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THE THEORY 
 
 OF 
 
 THE STEAM ENGINE; 
 
 THE INACCURACY OF THE METHODS IN USE FOR CALCULATING THE 
 EFFECTS OR THE PROPORTIONS OF STEAM-ENGINES, 
 
 AND SUPPLYING A SERIES OF 
 
 PRACTICAL FORMULAE 
 
 TO DETERMINE 
 
 THE VELOCITY OF ANY ENGINE WITH A GIVEN LOAD, THE LOAD FOR A 
 STATED VELOCITY, THE EVAPORATION FOR DESIRED EFFECTS, THE HORSE- 
 POWER, THE USEFUL EFFECT FOR A GIVEN CONSUMPTION OF WATER OR FUEL, 
 THE LOAD, EXPANSION, AND COUNTERWEIGHT FIT FOR THE PRODUCTION OF 
 THE MAXIMUM USEFUL EFFECT, ETC. 
 
 AN APPENDIX, 
 
 CONTAINING CONCISE RULES FOR PERSONS NOT FAMILIAR WITH ALGEBRAIC 
 
 SIGNS, AND INTENDED TO RENDER THE USE OF THE FORMULA 
 
 CONTAINED IN THE WORK PERFECTLY CLEAR AND EASY. 
 
 BY COMTE DE PAMBOUR, 
 
 FORMERLY A STUDENT OF THE ECOLE POLYTECHNIQUE, LATE OF THE ROYAL ARTILLERY, 
 
 ON THE STAFF IN THE FRENCH SERVICE, OF THE ROYAL ORDER OF THE 
 
 LEGION D'HONNEUR, ETC. 
 
 LONDON : 
 JOHN WEALE, 
 
 ARCHITECTURAL LIBRARY, HIGH HOLBORN. 
 
 1839. 
 
G. WOODFALL, ANGEL COURT, SKINNER STREET, LONDON. 
 
G8Z 
 
 INTRODUCTION. 
 
 IN the Treatise on Locomotive Engines, the first 
 French edition of which appeared in the beginning 
 of 1835, we developed a new theory of the steam- 
 engine. We then made the application of it merely 
 to locomotive engines, because the subject con- 
 strained us so to do ; but we announced at the 
 same time, that this theory was no less necessary 
 to calculate with accuracy, either the effects or the 
 proportions of other steam-engines of every de- 
 scription. 
 
 Immediately after the publication of that Trea- 
 tise, the theory we speak of was adopted in several 
 works. An illustrious member of the Institut, of 
 whom science has lately had to deplore the loss, 
 M. Navier, who had just published in the Annales 
 des Ponts-et- Chaussees of 1835, a paper on the 
 use of locomotive engines, immediately resumed his 
 work, and presented in 1836, in the same scientific 
 publication, another paper, in which he acknow- 
 ledged the accuracy of the theory we had just intro- 
 duced, and abandoned, in consequence, the ordinary 
 process of calculation to substitute ours in its place. 
 
 507 A * 
 
IV INTRODUCTION. 
 
 The English professor Whewell, whose name is so 
 well known in science, inserted also the principles 
 of this theory in the third edition of his Treatise 
 on Mechanics ; many engineers made it the basis of 
 their calculations for the use of locomotive engines 
 upon divers railways ; and finally, at the moment 
 when the present work appears, we find that M. 
 Wood, in the third edition of his Treatise on Rail- 
 roads, London, 1838, has just adopted it likewise, 
 (pages 555-557,) without mentioning there from 
 what source he has it, but afterwards repairing that 
 omission in a slip added at the beginning of his 
 work. 
 
 However, we had as yet developed our theory 
 only in its application to locomotives, and it had 
 been considered by those who had admitted it after 
 us, only as a theory specially applicable to those en- 
 gines. It remained, then, first to demonstrate it 
 in a general manner, and afterwards to apply it to 
 the different kinds of steam-engines in use. But 
 as the action of steam is extremely simple in loco- 
 motives, since those engines are rotative, non-con- 
 densing and unexpansive, much remained to be 
 done to establish formula suitable to all the other 
 systems of steam-engines. Such was the object of 
 a series of papers which we presented to the Aca- 
 demy of Sciences of the Institut, in the beginning 
 of 1837, and to which that illustrious body was 
 pleased to grant its approbation, recognizing that 
 the priority relative to the theory in question be- 
 longed to us. 
 
