HE
1037
N37
UC-NRLF
GO
o
ON THE
MEANS OF COMPARING
THE
RESPECTIVE ADVANTAGES OF DIFFERENT
LINES OF RAILWAY;
AND ON THE USE OF
LOCOMOTIVE ENGINES.
TRANSLATED FROM THE FRENCH
OF M. NAVIER,
il
INGENIEUR EN CHEF DBS FONTS ET CHAUSSEES, PARIS.
BY JOHN MACNEILL, CIVIL ENGINEER,
M.R.I.A., F.R.A.S., &c.
LONDON:
ROAKE AND VARTY, 31, STRAND.
1836.
LONDON:
ROAKK AND VARTY, PRINTERS, 31, STRANB.
RIGHT HON. SIR HENRY PARNELL, BART.,
TnEASURER OF THE NAVY, PAYMASTER OF TUB FORCES,
&C. &C. &C.
THIS ATTEMPT TO INTRODUCE INTO THE ENGLISH LANGUAGE
A TREATISE OF HIGH REPUTATION ON THE CONTINENT,
IS MOST RESPECTFULLY DEDICATED
BY HI3 OBLIGED
AND VERY GRATEFUL SERVANT,
JOHN MACNEILL.
January 1836.
755
THE little work of M. Navier, which I now
venture to introduce to the public, was trans-
lated during the intervals of professional
duties, and written out more as an exercise, for
the purpose of impressing the subject on my
memory, than with any view to publication.
Though always anxious to ascertain and pre-
serve the meaning of the author, I now feel
conscious that it may not be so well expressed
as it deserves, or as it might have been had I
been able to have devoted more time to the
undertaking. Those persons who may be desirous
of accurate information on a subject of such
increasing importance, (and it is for such I
principally intend it,) will, I am sure, forget
the inexpertness of the translator, provided he
be faithful, in the degree they become acquainted
with the ability of the author.
Two reasons may be given for supposing that
the publication of a work of this kind might be
patronized at the present moment.
1st. From the large capital already expended in
Vlli
the construction of Rail-ways, and the daily ex-
tension of the system through every part of the
kingdom, it was thought that any work which
elucidated its principles, or removed the difficul-
ties which attend a theoretical investigation of its
various properties, might be of considerable in-
terest and utility.
2nd. From the small amount of our real know-
ledge of the numerous circumstances which affect
the power of locomotive engines, particularly on
slopes, it was felt that any addition to the exist-
ing stock of information might be acceptable.
That this was necessary was very apparent during
the parliamentary examination of several Rail-
way bills in the last and preceding sessions, and
that it is equally so at the present moment may be
inferred from the more recent fact, that two emi-
nent mathematicians, who have studied the sub-
ject, have arrived at very different conclusions,
and have published opinions quite at variance
with each other.
The only formulae that I am aware of, which
relate to this branch of practical science, are
those of the Rev. William Adamson, published
in his Sketches of our Information on Rail-ways,
in 1826, and those of Mr. Tredgold, in his
Treatise on Railways, in 1825; the former are
mentioned by Mr. Wood, in his Practical Trea-
tise on Railways ; but he does not make use
IX
of them, and it is to be presumed that he would
have done so, had there not been some practical
objection to them. Those of Mr. Tredgold were
constructed before locomotive engines had ar-
rived at the perfection to which they have now
attained, and are therefore by no means appli-
cable to the present state of steam-power on
Rail-ways.
It is true that in Mr. Wood's work there are
two Tables, Nos. X and XI, pages 418 and 419,
which purport to give the gross weight that
an engine of given power can draw up slopes of
different rates of inclination, with different velo-
cities ; if these tables be founded on actual expe-
riment and practice, there can be no doubt but that
the velocity which he has supposed to be a function
of the weight of the train and rate of inclination of
the slope, maybe deduced from them ; this I have
endeavoured to do, and to apply the formula thus
obtained to a practical example of two proposed
lines of Rail- way.* The result would indicate
that an engine carrying a constant load of forty
tons, can traverse one of these lines in less time
than the other, but this superiority consists in
time only : we cannot predict that this line is in
other respects preferable or more economical
than the other. Various other circumstances be-
sides velocity should enter into the calculation,
* See Appendix 1.
before we can decide with certainty which is the
best line ; these tables therefore are inadequate
to the purpose, and we have not at present any
work in the language that I am aware of which
will enable us to arrive at a satisfactory compa-
rison.
During a fruitless search upon this subject,
I had the gratification of receiving from my
esteemed friend, M. Mallet of Paris, Ingenieur
enChefdes Fonts et Chauss6es under the French
Government, the work of M. Navier. On its
perusal, it soon appeared evident that the prin-
ciples and laws of the motion of locomotive
engines were investigated and explained with
that acuteness and ability which might be ex-
pected from a mathematician of such high repu-
tation, — that he had opened a road by which
future enquirers might enter upon this difficult
and important question and push the investiga-
tion to the degree now demanded by the matured
experience and highly improved apparatus of the
present time: and that he had pointed out not
only a proper mode of calculation, but also various
circumstances, which, modifying the process and
affecting the result, should be carefully attended
to in conducting all future experiments on the
motion of locomotive engines on inclined planes.
M. Navier's formulse will enable us to approx-
imate very nearly to a correct comparison be-
XI
tween two lines of Rail-way, and if data derived
from recent actual practice be substituted for the
assumed quantities in these formulae, there. is every
reason to believe that they will enable us to esti-
mate with great accuracy the absolute cost of con-
veying a given weight of merchandise or number
of passengers over any line of Rail-way, a point
apparently not hitherto attended to, but which
must be one of the principal facts ascertained
before any correct conclusion can be come to, as
to the advantage which a line of Rail-way is
likely to afford to the public, or before it can be
compared with another line of Rail-way in the
same direction.
The great advantage which Rail-ways possess
over every other mode of transport, consists
principally in the rapidity with which passengers
may be conveyed from one town to another ; and
in some cases merchandise, for the latter can be
carried cheaper by canals or even on roads, when
a velocity not exceeding two or three miles an
hour is considered sufficient. A Rail-way intended
for the quick transit of passengers, should in my
opinion be differently constructed from one in-
tended for the carriage of merchandise, and both
should be varied to suit the particular circumstan-
ces of the country they are intended to pass through,
and the extent and description of the traffic ex-
pected to pass over them. There is also a limit
Xll
which should be observed in forming the slopes on
a line of Rail- way, but no general rule will suffice ;
for as the expense of conveying a passenger or a
ton of goods along the line will depend on the
original outlay in the construction of the works,
the cuttings and embankments should be so ma-
naged, and the slopes so regulated, that the first
cost of the works and the engine-power should
be in proportion to the number of tons and
passengers expected to traverse the line, and the
importance attached to a rapid transit over it.
In discussing this subject, Mr. Tredgold
states, in his Treatise already referred to, page
15J, " It will readily appear from these equa-
tions that it is much less expensive to nearly follow
the undulations of the surface, than to make either
deep cutting or embankment beyond those limits,
which are easily determined by half an hour's
labour, in applying the equations to the case
under consideration. We have inserted numbers
and shown how to reduce it to the case of a road
of average expense ; and if a few examples be
added, it will assist in removing those extravagant
notions respecting cutting and embanking, which
sink the capital of the country in unprofitable
speculation."
It is to be hoped that this important part of
the subject will not be suffered to remain much
longer without a thorough examination, founded
Xlll
on actual observation of the working of loco-
motive engines on every possible degree of slope,
ascending and descending, and under every cir-
cumstance that is likely to take place in prac-
tice. But as practical men have seldom the
time, or the mathematical acquirements requisite
for such an intricate investigation, a series
of facts should be registered at all the dif-
ferent Rail-way establishments in the empire.
These should embrace, 1st, the cost of the en-
gines, their weight, a daily journal of their
repairs, and the number of miles travelled ;
the weight carried, the water and fuel con-
sumed ; the pressure of the steam on each par-
ticular part of the line, Sec. &c. If such facts
were collected, and then given to some of our able
mathematicians, there can be no doubt of their
being able to furnish formula that would be of
the utmost importance to the country ; or it
might be better to employ such persons to design
and carry on, under their own superintendence,
sets of experiments in any way they might think
most adapted to obtain the information desired.
Such persons might be easily found amongst our
mathematical professors, and the expense of the
experiments should be defrayed by the Rail- way
Companies in some certain proportion.
Some of the tables and formulae in M. Navier's
work, and all the weights and measures, I have
XIV
given both in French and English j and 1 have
transferred the notes which are at the bottom of
the pages in the original to the end of this work,
an arrangement which appears to me to facilitate
the clear understanding of the author, by allowing
his chain of reasoning to be pursued unbroken.
The note which I have added on the different
modes of estimating forces, and on the meaning of
the expression vis-viva (la force vive) is extracted
from the valuable work of Mr. Whewell on Dyna-
mics, and from the Elemens de Mecanique, par
M. Boucharlat, two authors distinguished for their
works on mathematical and mechanical subjects.
The note may, perhaps, by some be considered
irrelevant or unnecessary ; but I have thought it
might be acceptable to others, for the terms mo-
mentum and vis-viva have been sometimes con-
founded with each other.
P.S. — I have rendered the word pente by slope
in preference to inclination, inclined plane, or
gradient, considering the two former, though
generally used, as improper expressions, and the
latter, to say the least of it, as having so very
little to recommend it, that I hope it will have
an extremely short existence in our nomencla-
ture.
Judicious and appropriate terms are of the
XV
greatest importance in speaking and writing on
scientific subjects, particularly where technical
expressions must of necessity be introduced.*
A gentleman of high literary acquirements, to
whom I applied, and who has taken the trouble
to consider the subject, has suggested the term
clivity as one that is of more legitimate etymo-
logy than gradient, and more appropriate than
either slope, inclined plane, or inclination. I
regret that I was not in possession of this term
before I commenced the translation, the words
acclivity, declivity, which may be so regularly
derived from it, would have enabled me to have
given the sense of the original with greater
perspicuity.
* " It is highly desirable to keep scientific knowledge precise,
and always to use the same terms in the same sense." — Dis-
course on Natural Theology, by Henry Lord Brougham.
ERRATA.
Page 22, line 21, for °-°05 P?? read °-005 PT?"
26 — 2, /or 611.1 read 620.1.
27 — 5, for 637 read 641.
34—3, for irr read irr.
37 — 9, for velosity read velocity.
38 — 17, for 00.21148 read 0.021148.
42 last line, for 81163 read 81272.
45 last line but one, for 138 read 238.
55 and 56 in Note*, for 570 Farenheit read 25° Farenheit.
56, lines 5 and 12, and p. 57, last line but one, omit the initial C.
ON THE
COMPARISON OF THE RESPECTIVE ADVANTAGES
OF
DIFFERENT LINES OF RAIL-WAY,
AND ON THE USE OF LOCOMOTIVE ENGINES.
1 . General ideas relative to the establishment
of Rail-ways.
2. Principal elements in the comparison of
different lines of Rail- ways.
3. Determination of the power required to
draw a given train over a given Rail-way.
4. Determination of the weight of the train
which can be drawn on a given Rail-way by a
Locomotive Engine of a given power.
5. Examination of the uniform motion of the
train on the different ascending or descending
slopes which may form part of a Rail-way.
6. Examination of the motion of the train in
passing from one slope to another.
7. Summary. Comparative estimate of the
cost of transit on different lines of Rail-way.
B 2
4
The following observations are partly taken
from the course prepared for the students of the
Board of Bridges and Highways (Ponts et
Chaussees,) in France. It has been thought
that they would perhaps be interesting to such
engineers as may now be employed on Rail-way
projects ; and that they might throw some light
upon those difficult and complicated questions,
the examination of which has arisen from the
introduction of these works.
1 . General ideas relative to the establishment of
Rail-ways.
Rail-ways are generally considered under two
principal heads. 1st, As affording to commerce
a more economical mode of transport ; 2nd, as
giving the means of carrying goods, and more
especially passengers, with considerable speed ;
the mean rate of which may be stated, from what
has taken place in England, at about eight
leagues* an hour.
This great rapidity of transit being the cha-
racteristic property of Rail-ways, without which
they would lose their principal advantage, and
not produce the results that might be expected
from them, it has been considered necessary to
* Or about 20 miles, a French post league being 2,000
tpises, or 28£ to a degree.
employ almost exclusively locomotive engines
as the motive power. This system, moreover,
presents other important advantages in its sim-
plicity ; and in being able, after the Rail- way is
completed, to increase gradually the number of
engines as the demands of commerce require
it, and the number and the power of the appa-
ratus may always be proportioned to the work
which it really has to do, without the dan-
ger of incurring useless preliminary expenses,
and with the advantage of profiting by im-
provements as they occur in the progress
of the arts. One of the conditions therefore
which must not be departed from in laying out
great lines of Rail- way, is that these lines may
be traversed along ( their whole extent by loco-
motive engines, and, as much as possible, in
order to avoid interruptions and delays, that the
same engine draw throughout the same train.
The preceding condition shows that very
gentle slopes only can be admitted on Rail-
ways, and such that the differences which exist
between the powers required to draw the train
on different parts of the line, shall not affect the
working of the engines, or occasion any loss of
power.
The mean tractive power* required on an hori-
* In England at present this is taken at 91b. per ton, that is
9040
- — ' 249 or the 250 part of the weight.— 7V.
6
zontal part of the Rail-way may be estimated at
about the 200th part of the weight of the train, al-
though some experiments, made under favourable
circumstances, have given results rather less.
The weight of the train being taken as 1, this
tractive power is represented by 0.005, and each
millimetre in a metre (= 1 in 1000) of ascending
slope, increases this number by .001, so that in a
slope ascending 5 millimetres in a metre, (= 1 in
200) for example, the tractive force would be re-
presented by 0.01 ; that is to say, that it would
be double of what it is on an horizontal part. But
on a descending slope, on the contrary, each mil-
limetre in a metre of slope diminishes the tractive
power required by 0,001 ; so that the power re-
quired on a slope descending 5 millimetres in a
metre (=1 in 200) becomes nothing.
