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45, castle-street; 
amd sold by baldwin, cradock, and joy, and taylor 
akd bebsey, london. 



Liverpool, December 15, 1824. 


HA VING been requested by a friend, a 

Member of your Committee, to inspect the Locomotive En- 
gines and the Rail-roads near Newcastle and Sunderland, 
I have prepared the following Report, which he has desired 
me to publish and address to you. With my best wishes for 
the success of your most important and valuable under- 

I have the honour to be, 


Your most obedient Servant, 


To Charles Lawrence, Esq., 

Chairman of the Liverpool and Manchester 
Rail-road Committee, 





Liverpool, "SOth November, 1824. 

In every mechanical operation, whatever may be its 
natare, a certain eflfect or work has to be performed 
by some effort, called force or power, which is 
intended to overcome other forces, or resistances, 
opp6sed to the existing causes. If the whole of 
these caases are properly estimated before they are 
put ii^o operation, the result will be exactly fore- 
seen ; if, on the contrary, any of these causes, whe* 
ther impelling or resisting, have not been properly 
estimated, the result will be different from that which 
has been foretold. This previous estimation of effects 
has been vaguely called theory, and there is a gene- 
rally prevailing opinion, that theory and practice are 
oftentimes at variance; for we frequently hear such 
language as, things being good in theory and bad 
in practice." Nothing can be more absurd. Every 
thing attempted to be practised has been the result 
of preconceived notions of the causes which would 


interfere in favour of or against the result; but if 
the result be not what was expected, some of the 
causes have been wrongly estimated, or, perhaps, 
omitted altogether. Whichever of these may have 
been the case, we cannot call such a thing correct 
in theory, nor can any thing be correct in theory 
which does not hold«good in practice. 

Although, in this investigation, I have taken great 
pains to introduce all the circumstances for and 
against the accomplishment of rail-ways, I am well 
aware, that many have yet to be considered which 
must remain to be developed by experience. 

We have long been acquainted with the effects and 
advantages of rail-ways in lessening the friction and 
the consequent saving of horse power. These have 
undergone various improvements in their means of 
lessening the friction. My object, in this report, is 
to explain the principles of moving wheel carriages 
along rail- ways, whatever may be the power employed; 
The force applied I consider as a certain pressure, 
which I call the moving force, and which I shall, in 
my calculations, express in pounds : the weight to 
be moved along tiie plane I shall also express in 

With a view to make these principles better un- 
derstood, I will propose an hypothetical plane, or 
rail-way, destitute of friction, to go quite round the 
earth, keeping every portion of the surface of the 
rail the same distance from its centre. These 
would be the precise data to constitute a level 
plane. It will be clear to all who .are acquainted 
with central motion, that, if a carriage or other body 
be put in motion upon this hypothetical plane with 


any given velocity, it will continue to revolve round 
the earth with the same velocity, supposing it to have 
no friction, nor to be otherwise resisted. 

This hypothesis being admitted, I will now sup- 
pose, what we shall all readily allow, that this plane 
has a certain, but uniform, friction throughout, and 
that, in order to overcome this friction, we will sup- 
pose, it to have some power travelling with it for that 
purpose. This being effected, it will, I think, be 
granted, that, whatever force we add to that which 
overcomes the friction, the carriage will be put in 
motion, and its velocity will increase, equally in 
equal times, as long as this extra force is continued. 
If, at any period of its motion, this extra force be 
withdrawn, leaving that still in action which balances 
the friction, the body will go on with the velocity it 
had acquired, and, if the path reached round the 
earth, it would continue to revolve for ever. 

In order to apply ^this principle to practice, I have 
been anxious to get all the information I could on 
the subject of friction. For these valuable facts, I 
have been under much obligation to Mr. Stephenson, 
your engineer, and his friend, Mr. Wood, of the 
Killingworth Colliery. 

They have ascertained, by experiment, that an 
empty coal waggon, which weighs 23.25 cwt., requires 
a force equal to about 141bs. to keep it in motion, 
and they did not find, on varying the velocity, 
that this force was altered. When the waggons are 
loaded, the weight becomes 76.25 cwt., or 8,5401bs. 
If the axletrees of the waggon had been changed, 
according to the weight upon them, we should, doubt- 
less, find that the friction would increase as the 


weight ; but in this case it is not so, and the friction 
of the loaded carriage, instead of being SSlbs., as the 
weight would give, is only 49Ibs. In order to be 
rather over than under the truth, I have called itdOlbs. 
The engine is about eight tons, or 17,9201bs. I 
have stated the friction at about lOOlbs. 

In my examination of the locomotive engines at 
the Killingworth and Hetton Collieries, I have ascer- 
tained, that they are, in their present state, a great 
saving, compared with the employment of horse 
power ; and that they are capable of so much im- 
provement, as to put the matter beyond all doubt. 
My principal facts are taken from the Hetton Col- 
liery. The engines and waggons, which I saw, tra- 
versed a plane, very little inclined, in the direction of 
the load: its length was 2,541 yards, or nearly 
miles : the rise in this length about 22.75 feet 

I made two journeys with the engine, which drew 
16 waggons. In the first journey, we stopped for a 
short time on the way, but I found the average num- 
ber of strokes per minute of the engine to be 45, 
which, multiplied by the circumference of the wheel, 
9 feet, gives 405 feet per minute, or a little more than 
4J miles per hour. The second journey was per- 
formed in 15 minutes, which was at the rate of 52 
miles per hour. The pressure of the steam in the 
boiler was said to be 501bs. on an inch, and that 
upon the working piston 301bs. ; but, as there were 
no means of ascertaining this accurately, it must 
have been conjecture, particularly when the engine 
was going, for the safety-valve was then regulated by 
a spring, the force of which was very uncertain. 