INTRODUCTION. V 
 
 It is these papers which we now publish, after 
 having given them the development of which they 
 appeared to be susceptible. We shall there- 
 fore begin the work by proofs of the inaccuracy of 
 the theory employed before us ; and this part of 
 the work will be purposely established upon prac- 
 tical facts so simple, and explained in terms so 
 exempt from any scientific erudition, that it cannot 
 fail to be in the reach of all practitioners, and even 
 of persons the least acquainted with these questions. 
 
 Afterwards, the general properties of steam, or 
 the laws which regulate its action, will be deve- 
 loped. To the laws and formulae already known 
 on this subject, some will be added which will be 
 found very useful for calculating the effects of steam. 
 For instance, the successive formulse by means of 
 which the temperature of the steam generated under 
 different pressures was calculated, did not present in 
 their results a perfectly continued series : a new 
 formula substituted for that of Tredgold, will now 
 allow a complete and regular table to be offered on 
 this subject. Another formula will equally be found 
 in the work, which permits an immediate deter- 
 mination of the volume or density of the steam, by 
 knowing only the pressure at which it is generated, 
 and without recurring to the temperature. Finally, 
 a new and very important law will be explained, 
 according to which the steam, during the whole of 
 its action in the engines, remains at the maximum 
 density for its temperature ; and this law will make 
 it easy to take account, in the various calculations 
 
VI INTRODUCTION. 
 
 which may occur upon steam, of the changes of 
 temperature it undergoes during its action ; a pro- 
 blem which had not been solved as yet. 
 
 These preliminaries once established, the general 
 theory of the steam engine will be explained, and 
 its application will be shown to every particular sys- 
 tem of engines. High-pressure engines, locomo- 
 tives, Watt's and Cornish double-acting engines, 
 Woolf's, Evans's, Watt's and Cornish single-acting 
 engines, and finally atmospheric engines, will be 
 successively treated in special chapters ; and for 
 each of them practical formulae will be given, to 
 solve the divers problems which may offer them- 
 selves, either in the management or in the building 
 of those engines. 
 
 In this part of the work will be recognized the 
 advantages resulting from the new theory, as many 
 important questions which had not been solved by 
 the ordinary calculation, nay, some, the existence 
 of which could not even be admitted in that theory, 
 will now be solved in the most easy manner. Among 
 those questions we shall quote : the means of deter- 
 mining the friction proper to the engine itself; the 
 calculation of the velocity of the piston for a given 
 load, which remained impossible in the ordinary 
 theory ; the determination of the load to be adopted 
 for an engine, in order to make it produce its maxi- 
 mum of useful effect, a problem inadmissible in the 
 ordinary theory, since, by that theory, the engines 
 are always supposed to work with the load deter- 
 mined by the pressure in the boiler, that is, with 
 
INTRODUCTION. Vll 
 
 their maximum load ; the research of the expansion 
 and counterweight corresponding to the maximum 
 of useful effect in single-acting and atmospheric 
 engines, a question which as yet was a desideratum 
 in science. And the importance of these problems 
 will be recognized by merely announcing them, since 
 their immediate result is to make an engine work in 
 the most advantageous manner, an enquiry which 
 Watt and Robison acknowledged not to have been 
 as yet established upon principles *. 
 
 Among the practical results of the new theory, it 
 will be remarked that it explains quite satisfactorily 
 a question much debated at this moment among 
 English engineers : namely, the apparently sur- 
 prising effects of the Cornish engines. The fact 
 is that the effects of those engines have only been 
 put in question, because the calculation applied to 
 them could not agree with the practical facts de- 
 duced from observation. But when they are cal- 
 culated by the theory now presented, the results 
 are found to agree perfectly with actual facts. It 
 will be seen in the chapter relating to that kind of 
 engines, that in their mean working condition, they 
 may be expected to produce, per imperial bushel 
 or 84 Ibs. of Newcastle coals, a useful effect of 
 76,000,000 of pounds raised one foot ; and it is 
 known that the average of ten of those engines has 
 given lately an authentic result of 70,000,000 of 
 pounds raised one foot, for the above consumption 
 of coals. This coincidence is, therefore, a fresh 
 
 * Watt on the Steam-Engine, p. 81. 
 