It is evident from this, that the slopes on
Kail-ways must be very gentle, and it may be
said in general that the more the construction of
a road or Rail-way is improved so as to diminish
the tractive power required, the more is it ne-
cessary, in order to profit by the advantage thus
obtained, to reduce the slopes.
But there exists a special reason for not
forming, if it be possible, slopes above 1 in 200 ;
since upon slopes of greater inclination the action
of gravity becoming greater than the resistance
arising from friction, the motion of the train be-
comes accelerated.
On account of the danger which this accelera-
tion causes, it is necessary to prevent it by par-
ticular contrivances ; and even to cause the
train to descend at a moderate velocity. It is
necessary, therefore, to destroy that portion of
the action of gravity which produces acceleration,
and which exceeds the tractive power. If the
descending slope, for example, be 7 millimetres
per metre, (1 in 143,) so that the gravity tends
to cause the train to descend with a force repre-
sented by 0.007, \— ]43 / we employ only a part
of this force, represented by 0.005, \~ooo / which
balances the tractive power, and we are obliged
to destroy, by the use of breaks, or other means,
the part represented by 0.002, (=500') which
produces the acceleration. It results from this,
that a part of the power which is the effect of
the descent is lost. In general, descents on a
Rail-way will only produce a saving of power
proportionate to the height from which the train
has descended, when the descending slopes do
not exceed in inclination 5 millimetres per metre,
(=1 in 200,) the draught being supposed, as al-
ready stated, to be equal to the five thousandth
Part of the weight of the train.
These considerations point out in a general way
8
the view in which the establishment of Rail- ways
has been considered. The possibility of esta-
blishing a mode of conveyance exceedingly rapid,
the use of locomotive machines for the tractive
power, the reducing of the slopes to the least in-
clination that is possible, and as much as can be
to inclinations less than 0.005, (I in 200,) have
appeared to be considered the most essential
conditions. It is superfluous to remark, besides,
that the condition of diminishing, as much as
possible, the time of transit between two given
points, requires that we should endeavour to
reduce also the length of the Rail-way which
is proposed to be constructed between these two
points. It would be committing a great error to
suppose we may lengthen the line because the
velocity of transport over it is great. The same
principle which rendered the establishment
of a Rail-way desirable, in order to obtain a
mode of transport quicker than any other, re-
quires that the shortest lines be sought after,
and even to prefer them when sometimes they
appear to be disadvantageous in other re-
spects.
The setting out of the line on the ground,
when the country has been surveyed and laid
down by plans and sections, does not require any
new principles.
Suppose that it is intended to unite by a Rail-
way the point A and the point B, which is more
elevated than A ; the most advantageous direction
would evidently be in the right line A B ; having
one uniform slope. It is this line which ought
to be obtained, or the nearest practicable one to
it, both horizontally and vertically. If a uniform
slope is impracticable, or if it requires too great
a deviation from the direct line, it is necessary at
least to endeavour to rise progressively from
A to B, and never to ascend where we must
descend again, and vice-versd.
If such a line cannot be obtained, and there
exists between the points A and B one or more
lines of ridges and valleys which must be crossed,
it is always necessary to endeavour to rise or to
fall as little as possible, consequently to en-
deavour to cut the ridges in those points where
the height is a minimum, and the valleys in the
points where the height is a maximum, without
lengthening the line too much. And it is very
apparent that we shall generally be led to the
ridge lines by following the secondary valleys
which cut them, and which are always lines of
less slopes. But if these lines of less slopes are
still found to be too steep for a Rail- way, it will
then become necessary to pass through below
the ridge by means of a tunnel instead of ascend
ing to the summit.
It often occurs that between the two given
10
points A and B, several lines are to be found
which appear to agree with the principles here
laid down, and also that some one of these lines
presents advantages of some other kind, such as
that of passing near a considerable town, or
through a district where important manufactories
are carried on.
The choice to be made of these different lines,
and which should always be founded upon
considerations of the general interests of the
country, may be difficult. We shall endeavour
to explain some of the principal points which
should influence a decision of this kind.
2. Principal elements in the comparison of dif-
ferent lines of Rail- ways.
The interest of the country is, in this respect;
1st, The establishment of a very rapid mode of
transport, — a consideration which should give a
preference to the shortest lines, the velocity being
supposed to be the same on all ; 2nd, The in-
crease of wealth. The construction of a Rail-
way, like that of a canal, or new road, is favour-
able to the advancement of wealth, in the first
place, because the actual expense of transport
in this direction is diminished ; and in the second
place, because this diminution in the cost of
transport increases the value of the neighbour-
11
ing properties, facilitates the establishment of
new works, and increases production. The first
of these two effects, that is to say, the diminution
obtained on the actual cost of transport, is the
cause of the second ; so that this diminution is
the principal circumstance, and that which should
be especially considered.
We should even say that the rate of reduction
which is obtained upon the actual cost of trans-
port, by the establishment of a new communica-
tion, is almost the only circumstance which
should be thought of, if it were not necessary to
consider also the quantity of goods which is car-
ried, or may be carried hereafter, in this direc-
tion ; for it is evident that it may be less advan-
tageous to the country to produce a great
economy in the cost of transport upon a line
where there is little to carry, and more advan-
tageous to produce a less economy upon a line
where a large quantity of merchandise is carried.
It is therefore generally necessary to take into
consideration, in the comparison of different
lines, the quantity of traffic which may be es-
tablished on each, and even the increase in the
value of properties, and the developement of
production, to which the establishment of these
lines may give rise respectively, according to the
nature of the countries which they traverse.
We shall not here undertake to go minutely
into the influence of these last elements of the
question, which rather belongs to statistics and
to political economy, and with respect to which
we cannot offer at present any precise opinions ;
we shall therefore confine ourselves to the con-
sideration of the reduction which the establish-
ment of a Rail -way can effect upon the actual
cost of transport, a most important consideration,
to which, as already remarked, it is always neces-
sary to attend. This will form in every case
the principal element of the comparison which
is the subject of inquiry, and often lead to de-
terminations purely geometrical or mechanical,
and consequently exempt from arbitrary deduc-
tions.
The cost of transport upon a Rail-way, as upon
a road or canal, depends on two principal points,
which it is necessary to distinguish and consider
separately. The first of these is the expense of
constructing the Rail- way, and the second is the
expense of conveying the goods on the Rail-way
when it is constructed.
The expense of the construction of the Rail-
way is independent of the quantity of merchan-
dise or of passengers that will pass over it. The
expense of transport, properly speaking, upon
the Rail-way supposed to be constructed, de-
pends, on the contrary, upon the quantity of mer-
chandise or of passengers ; that is to say, upon the
13
tonnage ; all other things being equal, the expense
will evidently be proportional to the tonnage.
As to* the secondary expenses, such as the
annual cost of repairs and management, it may
be said that they are partly in proportion to the
expense of the construction, and partly to the
amount of tonnage.
We may therefore admit, without falling into
any serious error, that the annual cost of trans-
port on a Rail-way, is in all cases formed of two
parts, the one proportional to the expense of the
construction of the way, and the other propor-
tional to the amount of tonnage.
We should also observe, that the cost of trans-
port of one ton of merchandise cannot be specified,
unless the number of tons which shall be carried
annually from one extremity of the line to the
other be known.
Suppose, for example, that we know, in one
case, that, the road being constructed, the part
of the expense which is proportional to the ton-
nage will amount to 0.30 fr. per ton, per league ;
or [=ld,] per ton, per mile ; and in the other
case, that the part of the expense which is pro-
portional to the cost of construction, and which is
independent of the tonnage, represents a capital
of 1,200,000 fr. [=£48,000,] or an annual
expense of 60,000 fr. per league, [£800 per
mile.]
14
This annual expense, if the Rail-way has a
traffic of 100,000 tons per annum, will amount
to 0.60 fr. for each, and if the traffic be 200,000
tons, it will amount for each to 0.30 fr. ; so that
in the first case, the total cost of the transport of
one ton over one league, is 0.90 fr., or three-
pence per ton, per mile ; and in the second case,
it is only 0.60 fr. per league, or twopence per
ton, per mile.
The knowledge of the expense of construction
of a Rail-way, and even, to a certain extent,
that of the expense of repairs and management,
are subjects which do not differ from those in
which engineers are generally employed, and
which do not require any particular considera-
tion.
The knowledge of the expense of conveyance,
properly speaking, requires an investigation si-
milar to that which is made in the arts, for ascer-
taining the price of works executed by machines.
It depends upon mechanical principles, to which
we shall particularly apply ourselves.
3. Determination of the power required to draw
a given train over a given line of Rail-way.
Let us observe, that upon an horizontal line
15
the power required to draw a given weight, is
considered as being equal to almost the two
hundredth part of this weight, a result which we
shall here suppose, (conformable to what is ge-
nerally admitted,) to be independent of the
absolute velocity of transit, although there is
reason to believe that the tractive power in-
creases with the velocity. We conclude from
this, that in order to transport, with any velocity
whatever, constant or variable, a weight P, to the
distance represented by a, on an horizontal line,
it is necessary to employ a power represented by
p
~~ a ; that is to say, the power necessary to
raise the weight P to the height — . Thus, for
example, it is the same thing to transport the
weight P, to the distance of a league of 4,000 me-
tres, [= 4374 yards,] upon an horizontal Rail-
way, as to raise the same weight to a vertical
height of 20 metres, [=21.87] yards.
If the transit had been over an ascending
slope, forming with the horizon a very small
angle i; in such a way that i represents the rise on
a unit of the length ; the tractive power becomes
P (0.005 + ^,) and the power required to trans-
port the weight P, to the distance «, becomes
P (0.005 + t) a ;
and as the term i a represents the vertical height
16
to which the weight has been raised when it has
passed over the distance a, we see that the power
expended is here equal to what had been em-
ployed to transport the weight over a horizontal
line added to that which is necessary to raise the
weight to the vertical height to which it has
really ascended.
If the transit had been over a descending slope,
of which the inclination is in a similar manner re-
presented by i ; it is clear that the tractive power
is reduced to P (0.005 — i) ; and the power ex-
pended to travel over the distance a, to
P (C.005— 0 a.
From whence it results that the power expended
is in this case obtained by subtracting from the
power 0.005 P a, which is ^required to draw the
train over the horizontal line, that which is repre-
sented by the descent of the weight from the
vertical height i a, from which it has actually
descended.
But it must be observed, that in the statement
which has just been made, negative values of the
quantity P (0.005—^) a must not be admitted,
which would take place if the fraction i was
greater than 0.005 ; because the trains are not
allowed to accelerate their motion, which they
would have a tendency to do, and to acquire a
velocity greater than would naturally result
17
from their descent upon a slope more rapid
than 0.005. Whenever the descending- slopes
are greater than gob a negative value of
P (0.005 — i) a will be the result, in such cases
zero must be substituted for this value.
These observations lead to a rule, exceed-
ingly simple, for estimating the power which
is required to move a train over a line of Rail-
way. If we suppose that the same locomotive
engine draws the same train over the whole
line, and that there are not descending slopes
more rapid than ^ the power required to
effect the transport of the weight P, from the
point M to the point N, representing by A
the length of the line MN, and by H the height
of the point N above the point M, will evidently
be represented by
P ( —rr- + H ) in the direction M N, and
P ( ^7T — H ^ in the direction N M,
\ 2()0 /
whatever be the distribution of the ascending arid
descending slopes, which the line may present.
Consequently if the line MN were horizontal,
or, more generally, if the two extremes M and
N were at the same level, the transport of the
c
18
weight P, from one extremity to the other,
exactly equals the elevation of this weight
to a height equal to the 200th part of the dis-
tance MN.
When the points M and N are not on the
same level, it is sufficient to add or subtract
from this height their difference of level, accord-
ing as the transit is towards one extremity or
the other.
The preceding rule is rigorously correct,
when we suppose that the train starts from a
state of rest, at one extremity of the line, and
arrives at the other when the velocity is re-
duced to zero, and also considering the power
as sufficient to effect the transit without regard-
ing what is uselessly consumed by the friction,
and other deteriorating causes, inherent in the
working of the machinery of alocomotive engine.
This rule is, in fact, only a version of the
general principle of the preservation of the vis
viva,* from which it is known that a heavy body,
made to pass over any given line whatever,
always moves from one extremity to the other of
this line, acquiring or losing a velocity due to the
height from which it has ascended or descend-
ed, and in such a way that the body constantly
returns to the same velocity, when it moves
through points situated in the same horizontal
* See note A at end.
19
plane. We conclude from this that the length
of the line remaining the same, the amouut of the
quantity of power consumed to effect the transit
depends entirely upon the length of the line,
and the difference of the level of its extreme
points.
But if, as it is convenient to do, we wish to
value not only the power required to effect the
transit, but also the total quantity of power
really produced by the locomotive engine,
we should observe that when the train, after
being elevated to a certain height in passing
up an ascending slope, descends an equal
height in passing afterwards down a descending
one, the descent restores the power which had
been employed to raise the weight of the train ;
but not that which had been consumed by the
friction produced whilst the machine was exert-
ing the power necessary to effect the elevation.
Whence it results, that whenever there is a
useless ascent, that is to say, whenever an eleva-
tion is ascended, which must afterwards be de-
scended, or that an elevation is ascended, which
was before descended, it is necessary to keep in
view the fraction of the power which is required
to raise the train to the height in question, repre-
senting the effect of friction, and other resistances
of the locomotive engine. By paying attention to
this consideration, we perceive that the form of
20
the section between the two extremes of the line
is not unimportant, since there is a loss of power
whenever there is a useless ascent.