In order to put the principles, which I conceive to 


belong to rail- ways, to the test, I have applied them 
to the facts which I obtained from this experiment^ 
making the known data the means of finding what I 
did not get from observation, and comparing my re-^ 
suits with the facts given by the estimation of the 
superintendent of the works. I have, therefore, taken 
one common speed for the engine, viz. 45 strokes per 
minute, which also limits the velocity to 5 miles an 
hour, or 7J feet per second* 

Agreeably to the principles laid down in the com- 
mencement, when a force is applied equal to the fric- 
tion, the smallest force above that would, if continued, 
generate any required velocity. But it will be desir- 
able to have such a force at command as will gene- 
rate the necessary velocity in a short time, and, when 
that has been accomplished, to reduce this force, but 
still to leave it fully equal to the friction. If any part 
of the route has an inclination, there ought to be an 
extra force at command, above what would be re- 
quired for a dead level. The plane, on which this 
experiment was made, inclined, in the direction of 
the load, about i of an inch to a yard. This is as 
great, and perhaps a greater, inclination than any 
rail-road ought to have, where loaded carriages go 
up and down. The moving force ought, therefore, to 
be always greater than the friction added to the force 
which is required to overcome the inclination of the 
plane. The latter force assists the body to go down, 
and equally resists it in moving upwards. 
. On this account I have used, or supposed, a moving 
force, which will give the velocity of 5 miles an hour, 
or 7J feet per second, in the space of one minute. 


This will be performed down the above plane by the 
engine making 45 strokes per minute, with a pressure 
of 9.71bs. upon an inch, of each of the two cylinders, 
the area of each being 63.6 square inches. The weight 
of the engine and 16 waggons is equal to 154,5601b8., 
or nearly 70 tons. This velocity of 5 miles an hour 
being acquired at the end of one minute, the only 
force to keep the whole in motion, at the same rate, 
will be the difference between the gravity of the 
weight down the plane and the friction. The friction 
is 9001bs. ; the gravitating force of the weight down 
the plane 5401bs.; therefore 900— 540=3601bs. 

If the same weight, at that speed, had to move on 
a dead level, and acquired the same velocity in one 
minute as before, the moving force would require to 
be l,7811bs., which would require a pressure of 13.71bs. 
upon 1 inch. But, after the speed is obtained, it will 
require only Tibs, to keep it moving at the same rate. 
If the same load were required to move up the plane, 
it would require a moving force of 2,3281bs., or a 
pressure upon every square inch of 18.31bs. And 
this velocity would be kept up by a constant pres- 
sure of l,4471bs., which will be 11.31bs. upon every 
inch of the piston. 

In starting the engine, in the first instance, and 
giving the required velocity, it is probable the effects 
will agree very nearly with these calculations; 
namely, 154,S6(Hbs., moved at the rate of five miles 
an hour with a pressure of 9.71bs. upon every inch 
of the piston. Whether the pressure was reduced 
to the difference between the friction and the force 
upon the plane, which is calculated at 2.81b., it is 


difficult to say, as there was no steam-gaage to indi- 
cate the pressure when the engine was going. 

In observing the working of the engine, I should 
think the number of strokes should not be more than 
45 or 50 in a minute. This limits the speed of these 
engines not to exceed five miles an hour, for the cir- 
cumference of the wheels upon the rail being about 
nine feet, 45 x 9=405 feet per minute, or a little more 
than 41 miles an hour. Now, it would not be advis- 
able to increase the number of strokes much beyond 
this rate, say 50. If, with this, it were required to go 
nine miles an hour, or 792 feet per minute, then 
=15.8 feet for the circumference of the engine 
wheel, which will be about 5 feet in diameter. 

The weight I propose to be conveyed, by one 
engine, will be 88 tons, or 85,1201bs.; the friction of 
tbis, on a level plane, will be 494Ibs. Hien, the 
moving force to give liiis weight a velocity of nine 
miles an hour, in one minute will be l,5981bs. ; and, 
if we agree to have the same area, namely, 63.6, for 
each cylinder, the pressure upon an inch will be 


In order, however, to give some idea of the power 
required for different loads, I have given a table, in 
which the engine is constructed about the size proper 
for conveying the above weight at the rate of nine 
miles an hour. I do not think it would be econo- 
mical to make any engine much smaller than those 
used near Newcastle. The cylinders are nine inches 
in diameter, and the length of the stroke two feet. 
The boiler I would make about the same diameter, 
namely, four feet. I would make the' fire-grate and 


chimney of much greater area, and the wheels, instead 
of three feet, I would increase to five feet diameter. 