Vlll INTRODUCTION. 
 
 confirmation of the new theory, and similar facts 
 will be found in the work, relating to Watt's and 
 atmospheric engines. 
 
 The incorrect manner in which the effects or the 
 proportions of steam-engines have been hitherto cal- 
 culated, appears to us to have been the principal 
 cause of those continual disappointments expe- 
 rienced in the trying of steam-engines,, and of the 
 lawsuits which have often followed between the 
 purchasers and makers. Engine builders thought 
 they could assign beforehand the effects which 
 would be produced by an engine when completed ; 
 but experience soon demonstrated that they could 
 not be assured of accomplishing a determined ob- 
 ject, except the engine were precisely modelled after 
 another already tried ; and that, in all other cases, 
 they were in absolute uncertainty as to the real 
 effects. If then, apprehensive of furnishing an en- 
 gine too weak, they designedly gave it an excess of 
 power, the consequence was that power was ex- 
 pended in mere loss, and that the engine, therefore, 
 did not work with all the economy that had been 
 expected from it ; if, on the contrary, the engine 
 was found too feeble, then it was incapable of per- 
 forming its intended task, and it became necessary 
 to give up the use of it ; or else, the persons 
 charged with the management of the engine, know- 
 ing its insufficiency, sought to compensate the want 
 of force by increasing the pressure in the boiler, 
 that is to say, by loading, or even by making fast the 
 safety-valves, and dreadful explosions have often 
 
INTRODUCTION. IX 
 
 been the result. Thus, on board of steam-boats 
 too weak to make way against the currents on 
 which they were to navigate, the lives of many 
 travellers have been compromised. 
 
 On the other hand, the analytical calculations by 
 which we replace the old methods, are reduced to 
 forms so simple, that they cannot offer the least 
 embarrassment in the applications, nor have we 
 hesitated to risk some repetitions in order to render 
 the use of them still more clear and convenient. 
 We therefore think that these researches may 
 equally suit both engineers and practitioners, and 
 we sincerely hope that they may be found to ren- 
 der some service. 
 
CONTENTS. 
 
 Page 
 INTRODUCTION iii 
 
 CHAPTER I. 
 
 PROOFS OF THE INACCURACY OF THE ORDINARY METHODS 
 OF CALCULATION. 
 
 Sect. I. Mode of calculation hitherto in use for cal- 
 culating the effects of steam-engines . 1 
 Sect. II. Objections against that mode of calculation . 11 
 Sect. III. Formulae proposed by divers authors, to de- 
 termine the velocity of the piston under a 
 given load ; and proofs of their inaccuracy 16 
 Sect. IV. View of the theory proposed ... 18 
 Sect. V. New proofs of the accuracy of that theory and 
 
 of the inaccuracy of the ordinary theory . 25 
 Sect. VI. Comparison of the two theories in their appli- 
 cation to particular examples ... 33 
 Sect. VII. Of the area of the steam-passages . . 46 
 Sect. VIII. Of the differences which exist between the 
 
 theory proposed and the ordinary theory . 56 
 
 CHAPTER II. 
 
 * 
 
 OF THE LAWS WHICH REGULATE THE MECHANICAL ACTION 
 OF STEAM. 
 
 Sect I. Relation between the temperature and the 
 pressure of the steam in contact with the 
 liquid 59 
 
Xll CONTENTS. 
 
 Page 
 Sect. II. Relation between the relative volumes and 
 
 the pressures, at equal temperature, or be- 
 tween the relative volumes and the temper- 
 atures, at equal pressure, in the steam se- 
 parated from the liquid .... 70 
 
 Sect. III. Relation between the relative volumes, the 
 pressures and the temperatures, in the 
 steam in contact or not in contact with the 
 liquid ....... 73 
 
 Table of the temperatures, and of the volume 
 of the steam formed under different pres- 
 sures, compared to the volume of water 
 that has produced it .... 76 
 
 Sect. IV. Direct relation between the relative volumes 
 and the pressures, in the steam in contact 
 with the liquid 78 
 
 Sect. V. Of the constitutive heat of the steam in 
 
 contact with the liquid .... 84 
 
 Sect. VI. Of the conservation of the maximum density 
 of the steam, for its temperature, during 
 its action in the engine .... 89 
 
 CHAPTER III. 
 