Let us suppose that the point N is higher than
the point M, and that in the transit in the direc-
tion MN, there will be a useless expenditure of
power every time there is a descent. If, on the
contrary, the point N is lower than the point M,
there will be a useless expenditure of power
whenever there is a rise. Whence we may
conclude, that we should always endeavour to
avoid adopting an alternation of ascending or
descending slopes, but manage so that the line
should ascend or descend progressively from
one extremity of the road to the other. As to
the manner of estimating the effect of the useless
rises, it will always be easy to detect them by an
inspection of the section. It will be sufficient
for this purpose, to mark the points of maximum
and minimum height. Let h represent the sum of
these useless rises. The total power expended
by the locomotive engine being represented by
unity, we may represent by /* the portion of this
power which is riot used to effect the transit on
the elevation of the train, and which is consumed
uselessly by the friction and other resistances to
which the machinery is subject. It is evident
that the part of the total power employed to
effect the elevation of the weight P to the height
ht which is not restored by the descents, is ex-
pressed by p. P/i. Thus, in adding1 this quantity
to the expression which has been given before,
the formula
P (0.005 A ± H + fi A)
will represent for different lines of Rail-way,
a number proportional to the quantity of power
really produced to effect the transport of the
weight P. We should take the sign -f or — , ac-
cording as the point of arrival is higher or lower
than the point of departure.
The preceding result may be expressed by
saying, that the quantity of power expended
to produce the transit over any line whatever
is equal to that which shall be necessary to
elevate the weight of the train to a height ex-
pressed by
0.005 A ± H + ph; (I)
that is to say, to a height equal to the 200th part
of the length of the line, increased or diminished
by the difference of the level of the extremes ac-
cording to the direction of the transit, to
which is to be added the sum of the useless rises,
multiplied by the fraction expressing that part of
the total quantity of power furnished by the
steam engine consumed uselessly by friction, and
which is not employed to effect the transit.
This result is also limited by the two hypotheses
which we have before stated, that is to say, 1st*
that there is no part of the descending- slope more
rapid than 0.005 ; 2ndly, that the same loco-
motive engine draws the train over the whole
line. It is therefore necessary to consider the
particular cases which do not coincide with these
conditions.
1st. If there is in some part of the line a
descending slope i, more rapid than 0.005, and of
which the vertical height is r/, we may remark,
agreeably to what has been said before, that we
only make use of the fraction 0.005 of the action
of gravity in descending the slope, and that the
part ^—0.00.5 of this action is entirely lost, since
we cannot allow the train to acquire the velocity
which this would tend to give it.
The descent of such a slope cannot therefore
be considered as producing the power Pr/ corres-
ponding to this descent, but only to a power
equal to — ' — - > and the quantity of power
( *"- °;OQ5) P^ is lost. We therefore conclude that
after having expressed as above by formula (1),
the height to which the weight of the train is
raised by the power which produces the transit, it
is necessary to add to this height the quantity
is
— 0.005
n (2)
whenever there is a descending slope i more
rapid than — , the difference of level of the two
/&00
extremes of this slope being r/.
If there is upon the line an ascending slope so
steep as to require the use of an auxiliary engine,
it is necessary, in addition to the quantity of power
already determined, agreeably to what has been
said above, to add the quantity of power necessary
to raise the auxiliary engine from the lower to
the upper extremity of the slope. Let us desig-
nate by a the length of the slope, and by r/ its
vertical height. Let us further suppose, that the
weight of the auxiliary engine is equal to the
fraction k, of the total weight of the train re-
presented above by P. It is evident, that the
quantity of power necessary to transport the
auxiliary engine will be represented by k P
(0.005. a + r?). Whence we conclude that we
arrive at this quantity of power, by adding to
the height expressed by the formula (1) the
quantity
k (0.005.^ + 0 (3)
Further, there will not require any deduction
to be made, for the descent of the machine when
it returns along the slope in a contrary direction,
the quantity of power which this descent would
produce being entirely lost.
We can always, by means of these principles,
easily estimate the amount of power necessary
to effect the transit of a train from one ex-
tremity to the other of a Rail-way, an amount
the value of which is thus expressed in extremely
simple terms. However evident these principles
may be, it will not, perhaps, be useless to give
an example of the calculations to which they
refer.
Let there be a line of Rail-way defined by the
following section, of which we can construct the
figure.*
References to the
Distances in
Heights in metres of
points of the
metres between
these points above
section.
these points.
the point M.
M
_
_
a
4,000
8
b
5,000
50
c*
6,000
57
(l*
10,000
51
e
11,000
58
f*
8,000
62
<7*
N
32,000
25,000
23
32
101,000
In the above table the points marked with an
* See note B.
25
asterisk, are those in which the heights are
maxima or minima. In applying the princi-
ples already stated, we observe that the length of
the line being 101,000 metres [=110.457 yards,]
the transit along this line, supposing it to be hori-
zontal, is in the first place equal to the elevation of
the weight, to a height equal to - — — , or 505
metres [=552 3 yards].
Further, if the transit is made in the direction
MN, the weight must be elevated to the height
of 32 metres, [= 35 yards,] which is the quan-
tity, the point N is more elevated than the point
M. The power, therefore, expended to produce
the transit will be equal to that which is neces-
sary to raise the weight to a height of 537 metres,
[=587.3 yds,] if there were no (contre pentes)
descending slopes on the line.
But in consequence of these descending slopes,
there is a useless ascent before the first maximum
c, of 57 metres, [=62.3 yards,] less 51 metres,
[=55.8 yards,] or 6 metres, [=6.5 yds.] and
before the second maximum f> of 62 metres —
23 metres, or 39 metres [=42.7 yds.] Total of
the useless ascents 45 metres [=49-2 yds.]
If we admit then that two thirds of the power
produced by the locomotive engine is required
to overcome its friction, and other resisting
26
causes,* we must add 30 metres to the number
before found, which will give 567 metres [=61 1.1
yards.]
We must also observe, that in the interval
between the point a and the point b, which is
a length of 5,000 metres, there is a rise of 42
42
metres which gives a slope of — or 0.0084,
5,000
=1 in 119, sufficiently steep to make it desirable,
and, perhaps, necessary to place upon it an auxi-
liary engine.
Attending to this circumstance, we must spe-
cify in what proportion the weight of this engine
increases the weight of the train, and this will
depend upon the absolute power of the engine
used, and the velocity with which it is proposed
to ascend the slope.
If we admit, for example, that the weight of
the auxiliary engine, is the fourth part of the
weight of the train, we should increase this
weight in the proportion of 5 to 4, and we get
the amount of power required to move this en-
gine, by adding to the number before found,
1st, the fourth part of ~ — or 6.25, metres
[=6.8 yards,] for the transit over that part of
the line supposed to be horizontal ; 2dly, the
* Note C at the End.
27
fourth part of 42 metres, or 13 metres, [=14.2
yds.] for the elevation of the engine from the
bottom to the top of the slope ; these two sums
added together make 19-25 metres [=21.0 yds,]
which increases to 586.25 metres [=637 yds,]
the height to which the weight of the train
should be raised by expending a quantity of
power equal to that which would effect the
transit in the direction MN of the proposed
line.
If now we examine the transit in the opposite
direction NM, we have in the first place, as
above, the height of 505 metres, [=552.3 yards,]
which represents the transit along the line, sup-
posing it horizontal, from which it is necessary to
subtract the 32 metres, the amount of the descent
from the point N to the point M, and this gives
505-32=473 metres [=517.3 yards.]
In the next place we observe that the amount
of the useless ascents is expressed by the same
number as before, (which is always the case,) so
that we must take account of it in the calculation
in the same way by adding 30 metres [==32.8 yds,]
to the preceding number, which will give 503
metres [=550.1 yards.]
Finally, by observing that the train descends
from the point b to the point a, a height of 42
metres, on a slope of 0.0082, which is greater
than 0.005, we perceive that there is a part of
the action of gravity which is not made use of
during" this descent, and which is represented by
the fraction go? from which it results that we
have taken the useful descent too much in the
preceding number by a height equal to 42 metres,
32
multiplied by the fraction ^ that is to say, to
16.39 metres [==17-9 yds.] Adding then this
last number, we shall find the total to be 519.39
metres, [=568 yds,] the height to which the
train would be raised by the power which would
effect the transit in the direction N M.
There is nothing to be reckoned for the de-
scent of the auxiliary engine on the slope b a,
so long as the mechanical value only of the
quantity of power consumed is taken into ac-
count, but it is evident that the useless ex-
penditure which is incurred by having this
machine upon the line, during the whole time
it is not employed to aid the ascent of the train,
should be reckoned.
In summing up the considerations which have
been just set forth, we perceive that the valuation
of the power which it is necessary to provide
to effect the transport of the weight P, from one
extremity to another of a Rail-way, is reduced
to determine, by means of the formula (1), and,
if there be occasion, by employing the formulae,
29
(2) and (3,) a certain vertical height Z, to which
the weight P may be raised by employing the
said quantity of power.
This power is then expressed by the product
P Z ; and if we represent by J the ratio of the
height Z, to the length A of the line, or if we
suppose
the product PJ will express very nearly the
mean tractive power, which the locomotive engine
should exert to accomplish the draught of the
train.
4. Determination of the weight of the train
which can be drawn upon a line of 'Rail-way ',
hy a locomotive engine of a given power.
A locomotive engine being given, the power
which it can exert is limited principally by two
circumstances. 1st, By the quantity of steam
which can be generated in a given time ; 2ndljr,
by the tractive power which the machine can
exert without slipping upon the rails.
It is evident that, in all possible cases, there
will be a loss, if all the vaporating power of the
fire-place and boiler were not employed ; that is
to say, if all the steam is not produced which
could possibly be obtained. Thus the first con-
30
dition, in the proper management of an engine,
is to produce constantly the same quantity of
heat. It results from this, as will be shown
further on, that the weight of the train being
given, there is for each slope a certain velocity
which should be adopted, and vice .versa. Fur-
ther, the weight of the train cannot exceed the
limit corresponding to the adhesion of the engine
on the rails.
The action of the engine, the motion of the
train being supposed uniform, is also subjected to
the condition, that the pressure of the steam should
be in equilibrium with the tractive power, and
which latter we should consider as applied to the
circumference of the wheels of the locomotive
engine. This condition determines the pressure
which the steam must have in order to draw a
given weight. If we previously fix a limit,
which this pressure shall not exceed, the con-
dition which this involves fixes a limit to the
action of the locomotive engine.
The influences of these different conditions,
and the results to which they lead, cannot well
be explained without expressing them by means
of formulae.
Let us suppose, for a moment, that the ex-
pense of the heat of the fire-place is constant
when the same weight of steam is* produced, in
each instant of time, — a supposition which is pro-
31
bably so near the truth, that we may admit it
without inconvenience in calculations of this
kind. Moreover, with the view of obtaining
formulae as simple as it is possible, we shall re-
mark that within the limits of the pressure, under
which we suppose the steam produced, we may,
without committing any very great error, take
0.5 n + 0.09 (5)
for the expression of the weight in kilogrammes
of the cube metres of steam produced under a
pressure of n atmospheres.*
This being supposed, we may represent by P
the total weight of the train, comprising the
engine, the tender, the waggons and their load,
expressed in kilogrammes.
Q. The area of the two pistons of the loco-
motive engine.
c. The length of the strokes of the pistons.
r. The radius of the wheels.
F. The pressure exerted by the steam formed
in the boiler, on a square metre, expressed in
kilogrammes.
Y. The volume of the steam generated in a
second.
II. The weight of the steam.
U. The velocity of the train, or the distance
it runs in a second.
* Note D.
3%
The linear dimensions are supposed to be
expressed in metres, as well as the velocity U.
The weights and pressures are expressed in
kilogrammes. We may observe, in the first
place, that the pressure of one atmosphere on a
square metre being 10,330 kilogrammes, the
number of atmospheres, represented by n, in for-
F
mula, (5,) becomes n = — — *
We may, therefore, consider the weight of a
cube metre of steam generated in the boiler un-
der the pressure F, as represented by the follow-
ing formula.
0.09 + 0.5
10330
or more simply,
0.09 -f- 0.0000484 F ..... (6)
And from this will result between the numbers
y, IT, the relation
_ ___ IT
7 "" 0.09^j- 0 0000484~F '
The question which we propose to resolve from
these data is the following,
The weight, power, &c., of the locomotive
engine are supposed to be given, and consequently
* See note E, where these formulae are given in English mea-
sures.
33
we know, besides its dimensions, the weight
IT, of the steam which it can generate constantly
in a unit of time. The mean velocity U, with
which it is desired to travel over the Rail-way is
determined. And, finally, the line of Rail-way is
supposed to have been submitted to the inves-
tigation which has been explained in the preced-
ing article ; by means of which we know the ratio
J given by formula (4), and the mean tractive
power, represented at the end of the same
article by PJ, a power required to be pro-
duced to effect the transit of the weight P. It
is required to determine the amount of the
weight P, which can be drawn along the line by
the engine.
This determination rests upon the two follow-
ing circumstances : 1st. That the volume of steam
expended in a unit of time, is equal to the
volume of space passed through by the pistons.
Sndly. That the pressure of the steam in the
boiler is such, that the power transferred to the
circumference of the wheels is equal to the
mean tractive power PJ.
The first condition is expressed by the equa-
tion.
y = — ftU (8)
7T/*
in which TT represents the ratio of the eircum-
34
ference to the diameter, and which by substitu-
tion in equation (7,) is changed into
II
0.09 + 0.0000484F KT' '
To express the second condition, it is neces-
sary to fix the relation which exists between the
pressure F, produced by the steam in the boiler,
and the power transmitted to the circumference
of the wheels ; which effects the transit.
It follows from the principles explained in the
Treatise on Steam Engines, by Tredgold, that
the power transmitted by the piston rods, in high
pressure engines without condensation, may be
calculated by representing by 0.6n — 1, the num-
ber of atmospheres corresponding to this power, n
representing, as before, the number of atmo-
spheres corresponding to the pressure which
takes place in the boiler. In the engines here
spoken of, it is necessary to observe that the
steam is thrown into the chimney with consider-
able velocity, with the view of urging the com-
bustion, and to some other circumstances, which
lead us, after an attentive examination of known
facts, to consider the power which is transmitted
to the circumference of the wheels as answering
to a pressure exerted on the pistons expressed
by a number of atmospheres equal to O.dn — 1.
35
The amount of this pressure expressed in kilo-
grammes, on a square metre, will therefore be *
0.5F— 10330; . . . , . . (10)
and consequently the equilibrium which should
exist between the pressure F, under which the
steam is formed in the boiler, and the tractive
power, will be expressed by the equation
— (o.5F-1033o)n=PJ. . . (11)
7T7* X /
The two equations (9) and (11) express the
conditions of the movement of the locomotive
engine when the whole of the power which this
engine can produce is employed.