In the principle laid down, the friction is supposed 
to be the same with any velocity which will be 
required. The resistance of the air, in the velocity 
of nine miles an hour, would be a little more than 
six ounces upon every square foot of surface ex- 
posed directly to the front of the moving body. 
This eflFect would vary with the direction and power 
of the wind ; it will, however, be so little, that it may 
be safely neglected in this calculation. 

Although the experiments made by Professor 
Vince, and by the French philosopher Coulomb, are 
sufficient to establish the law relative to the friction of 
bodies moving on planes, and that they also agree 
with the experiments by Messrs. Stephenson and 
Wood, at the Killingworth Colliery, it would, I 
think, still be desirable to repeat these experiments 
with greater velocities than have hitherto been tried. 
For this purpose, I would recommend a course of 
experiments upon a small scale, which would be of 
great value, and would save much expense in the 
construction of the real working machines. The 
scale on which I would make this experiment would 
be similar to that on which Mr. Smeaton made his 
experiments on water-wheels and wind-mills. This 
would consist of an upright revolving shaft, with an 
arm or moveable radius similar to that of a horse 
mill. The length of this I would make about 15 feet; 
the end of it would, therefore, move in a circle of 30 
feet in diameter. At the extremity of this circle 
I would make a rail-way quite round the circle. 


constructed in every respect in proportion to the pre- 
sent iron rail-ways, on the most modem plan. I would 
also have a carriage, or rather a series of carriages, 
having wheels suitable to the rails, and loaded in the 
same proportion. Round the upright central shaft I 
would coil a rope, which should go over a pulley, 
and have a weight to act at the end of the rope. This 
weight, by its gravity, would give motion to the re- 
volving arm, which would be connected with the car- 
riages to be drawn along the plane or circular rail- 
road. In forming the connexion between the arm 
and the carriages, I should place in the connecting 
chain or rope a spiral spring, similar to a spring 
steelyard. The carriages bein^ loaded, and the mov- 
ing force applied, it would in the first instance acce- 
lerate, as was stated in the beginning, the velocity, 
increasing with the time. I should have first observed, 
that a force or weight should have been first applied, 
which would just overcome the friction of the shaft 
and pulley, aad then a weight added to give the car- 
riages a certain velocity in a given time. When the 
body has acquired what may be deemed a sufficient 
velocity, the moving force may be diminished till the 
motion becomes uniform, which may be easily ascer- * 
tained by a stop watch. During the time the velocity 
is increasing, the spiral spring must be carefully 
observed. Previous to making the experiment, this 
spiral instrument, which we may term a dynamometer, 
should be graduated into pounds and parts similar to 
a steelyard, by which means it will be known what 
force the friction is equal to in pounds, and whether, 
and how much, it increases whilst the velocity is in* 


The weight used as a moving force in this appa- 
ratus, will have the same effect with the force of the 
steam on the piston of the engine, and may in every 
respect be compared with it, after making proper 
allowance for its own friction and the rigidity of the 

If these experiments should confirm this law rela- 
tive to rail-ways, and even common roads, it may be 
turned to greater practical advantage than has as yet 
entered into the views of the present proprietors of 
rail-ways. We have hitherto only seen the force of 
animals applied to carriages, and the great weight 
conveyed by coaches in proportion to their speed 
has frequently been observed with surprise; this, 
however, would create much more wonder, if a force 
equal to the first power of the horse could be kept up 
for a length of time. This will be best explained by 
a statement of the decrease of a horse's power as the 
speed increases. If a horse, standing still, can by 
his strength keep a weight of 1691bs. from falling, 
when suspended over a pulley, he will exert 1211bs. 
when he goes 2 miles an hour, lOOlbs. when he goes 
3 miles an hour, 811bs. with 4 miles an hour, 641bs. 
with 5 miles, 491bs. with 6 miles, 361bs. with 7 miles, 
251bs. with 8 miles, 161bs. with 9 miles, 91bs. with 10 
miles, 41bs. with 11 miles, lib. with 12 miles, and at 
the speed of 13 miles he is not able to exert any power. 

It will be evident, that if any power be applied, of 
the same energy as that of the horse in the first in- 
stance, and is not diminished by the increase of 
speed, the result will be very striking when com- 
pared with the effects of horses. . The force of the 
engine is applied to the wheels to give them, a rolling 


motion, and on that motion depends the progressive 
speed, this power not being diminished by the speed, 
as is the case with the horse. 

Having shown the advantages which may be de*^ 
rived from the application of moving force on rail- 
ways, I shall give a comparative view of these with 

The common roads differ from rail-ways only in 
their quantity of friction being about 1& times greater 
than the. best rail-ways. When this friction is over- 
come by any power, supposing the road uniform, any 
additional force applied will uniformly increase the 
velocity of the carriage to any extent. The great 
irregularity of all common roads will not allow this 
fact to be verified, and we can only expect this law 
to be realized in rail-ways. With respect to canals, 
they are governed by a very different law from that of 
rail-ways. The resistance, instead of being constant, 
as in common roads or rail-ways, increases, at least, 
as the square of the velocity. Whatever power is re- 
quired to move a floating body with any given velocity, 
it will require four times that power to give it twice 
that velocity, and nine times the same to give a treble 
velocity. , ■ 

In order to give a more precise view of the relative 
advantages of rail-ways, common roads, and canals, 
I have arranged them in Table II. It appears from 
this table, that, at the rate of two miles an hour, the 
same moving force being applied to a canal and a 
rail- way, the canal has the advantage, as two to one, 
but, at the rate of three miles an hour, the rail- 
way has the advantage over the canal as 22,400 to 
19,911, and at the rate of 2.82 miles they are equal. 


