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. 
 343, 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 57 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 
 
 P ar t 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 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 ne g a tive 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 
 
h t which is not restored by the descents, is ex- 
 pressed by p. P/i. Thus, in adding 1 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.^ + (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 ; 2ndlj r , 
 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- 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^? + 103 3o 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 O m 28 ; the surface 
 of the two pistons O m ' q - 12315=Q. 
 
 Stroke of the pistons O m . 41=c. 
 
 Radius of the wheels, O m . 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 20 m .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 = 6 m - 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 5012, 
 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.ltol. 
 
 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 
 6 m 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 6 ra 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 9 m -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 6 m .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 
 
 * 291ofFarei)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 158f 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 9 m .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 6 m .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 9 m .8 to 
 6 m .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 6 m .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 9 m .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 p 9 
 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 
 Jdp.x 2 =pr$, p being the weight of the wheel, 
 then the preceding expression of the vis-viva of a 
 
 wheel may be written u i 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 u o 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. 
 
 r f 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 \ 
 V 200 ' 
 
 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=A 9 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 O m .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 wmc h 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 + X .oooo484F e( l ua]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 n2, 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 O k .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 O k .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 0.479 k 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 Po, in formulae 
 (20), U is found -17 m .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 17 m .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 13 m .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 resi stance 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 * 
 
 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 -- ^ - X r+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 v 1 . 
 
 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 ; v l = 12 ; 
 C= 300 - 10 ; this formula is reduced to 
 
 12 X 450 + fc !2 x 300 v 300 r 
 
 100 " x ;+ii- 
 
 540 + 360-080 r 
 
 v " x r -i-224? vvm ch will give the numbers 
 
 in Table 10. Thus suppose r, the rate of inclination, to be 200, 
 and the velocity v = 12 t , the above formula becomes 
 
 540+360360 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 v l = 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. l h . 53 m . 
 53*. and the line AB in 13 h . 57'. 37 s ; 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 M 1 . 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 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& = Mv 2 . 
 
 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 Mv 2 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 Mv 2 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 
 
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 0,0- o.g 
 
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 P. GO * -o 
 
 ^ 
 
 o * 
 
 2 
 
 
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 '&& S o 'bo 
 
 C .- 3^> = . 
 
 
 p o 
 
 3^ TH 
 
 II 
 
LONDON: 
 
 ROAKE AND VAUTY, PRINTERS, 31, STRAND. 
 

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