The value of F drawn from equation (9), is
~" 0.0000484
which represents a pressure in the boiler, that
cannot be exceeded when the train is moved with
the velocity U.
The value of F, taken from equation (11), is
F= l (— !7r+ 10330 A .... (13)
\ c ii /
which represents the pressure that should be in
the boiler, to enable the engine to draw the
weight P.
By equalizing those two values of F, we get
the equation,
* Note F.
D %
36
P_ L. f M_ 5L_ (o^x^? + 1033o Y-51
J ^0.0000484 U ^0.0000484 ArJ
or more simply,
P = \-( 10330-5 — 11260— );. . . (14)
J \ U 7T TS
U
and reciprocally,
U
PJ+ 11260 -
TTT
Equation (14) expresses the greatest weight P,
which can be drawn by the locomotive engine
with the velocity U ; and equation (15) ex-
presses the greatest continued and permanent
velocity which can be given to a train, the weight
of which is P.
If we establish between P and U, the relation
expressed by the equations (14) or (15), the
action of the engine will be so regulated as to
produce all the effect of which it is capable.
By multiplying equation (14) by U, or the
equation (15) by P, we get
PU =-
*l
or
10330II
....... (17)
71TP
for the expression of the useful effect which can
,be obtained in the unit of time, (in reference to
37
the gross weight of the train) expressed in func-
tions of the velocity of the transit, or of functions
of the weight P of the train moved. It is seen,
agreeably to known results, that this useful effect
is increased, by increasing the radius of the
wheels of the locomotive engine ; or more gene-
eft
rally, by diminishing the ratio — . Further, it is
evident that the useful effect is greater, in propor-
tion as the velocity of the transit is less, or as the
weight carried is greater. But it must not be for-
gotten, that the amount of weight which can be
carried is limited both by the condition, that
the wheels of the locomotive engine do not slip
upon the rails ; and also by the condition of not
subjecting the pressure F, under which the steam
is formed, and of which the value is given by
formula (13), to exceed a given limit.
It is not difficult to perceive, after this remark,
that the load, which it is proposed to draw by a
given engine, and the velocity with which the
transit is proposed to be effected, cannot be
varied to any great extent.
When it is desirable to change, in any con-
siderable degree, these two conditions, it is ne-
cessary to change the proportions of the appara-
tus in such a way as to preserve very nearly the
ratio — - 5 so that the useful effect obtained, in
38
reference to the weight of the train, also pre-
serves constantly the same value, or nearly so.
Equation (14) will give immediately, in each
particular case, the solution of the question
which is proposed in this article.
To give an example, suppose a locomotive
engine is used similar to the machines at
present used upon the Manchester and Liver-
pool Rail-way, and of a medium power. The
weight of such an engine may be reckoned at
8 tons, that of its tender at 4 tons, and we shall
have the
Diameter of the cylinders Om 28 ; the surface
of the two pistons Om'q- 12315=Q.
Stroke of the pistons Om. 41=c.
Radius of the wheels, Om. 76=r.
From which we deduce — =00.21148.
irr
Weight of the steam formed in a second,*
Okil. 4 = 11.
These values being substituted in the formula
(13,) (14,) (15,) will give respectively
F = 94.57 JP + 20660, . . . (18)
P= 1(112? -238) . . . (19)
Note G.
39
Formula (19) is the expression of the total
weight which can be drawn by this engine.
Formula (2C) is the expression of the perma-
nent velocity * which the train can acquire on a
slope which requires a tractive force, JP.
Formula (18) shows the pressure under which
the steam should be formed.
Let us suppose that we had found by a process
similar to that which has been described in the
preceding article, that the mean tractive force
which should be employed on a given Rail-way
is represented by the fraction 0.006 of the
weight to be moved, which is the same thing as
to admit that the power expended in moving the
train, is the same as if the Rail-way presented
throughout its whole extent an uniform ascend-
ing slope of 1 in 1000 : we should then make
J = 0.006.
If we suppose also that the transit is to be
made with a mean velocity of about 8 leagues
per hour, we must put U = 9 metres [== 9.84
yards.]
These last values substituted in formula (19,)
will give P=3685Q kilogrammes or 81272 Ibs.
for the amount of the total weight of the train,
or about 37 tons. Subtracting 12 tons for the
weight of the locomotive engine and its tender,
there remains 25 tons for the weight of the
* Note H.
40
wagons and their loads. The load or useful
weight is about the f of this last weight, or 17
tons.
If we substitute in formula (18) for JP the
value 221.1 kilogrammes, F will become = 41570
kilogrammes, for the pressure exerted by the
steam on a square metre, which is a little more
than 4 atmospheres, or a little more than 3
atmospheres, above the exterior atmosphere.
5. Investigation of the uniform motion of the
train, on the various ascending or descending
slopes, which may constitute part of a Rail-
way.
The whole weight of a train supposed to be
drawn by a locomotive engine of a given power,
having been determined in the manner which
has been just explained in the two preceding arti-
cles. There is no doubt that the results obtained
might be realized in actual practice, if the line of
rail-way presented auniform slope requiringa con-
stant tractive power JP, to draw the weight P, in
which case the velocity of the motion would be
also constant. But as a line of rail- way presents
generally unequal slopes, it is necessary to exa-
mine in each- particular case, if the existence of
these slopes do not alter these results, and in
what limits the slopes should be confined, in
order that these results may be applicable.
41
It may be said in general, 1st, that the result
of the preceding article is applicable, or, which is
the same thing, that any loss upon the action of
the locomotive engine will not take place in con-
sequence of the existence of an ascending slope,
when the engine can draw the train upon this
slope ; that is to say, when the tractive power,
which is required on the slope does not render
it necessary to raise the pressure under which
the steam is formed too high, or will not cause the
wheels of the locomotive engine to slip on the
rails.
^ndly. That also there will be no loss in
consequence of a descending slope, when the
action of gravity on the train does not surpass
the resistances, including the power necessary to
propel the locomotive engine when unattached
to the train.
If, therefore, we represent in general any ascend-
ing slope whatever by i, so that the power neces-
sary to draw the weight Pupon this slope should
be expressed by (0.005 -f i) P, we must make
J = 0.005 + i in formula (13), in order to verify
if the value which results from it for the pressure
F does not surpass the proper limit. Further,
we must also examine if the power (0.005 + i) P
would not surpass the power which the locomo-
tive engine can exert without slipping, and which
may generally be estimated at the twentieth part
of the weight of the engine.* If the elope i
satisfies these two conditions, it will not im-
pede the transit of the train, or cause any loss
of power.
If now i represents any descending slope what-
ever, so that the power necessary to draw the
weight P upon this slope will become (0.005 — i).
P, we must first examine if this quantity, when
put in the place of J P in formula (15), gives to
the denominator of this formula a negative or
zero value, or only leads to a value of U, sur-
passing the greatest velocity which it is proper
to allow the train to take. In this case the de.
scending slope requires the use of the break, and
will occasion a loss of power ; in all other cases
it produces none.
To make this more clear, we will now apply
these principles to the example which has been
given at the end of the preceding article, and to
which the formulee (18), (19), and (20), relate.
The following table is formed on the principle
that the values of F are calculated by formula
(18), and the values of U by formula (20), in
giving to P the value P= 36850 kilogrammes,
or 81163lbs. as before found.
* Note I.
43
Ratio of the
tractive force
Pressure in
Indication of the
to the weight
the boiler on a
Permanent
slopes.
drawn
square metre
velocity.
= J.
= F.
= U.
Kilogrammes.
Metres.
Descending 0.006
— 0.00 1
17,170
20,5
0.005
0.000
20,660
17.4
0.004
+ 0.001
24,150
15.0
0.003
0.002
27,630
13.3
0.002
0.003
31,120
11.8
0.001
0.004
34,600
10.7
Zero 0.000
0.005
38,090
9.8
Ascending 0.001
0.006
41,570
9.0
0.002
0.007
45,060
8.3
0.003
0.008
48,540
78
0.004
0.009
52,030
7.3
0.005
0.010
55,510
6.8
0.006
0.011
59,000
6.4
We see by this table, that on an ascending
slope of 6 millimetres per metre, or 1 in 166,
the pressure of the steam in the boiler would not
amount to 5 atmospheres beyond the exterior
pressure, and that the train would maintain upon
this slope a velocity of more than 6 metres per
second, or nearly six leagues, or 14i miles per
hour. We also perceive that on a descending
slope, having the same inclination, the velocity
of the train would not exceed 20m.5 per second,
or about 18 leagues or 45 miles per hour. Also
the ascending slope of 6 millimetres per mdtrei
or of 1 in 166, requires for a weight of 36850
kilogrammes, a tractive force of 405 kilogram-
44
ines, which is a quantity very nearly equal to the
20th part of the weight of the locomotive engine,
which is 8 tons. It appears then, from these
results, that in the circumstances which have
been supposed, that is to say, in admitting that
the mean tractive force on the line of Rail- way
corresponds to an uniform ascending slope of
1 in 1000, and a mean velocity of 8 leagues per
hour, the motion of a train of about 37 tons
gross, drawn by a locomotive engine of 8 tons
weight, should not be impeded by ascending or
descending slopes of 1 in 200, or even 1 in 166,
and that the existence of these slopes would not
occasion any loss of power. But it is also evi-
dent, that there is a reason that the slope of
1 in 200 has been considered as the limit beneath
which the slopes should be endeavoured to be
kept.
It is necessary also to remark, that the results
to which we have just come are derived from
data which have been used, and particularly to
the mean velocity of transit which has been as-
sumed.
If we suppose a less velocity, the same engine
will be capable of drawing a greater weight, and
the tractive power augmenting in consequence,
shows that there will be a slipping of the wheels,
on a slope less than 0.005 or 1 in 200.
When the same engine draws a greater
45
weight, with a less velocity, the expense of trans-
port is diminished, which arises principally from
this, that the useful weight carried forms in this
case a greater portion of the total weight of the
train. It is useful to form a correct idea of the
amount of this diminution.
Let us suppose, for example, that a line
of Rail-way does not present any slopes
steeper than 0.003, we can allow to the train,
drawn by the locomotive engine, such as that
already spoken of, a weight corresponding to
the limit fixed by the slipping of the wheels on
such a slope ; that is to say, a weight equal to
Q-QOg =50,000 kilogrammes, or 50 tons.
If at the same time we are desirous that the
traction of such a weight should not raise the
pressure of the steam higher than in the preceding
case, where the weight was only 37 tons, it will
be necessary to increase the length of the stroke,
or the diameter of the pistons, or to diminish the
diameter of the wheels of the locomotive engine,
so that the quantity -5 should preserve the same
value. It will result from this that the number
138, which is found in the denominator of for-
mula (20), will become 238-^=
This formula (20) is then changed into
4132
-JP +322 '
and if we make P = 50000 kilogrammes and
J = 0.006, it will give U = 6m- 64 for the mean
velocity with which the locomotive engine will
draw over the whole extent of the line, the
new train of 50 tons. To compare the expense of
the transit in these two cases, we shall consider
therefore in the case of the train of 37 tons that
the net weight is proportional to 27 — 12, or 25,
and in the case of the 50 tons that it is 50—12,
or 38.
Besides, in the first case the mean velocity
was 9 metres, or 9.84 yards per second, and in
the second case that it was 6.64 metres, or 7-26
yards. The expense of conveyance is therefore
greater in the first case in the proportion of
|. «*orl.l«tol.
When Rail- ways of a certain extent are under
consideration, it can scarcely be hoped to reduce
the greatest slope to less than a 3 thousandth, or 1
in 333 ; and consequently, if we wish that the
same engine should draw the same weight over
the whole line, we cannot make an engine weigh-
ing 8 tons draw a train, the total weight of which
exceeds 50 tons.
47
The moving of such a train with a velocity of
6m 64, found above, or nearly 7 leagues an hour,
is therefore the most economical arrangement that
can be adopted. Now it is shown, by the pre-
ceding calculation, that in carrying the velocity
to 9 metres per second, or a little more than 8
leagues per hour, which requires the weight of
the train to be reduced to 87 tons, is the expense
of carriage increased, upon the line of Rail-way
which has been taken for the example, by about
12 per cent. But then this advantage is ac-
quired, which in certain cases may be very im-
portant, that the train will overcome without
difficulty, not only the slopes of 1 in 333, but
even slopes exceeding 1 in 200.
All these circumstances lead to the establish-
ment of this fundamental principle, that the essen-
tial character of Rail-ways is to afford the means
of a very rapid conveyance.
In order to render it convenient and advan-
tageous to affect the transit at a low velocity, it
would be necessary that the line should not have
any ascending slope in the direction of the
transit.
Let us remark that the velocity of the tran-
sit is here considered as given and fixed at 9
metres per second, so that we should deviate
from the conditions laid down, if we reduced the
velocity to 6ra 64, with a view of increasing the
load, and obtaining- a saving of 12 per cent
mentioned above. It is true that the velocity of
9 metres per second, or 20 miles an hour, is
considered as necessary only for the conveyance
of passengers, and that if the conveyance of mer-
chandize is carried on separately, the velocity
of the trains loaded with the latter might without
inconvenience be less.
This circumstance leads to the conclusion that
there will be a certain advantage in not having
slopes greater than 1 in 333, which would permit
the conveyance of merchandize with a less velo-
city, and by allowing a greater weight to be
moved, would produce a certain saving in the cost
of carriage, the amount of which can be ascer-
tained from the preceding calculations.
We should consider, also, that if we are not
rigorously tied down to the condition, that the
same locomotive engine should draw the same
weight over the whole line, we may in most
cases, much diminish the injurious influence of
slopes greater than 1 in 333, or even 1 in 200,
by diminishing the weight of the trains on these
slopes only, or in placing there auxiliary engines.
They could then be passed over with velocities
much greater than what has been here stated, a
circumstance which would compensate in part
for the increased expense which would arise from
the plan here suggested.