At the rate of nine miles an hour, for which the 
first table is constructed, the canal would only take a 
weight of 2,2121bs., which is less than v^th of the 
weight conveyed on a rail-way, with the same power. 

When a powerful horse commences his draught 
from a state of rest, he begins with exerting a force 
equal to I74]bs. ; but his power to draw decreases 
with the speed, and, as the table shows, he exerts a 
power of 1251bs., at two miles an hour, by which he 
conveys 20 tons, including the vessel. 

It will be evident, that the speed by means of 
horses, whatever may be the number, can never ex- 
ceed 12 or 13 miles an hour, for at this speed they 
can exert no power. It vnll, therefore, be necessary, 
in order to travel at the rate of 9 to 10 miles an hour, 
to empldy the power of steam, and this will be best 
performed by the locomotive engine. Althoagh it 
would be practicable to go at any speed, limited 
by the means of creating steam, the size of the 
wheels and number of strokes in the engine, it 
would not be safe to go at a greater rate than 9 
or 10 miles an hour. If the number of double 
strokes of the engine could be as great as 60 per 
minute, and the wheels on which it moved were the 
enormous size of 6 feet diameter, the speed would 
not be quite 13 miles an hour. If, by any chance, 
the wheels of the engine should get off the rails, 
which is sometimes the case^ a greater speed than 
that above recommended would be attended with 
proportionate danger. It will appear, from the prin- 
ciples laid down, that any power greater than the 
friction being applied will cause the vehicle to gain 
equal velocities in equal times, as long as that excess 


of power is continued. This may be shown from 
the theorems that are given at the conclusion. With 
a view to render the subject suflSciently intelligible 
to those who do not read algebraic formulae, I shall 
give a short analysis of the principle in common 
numbers, which, for the sake of greater clearness, I 
will make even numbers. Suppose 40 ,tons have to 
be moved by an engine weighing 8 tons, which will 
leave 32 tons for loaded carriages. The friction of 
this weight will be about SOOlbs. If a moving force 
equal to this be added, the body will be in a state to 
move with any additional force with the same effect 
as if it had no friction. Let this additional force be 
SOOlbs. The whole weight being 40 tons, or 89,6001bs., 
the accelerating force will be to that of the force of 
gravity as 500 to 89,600, or ^^=about -j^ijth that of 
gravity. Now by gravity a heavy body falls through 
a space ofl6rz feet in a second, and at the end of 
tiiat time it will have attained a velocity equal to 
double that space, or 32 feet per second. It is shown 
by writers on mechanics, that the velocity is doubled 
in 2 seconds, and trebled in 3, so that the velocity of 
a falling body may always be known bj multiplying 
32 by the time of its fall. But the force here stated 
is only T+gth that of gravity: hence, if the velocity of 
gravity be multiplied by tW» or divided by 179, the 
quotient will be the velocity which the moving weight 
vnll have acquired in the same time. Suppose we 
wish the weight to acquire the velocity of 9 miles an 
hour, or 13.2 feet per second, and it is required to 
find ih6 time in which the weight with the nett force 
of SOOlbs. will arrive at the above velocity. This will 
be obtained by multiplying 179 by 13.2^ and dividing 


179 X X3 2 

the product by 32. Thus, ^ — =74 seconds, tfie 

time required. To give this in words, as a general 
proposition, multiply the whole weight to be moved 
by the required velocity in feet, and divide the pro- 
duct by 32 multiplied into the difference between the 
friction and the moving force. This will give the 
time required to gain the given velocity. 

This rule applies only to a level plane. The force 
which an inclined plane gives to a weight will be ob- 
tained by multiplying the weight by the height of the 
plane, and dividing the product by the length. This 
force requires to be subtracted from the moving force 
like the friction, when the weight ascends, and added 
to it, when the weight descends. 

Although a locomotive engine will move up a plane 
a little more than Jth of an inch'*^ to a yard, it will be 
found, in practice, very desirable to have the line 
divided into dead levels and very short inclined planes, 
if there is any difference of level between the two 
places. The length of an inclined plane ought never 
to be such as will prevent a person from seeing the 
whole course of it from the top or bottom. As these 
planes would require to have fixed engines on their 
summits, it would be desirable to have as few as 
possible. An apparatus may be connected with these 
fixed engines, which will confine their office to the 
difference of weight between the goods going different 
ways. It will be evident, that the carriages going up 
these planes will require to be drawn by a rope or 

* When the weight upon an inclined plane becomes greater than the 
friction, the wheels will turn, but no progress will be made. Agreeably 
to the data given for friction, the inclination at which this will take 
place is one-fifth of an inch to a yard. 


chain,, and it would be found, as is at present the 
case, that sometimes this rope breaks, and the car- 
riages are precipitated and totally destroyed. Now 
such an accident as this with passengers would be 
fatal to the whole scheme. 