 GENERAL THEORY OF THE STEAM-ENGINE. 
 
 ARTICLE I. Of the effect of engines, in the case of a given 
 expansion with any velocity or load whatever. 
 
 Sect. I. Of the divers problems which present them- 
 selves in the calculation of steam-engines . 96 
 
 Sect. II. Of the velocity of the piston with a given 
 
 load . ...;.. . 99 
 
 Sect. III. Of the load of the engine for a given ve- 
 locity ^y in 
 
 Sect. IV. Of the evaporation of the engine to produce 
 
 given effects . . . . . .113 
 
 Sect. V. Of the different expressions of the useful 
 
 effect of the engines . . .114 
 
 Sect. VI. Table for the numerical solution of the 
 
 formulae, for rotative engines . . .118 
 
CONTENTS. Xlll 
 
 ARTICLE II. Of the maximum useful effect with a given 
 
 expansion. 
 
 Page 
 
 Sect. I. Of the velocity of maximum useful effect . 121 
 Sect. II. Of the load of maximum useful effect . . 127 
 Sect. III. Mode of determining the friction of unloaded 
 engines, and their additional friction per 
 unit of the load ; derived from the pre- 
 ceding enquiry . . . . .128 
 Sect. IV. Of the evaporation of the engine . . . 131 
 Sect. V. Of the maximum useful effect of the engine . 131 
 ARTICLE III. Of the absolute maximum of useful effect . 134 
 
 CHAPTER IV. 
 
 HIGH-PRESSURE ENGINES. 
 
 ARTICLE I. Theory of high-pressure engines and of unexpansive 
 engines in general. 
 
 Sect. I. Of the effects of the engine with any given 
 
 load or velocity . . . . .141 
 Sect. II. Of the maximum useful effect of the engine . 147 
 
 ARTICLE II. Practical formula for the calculation of high- 
 pressure engines, and example of their 
 application . . . . . .148 
 
 CHAPTER V. 
 
 LOCOMOTIVE ENGINES. 
 
 ARTICLE I. Theory of Locomotive engines. . . .156 
 ARTICLE II. Practical formula for the calculation of loco- 
 motive engines, and example of their appli- 
 cation . . . . . . .160 
 
 CHAPTER VI. 
 
 WATT'S ROTATIVE OR DOUBLE-ACTING ENGINES. 
 
 Sect. I. Practical formulae for the calculation of these 
 
 engines, and example of their application . 169 
 
 Sect. II. Considerations on the application of the 
 ordinary mode of calculation to Watt's 
 engines and to stationary engines in general 1 76 
 
XIV CONTENTS. 
 
 CHAPTER VII. 
 
 CORNISH DOUBLE-ACTING ENGINES. 
 
 Practical formula for the calculation of these engines, 
 
 and example of their application . .187 
 
 CHAPTER VIII. 
 
 WOOLF'S OR EDWARDS'S ENGINES. 
 
 ARTICLE I. Theory of Woolf's engine .... 201 
 ARTICLE II. Practical formulae for the calculation of these 
 
 engines . . . . . . .211 
 
 CHAPTER IX. 
 
 EVANS'S ENGINES. 
 Practical formula for the calculation of these engines . 217 
 
 CHAPTER X. 
 
 WATT'S SINGLE-ACTING ENGINES. 
 
 ARTICLE I. Theory of Watt's single-acting engine. 
 
 Sect. I. Of the regulation of the engine . . .223 
 
 Sect. II. Of the effects of the engine, with a given 
 counterweight, and an indefinite load or 
 velocity .232 
 
 Sect. III. Of the velocity of the engine, with a given 
 
 load ... . 243 
 
 Sect. IV. Of the load of the engine, with a given ve- 
 locity ... ... 244 
 
 Sect. V. Of the evaporation necessary to produce de- 
 sired effects 246 
 
 Sect. VI. Of the useful effect of the engine . . 246 
 
 Sect. VII. Of the load and the velocity which corre- 
 spond to a given expansion . . . 247 
 
 Sect. VIII. Determination of the friction of the engine 
 when unloaded, and of its additional friction 
 per unit of the load .... 248 
 
 Sect. IX. Tables for the numerical solution of the for- 
 
 mulse, for single-acting engines . . 252 
 
CONTENTS. XV 
 
 Page 
 Sect. X. Of the maximum useful effect with a given 
 
 counterweight, and of the absolute maxi- 
 mum useful effect of the engine . .257 
 ARTICLE II. Practical formulae for calculating Watt's 
 single-acting engines^and example of their 
 application ...... 260 
 
 CHAPTER XL 
 
 CORNISH SINGLE-ACTING ENGINES. 
 