49
We may presume, from what has been stated,
that there is often little to gain, by increasing to
any great extent the expense of constructing a
line of Hail-way, with a view of reducing the
limit of the slopes from 1 in 200 to I in 333. It is
impossible to establish on this point absolute and
general rules, but the principles which have been
here stated, afford the means of choosing in each
particular case, that arrangement which is the
most advantageous.
It is easy to perceive to what extent the diffi-
culty and expense of carriage rapidly increases
on long ascending slopes, which exceed 1 in 200.
The following table*
Total weight of
Rate of
ascending
slopes.
Ratio of the
tractive pow-
er to the
weight drawn
== J-
the train which
the engine
weighing 8 tons
can draw with-
out slipping
= P.
Net weight
carried
= f(P-
12,000.)
kilogrammes.
kilogrammes.
0005
0.01
40,000
18,667
0.0075
0.0125
32,000
13,333
0.01
0.015
26,667
9,778
0.0125
0.0175
22,222
6,815
0.015
0.02
20,000
5,333
0.0175
0.0225
17,778
3,852
0.02
0.025
16,000
2,667
* See Note K.
50
points out the gross weight of a train, that the
engine, to which the preceding calculations have
been applied, can draw without slipping upon dif-
ferent slopes, and of the net weight which can
be carried. As the tractive power is in all
cases supposed equal to the — of the weight of
the engine, that is to say to 400 kilogrammes, or
882 Ibs. the pressure under which the steam ought
to be generated, should be the same in all cases,
and about 4.8 atmosphere, independent of the ex-
terior atmosphere. The velocity of the train
should then also be the same in all cases, and about
6i metres or 7 yards per second. Thus the use-
ful effect obtained on each slope is proportional
to the corresponding number found in the last
column. Upon a slope of 0.02 or 1 in 50, the
engine cannot draw more than a waggon.
We have seen in the preceding article that the
train of 37 tons, drawn by the engine on an
ascending slope of 1 in 1000, with the velocity
of about 8 leagues per hour, carries a net load of
about 17 tons. As the net load corresponding
to a slope of 1 in 100, in the preceding table, is
not much more than the half of this, we see that
the passage of the train over this slope, requires
an auxiliary engine of almost an equal force ; this
agrees with known facts,
If, besides, the pressure of the steam is not
51
raised in the two engines to about 4i atmo-
spheres, above the exterior atmosphere, and if,
for example, we do not exceed the ordinary
limit of 3i atmospheres, the tractive power being
inferior to the resistance, the velocity with which
the train commences to ascend the slope will
progressively diminish, towards a state of rest.
But by raising the pressure to 44 atmospheres,
the velocity would not be reduced to less than
about 6^ metres, or 7 yards per second.
The principles which have been just explained,
appear to indicate, that on great lines of Rail-
way, where we are confined not to surpass
slopes of 1 in 200, and where it is proposed to
establish a rapid conveyance, the use of auxiliary
engines will not generally be required. Besides,
we can always, by means of these principles,
perceive distinctly if the use of auxiliary engines
be necessary, or if the descending slopes require
the use of a break \ and consequently if the case
be one of those which have been pointed out as
exceptions in Art. 3, in which we cannot employ
alone the formula (1) for the calculation of the
power consumed by the transit, and in which
this calculation requires the use of the additional
terms given in the formulae (2) and (3).
6. Examination of the velocity of the train
during its passage from one slope to another.
The principles laid down in the two preceding
articles, are founded on a consideration of the
permanent velocity > which can be produced and
maintained by the locomotive engine on each
slope, and the trains have been considered as
passing over the entire lengths of the various
slopes with this velocity. This supposition being
adopted, the conditions expressed in Art 5, to
show whether an ascending or descending slope,
shall occasion or not a loss of power in the
working of a locomotive engine as determined
by Art. 4, may be admitted without fear of error.
But the supposition of which we have just spoken
is not altogether conformable to natural effects,
for the train cannot at once change the velocity
due to one slope to that of another, whether on
account of the inertice, or because that on every
slope the permanent velocity given by formula
(15), (and for the particular case which has been
taken for an example by formula (20),) supposes
the existence of a certain pressure of the steam,
a pressure the value of which is given by formula
(12). Now we cannot, all at once, increase the
pressure under which the steam is generated,
since this increase is inseparable from an eleva-
tion in temperature of the water in the boiler.
53
The train must pass gradually from its actual
permanent velocity to the permanent velocity
due to the new slope upon which it is about to en-
ter, in the same time that the water and the steam
contained in the boiler passes gradually from their
actual temperature, to the temperature under
which the steam should be generated, in order that
its pressure be in equilibrium with the tractive
power which is necessary to this new slope, and
thus maintain uniform velocity.
We must now investigate if these necessary
changes can take place, without occasioning on
the one part a greater consumption of fuel, and
on the other part without diminishing the velo-
city of the train.
In the first place we may remark that there
is not generally any loss of power in the system
which we are considering, when there is no es-
cape of steam from the safety-valves. Whenever
the steam generated does not leave the engine
until after having acted on the pistons, the heat
which it has been necessary to transmit to it
has been employed to overcome the resistances
which are opposed to the motion of the train,
or to give to the mass of this train the vis viva
equivalent to the effect of these resistances.
It is, besides, very evident that we continue
to neglect here, as a secondary object, the con-
sideration of the loss of heat which takes place
54
at the exterior surfaces of the heated parts of the
the apparatus, or rather of the slight differences
which these losses may occasion according- to the
elevation of temperature in the water of the
boiler. After this remark the question proposed
is simply to examine if the passage from one
slope to another can be made without a loss of
steam.
This passage may be considered under two
heads. 1st, when a slope is arrived at where
the resistance to motion will be greater ; Sndly,
where the resistance will be less.
Let us admit, for the sake of example, that
the train travelling over a level, is just about to
ascend a slope of 0.005. In continuing to take
for an example the train, of which the conditions
of its motion have been determined in Article 4,
it is seen by the table at page 49, that this train
will travel upon the level line with the velocity
of 9m-8, [ = 10.72 yards] the steam being gene-
rated under a pressure of about 3.8 atmospheres,
and consequently at the temperature of about
144°.* When its motion is regulated by the
slope of 0.005, it will move upon it with the ve-
locity of 6m.8, [= 7.44 yards] the steam being
generated under a pressure of about 5.6 atmo-
spheres, which answers to a temperature of about
I58°.t
* 291°ofFarei)heit. f 317 of Fareriheit.
55
The temperature should therefore be elevated
14°* in the boiler ; and we may suppose that this
elevation of temperature may be effected in
about 6 minutes, if all the heat transmitted by
the furnace was employed, since we know that
it requires an hour or more to elevate the tem-
perature of the water in the boiler to 150°.t
But as it is necessary to furnish at once the heat
required to form the steam employed to keep up
the motion, and to obtain the elevation of the
temperature required, we ought to suppose that
even in urging the fire it will require more than
6 minutes to produce it.
Be this as it may, there is nothing to prevent
us, as long as we please, before entering upon
the slope, 1st, to increase the fire; 2ndly, to load
the safety-valves, so as to obtain a pressure of
5.6 atmospheres ; 3rdly, to diminish gradually
the size of the orifice of communication by which
the steam passes from the boiler to the cylinders.
The first disposition tends to raise the tempe-
rature ; the second establishes the limit, suitable
to the pressure which the steam should acquire
in consequence of increased temperature ; the
third has for its object to regulate the quantity
of steam sent to the cylinders, so that although
the pressure is raised in the boiler, the action on
the pistons remains nevertheless always the same,
in such a manner that the velocity is not accele-
* 57° Farenheit. f 302 Farenheit.
.56
rated. In operating- thus, the steam expended
only carries with it the same quantity of heat,
and the extra heat produced is entirely employ-
ed to raise the temperature. When the tempe-
rature shall be thus raised 14° * C. in the boiler,
it will not be necessary to urge the fire further.
The apparatus may be thus made ready and kept
indefinitely in this state ; that is to say, the tem-
perature being 158°f in the boiler, and neverthe-
less by a proper diminution of the orifice of com-
munication, the action of the pistons is not
greater than when this temperature was 144°^ C.
The train will thus arrive at the commencement
of the slope ; it will begin to ascend With its Telo-
city of 9m.8, [= 10.7 yards] which will diminish
gradually, towards a state of rest, if the orifice
of communication was not then progressively
opened, in such a way, that at the instant the
velocity is reduced to 6m.8, [= 7.44 yards] the
cylinders would receive the whole of the steam
which the fire could generate, under the pressure
of 5.6 atmospheres.
Now it is evident, that unless the fire has been
urged more than was necessary to promote the
elevation of the temperature to the required
degree, there will not be any loss of steam
through the safety-valves, since the velocity of
* 67° of Farenheit. f 317" Farenheit.
t 292° Farenheit.
the train diminishing progressively from 9m.8 to
6m.8, is constantly more than is necessary to en-
able the pistons to use all the steam which the
machine can generate under the pressure of 5 6
atmospheres, regulated by the weight on the
safety- waives. If such a loss should be feared, it
is necessary that the velocity of the train be
diminished below the speed of 6m.8, which accords
to the slope to be passed over : this can only
take place through the negligence of the con-
ductor, who has not opened the orifice sufficiently
soon after the entry of the train upon the slope.
Suppose, now, that the train has arrived at the
summit of the slope of 0.005, and just about to
enter a level line. As the safety-valves remain
loaded for the pressure of 5.6 atmospheres, the
fire should be lowered for some minutes. As
soon as the train is on the level line, the actual
pressure of the steam exceeding the resistance,
the velocity is immediately increased. From
whence we see in the first place, that the steam
cannot escape by the safety-valves, even when
the fire is not slackened, unless the orifice of
communication is not so regulated as to allow
all the steam which can be generated to go to
the pistons. But in thus lowering the fire we,
in order to form steam, avail ourselves more
certainly of the heat which was given to the
water and the boiler to raise them to 1 4<° C. of
more elevated temperature.
58
The velocity of the train will cease to be ac-
celerated, 1st, when the pressure of the steam
shall not be greater than is necessary to balance
the resistances ; 2nd, when the motion of the
pistons takes all the steam that can be formed by
the furnace restored to its ordinary state. These
two circumstances taking place when the tempe-
rature is 144° in the boiler, and the velocity of
the train 9m.8, this state will establish itself
spontaneously with the single precaution of
giving a sufficient passage for the steam from
the boiler to the cylinders. The safety-valves
can then be unloaded if it be desired, and regu-
lated to the pressure of 3.8 atmospheres.
The remarks which have just been made are
applicable to all analogous cases. We ought to
observe, that by the passage from one slope to
another, it does not at all follow that there is
a necessity for a loss of steam, and that such a
loss takes place only through the fault of the fire-
man or the conductor, which is equally liable to
occur in the ordinary working of the engine.
From this it may be concluded, agreeably to
what has been said above, that this passage does
not occasion any loss in the action of the engine,
and in fact, we see distinctly the compensations
which take place, the heat which had been em-
ployed to elevate the temperature in the boiler
being given out again when the temperature is
lowered, (except a slight difference due to the
59
effect of the loss by the exterior surfaces ;) and
the diminished velocity which takes place after the
train has overcome the superior extremity of the
slope being compensated by the greater velocity
with which it commenced to ascend the slope.
The exact determination of the velocity of
the train in the circumstances here considered,
presents a very complex question, of which a
part of the elements is arbitrary, or cannot be
exactly appreciated. In even seeking to sim-
plify this question by convenient hypothesis, it
still remains very complicated. The approximate
solutions which can be obtained from it, require,
besides, some developements which would extend
this paper to too great a length.
We shall confine ourselves to giving the for-
mulae by which may be determined the varied
velocity of a train, in the particular case of the
pressure of the steam in the boiler being kept
constantly the same : formulae which it may be
useful to have at hand.
The condition which regulates this velocity is,
that the vis viva varies in each element of time
by a quantity equal to double the quantity of
action exerted by the power resulting from the
production of the steam, diminished by the quan-
tity of action destroyed by the resistances.
We may represent, in preserving the denomi-
nations of the preceding articles, by p, the weight
of the wheels belonging to the waggons, and
60
to the locomotive engine which form the train,
and by u, the velocity of the train at the end
of the time t. The vis-viva of the part P — p9
of the weight of the train will be expressed
by u?\ as to the vis-viva of the wheels, it
must be observed that every point in a wheel may
be supposed to turn round that point which rests
on the rail, the centre of the wheel turns round
this point with the velocity u. Then let x re-
present the distance of the element dp, of the
weight of the wheel at the point of its circum-
ference which rests upon the rail, and r, the
radius of the wheel ; the actual velocity of the
?/•!?
element dp is - ; and consequently the actual
vis-viva of the wheel is — , in taking- the
gr<2
integral fdp.x*> for the whole wheel. Now
we know that in calling S the length of the
simple pendulum, the oscillations of which are
of the same duration as that of a wheel suspended
from a point in its circumference, we have
J°dp.x2=pr$, p being the weight of the wheel,
then the preceding expression of the vis-viva of a
wheel may be written — — ui From whence we
r g
conclude, in supposing that the ratio- is the same
value for all the wheels of the train, the total
vis-viva of this train is expressed by
61
p representing, as stated before, the weight of
all the wheels.
On the other side, the power exerted by the
steam, diminished by the resistances is expressed
by
~ (Q.5F — 10330) — (0.005 ± «) P,
i being the slope on which the train is moving,
and the sign + or the sign — being taken ac-
cording as this slope is ascending or descending.
We therefore have for the equation of velocity
- -T=~ (°-5F - 10330)— (0.005 ± HP,
g at wr
...... (21)
of which the integral (F being supposed constant
as well as i), is
a*
p — P + -P
...... (22)
in representing by uo the initial velocity, or that
which exists at the moment we reckon t=o.
According to what takes place in practice, the
weight of the wheels is very nearly the half of
the weight of the waggons when empty, and
consequently very nearly the £ of the total
weight of the train. On the other hand the
ratio - differs very little from - • Thus we have
62
very nearly
S 13
?_„ + _„= rJ+j? L~|P=--P.
rf L_6 2'6J ]2
We can therefore, without much error, apply the
approximate formula.