Having given this subject some consideration, I 
shall propose a means which perfectly obviates this 
evil, and, what will still more recommend it, it will 
be seep by every person that there is no danger. 

The carriages and the engine, by this plan, vsrill 
require to be propelled up or let down the plane, 
the rope that comes from the fixed engine being at 
the bottom of the plane. The hook being in the rear 
of the whole and the rope under the carriages, I 
would now fix this hook to a separate carriage, which 
would be merely to propel them upwards. Between 
every pair of bearers of the rail, which are about a 
yard in length from each other, I would have pieces 
of cast iron quite across the road, of sufficient strength 
to resist the weight of all the carriages in the case of 
the rope breaking. The hook at the end of the rope 
should be fastened to one end of a bended lever 
of the propelling carriage. When the whole is in 
motion, the force of draught will act upon the bended 
lever, and raise another part of the same above the 
cross pieces of cast iron, and this will be kept in that 
position by the tension of the rope. If now the rope 
were to break, the end of the lever, which had been 
kept up, will fall, and instantly butt against the cast 
iron bar it last passed. Supposing the rope to break 
immediately before it passes a bar, the weight can 
never accelerate more than a yard, which would not 
give a considerable shock ; at any rate, no danger 
could be experienced, nor would any thing but the 
rope be injured. 


The propelling carriage here alladed to will have 
to go in the rear of the carriages op the plane, and 
in the front when they descend. 

The fixed engines should be placed nnd^ the 
road, as much short of the commencement of the 
descent as will be equal to the space occupied by 
the line of all the carriages. 

When the locomotiTe engine, about to descend, 
arrives at this station, the propelling carriage will, 
have the hook of the fixed engine attached to it, and 
by its 0¥m power go on with the rope and propelling 
carriage. The. locomotive engine will continue to 
work, till as many carriages are upon the inclined 
plane as will drag the rest forward. At this period 
the descending load will begin to n|ise a weight con- 
nected with the other end of the rope, which will jnst 
allow the weight on the plane to descend with a < 
proper and uniform velocity. If the locomotive 
engine works all the time, a greater weight will be 

When the motion is the contrary way, and an 
engine with its load comes to the foot of the plane» 
the engine, in this direction, is supposed to propel 
its load, and the propelling carriage is now brought 
behind it The hook of the rope from the fixed 
engine being attached, the whole goes up the plane, 
and the weight, which was raised by the descending 
load, now assists, or may be equal, to draw this load 
up the plane. The locomotive engine may keep 
working, if required. 

The weights to be employed in raising the load, 
or being raised themselves by the descending load, 
may be so contrived as to admit of an exact adjust- 
ment to suit the different loads, whether asc^iding or 


By this means it will be clear, that the power 
required by the fixed engines will be the excess 
which the ascending load has above the descending, 
and this can in general be known, very nearly, before 
a rail-way is begun ; so that the power of the fixed 
engine may be known beforehand. In a future work 
on this subject, I shall give a more detailed descrip- 
tion of the reciprocating plan, with proper drawings 
for executing the work. 

I hope by means of the tables, and the best descrip- 
tion I have been able to give in this report, I have 
rendered the principles of rail-ways as clear, as our 
present experience will admit. I have avoided as 
much as possible every thing technical, and have not 
as yet used any jalgebraic formulae. As, however, the 
theory of this important subject cannot be completely 
demonstrated and made general without these for- 
mulae, for the sake of those skilled in the application 
of algebra to mechanics I shall subjoin the algebraic 
investigations, which have led to the deductions given 
in the former part of this report. 

Each of the quantities, which will enter into this 
investigation, will be represented by appropriate let- 
ters. Let 

IF:zThe whole weight in pounds to be moved, including the 

Ill := The force, also in pounds, applied to noiove the weight 
vzzThe velocity in feet per second with which IF is to be 

moved when at its full speed. 
t^The time required for the weight W to get the velocity v. 
aizThe area of the steam cylinders. 
/=The length of the stroke. 
N^fhe number of strokes per minute. 


l^The diameler of the wheels of the engine which roll upon 
the rails. 

J'^The amount of friction of IT in pounds, which is equal to 
the force that will keep it in motion with the velocity «. 

JSr=:The height of the plane, which, for the sake of simplicity, 
we will call i. 

Xi=:The length of the plane, which, in all rail-ways, should 
never be less than 360 times the height, 

puzThe pressure upon a square inch of the piston, and 
ap=the whole pressure on both pistons. 

/*= 3.1416, the circumference of a circle, the diameter of 
which is 1. 

gz^lS^ feet, being the space a heavy body falls through by 
gravity, in one second. 

l>/=bThe circumference of the wheel. 
DJk^^The number of feet per minute, and 


Since the engine makes a doable stroke for one 
revolution of the wheel, the speed of the piston to 
that of the carriage will be as 2 / to Df; hence, the 
moving force will require to be greater in liie ratio of 

If the crank of the engine turned the wheel 

upon a fixed centre, the inean effect of the same, as 

a lever, would be about , .6 being the mean of 

all the sines in the quadrant. Since, however, the 
centre is moveable, and the fulcrum of the lever at the 
ground, it becomes similar to a pulley, in which the 
weight is to the power as 2 to 1. The power of this 

lever will, therefore, be doubled, and -^-must be 


added to,^ ; hence we have ^ .6 =(D + I) '3. 


