energy, or energy of position. In all locomotive systems now in operation, no attempt is made to utilize this potential energy after it has been attained. On the contrary, outside of the mere descent, it is a troublesome element which has to be overcome by brakes. If it can be put to practical use its superiority over kinetic energy, which must be generated for the occasion, is manifest.

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Let us now investigate the load which a locomotive will draw on a level, and then extend the investigation to inclines suitable for systems intended to overcome great elevation.

Let W" total weight of train.
"T

=traction of locomotive, on
level.

Let us suppose that we have a train of loaded cars, whose weight is P, at the summit of a steep incline, and let it be connected with a counter weight, enough lighter than itself to provide for friction. The train descends and the counter weight ascends, and is then in position. to eert its potential energy in elevating another train, provided it is lighter than the first. When the traffic down the plane is greater than in the opposite. direction, the system may be operated without the aid of an outside motor. is, indeed, a method very often used in mining operations where ore or coal is to Projecting, all the forces upon the axis

It

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a

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angle of inclination.

f coefficient of friction, of

train.

T' traction of locomotive on plane.

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or

8

W"sin a+W", cos=p G cos a

W'tan a+

2240

8W" PG

=

2240 5

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Y=weight of counter weight.
G =
66 "locomotive and tender.
Ffriction "
F'= 66

etc.

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"sheave journals, ropes,

W" total load which locomotive will draw upon the incline. W' may be determined from eq. (2) by making tan a=

or

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W"

.6

W" +

=

G

280

W" 1.05 G.

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In determining the last member (F') it is only necessary to obtain the initial (2) friction, or that which must be overcome in order to set the system in motion. When a=0 and p=.6 (which will be Since the pressure on journals is 2 sin a the case if the locomotive has four driv-.14 Y, and the coefficient of axle fricing wheels) tan a=0. Then (from eq. 2)

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Y=G+1.05 G—.007 G-.0105 Y
Y=2.02 G.

consideration either the weight or rigidi-
ty of the rope, but is sufficiently accu-
rate to show that the system will be un-
der full control of the locomotive, provi-
ded the inclination does not exceed one
heavier than 2 G, for it can always pro-
in nine, and the counterweight is no
ceed up the plane and manage the coun-
mind, the reasons for the following plan
terweight unassisted. Bearing this in
for a lateral road will be manifest:

This calculation does not take into

built on an ordinary supporting grade, Let the mean portion of the line be but at such places as the character of the country may warrant, let occasional planes, not exceeding one in nine, be thrown in, and let each of these be provided with suitable counterweights, weigh(3) ing no more than 2 G. Then all trains

weighing less than 3 G may proceed over dry rails it rises as high as G cos a. the planes without delay, but should To obtain the maximum utility it would, they exceed this limit it will be necessary of course, be inexpedient to assume a to divide them, and make two trips over the plane. If, however, the traffic can be arranged so that the counterweight will always be in position for the next train, that is, if the trains alternate in direction, it may be made much heavier and the angle of inclination may also be increased.

To illustrate this let us take an extreme case. For instance, take a=45° then tan a=1, and from eq (2) we obtain the value of W".12 G, which may be interpreted thus: On a plane whose inclination is 45°, a counterweight weigh ing .88 G will be required to elevate a locomotive weighing G, provided the full traction of the engine is utilized. As soon as the inclination has passed the limit one in nine, the system is no longer under control of the locomotive, and there is no further necessity of limiting the weight of counterweight. But (in the case a 45°) the trains passing down the plane must weigh at least Y-.12 G, and those passing up the plane will be limited to Y+. 12 G.

=

In most cases these conditions would be found inconvenient if not impossible, and the use of inclinations greater than the angle of adhesion would be expedient only under very exceptional circum

stances.

The construction and equipment of planes suitable for this system would be, in most cases, neither difficult nor expensive. Perhaps it will be well in this connection to give a few details. However, before doing so it will be necessary to assume data for a particular case.

Since the value of exerts so great an influence upon the maximum grade which may be given to the plane, it is very essential to use a locomotive, the weight of which is well concentrated upon the drivers. This effect is best produced in what is known as a "tank engine;" and in the following details it will be assumed that such an engine is used.

Now, since the track is exposed to all kinds of weather, a poor, though not absolutely the worst condition, of the rails must be taken to compute the inclination of the plane. For frost, the adhesive power is G cos a, while for thoroughly wet rails it is G cos a and for

factor of adhesion providing for snow, for since the planes are short it would, doubtless, be more economical to clear the rails than submit the system to such unfavorable and extreme conditions. In the previous discussion the factor was used, and the value of p was taken at .6, thus providing for frosty rails and a tank locomotive. Under these suppositions, the inclination was found to be about one in nine. This is a somewhat smaller inclination than that assumed by Mr. Handyside and others, which is probably due to the fact that it was derived under the assumption that the rails are in poor condition.

In order to illustrate the problem let it be proposed to overcome an elevation of 200 feet, and assume available ground for a straight incline which will be 1800 feet long, since the inclination is one in nine.

From what has proceeded it will appear that when the rails are covered with frost the weight of the counterweight is limited to 2 G, or in this case, 80 tcns. If, however, the rails are thoroughly wet this weight may be increased to 120 tons, and for dry track, to 125 tons. To provide for this difference and always obtain the maximum effect of the system, it is only necessary to divide the counterweight into sections weighing respectively 80, 40 and 5 tons. In good weather the total weight may be elevated by the locomotive, but when the rails become slippery the sections must be uncoupled, and the 80-ton weight alone used. A good form of counterweight, and one which would combine economy in first cost, compactness, and durability, would be simply pig iron cylinders, roughly cast around axles and fitted with ordinary car wheels. These cylinders might be connected together in groups of two, by means of side beams carrying suitable boxes for the axle journals, and the groups, in turn, coupled into a train of the requisite weight.