1? £ (u— MO) = R2(0.5F — 10330) — (0.005±«)P~|/,
12 g L.7rr
or if we wish
»/-. Mr - Igpfl a5F — 1033Q _ (n no* -4- i)"V'- • (23)
13l-7rr P
This formula may serve to determine the ve-
locity of the train, at all times that the apparatus
is supposed to be so regulated that the pressure
F of the steam in the boiler is kept constant,
which requires the condition that the heat trans-
mitted in every unit of time increases with
the velocity, in order that the boiler may always
furnish a supply equal to the variable consump-
tion of the steam.
We could, for example, employ the formula in
question in order to ascertain if a train, by means
of a certain acquired velocity, could surmount a
steep slope of a given length. In fact, the equation
(23) shows that a train commencing to move from
a state of rest should have acquired the velocity
u, after having run over a space expressed by
W2
12,-cn 0.5F- 10330 1 ; ' ' (24^
13 L** P
and reciprocally that the velocity u will be re-
duced to zero, when the denominator of the
formula (24) becomes negative, after the train
shall have run over a space expressed by this
formula.
If we still take for example the train as de-
Cfi
termined in Art. 4, we have -=0.021148,
7TT
and P = 36850. The expression (24) becomes
-^[0.005928 (0.5/z-l)- (0.005±
n being the number of atmospheres which ex-
presses the constant pressure of the steam which
is generated. Suppose, for example, that this
pressure being extended to 5 atmospheres, and
that the combustion is increased sufficiently to
allow the consumption of steam which should
take place at a velocity of 15 metres [= 16.4
yards] per second, which supposes a quantity of
heat transmitted in a unit of time, more than
double of the mean quantity which has been
taken in the preceding calculations. Admit also
that the train commences with this velocity of
15 metres, [= 16.4 yards] to ascend a slope
of a centimetre per metre, [= 1 in 100]. Put-
ting in the formula (25) n = 5, u = 15
metres, i = 0.01, we shall have 2034 metres
[= 2224 yards] for the distance which the train
64
can run up the slope before it stops and descends
again in a contrary direction.
If we suppose the slope 2 centimetres per
metre, [= 1 in 50], or if we make i=0.02, the
same formula will give 771 metres [= 843 yards]
for the distance in question.
If besides we wish to know what distance the
train, starting from a state of rest, ought to run
upon a horizontal line to acquire a velocity of 15
metres [= 16.4 yards] per second, we must
make i=o, in the formula (25), and we shall have
the distance 3192 metres [= 349 1 yards].
7th. Summary. Comparative Estimate of the
Cost of Transit on different lines of Rail-
way.
We have shown in Article 2, that the degree
of advantage which a line of Rail-way can
present, depends in a great measure on the re-
duction that can be effected in the actual price of
the cost of transport.
We have further remarked that the price of
transport upon a line of Rail-way, results from
two principal elements ; that is to say, 1st, the
amount of the cost of construction, to which
is to be added a part of the cost of manage-
ment and repairs. 2nd, The cost of transport,
65
properly so called, to which is also to be added
a part of these same secondary expenses.
The annual sum produced by the interest
and the first installment of the expense of con-
struction, increased by the expense of repairs and
management which relate to it, being divided by
the number of tons of merchandise supposed to
pass annually by the road, will give the propor-
tionate expense of each ton.
As to the cost of transport, properly so called,
we distinguish two parts; 1st, the expense
of the locomotive engine, comprehending the
purchase and the repairs of this engine and its
tender, the fuel, and the water consumed, the
workmen which attend it ; 2nd, the expense
of the wagons comprehending their purchase
and repairs, and the workmen or attendants
employed in the management and care of the
train.
To these expenses should be added that of
warehouse and offices, as well as that of the
workmen and other agents employed for ware-
housing, loading, unloading, and the regulat-
ing the transport of the goods and passengers.
The 3rd article, and those which follow it,
have for their object the determination of the most
important part of the expense of which we have
just spoken, that of the locomotive engine.
66
We have given in the 3rd article a general
rule for finding the amount of the quantity
of action necessary to effect the transport of
a given weight, upon a given line of Rail-way ;
in article 4, the manner of deducing from the
result obtained the total weight of the train,
which a locomotive engine can draw upon this
line with a given volocity, and consequently the
weight of payable merchandise which should be
carried by this engine. We have afterwards, in
articles 5 and 6, exhibited the use of this rule by
a particular examination of the motion of the
train upon unequal slopes which might appertain
to the line of Rail-way, and shown in what limit
of slopes it may be applied without error, or
what the slopes should be which would require
the employment of auxiliary engines.
The result in question may be summed up in the
following manner. In preserving the denomi-
nations employed in the preceding articles, (see
page 31,) we sLall designate by
A, the length of the line of Rail-way, valued
in metres.
U, the mean velocity with which this line
should be traversed, expressed in metres per
second.
P, the total weight of the train, determined
conformably to what has been said in article 4,
67
which can be drawn by the locomotive
engine, with the mean velocity U, expressed in
tons.
Q, the weight of this locomotive engine and
its tender, also expressed in tons.
A, the expense of working this engine in each
unit of time, (which we suppose to be one
second.)
Observing also that if the weight of the
payable merchandise, is about J (P— Q), we
shall evidently have
for a close approximate expression of the expense
of the locomotive engine for every ton, trans-
ported from one extremity of the line to the
other.
It does not appear possible to give a more
simple rule, if, as it is convenient to do, we con-
ceive that the locomotive engines shall be con-
stantly managed so as to obtain from them all
the mean action of which they are capable.
We should, in fact, perceive, that if we do not
restrict ourselves to this condition, losses would
follow in some cases, upon the purchase and
repairs of the engine, and upon the labour of
the workmen if not upon the value of the fuel
consumed, by the effect of which the compari-
68
sons we have in view would cease to present the
requisite accuracy.
This accuracy cannot be obtained, but by
a calculation of the kind which is presented
in this work, bringing into the calculation in
every particular case, the proportion of the
weight of the locomotive engine, to the total
weight of the train which it draws, and always
keeping in view this proportion, as an essential ele-
ment of the result
As to the second part, which forms the cost of
transport, properly so called, that is to say, the
expense of the wagons, it appears that we
might regard it, as being for every tori of
merchandise transported proportional to the
length of the line.
This expense will be so much per ton per mile,
and estimated accordingly.
Finally, as to the secondary expenses, of
warehousing and despatching, it may be said that
they differ little for two lines of which the lengths
are not very unequal ; but we cannot doubt,
that in general they increase with the length of
the lines, and it appears convenient, when Rail-
ways of a great extent are considered, to appre-
ciate them, as well as the preceding expense, at so
much per ton per mile.
In recapitulating what has been just said, we
69
see that the total price of transport of a ton, from
one extremity to another of a Rail-way, will
consist of,
1st. The annual interest of the expenses of
construction, and the annual expenses of manage-
ment and repairs, divided by the number of tons
transported annually.
2nd. The expense of the locomotive engine ex-
pressed by the formula (#6).
3rd. The expense of the wagons, which is pro-
portional to the length of the Rail-way.
4th. The expense of warehousing and despatch-
ing which we shall also consider as being propor-
tional to the length of the Rail-way.
We then see that the valuation of this total
price is thus reduced, in each particular case,
to the determination of a very small number of
elements, that is to say, the expense of construc-
tion and repairs, for which data is given by the
formation of the project, the estimate of the
annual tonnage, the determination of the weight
of the train which should be drawn by a locomo-
tive engine of a given power ; and lastly, the
length of the line of Rail- way.
We see further, and sometimes this remark
will be very important, that if, on making a com-
parison between two lines, all the elements
which we have just specified are found in
favour of one of them, the preference which it
70
merits, in respect to the economy which it would
produce into the cost of transport is evident,
without having occasion to value in money the
relative influence of each of these elements, a
valuation which always presents some uncer-
tainty, seeing the little extent of information
which is possessed on this subject, and the diffi-
culty of knowing exactly the annual tonnage.
Thus, 1st, if a line is less expensive in con-
A
struction ; 2nd, if the ratio — is less ;
3rd, and lastly, if the length of the line is less ;
we are certain that the expense of transport will
be less on this line. The result of the compari-
son depends therefore entirely upon the deter-
mination of geometrical or mechanical quanti-
ties, in estimating which there is nothing arbi-
trary or uncertain. But if the three elements upon
which the comparison rests, that is to say, the
annual expense of the cost of construction, the
repair, and the management, the quantity of action
necessary to transport a given weight from one
extremity of the line to the other, lastly, the
length of this line do not all give a favourable
result to one of the lines, which are compared,
it becomes necessary, in ;order to decide the
question, to value in money each of the parts of
expense of transport, and consequently to specify
71
the quantity of merchandize, and the number of
passengers, which may be presumed would be
carried annually along the lines.
Since the foregoing observations went to
press, the following additional remarks by M.
Navier have appeared in the Annales des Ponts
el Chaussees.
According as we examine with more care the
circumstances depending on the use of locomotive
engines on Rail-ways, our ideas become more
extended and correct. I shall therefore return
to the subject, on account of a remark which M.
de Prony has been pleased to communicate to
me upon some parts of the observations published
in the preceding work.
There has been given, page 17, a rule for the
calculation of the quantity of power required for
the transit of a train of the weight P, from one
extremity to another of a Rail-way, acording to
which this power is very simply expressed by the for-
mula P/_^_±H\
V200 '
P being the weight of the train, A the length of
the line, H the height of the extremity N, above
the extremity M and the signs + or — are used
72
according as the transit is made in the direction
MN, or in the direction NM. This expression
is independent of the figure of the section between
the two extremities M and N of the line.
It has been further observed, at p. 19, that
where it is wished to comprehend in the cal-
culation the power employed to surmount
friction, the expression of the whole quan-
tity of power employed in the transport,
must then depend on the figure of the sec-
tion, and that the influence of the descending
slopes should be considered. It is with this
view that the term fjJi has been introduced in
formula (1) of page 21 : the addition of this
term was necessary, because in the locomo-
tive engine a part of the power expended is
constant, and will exist even when the effort to
be overcome is reduced to zero, so that the
power expended is actual loss, and the useful
effect does not always remain in the same ratio.
Nevertheless, in examining this subject more
attentively, it will be perceived that we have
attributed too great an influence to the effects of
the descending slopes, and even that within the
limits which comprehend the effects of this kind
submitted to calculation, the term p.h may be
entirely suppressed in the formula which we
have just mentioned. This is seen immediately,
from the constitution of the approximate ex-
73
pression which ,is employed for estimating the
work performed by the locomotive engines.
Let us take again the equation (15) page 36 ;
according to which the permanent velocity U,
which can be given to a locomotive engine
drawing a train of the weight P upon different
slopes, is represented by
10330II
U=
PJ+ 11260 ™-
irr
In this formula PJ expresses the effort which
it is necessary to produce to draw the train ;
~r is a constant, of which the value is given for
every locomotive engine. The quantity n, which
is the weight of the steam produced in a second
is also supposed constant.
We may then write more simply
m and n being constants.
This granted, let there be a line of Rail-way
MN, and let us represent by A its length, and
by H the height of the extremity N, above the
.extremity M ; let us admit at first, that the slope
is uniform from one extremity to the other.
The velocity which the locomotive engine will
acquire on it, will be expressed by
U =
TT
P(0.005+—
Let now T represent the duration of the
transit, or the duration of the work of the loco-
motive engine. We shall have
T=— , orT__P(Q-OQSA + H)+rcA
U m
Let us further admit that the length A
is composed of several parts of which the re-
spective lengths are a, a', a", &c ; and that in
passing the length a, a height h is attained ; that
in passing the length a', a height h ; in the
length a" a height h" ; and so on. The duration
of the transit for each of these intervals will be
respectively,
t= P
m
t' =
= P(0.005a"
m
&c. &c. &c.
and the sum of these durations, or the total time
of the transit, will b
P [0.005(a + a' + a"_+&) +& + # + #'&] +n
m
75
Now we have a + a+a" + 8t=A9 and
+ & =H, then
It is superfluous to remark, that if a descent is
made in running over some of the parts, a, a',
a", &c. it will be necessary to give the sign — to
the differences of the level, A, A,' A'', &c. of the
extreme points of these parts. The preceding
result would still subsist.
From this result, which is important for the
establishment of Rail-ways, the duration of the
work of the locomotive engine is assumed, and
consequently the portion of the expense of the
transport which arises from it, will always be the
same, whatever be the figure of the section be-
tween the two extremities of the same line, pro-
vided that the length of this line is not altered.
This conclusion supposes, moreover, that the
same locomotive engine draws throughout the
same train, and that there are no descending
slopes sufficiently steep to require the use of the
break.
It is seen, by what precedes, that in tracing a
line of Rail-way, there is no inconvenience in
rising higher, to re-descend afterwards, as long as
that does not make it necessary to extend
the limit of the slopes. Thus, for example,
several lines uniting two given extreme points,
upon which it is admitted that the same locomo-
76
tive engine draws throughout the same train,
will be perceptibly equal in respect to the ex-
pense of the transport, whatever be the heights
to which they rise or from which they descend,
if their lengths be equal ; and if upon any of these
lines the steepest slopes do not surpass Om.005.
(= 1 in 200.) But a line, where the limit of the
slopes should be less, would present an advantage
conformable to what has been said page 9>
in permitting, without losing any thing on the
action of the engine, heavier convoys to be
drawn with less velocities. It appears then,
in tracing lines of Rail-ways, that especial care
should be taken to diminish the length of the
transit, and to lower the limit of the slopes.
The suppression of the term juA, in formula
(1) of page 21, will moreover, simplify the ex-
amination and calculation necessary to compare
different lines in making use of the ideas which
have been suggested.
NOTES.
NOTE A.
The vis-viva of any system of points, is the product of the
mass of each point into the square of its velocity.