This, compounded with the. quantity gives 

' ^JDf ^ ^ multiplier of the- moving force 

to give the' velocity ; but this has also to be multiplied 

by the velocity which gravity would give, in order to get 

the velocity of the moving weight For the sake of 

getting the time in which this velocity is acquired;'we 

will express the velocity which gravity would* give in 

the time by 2 gt Hence we have for the multiplier of 

. + .6 X 2 gtl (1> .2grt 

the moving f6rce — ^ sr— — ^ — ^— 

Now m is stated to be the moving force to give the 
velocity v in the time ^ if the body had no friction, and 
supposing it to move on a level plane, such as the 
hypothetical plane we imagined to go round the 
world. But I have also given a quantity for that, 
and must prbVide an additionid moving' fbrce just 
equal to it. For the sake of simplicity, JP may be 
used for that force. The true^ moving force, on a 
level plane, will be m — JF, and the accelerating force 

"j^ Now, 2 gt is the velocity which gravity 

would give to a heavy body in the time t; hence v = 
^^^2 gt. In the application of steam power the 

true value of v will be 

1st, v-y ^ ^ 

w— g _ v D f „ W vDf 

W - (D + 01 .3 gtl "^-^ -(!>• + 01 .2 gtl. 



2d, m=. 4- F 

1 .2 (1> + /) Sf« 
OH ^i7-_ a -2 (D 4- /) g^O JF ) 

In order to make these theorems universal, when 
the plane is, or is not level, vre must use the quantity 

-J-- with a plus or minus sign as the load ascends or 

descends. The theorem will then stand thus : 

6th, m— F 


iD+l)l'2gtl - L 
Id order to find t, from theorem 4th, we have 

1-2 gtl) ^^_q:___J= vLf 

W^L W 
7th. «^ 

— T I 

|(2>+/)l-2 g/ 

If the plane be level, then H=o^ and in this case 
^ vanishes. 

In beginning to calculate the size of the engine to 
convey a given weight TT on a rail-way, the rate, or 


speed, is the first thing to be fixed, which is t;. Then 
the number of strokes per minute, or n, which should 
not exceed dO« Then find the diameter of the wheels, 
which will be got by the following reasoning: Since 
Df is equal to the circumference, nDf vfill be the 
space the engine passes over in 1 minute, therefore 

^^v, and consequently D =^^- Th© length of 

the stroke should not exceed 2 feet. Since then n 
and / are constant, the only way in which diffierent 
effiects can be produced is in the alteration of a 
and p, and as it will never be advisable to let p 
much exceed 151bs., this will also be limited, and a 
must be so taken as to accord with the above limita- 
tions. It will also be advisable not to allow D to be 
more than 5' or 6 feet. 

In order to find the area of the cylinders, it will be 
remembered, that nt, being the nett moving force 
under the piston, will be equal to the area into the 
pressure upon an inch, or op. If this be substituted 
for m theorem 5th, we get 

^ \{P+l) l-2gtt) - L f"^^' 

when w=o, pa^F±i 


and p=- 

* If p be in ponndi on an inch, a will be in inches, and for two 
cylinders must be taken as eqoal to double the area of one piston. 


When B=o, p=— and if a force equal to F±—-=r- 

were to be in equilibrio with p, and act at the 
paiiph^y of the wheel, when v=o, 

P==*(f± a{^^lj ^^^^ *® engine moves 


uniformly with p= — the same as if at rest, 


and in equilibrio with Ihe resisting forces, supposing 
the friction not to increase with, the velocity^ 

Upon the whole, the advantages of a rail-road^ on 
wJUcIl the locomotive power is used, are so striking 
that it is matter of surprise this mode of conveyance 
has not been resorted to earlier. Its adoption, how- 
ever, is now inevitable; and, when applied in proper 
places, and under judicious management, cannot Ml 
of becoming highly beneficial, to the propiictots and' 
to. the. public. But noticing can be mora^ delusive 
than to suppose, that because rail-roads: are in prin^ 
ciple better than canals, or high jroads, they ,^ill an- 
swer everywhere ; and yet the: e^dting rage for them 
would seem to justify such an opinion. The preten- 
sions held put by some of the projectors in various 
places, do appear to me unwarranted, either by facts 
or theory ; and I have no doubt, but that when the 
public mind becomes more sober on the subject, the 
real importance of the rail-road system, great as it 
undoubtedly is, will be more correctly estimated. 

* If the fall effect of these levers were to be in operation at the 
same time, this denominator would be "^itLbut since the cranks are 
at right angles to each other, the latter will require to be multiplied 
by . 6, wfaichiwillifive .S(I>xO. 


This new application of locomotive power is of in- 
finite importance to the country, and I should regret 
to see it abused. 