 Practical formulae for calculating these engines, and ex- 
 ample of their application . . . 270 
 
 CHAPTER XII. 
 
 ATMOSPHERIC ENGINES. 
 
 ARTICLE I. Atmospheric engine, with condenser. 
 
 Sect. I. Of the regulation of the engine . . . 280 
 
 Sect. II. Of the effects of the engine with a given 
 counterweight, and an indefinite load or 
 velocity 283 
 
 Sect. III. Of the velocity of the engine with a given 
 
 load 292 
 
 Sect. IV. Of the load of the engine with a given ve- 
 locity 294 
 
 Sect. V. Of the evaporation necessary to produce de- 
 sired effects 295 
 
 Sect. VI. Of the useful effect of the engine . . 296 
 
 Sect. VII. Determination of the friction of the engine 
 unloaded, and of its additional friction per 
 unit of the load 297 
 
 Sect. VIII. Tables for the numerical solution of the 
 
 formulae, for atmospheric engines . .301 
 
 Sect. IX. Of the maximum useful effect with a given 
 counterweight, and of the absolute maxi- 
 mum useful effect of the engine . . 305 
 
XVI CONTENTS. 
 
 Page 
 ARTICLE II. Atmospheric engine without condenser. 
 
 Sect. I. Modifications to be made in the preceding 
 formulae for the case of engines not fur- 
 nished with a separate condenser . . 307 
 
 Sect. II. Of the difference between the total and the 
 effective evaporation of the engine, arising 
 from the refrigeration of the cylinder at 
 each stroke of the piston . . .318 
 
 ARTICLE III. Practical formulae for atmospheric engines, 
 
 and example of their application . . 329 
 
 APPENDIX. 
 
 Concise rules designed for persons not familiar with alge- 
 braic signs, and intended to render the use of the for- 
 mula contained in the work perfectly clear and easy . 341 
 
THEORY 
 
 STEAM-ENGINE 
 
 CHAPTER I. 
 
 PROOFS OF THE INACCURACY OF THE ORDINARY 
 MODE OF CALCULATION. 
 
 SECT. I. Mode of calculation now in use, to 
 calculate the effects of a steam-engine. 
 
 THE object of this work is to demonstrate that the 
 calculation of the effects or the proportions of 
 steam-engines, either as it is practically used or as 
 it is found indicated by authors who have treated 
 this subject, is completely erroneous, and to develop 
 a new theory which leads to accurate results. Our 
 first chapter, then, will necessarily be devoted to 
 proving the inaccuracy of the ordinary methods 
 of calculation. From that we shall pass successively 
 to the development of the theory proposed, and its 
 application to the different systems of steam-engines 
 in use. 
 
2 CHAPTER I. 
 
 The effect produced by a machine consists of 
 two elements : the resistance set in motion, and 
 the velocity communicated to that resistance. Hence 
 results, that the calculations which first occur in 
 the application of machines refer to the two fol- 
 lowing problems : 
 
 1st. The machine being supposed constructed, and 
 the velocity of its motion given, to determine what 
 resistance it can move. 
 
 2d. The machine being supposed constructed, and 
 the resistance it has to move being known, to deter- 
 mine the velocity it can communicate to that re- 
 sistance. 
 
 A third problem then presents itself as a natural 
 consequence of the two preceding, viz. : 
 
 3d. The resistance being known as well as the 
 velocity to be communicated to it, to determine the 
 dimensions that ought to be adopted in the con- 
 struction of the machine, in order to produce the 
 effect proposed. In steam-engines, this third pro- 
 blem reduces itself to determining the size of the 
 boiler, or, in other words, the evaporation it should 
 be capable of, in order to obtain the effect proposed. 
 