In cases where the proper excavations could be cheaply made, counterweight might be made less with a somewhat narrower tread than the cars, and run on a track in a covered trench, underneath the main track, thus reducing, in a great

measure, liability to accident, and securing the counterweight against all obstructions.

COMPARISON.

90.4 feet per mile, and the distance re-
quired to overcome 200 feet elevation is
2002.2 miles. This means that over-
coming an elevation of 200 feet and a
total curvature of 180° per mile is equiv-
alent to overcoming 100×2.2=220 feet
on a straight track. We may therefore
consider our grade as 100 feet, and dis-
tance as 2.2 miles, and pay no further
attention to curvature.
In this case the
length of the supporting grade is
2.2 X 5280

In the absence of accurate data it
would be a difficult matter to make a
close comparison between the cost of a
road built on this plan, and one in which
the same elevation is overcome by an or-
dinary supporting grade; but the follow-
ing is a rough approximation to the com-
parative merits of one plane of the
gravity system, and a corresponding por-equivalent gravity plane.

tion of the common road.

1800

6.45 times as long as the

This will enable us to roughly approx

tems, but in order to make a fair comparison of their merits, we must also take into consideration the amount of work which can be done on each in the same time.

To arrive at this result, let us first determine the time required to take the maximum train over the plane, and also the time in which the same locomotive, drawing the same load, can climb the grade.

As before, let it be required to overcome an elevation of 200 feet, and as-imate to the relative cost of the two syssume available ground for incline of one in nine, and also for the development of a supporting grade. The length of the former will be 1800 feet, while that of the latter will depend upon the grade assumed. As a gravity system would be adopted only in mountainous districts, curvature would exert considerable resistance, and should be taken into account. This is easily done, in case of the plane, by assuming the weight of counterweight to be enough greater than that of the maximum train, to overcome all resistances. Five tons difference, which has been assumed in the following discussion, is not only sufficient to do this, but would also render some assistance to the locomotive. In the case of the support-H cos a) ing grade, however, these resistances must be overcome by the locomotive and must be considered.

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There is a custom of equating or compensating for curvature on supporting grades, by deducting from the rate of inclination an amount equal to from 150 to 8 of one per cent. of the distance for each degree of curvature. Thus, if we assume a nominal grade of 100 feet per mile (1.9 feet in 100 feet, or a 1.9 per cent. grade) it must be flattened to 18 (an average of .06) feet in every 100 feet for each degree of curve, or .06 X52.8=3.168 feet per mile, for each degree in 100 feet. Now, let us assume something like the ordinary characteristic of alignment in a mountainous country, say a total deflection of 180° in each mile of distance or 3.03° in each 100 feet. This will call for a reduction of 3.03 X 3.168 9.6 feet per mile.

The absolute grade is then 100-9.6=

Let Hthe absolute horse power which the locomotive is capable of imparting on a level road. (If the cylinder power is large enough, H is dependent upon the adhesive power, which varies as the cosine of the angle of inclination. On an incline, therefore, the power

Let a

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s

t

=

inclination of plane.
length of plane, in feet.
time required to pass over
the distance s.

"8lbs. friction per gross ton, on
level track.

W load, including locomotive,

in tons.

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have for the force which the locomotive us then assume that the entire counter

must overcome

W sin a +

W cos a
280

weight (125 tons) is used, and that the maximum train weighs 120 tons, exclusive of locomotive. The extra five tons of the counterweight, will easily overthe train pressing against the locomotive. come the friction of the system and keep Therefore, in the case of the plane, it is only necessary to find the time in which the locomotive, uncoupled from its load, ascent requires t will ascend the plane. This may be

Now since work is pressure into dis tance, the work absorbed in taking the load up the plane is,

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W cos a

280

a)

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Since one horse power = 33000 foot

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found in terms of H, by substituting in eq (a), W=40 tons S=1800 ft. and tan a=}.

Making these substitutions and reducing we find t=8257 A.

In the case of the supporting grade, however, the locomotive must elevate, not only itself, but also the train (120 tons), making a total load of 160 tons. Therefore,

W=160 tons—tan a=;

1.9 100'

and s=2.2X5280. From these values we obtain t,=41957

horse power to draw the load up the A. plane. But the actual horse power ex- Hence pended by the locomotive is H cos a.

W cos 280

Consequently

2240 s (W sin a+

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=H cos a

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t,

t

41957 A
=5.08 or t1=5.08 t
8257
A

That is the time required to ascend the grade is 5.08 times as much as that required to ascend the plane. Now let us suppose that the descent of the plane occupies just as much time as the ascent, and that 2.08 t is consumed in delays at the top and bottom. The train can proceed up the plane, the locomotive descend, elevating the counterweight, and re-ascend with another train, in the time that it would require for one train to pass over the grade.

Thus, at least twice as much weight can be transported over the gravity plane, in the same time, as over the supporting grade.

Let us now assume an average cost of construction, exclusive of equipment, in a mountainous country, say $30,000 per mile, and endeavor to arrive at the relative expense of the two systems. At this rate the cost of the supporting grade would be $66,000. If we assume that the cost of grading and tracking the gravity plane to be one and a-half times as much as the same length of main line, we will have 1,800 feet of inclined plane to construct at a cost of $45,00

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