P.S. Since commencing this report, these principles 
have been given in the newspaper called the Scotsman. 
The author speaks of it as a new idea, at least as it 
applies to rail-ways, although it is founded upon the 
facts given very long ago by Coulomb and Vince. 

Whatever may be the claim to originality in this 
application, I have at least an equal claim with this 
author, as my introduction, which developes these 
principles, was read by several of my friends here, 
before the above articles were niade public. 






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1 oontains the engine and number of carriages. 

2, the weights to be moved at die rate of 9 miles an how. 

3, the moving force to be applied, capable of giving that 

velocity, in addition to that force which just over- 
comes the friction. This is expressed by vDfW 
^ , (D+l)1.2gtl. 

4, the force which the weight W would exert down a plane, 

the length of which, X, is equal to 360, when the 
height IT is equal to 1 or ^^th of an inch to a yard- 

It is expressed by — J- . This force would, of coarse, 
be nothing on a level plane. 

5, Hie moving force in pounds equal to the friction, repre- 

sented by F. 

6, the moving force including that to overcome the friction, 

being the sum of column 3 and 5, or the force required 
on a level plane to generate a velocity of 9 miles an 
hour, by keepingthe force in action one mmute. When 
this velocity is attained, all the force in the 3d column 
4nay be withdrawn, leaving that in the 5th column 
which will be sufficient to keep up the velocity required* 

7, the moving force required, when the weight goes down 

the plane, to give the required velocity in one minute. 
This will be found equal to the difference between 
columns 4 and 5 added to column 3. After this 
velocity is acquired, the last force may be withdrawn, 
leaving the difference between columns 4 and 5. to 
maintain the velocity required. 
-6, the force required to generate the above velocity in one 
minute, up the plane, being the sum of columns 3, 4, 
and 5* When the rate of 9 miles an hour is acquired, 
the force of column 3 is withdrawn, leaving the other 
two forces to keep up t|ie speed. 


ITaB Weight of the engineer 17»020lb.y also of the carriages, 

J'sFiictioii of the enginet^lOOlb., also of the carriages, each 

vssVelooity at the rate of 9 miles per hour, or 13'2 feet per 

Dss Diameter of the engine wheel asd'OS feet. 
/=Length of the stroke=2 feet. 
#±=Time, or 00 seconds. 

/xsCircamference of a circle whose diameter is 1=8*1416; 

therefore 2>f, the circumference of the engine wheel =s 

15*8 feet. 
J9r= Height of the plane=l to 
X=Ito length 360« 

^ssSpace which a body falls through in a seconds Id^ feet. 


Moving ForoeF to 



In Pounds. 

In No. of Horses. 














com moo 








On a 

for thar 

Off a 
for its 

For a 



Moving force 
to produce the 

in one minute 

on the 
Common road 
and Railway, 
in Pounds. 










1 CO 












' 1.48 





• • 















• • 





349 ^ 
























1 18 the velocity in miles per hour with which the load is 
to moTe. 

2, the same in feet per second. 

3, the weight to be moved on a common road supposed 

to be level. 

4, the same on a level rail-way. 

5, the weight which can be moved on a canal. 

6, the moving force to move the respective weights on the 

common road and rail-way. 

'7, the moving force required to take its weight on a canal. 

8y the same expressed in number of horses on a common 
road and rail-way. 

9y the same for the canal. 
10, the moving force required to give the velocity on the 
common road and rail-way in one minute. The same 
may be expressed in pounds upon an inch^ by dividing 
these numbers by the area of the steam cylinder. 



As I have in this report recommended the locomotive steam- 
^bgine^ as the most economical power, for every part of a 
rail-way in which the rise is not more than 7^ inch to a yard*, 
and since this engine must be so far of the high pressure kind, 
as it is destitute of the means of condensing the steam by the 
air pump and cold water in a condenser^ I have deemed it 
useful to give a comparative statement of the merits and de- 
merits of the high and low pressure engines. This will not 
be with any idea of the latter ever being used on rail-ways, 
but to show how far the high pressure engine is dangerous, 
under what circumstances it has been so, and in what degree 
it differs from the low pressure engine in point of economy. 

The low pressure engine works with steam of a pressure 
equal to 34 inches of mercury, or 17lbs. upon the inch, the 
effective power being 20 inches of mercury, or lOlbs. upon 
an inch of the piston. Of this 7lbs., about one is caused by 
the imperfection, of the vacuum, which is generally 2 inches 

* On examination of the plan and section of the intended rail-road 
between Liverpool and Manchester, I find the greatest rise, and that in 
only one instance, is not more than ^ part of an inch ; therefore, it i» 
the most favourable which could well be constructed. 


of mercury. The other six have to be divided between the 
fiiction of the steam piston and that of the air pump. ~ The 
area of the latter to that of the former is about 6 to 1, and 
their diameters will be as 2*45 to 1, which is the ratio of their 
friction. Hence, if the 6lbs. has to be divided between the 
two, we get l*74lbs. upon an inch for the frictiofi of the ahr 
pump and 4*26 for the steam piston. . Thus we allow for the 
low pressure engine 7lbs. to an inch for the loss of power and 
4*26lbs. for the high pressure. • 