 These three problems are the basis of all calcu- 
 lations on steam-engines. They may take divers 
 modifications and give rise to several questions, 
 which hereafter we shall notice, but whose solution 
 will entirely depend on that of the three funda- 
 mental problems just mentioned. Thus, for in- 
 stance, to find the useful effect of an engine of 
 which the number of strokes of the piston is counted, 
 
OF THE ORDINARY MODE OF CALCULATION. D 
 
 that is, whose velocity is known, amounts to deter- 
 mining the resistance it can move at that velocity ; 
 since that resistance being once known, it suffices 
 to multiply it by the given velocity, to have the 
 useful effect required. The horse-power of an 
 engine, and its effect for a given weight of fuel, 
 being nothing more than the useful effect of the 
 engine referred to particular units ; that is, to the 
 power of the horse considered as the unit of force 
 produced, or to the consumption of a certain 
 quantity of fuel considered as the unit of force ap- 
 plied ; it is plain that these questions merge into 
 that of the useful effect. This point, however, will 
 be readily recognized when we come to treat spe- 
 cially of these questions. 
 
 Thus all enquiries relative to steam-engines re- 
 duce themselves finally to the three just announced: 
 to find the load, to find the velocity, to find the 
 evaporation. 
 
 The only mode of calculation hitherto in use to 
 estimate either the effort of which a steam-engine 
 is capable, or the useful effect it can produce, is 
 first to perform the calculation under -the suppo- 
 sition that the steam acts in the cylinder with the 
 same elastic force as in the boiler, and without re- 
 garding the friction of the engine ; then to reduce 
 the result in a certain proportion indicated by a 
 constant coefficient. This method, which we shall 
 name the method of coefficients, was resorted to, 
 because there being no means of knowing, d priori, 
 the pressure of the steam in the cylinder, it was 
 
 B 2 
 
4 CHAPTER I. 
 
 naturally enough at first concluded to be equal to 
 that of the boiler. But as the result thus obtained, 
 and which was called the theoretic result, was in- 
 variably found much higher than the practical 
 results compared with it, a reduction was found to 
 be necessary. 
 
 The necessity of this reduction was attributed 
 to two causes : 1st, to the friction of the engine 
 having been neglected in the calculation ; and 
 2nd, to no account having been taken of losses 
 resulting from the five following circumstances : the 
 contraction of the passages through which the 
 steam has to pass, the changes in the direction of 
 the conducting pipes, the friction of steam in the 
 steam-pipes, the waste of steam, and its partial 
 condensation. And as these causes seemed likely 
 to act similarly, not only in the same engine, as 
 long as the passages of the steam were not varied, 
 but in all engines of the same system, it was natural 
 to suppose that they would produce, on the definitive 
 result of the calculation, a reduction proportioned 
 to its total value. 
 
 It was in consequence judged, that the real result 
 might be attained by reducing the theoretic result 
 in a certain constant proportion. It had been 
 observed, also, that the ratio of the theoretical and 
 practical effects was not the same in the different 
 systems of steam-engines ; this consequently led to 
 the admitting of a different coefficient for each 
 system. 
 
 This mode was the most natural, it was even 
 
OF THE ORDINARY MODE OF CALCULATION. 5 
 
 the only one possible, so long as means were want- 
 ing to determine beforehand what would be the 
 real pressure of the steam in the cylinder under 
 given conditions. And notwithstanding we have 
 undertaken to demonstrate the errors which result 
 from the application of that calculation, and to sub- 
 stitute another in its place, we are far from wishing 
 to depreciate the works in which that calculation is 
 developed. It was felt, no doubt, that that method, 
 from the very circumstance of its consisting in the 
 use of a coefficient to represent in total, various 
 effects which had never been submitted to direct 
 admeasurement, could be but an approximation, a 
 mere provisional method. It was used as we use a 
 bad instrument, till we can get a better. Many of 
 the works wherein it is explained, acknowledge at the 
 same time that the theory of the steam-engine is as 
 yet unknown or imperfectly studied. Moreover, 
 these works do not all treat the subject in a manner 
 perfectly similar, and therefore the observations we 
 are about to make, cannot be addressed equally to 
 all. We wish it, then, to be clearly understood, that 
 in comparing the two calculations together, when it 
 shall be needful so to do, our end is to establish 
 the accuracy of the method we propose, and not to 
 attack the writings of others. 
 