The moving force, or quantity of motion, of a body, is ge-
nerally understood to mean the product of the mass into the
velocity, and is the same as the momentum : and the conserva-
tion of the quantity of force thus measured is proved by proving
the conservation of the motion of the centre of gravity. But if
the force of a body in motion be measured by the whole effect
which it will produce before the velocity is destroyed, or by the
whole effort which has been exercised in generating it, without
regard to the time, it must be measured by the mass multiplied
into the square of the velocity. Thus balls of the same size
projected into a resisting substance, as a bed of clay, will go to
the same depth, so long as their weights, multiplied into the
squares of the velocities, are the same. Force thus measured
is called vis-viva, in opposition to force measured by momentum,
which is proportional to the pressure, or dead pull, producing
it. And it will appear, that forces will always produce a certain
quantity of vis-viva, by acting through a given space, whatever
be the manner in which the bodies are constrained to move. —
On the Motion of Points constrained and resisted, and on the
Motion of a Rigid Body. Second Part of a Treatise on
Dynamics, by W, Whewell, M.A. See Appendix 2.
78
NOTE B.
Reference to
the points of
Section.
Distances in
yards between
tbese points.
Height in feet
of these points
above the point
M.
M
__
__
a
4,374.5
26.24
b
5,468.0
164.01
c*
6,561.6
186.97
d*
10,936.0
167.29
e
12,029.6
190.25
f*
8,748.8
203.37
r
34,995.2
27,340.0
75.44 ,
104.97
NOTE C.
IT will be seen further on, that we adopt the following rule for
estimating the power of a locomotive engine : n being the num-
ber of atmospheres corresponding to the pressure under which the
steam is generated, and consequently the power resulting from
the production of this steam being proportional to n — 1, we
suppose the effort which is transmitted to "produce the draught
is proportional to 0.5w — 1.
Hence the fraction 'Jn expresses the part of the power
produced which is used, and the part of this power which is lost
by the effect of resistances is expressed by 1 —
O.on — 1
n— I '
or
0.5«
^-r. The value of this expression is different, according to
the magnitude of the number n. If we suppose w=4, which
agrees with the pressure that most commonly takes place in the
working of locomotive engines, this value is §.
In taking this number to estimate in an approximate manner
79
the effect of the descending slopes, we shall estimate rather above
than helow its true value, the quantity of power which they
destroy.
NOTE D.
The degree of exactness of this formula may he judged of by
the following table.
Temperature in
degrees of the
Pressure in
Weight of a cube metre of
steam, calculated
centegrade
thermometer.
atmospheres.
by the known
formulae.
by the proposed
formulae.
kilogr.
kilogr.
100
1
0.590
0.59
121
2
1.U7
1.09
135
3 «
1.615
1.59
145
4
2.101
2.09
153
5
2.574
2.59
160
6
3047
3.09
This table, reduced to English measures, is as follows : —
Temperature in
degrees of
Pressure in
Weight of 61,023 cube inches of
steam calculated
Fahrenheit's ther-
mometer.
atmospheres.
by the known
formulae.
by the proposed
formulae.
Ibs.
Ibs.
212
1
1.301
1.301
249
2
2.463
2.404
276
3
3.563
3.507
293
4
4.634
4.609
307
5
5.677
5.712
319
6
6.720
6.815
80
NOTE E.
p
W = 122783' *n wmch F represents the pressure in Ibs. on a
square metre, or 1,550 square inches.
We may therefore consider the weight of a cube metre or
61,027 cube inches of steam generated in the boiler under the
pressure F, as represented by the following formula :
F
0.1985 -f 1.10275
22783
or more simply, 0.1985 + 0.0000484 F ..... (6)
and y = _ 5 _ (7)
' 0.1985 + 0.0000484F.
in which II equals the weight of the steam in Ibs. And y —
cube metres, or y x 60,027 = Otl985 +X0.oooo484F e(lua]s thc
volume in cube inches of steam generated in a second.
NOTE F.
The expression which we here adopt, (from Tredgold) for the
value of the power transmitted, differs from the rule very gene-
rally admitted, of considering the power used as being a deter-
minate fraction of the power represented by the production of
the steam, and which will lead us to express the power trans-
mitted by k (n — 1), k being a fractional coefficient. But the
formula proposed by Tredgold seems better adapted to the
nature of the question, since it is necessary to pay attention to
this circumstance, that even whilst the tractive power is nothing,
81
as it would in fact be on a descending slope of 5 millimetres
per metre, [ = 1 in 200,] the pressure of the steam in the
boiler should nevertheless produce the power necessary to make
an equilibrium with the exterior atmosphere and to propel the
apparatus without a load.
Now the expression k (n — 1) being equalized to zero, gives
7i=l ; a result which does not satisfy the condition of which
we have just spoken ; whilst the expression Q.5n — 1, equalized
to zero, gives n—2, which on the contrary is a satisfactory
result.
The formula 0.6n — 1, besides, does not differ, for the values of
the number n, which correspond to circumstances the most
common in the working of locomotive engines, from results
known by experience, and from the mean values admitted by
engineers. Mr. Stevenson, (Observations on the Comparative
Merits of Locomotive and Fixed Engines, <5fc. page 29,) takes
the pressure which produces the draught equal to the half of
the pressure which is in the boiler independent of the exterior
atmosphere, which makes k=6.5 in the expression A: (n — 1).
But this valuation seems too much, especially when the pres-
sure of the steam is not very great. Mr. Wood admits that the
o
useful effect produced by locomotive engines is the — of the
power produced by the formation of the steam, (Practical
Treatise on Rail-ways, see pages 227 and 231 of the French
translation,) which makes ArzzO.3, a value which on the con-
trary appears a little low, when we examine with attention all
the experiments stated by the author, and especially when we
mean to express by n the true pressure under which the steam
is produced, a pressure which is generally below that which
corresponds with the load on the safety valves. In making k= J
the formulas k( n — 1) and 0.5 n — 1, will agree in the case where
w=4, that is to say, for the pressure which most commonly takes
place in the working of the engines.
NOTE G
The quantity of steam produced in a unit of time, is the
principal element which determines the power of an engine.
Mr. Wood's work, in general so useful, still leaves, perhaps,
more precise determinations on this point to he desired. From
what is said at page 231 of the French translation, the author
appears to consider the engine, to which tahle 1 1 in the pre-
ceding page refers, to be like the Planet, which, according to
table 9, page 226, vaporates 1249 kilogrammes = 2755 Ibs. of
water per hour, or Ok.347 = .767 Ibs. per second.
The dimensions of the Planet are besides conformable to
those of the machine which we take for an example, and ac-
cording to Mr. Wood, page 228. The Planet weighs from 5
to 6 tons. We have taken the weight of the steam carried off
in a second by the motion of the pistons, as much as Ok.4 =
.88 Ibs., and this estimate does not appear to us to be too high,
it being remembered that we have given the engine rather a
greater weight. The estimation of which we speak is interme-
diate between the result which Mr. Wood seems to admit, and
that which is taken by Mr. Pouillet for the base of the calculations
which he has given in the 3rd No. of the Portefeuille Indus-
triel du Conservatoire des arts et metiers. M. Pouillet takes
the weight of steam at 28.75 per min. or 0.479k as the pro-
duct of an engine ; the weight of which is 6.5 tons ; and with the
water in the boiler and the coal on the bars is 8 tons.
It has appeared to us preferable to keep nearer to the result
admitted by Mr. Wood, in order to avoid exaggeration on the
power of the engines.
It seems to us besides, that we have not estimated too low,
the action of the engine taken as an example, when we remark
that the results to which the proposed formulae lead, correspond
to some useful effects which surpass considerably the results
given by Mr. Wood in his table 1 1 , page 230 of the trans-
lation.
83
NOTE H.
It may be remembered, that in making P—o, in formulae
(20), U is found -17m.36,- 18.98 yards for the last limit of
the permanent velocity which the locomotive engine can take,
unless on a descending slope so steep that the action of gravity
overcoming the resistance of the friction, the term JP may be-
come negative.
As it is known that the locomotive engines have sometimes,
run over extensive portions of the Rail-way from Liverpool to
Manchester, or even the entire length, with velocities which were
not much below 17m.36 — 18.98 yards per second, this circum-
stance might give rise to some doubt, as to the exactness of the
expressions which we here employ. But it must be remarked*
that the extraordinary velocities in question have been certainly
obtained by increasing the combustion, much more than was
done in the regular work of the engines, and consequently
producing in a unit of time, a quantity of steam greater than
we have supposed. Formulae (15) shows, that the velocity is,
all other things being equal, proportional to this quantity of
steam.
The greatest velocities cited are those of a journey made by the
Planet with some electors from Manchester to Liverpool, in 58
minutes, which corresponds to a velocity of about 13m.8n 15.09
yards'per second ; and that which took place when Mr. Huskisson
was carried wounded to Manchester, after the accident which
caused his death. They then ran, it is said, a space of 15 miles
in 25 minutes, which is equal to about 16 metres =17. 5 yard per
second.
This last observation is, perhaps, not so much to be depended
on as the other.
NOTE I.
According to the result given in the Practical Treatise on
Rail-ways, by Mr. Wood, (page 169, and following, of the
translation,) the limit of the effort which an engine can exert
without slipping on the wheels, is fixed at — of its weight in
the case where the four wheels are acted on by the pistons,
and at — in the case where the pistons only act on the two
wheels. It may also be perceived that the limits relate to a
particular state of the rails, which seldom occurs, and but
for a very short period in France. Further in the calculations
which lead to the results of which we speak, Mr. Wood esti-
mates the tractive power at 10 English pounds for each ton of
the weight drawn, that is to say, at — - of this weight, whilst we
estimate it here at — . Finally, Mr. Wood supposes the wheels
to be 4 feet in diameter, whilst the wheels that we take are 5
feet ; and it is known, that the resistance to slipping increases
with the dianreter of the wheels. It appears, therefore, that the
estimate which we have adopted may be admitted without fear.
We find also among the experiments mentioned by Mr. Wood,
many cases where the draught has surpassed perceptibly this
estimate. For example, we see (page 219 of the translation)
that the Fury ha? drawn 46.25 tons over a space more than a
league. As her weight was 4.25 tons, and the tender 3.2 tons, the
total weight of the train being 53.7 tons, gives a tractive power
equal to 0.2685 tons, in reckoning it at g— of the weight drawn,
notwithstanding which the 20th part of the weight of the engine
is only 0.2125 ton. We see also, (page 250,) that the Planet
has taken from Liverpool to Manchester a train weighing in all
86 tons, of which we consequently reckon the tractive force
on the level parts at 0.43 tons; nevertheless the engine
weighing 6 tons, the 20th of its weight was only 0.3 ton.
NOTE K.
The table given at page 49, converted into English measure,
is as follows.
Rate of as-
cending slopes.
Ratio of the
tractive power
to the weight
drawn
= J.
Total weight of
the train which
;he engine weigh-
ng 8 tons can draw
without slipping
= P.
Net weight
carried = §
(P -12.000).
Ibs.
Ibs.
1 in 200
0.01
88,219
41,170
1 in 133
0.0125
70,575
29,406
1 in 100
0.015
58,814
21,565
1 in 80
0.0175
49,010
15,030
1 in 66
0.02
44,110
11,362
1 in 57
0.0215
39,209
8,496
1 in 50
0.025
35,288
6,882
See table at the end calculated by the late Mr. G. Dodds,
Engineer of the Monkland and Kirkintilloch Rail-ways.
APPENDIX I.
The tables alluded to in the preface are those of Mr. Wood,
pages 418, 419, of his work on Rail-ways. These tables
are intended to show the gross load which locomotive engines,
capable of taking a certain number of tons at a certain number
of miles an hour, will drag at different velocities in miles
an hour, on different ascending slopes.
The formula by which these tables are calculated is not given,
but it appears to be derived in the manner hereafter stated ; and
as a good deal of discussion has arisen on the subject, I take
the liberty to give it at length, with the following extract from
Mr. Wood's work, which appears to bear on the subject, and a
table, which I have calculated from the aforesaid formula,
showing the time required to traverse two given lines of Rail-
way, with an engine supposed capable of drawing 40 tons at
15 miles an hour on the horizontal.
Mr, Wood says, " in the construction of these tables ( 10 and
11), we have supposed the power of the engine to be constantly
the same, or the production of steam in the boiler to be con-
stant and regular, or equal quantities in equal times; but a
little consideration of the mode by which the steam is generated,
will show this is not the case.
" The supply of air for the support of combustion, is almost
entirely produced by the exit of the steam into the chimney ;
8?
if this is capable of producing a certain effect at twelve miles
per hour,, when the engine is moving at twenty-four miles per
hour, double the same nu;nber of cylinders, full of steam, pass
into the chimney in the same time, and therefore the draught
will be greater, and the generation of steam more rapid; and
hence we should have, at greater velocities, an increase of effect
with the same engine. But as in this case, the piston moves
at a correspondingly increased velocity, thereby producing a
diminution of effect; and there being also an increase of re-
sistance from the air at greater velocities, perhaps in the absence
of experiment, to prove the amount of all these different forces,
we ought in practice to suppose the power of the engine con-
stant, and capable of producing, at the higher velocities, an
effect equal to that shown in Table XI.
" These tables are formed on the supposition that the load
is equal in both directions, as they show the load which the
engines are capable of drawing up planes of the inclination
given ; the load down the respective planes will be as much
greater as the assistance which gravitation affords."
The formula by which these tables are supposed to be calcu-
lated may be thus derived. Let x — the gross weight in Ibs.,
and r, the denominator of the fraction which represents the ratio
of the length of the plane to its height. Then
— = the gravity down the plane.
= gross weight in tons.
~~Z xW = ^= tlie resistance in Ibs. Then
2>&QpJ 6&*x
T X
—— + r — power required up the inclined plane,
— — ~ = power required down the same plane.
This formula becomes rx ± 224 x = 224 pr ; by putting p—
224 pr 224 V 450 r . ,,
power and x — — 7^- = in Ibs.
r-f-224 r-4-224
88
or by dividing by 2240
x 45r
tOI1S>
Let C= the constant resistance arising from the friction of
the working parts of the engine, and which is not influenced
by the velocity of the engine. If the absolute power of the
/pi PN * 0
engine = P + C. at twelve miles an hour. Then — =
absolute power of the engine at the velocity v, and
/p I Q\ l<£
- - -- C= the effective power or disposable force of the
, (P+C)12-Cv
engine at the velocity v, and - -r — — = effective power in
tons, on the horizontal at the velocity v ; this value substituted
- 12P + 12C — vC r
for 45, in equation (1) gives -- ^ - Xr+224~ tons
that can be drawn up the inclined plane r, with the velocity v,
t) _V— v r
and generally - — — - x tons, ... (2)
that can be drawn up the inclined plane represented bv r, with
the velocity v, by an engine that is capable of exerting an effec-
tive power of P, with the velocity v1.