In the low pressure engine, we have, dierefore, steam of 
17lbs. acting against a pressure of 1, which is the value of the 
vacuum, and against 6lbs., the friction of the piston. In the 
high pressure engine, we have to act against 15lbs., the pres- 
sure of the atmosphere, and 4^26, the friction of the piston. 
Hence, in order to have a nett pressure of lOlbs. upon an inch, 
as in ihe low pressure engine, we shall require steam of ti 
pressure equal to 15+4*26+10=:29'26; and if the steam, in 
both cases, is lost, iheir relative economy, in the consumption 
pf fuel, will be in the ratio of their quantities of steam, or as 
17 to 29-2d, or 1 to 1-72, 

As there will be a difference of about 20"* of temperature, 
there will be a somewhat greater loss of heat radiation ia 
the high pressure engine. This, it must be observed, is the 
greatest difference, in point of economy, that can exist 
between a high and low pressure engine, since they are sup- 
posed of equal size and work with the same power. 

If we weie now to double the power of these engines, by 
doubling the area of the cyhnder of the low pressure engine, 
and doubling the working pressure of die high pressure, the 
first will now be 2(1+10)+6X ^/2z=30*46, and the second 
will be 20+14+4*26=38'26. The ratio is now as 1 to 1*26. 

If the area of the low pressure were trebled, then the num- 
ber for this ratio would be 3X(l+10)+6x >/3=43-4, that 
for the hi^ pressure 3X104-14+4*26.=i 48*26. Hence the 


iratib in which the pressure of the st^km upon ike high 
engme is 4B*24» and on tSie inch will be 4^ to 4&'1i6, or as 
1 to I'll. The economy til the high pressure will still ap- 
proach nearer to that of the low pressure by making the Area 
of Hie latter 4 times, stHl keeping the pressure on an inch 
the same, and increasing the high pressure in the same propor- 
tion by increasing the pressure only, the cylinder and length 
jbf stroke remaining the same. In this case, the number will 
be for the low pressure 4(1 +10) +6 x >/4=66, and for life 
high pressure 4x10 +15+4^26= 59*28, or as 1 to 1S[)58 
nearly. In all these cases the temperature of the steam cyBn- 
der win be the same in the loW pres^re elighie, not much 
exceeding 212'', the temperature of ordinary steam; that of 
the high pressure engine increasing with the pressure. In 
the first case, they are as 217^ to 248". In the second, the 
temperature of the high pressure cylinder will be 264% that 
of low pressure remaining the same. In the third case, the 
temperature will be 280^, and in the fourth it will be a few 
degrees higher. This will cause an increased waste by ra- 
diation, but if the boiler and cylinder be cased, this loss may, 
hi a great measure, be prevented ; and in the last case, the 
high pressure engine comes very near the low pressure in 
point of economy* 

In this comparison of the high pressure engine with the low 
Pressure, we have seen, that the economy increases with the 
pressure of the latter; but it must also be remembered, that 
the whole of the steam is consumed. This must necessarily 
be the case in the mode of condensing the steam in low pres- 
sure, but it need not be the case in . that of the high pressure 
engine. Instead of discharging the steam into the open air, 
as is the common practice, it may be discharged into a large 
Tetsei, placed in the front of the enghe, with a view to give it 
^e best chance of being cooled by the surrounding air. This 
I should make of plate-^iron, about 4lb8. to the square fbot. 


Its diameter might be three feet^ and its height a litde less 
than the smoke chimney, which would be about 15 or 16 feet. 
The capacity of this vessel would, therefore, be about .112 
cubic feet, and its surface 144 feet. If, in throwing all the 
steam into this cavity, its temperature should be kept up to 
ISO**, which, I think, would be the maximum in the warmest 
weather, the pressure in the interior would be equal to TJlbs. 
upon an inch, supposing the vessel to be made perfectly air- 
tight, and provided at the top with a valve opening outwards, 
which would let steam or air out when the pressure was out* 
ward, and preserve the rarification, when *the pressure was 
inward. This vessel would have to be made with great care, 
to be sufficiently air-tight. In the examples given, the high 
pressure steam has 15lbs. upon an inch of reaction. In the 
use of the condenser, in its perfect state, all the water would 
be 'returned to the boiler, with the exception of what escaped 
by accidental imperfection in the apparatus, and the strength 
of the steam employed would be reduced T^lbs. upon an 

The comparison of the high pressure engine, under tiiis im- 
provement, with the low pressure, would be as follows : In 
the first case, when the engines are of the same size, we have, 
as before, for the low pressure 10+1+6=17, and for ibe 
high pressure 10+7'6+4*26=21*76. If, as before, we 
double the area of the low pressure and the pressure of the 
other, we get 2(10+i)+6 x >/2=3b.^ for the low, and 
2+10+7-6+4-26=31.76. We here see that with this addi- 
tional vessel, when the pressure of the high pressure boiler is 
not more than 31*76lbs. upon the inch, the economy of the 
two is nearly equal, and very nearly all the water will be 
saved. To provide, however, for the littie that may be lost, 
there may be that quantity put into the vessel when starting, 
and die bottom of this large vessel being connected with the 
small pump for supplying the boiler, the same will be mixed