 To return to the method in use, this was the 
 proceeding. The force applied to the piston was 
 computed, in supposing the pressure of the steam 
 in the cylinder equal to that of the steam in the 
 boiler : that is to say, the area of the piston was 
 
O CHAPTER I. 
 
 multiplied by the pressure of the steam in the boiler, 
 which gave the force exerted by the engine ; this 
 result was then multiplied by the velocity of the 
 piston, and thus was obtained the theoretic effect of 
 the engine. But the result of this calculation having 
 been compared with that of some experiments made 
 on engines of the same kind, the ratio between the 
 two results had furnished a fractional coefficient, 
 which was regarded as the constant ratio between 
 the theoretical and practical effects of all engines 
 of the same system; therefore, in multiplying the 
 number expressing the theoretic effect by this frac- 
 tional coefficient, a definitive product was obtained, 
 which was the practical effect that could be ex- 
 pected from the engine. 
 
 Supposing, for instance, an engine without con- 
 densation, and expressing the area of the cylinder 
 by a, the pressure of the steam per unit of surface 
 in the boiler by TT, and the velocity of the piston by 
 v ; a-TT was the force applied by the engine, and airv 
 the theoretic effect it ought to produce. As then 
 some experience, instead of giving an effect equal to 
 aw, had given but a certain fraction of it, which we 
 will express by k, the coefficient k was admitted as 
 representing the constant ratio between the theo- 
 retical and practical effects. So that the useful 
 effect of a non-condensing engine was represented by 
 
 kanrv ; 
 
 or the theoretic effort of the engine being expressed 
 oy air, its useful effort, or the resistance the piston 
 could move, was represented by 
 
OF THE ORDINARY MODE OF CALCULATION. 7 
 
 a R = kanr 
 
 R expressing that resistance supposed to be divided 
 per unit of the surface of the piston. 
 
 The coefficients indicated by Tredgold, in order 
 to pass from the pressure in the boiler to the part 
 of it which is applied to produce the useful effect, 
 are the following : 
 
 Non-condensing unexpansive engines .... *60 
 Non-condensing expansive engines .... "60 
 Non-condensing expansive engines, with two cy- 
 linders '47 
 
 Watt's single-acting engines *60 
 
 Single-acting expansive engines *60 
 
 Watt's double-acting steam-engine .... '63 
 Double-acting engine, with two cylinders . . '48 
 Atmospheric engine (coefficient to be applied to 
 the atmospheric pressure, and including a de- 
 duction for the pressure of the uncondensed 
 steam on the opposite side of the piston) . . "52 
 Atmospheric engine, with a separate condenser 
 (coefficient to be applied like the preceding 
 one) *54 
 
 It must be understood that these coefficients, ex- 
 cepting those for the atmospheric engines, are to 
 be applied to the total pressure of the steam in the 
 boiler ; that is, before any deduction is made, either 
 for the pressure of the uncondensed steam or for 
 the atmospheric pressure, on the opposite side of 
 the piston. If, on the contrary, before applying the 
 coefficient, the pressure in the boiler had been 
 diminished by the pressure subsisting on the other 
 side of the piston, in order to deduce first what is 
 
8 CHAPTER I. 
 
 called the effective pressure of the steam, then other 
 coefficients, smaller than those indicated here, ought 
 to be u$ed. For instance, if we suppose a high 
 pressure steam-engine working at the total pressure 
 of 65 Ibs:. per square inch, and that we apply to 
 that pressure the coefficient *60, we shall have for 
 the useful part of the total pressure 
 65 x -60 = 39 Ibs.; 
 
 and if we deduct from this force, the atmospheric 
 pressure which acts on the other side of the piston, 
 the practical effort applied by the engine will be 
 
 65 x -60 -15 = 24 Ibs. 
 
 But, as the effective pressure of the steam in the 
 boiler is 
 
 65 -15 = 50 Ibs., 
 
 it is clear that, if we had had to pass from the 
 effective pressure of the steam to the practical effort 
 applied by the engine, a coefficient of *50 ought 
 then to have been used, instead of the coefficient 
 60 applied to the total pressure in the boiler. 
 
 It is to be remarked also, that, as the theoretic 
 effect of an engine is known only after deduction of 
 the pressure on the other side of the piston, it fol- 
 lows that if the theoretic effect of the engine had 
 been definitively calculated by this mode, and that 
 we were to pass from it to the practical effect, it 
 is the coefficient *50 which ought then to be em- 
 ployed. Therefore, this last coefficient indicates 
 in reality the reduction operated upon the theoretic 
 effects, to conclude from them the practical effects 
 
OF THE ORDINARY MODE OF CALCULATION. 9 
 
 of the engine ; but the calculation comes to the 
 same either way, provided a suitable coefficient be 
 used. 
 