Supposing, as in Table 10, that the engine is capable of draw-
ing forty-five tons on the horizontal at a velocity of twelve miles
an hour, the power it then exerts is 4501bs. taking the resistance
to be ~ part of the weight. Then by making P=450 ; vl= 12 ;
C= 300 - 10 ; this formula is reduced to
12 X 450 +fc!2 x 300— v 300 r
100 "x;+i£i-
540 + 360-080 r
— v — " xr-i-224? vvmch will give the numbers
in Table 10. Thus suppose r, the rate of inclination, to be 200,
and the velocity v = 12t, the above formula becomes
540+360—360 200 200
lg • - X -^ - 45 x 4— = 21.22 as given m the tables in
the column of velocity 12 and opposite the rate of inclination
200. From the formula- - r — — * ~~ tons = T, we
89
have the velocity „ = ^— . (3)
and by putting vl = 15, P = 40, c = 400, this formula becomes
and the time required to traverse each plane will be expressed
(Hg+r)
by .750 35r = »*; ....... ... (5)
in which D = the length of the plane in chains.
From these formulae the time required to traverse each of two
proposed lines of Rail-way between the points A and B, but by
different routes, have been calculated. See Tables 1 and 2.
From those tables it appears that an engine with a given
load will traverse the line (AD) in both directions in 14. lh. 53m.
53*. and the line AB in 13h. 57'. 37s; the difference in time
will therefore be 14. 1. 53 — 13. 57. 37 = 4' 16" in favour of
the line AB.
As far as time alone is concerned, this will probably be a
very near approximation ; but no further : it does not enable us
to say which line is actually the best, or over which a given
weight can be carried at the least expense.
TABLE No. 1.
Showing the Time that an Engine capable of drawing 40 Tons on the
horizontal at 15 miles an hour will traverse the Rail-way A D,
with the same load.
Distances
in
Chains.
Differences in Level,
A to D.
Rate of Incli-
nation.
Velocity in miles
per hour.
Time occupied on each
plane in minutes.
Rise in
Feet.
Fall in
Feet.
A toD.
D to A.
From A to D. From D to A.
169
397
263
305
534
488
257
326
588
528
202
501
61
290
110
393
170
152
40
392
600
113
268
136
75
230
54
436
40
87
96
186
Level
1 iu 1746
1 - 560
Level
1 in 202
1 - 500
1 - 264
Level
I in 250
Level
1 in 330
Level
1 in 400
Level
1 in 330
Level
1 in 330
Level
1 in 660
Level
1 in 330
1 - 440
Level
I in 449
1 - 610
Level
1 in 715
Level
1 in 330
Level
I in 330
1 - 500
A toD .
D to A .
15-00
14-09
12-5
15-00
9-64
12-25
10-53
15-00
27-17
15-00
22-70
15-00
20-83
15-00
22-70
15-00
22-70
15-00
18-06
15-00
22-70
20-12
15-00
19-98
18-37
15-00
12-96
1500
22-70
15-00
22-70
19-33
li. in. se
6 53
7 8 J
15-00
16-02
18-75
15-00
33-66
19-33
26-05
15-00
10-35
15-00
11-19
15-00
11-71
15-00
11-19
15-00
11-19
15-00
12-82
15-00
11-19
11-95
15-00
12-00
12-67
15-00
17-78
1500
11-19
15-00
11-19
12-25
Sum.
c.
14
J9
8-45
21-13
15-78
15-22
41-54
29-86
18-30
16-3
16-23
26-4
6-67
25-05
2-19
14-5
3-63
19-65
5-63
7-6
1-66
19-6
19-82
4-04
13-4
5-10
3-02
11-5
3-12
21-8
1-32
4-35
3-17
7-21
8-45
12-34
10-52
15-22
11-89
18-62
7-39
16-3
42-60
26-4
13-53
25-05
390
14-5
7-37
19-65
11-39
7-6
2-84
19-6
40-21
7-09
13-4
8-5
4-38
11-5
2-27 .
21-8
2-68
435
6-43
11-38
15
31
. . .
174
64
46
. . .
155
. . .
40
. . .
10
22
. . .
34
4
. . .
120
17
. . .
20
8
5
8
19
24
From
413-21
h. in. sec.
6 53 14
428-65
h. in. sec.
7 8 39
14 1 53
TABLE No. II.
Showing the Time that an Engine capable of drawing 40 Tons on
the horizontal at 15 miles an hour, will traverse the Rail-tvay
A B, with the same had.
Differences iu Level
Distance
A to B.
Rate of Incli-
Velocity in miles
Time occupied on each
in
Chains.
Rise in
Feet.
Fail in
Feet.
nntion*
per hour.
plane in niiiintt s.
A to B.
B to A.
From A to B
From B to A.
21
4-6
in 310
23-48
11-01
0-66
1-43
30
0-6
.
3960
15 17
12-13
1-48
185
382
43-0
590
18-51
12-60
15-47
22-73
196
121-6
107
33-66
7-32
4-36
20-00
165
c
22-3
490
19-44
12-20
6-36
10-14
260
.
36-3
473
19-65
12-12
9-92
16-06
195
0-3
Level
15-00
15-00
9-75
9-75
48
*l-3
in 2530
14-36
15-69
2-50
2-25
1096
147-6
B
490
12-20
19-44
67-37
42-28
707
.
*49-0
950
17-00
13-42
31-19
39-51
727
68-0
700
17-85
12-93
30-54
42-10
42
1-0
2600
15-67
14-38
2-01
2-19
187
133
930
16-68
13-09
8-40
10-71
324
,
31-9
670
18-01
12-85
13-49
18-91
872
21-0
2740
15-63
14-41
41-20
45-38
82
4-9
1140
16-63
13-65
3-69
4-50
433
.
109
2662
15-65
14-39
20-75
22-56
149
*6:9
.
1460
13-93
16-25
8-02
6-87
289
4-3
4700
15-36
14-65
14-11
14-79
eii
42-0
960
16-98
13-43
26-98
34-12
677
.
3-6
Level
15-00
15-00
33-85
33-85
203
15-0
1 in 890
13-32
17-15
11-43
8-87
41
Level
15-00
15-00
2-05
2*05
159
11-0
1 in 960
16-98
13-43
7-02
8-88
75
.
0-6
Level
15-00
15-00
3-75
3-75
352
iV-o
. . .
I 1360
13-85
16-34
19-06
16-15
Sum ,
395-41
441-80
h. m. sec.
h. in. sec.
6 35 25
7 21 48
h. m. sec.
From A to B 6 35 25
B to A 7 21 48
13 57 37
APPENDIX II.
On Hie different Methods of Estimating Forces, and on what is
understood by la force vive, (vis viva.)
It has already been shown that two forces, F and F, ap-
plied to the same moveable body, were to each other as the
velocities they communicated to that body. Let us now consider
these forces when they are applied to different masses, and let
us suppose at first that two equal forces directly opposed to each
other, act on two masses, M and M' equal and spherical ; they
will communicate to these masses two velocities v and v ', which
will be equal.
M and M' in meeting will press mutually, and will be in
equilibrium, because every thing is equal on each side. But if
we had M = riM.', and that V was greater than V, we should
consider M as composed of the masses m', m", m", . . w(n),
each equal to M1. It is certain that in virtue of the mu-
tual connexion of the parts which compose solids, one of these
masses cannot move without drawing the others along with it,
so that if the body M should move, for example, through three
metres, per second, each of the masses m', m", m", &c., should
also move through three metres per second, which is the same
thing as to say that if V is the velocity of M, the masses m,
m", m", &c., shall each have the velocity V. Now if m', moving
with the velocity V, impinges against the mass M', which, by
hypothesis, is equal to it, it will destroy in V a velocity equal
to V ; and if in the same time m'', acting by the interposition
of the other masses strike also against M', they will again de-
stroy a part V, of V, and so of the rest. So that all these masses
united will spontaneously destroy in V, a velocity equal to nV ;
and in supposing that the velocity V is then exhausted, there
will be an equilibrium : it is necessary then that, in this case,
V = nV. Eliminating n, between this equation and the equa-
tion M — wM , we shall have the proportion, M : M' : : V' : V,
93
which shows that when a mass M contains n times the
matter of another, these masses ought to be in the inverse
ratio of their velocities, in order that they may he in equi-
librium.
This proposition still holds good when M does not contain
M' an exact number of times, which would be easily demon-
strated by infinitessimals, or the method of limits.
Consequently from what precedes that, since the velocities of
two bodies are in the inverse ratio of the numbers of the material
particles they contain, it follows that when these bodies are of
the same volumes and of different densities, that their velocities
are in the inverse ratio of their densities.
Suppose now that the force F, which communicates to the
mass M the velocity V, acts upon a mass which is M times less,
and which may consequently be represented by — = 1, this force
will communicate to the mass 1 a velocity which is equal to M
times that which F will communicate to M ; this velocity will
then be expressed by MV.
For the same reason, the force F', which communicates to M'
M'
the velocity V, will communicate to the mass -^ = 1, a velo-
city M' V.
The velocities MV and M' V, being those which are commu-
nicated by the forces F and F', to the same mass 1, it follows
from the principles of the velocities being proportional to the
forces, that we have F : F : : MV : M' V.
The expressions MV and M'V, are what we call the quantity
of motion communicated by the forces F and F' to the moving
bodies M and M'.
The unit of force being arbitrary, we can represent it by the
quantity of motion which it communicates to the moving body.
Thus, in supposing that F' is this unit of force, we shall replace
F' by M'V', in the preceding proportion, and we have by it
F = MV.
94
If we consider the force 0, which acts instantaneously, it has
been shown that this force is represented by the velocity which
it will communicate to the moving body in the unit of time, if
the movement should become all at once uniformly accelerated,
we shall then have, in putting for V its value <£» F = MQ.
This equation shows further, that <p is the force which should
act upon a unit of the mass ; for if we make M = 1, we have
F = 0. <f> being the accelerating force, F is that which is
called the moving force.
It has been shown, that in calling g the force of gravity, P
the weight of the body, and M its mass, we had P = Mg; if,
between this equation and the preceding, we eliminate M, we
shall find F — P J£ ; so that when the accelerating force 0 is
that of gravity, <j> = g, and the preceding equation is reduced to
F = P, in this case the accelerating force is therefore estimated
by the weight of the body on which it acts.
There was formerly a celebrated dispute amongst geometri-
cians on the measure of forces. This dispute, as many others,
originated from a misunderstanding of the definition of the
words.
A force being known to us only by its effects, may be mea-
sured in different manners, according to the use to which it is
desired to appropriate it. For example, if it be proposed to de-
termine the burden which a man can sustain for an instant, it
is evident that the force of this man will be proportional to the
weight which he will be capable of sustaining, and consequently
can be represented by this weight ; but if it be desired to mea-
sure the force of this man by the work he can execute in a given
time, there will be another manner of estimating his force ; and
which will be entirely different from the other, because we feel
that a weaker man with a greater disposition to sustain a longer
fatigue might produce in his work a greater result, and in this
point of view be considered as endowed with a greater force than
the other.
In this second mode of considering the power of a man, we
95
shall regard it as proportional to the weight which he lifts, and
to the height to which he shall have raised it in a given time ;
it being understood that we do not suppose that the effort varies
in proportion to the height, because, in fact, this height only
represents the number of times that a certain work is repeated.
Thus, in supposing that two men raising in one day's work the
same weight, the one 600 metres high, and the other 200 metres,
in this manner of estimating the force, we shall look upon one
of these men as having three times as much power as the other.
It follows further from this, that if, in the same day's work,
two men raising the first 20 kilogrammes to 200 metres, and the
second 10 kilogrammes to 400 metres, we should regard them,
according to the present hypothesis, as having equal powers, al-
though really the intrinsic power of these men may be very dif-
ferent ; but here we regard them only in proportion to the work
done.
It is in this manner that Descartes estimated the power of a
man, or of any other motive power, the dispute which caused
a difference of opinion between him and other geometricians,
only rested upon the definition of the word power. He main-
tained that a power should be measured by the mass into the
square of the velocity. We proceed to show how, when we con-
sider bodies in motion, that this definition of Descartes of the
word power (force), leads to this consequence.
Suppose P the weight raised, and h the height to which it
ought to be raised in a given time ; the power in the hypothesis
of Descartes will then be measured by the product P x h.
We can in this expression replace P, by its value M#,* arid
we shall have PA — M.gh \ multiplying by 2, it will become
2PA = M X 2gh; observing that the square of the velocity v due
to the height A, has for expression 2gk, we can replace 2gk by
02; this gives 2P& = Mv2.
Having defined the word power (force) differently from Des-
* In a former part of this work the author shows that the coefficient g
represents the force of gravity.
96
cartes, we shall not say as he does, that Mv2 is the measure of a
power (force], because we have shown that a power (force}
should be represented by the quantity of motion Mv that it pro-
duces. Thus, to avoid all ambiguity, we shall employ a new
denomination in giving, according to usage, the name of force
vive, to the product Mv2 of the mass by the square of the
velocity.
La force vive (vis-viva) is of great utility when the effect
of a machine is desired to be known. If it be required, for ex-
ample, to apply a flow of water, to move a carriage on a
given road, to compress a mass of air, to draw from a mine a
certain quantity of coals, &c., in all these cases we can compare
the effect of the moving power, to the product of a certain weight
by a given length ; consequently, to an expression of the form
Ph, of which the double, as we have just demonstrated, becomes
the product
97
13 il
s So
« 0,0- o.g
,
8
a *-. i •«.
P. GO •* -o
^»
o •*
2
..
'&»•& S o 'bo
C .-° 0 3^> = .
p o
3^ TH
II
LONDON:
ROAKE AND VAUTY, PRINTERS, 31, STRAND.
WAN DEPT.
M> 2lA-60m-3,'65
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