 Such was the solution of the first of the three 
 problems above mentioned. The second, which 
 consists in determining the velocity, had not been 
 the object of any enquiry, by the mode of reasoning 
 we have just exposed. 
 
 The third problem, or the evaporation of water 
 necessary to produce a given effect, had been 
 solved in a manner similar to the first. The rule 
 consisted in calculating the volume described by 
 the piston, and in supposing that volume to have 
 been filled with steam at the same pressure as in 
 the boiler, and then applying to it a constant co- 
 efficient. That determined in the preceding pro- 
 blem was usually employed, but it was applied as a 
 divisor, with a view to augment the evaporation in 
 proportion to the losses represented by that co- 
 efficient. 
 
 Thus, retaining the foregoing notations, and 
 expressing by m the volume of steam formed at the 
 pressure of the boiler referred to the volume of 
 water that produced it, we perceive that the volume 
 described by the piston during the unit of time, 
 was av. From the signification of the letter m 
 this volume of steam represented a volume of water 
 expressed by 
 
 av f 
 m ' 
 but as it was deemed subject to a loss represented 
 
10 CHAPTER I. 
 
 by the fraction or coefficient k, the volume of water 
 really necessary to supply the expenditure , be- 
 
 av 
 
 m 
 
 came 
 
 S _ av 
 km 
 
 Such was the calculation in use ; it solved, as 
 we have seen, only two of the three fundamental 
 problems. We shall, at a future moment, return 
 to what regards the velocity of the piston under a 
 given load. 
 
 Besides what has been said, the received ideas 
 relative to the pressure of the steam in the cylinder, 
 consisted in deeming that, the pressure in the boiler 
 being given and fixed, it were possible at pleasure 
 to vary the pressure in the cylinder and to produce 
 there any desired pressure, provided it were in- 
 ferior to that of the boiler, by contracting more or 
 less the orifice of the steam-passages ; and it was 
 thought that, when this orifice was entirely open, 
 with the area usually given to it in fixed engines, 
 to wit, -^ of the area of the cylinder, the pressure 
 of steam in the cylinder could differ but in an in- 
 considerable quantity from the pressure in the boiler. 
 
 However, as the indicator of Watt, applied to 
 the cylinder of several engines, had demonstrated a 
 certain diminution of pressure therein, when com- 
 pared with the pressure in the boiler, the authors 
 who took this fact into account, without perceiving 
 its real cause, still attributed it to the circum- 
 stances already explained, and it became merely 
 
OF THE ORDINARY MODE OF CALCULATION. 11 
 
 one of the elements in the explication of their 
 definitive coefficient. Thus, in all cases, the pressure 
 in the cylinder was considered as being equal or 
 proportional to that in the boiler, and therefore 
 constant, so long as no change took place in the 
 pressure of the boiler; but in no wise as being 
 regulated by the resistance, or as variable with the 
 resistance, independently of all pressures in the 
 boiler, which we shall demonstrate that it in reality 
 is. 
 
 SECT. II. Objections against that mode of calcu- 
 lation. 
 
 The objections which first present themselves 
 against that mode of calculation are the following : 
 
 1st. The coefficient adopted by many to represent 
 the ratio of the theoretic effects to the practical, 
 in high-pressure engines, was '33 ; which was ex- 
 plained by saying that the remainder, or *66 of the 
 total force developed, was absorbed by the frictions 
 and losses. Not that these frictions and losses 
 had been measured and found such ; but merely 
 that the calculation, which might be inexact in its 
 very principle, wanted so much of coinciding with 
 experience. 
 
 To obtain conviction of the impossibility of 
 justifying such an assertion as to the value of the 
 frictions and losses, it suffices to peruse the expla- 
 nation of it attempted by Tredgold, who follows 
 this method. 
 
12 CHAPTER I. 
 
 He indicates that a deduction of 4 tenths should 
 be made on the total pressure of the steam, (in- 
 cluding the atmospheric pressure,) which amounts 
 to making a reduction of '5 on the ordinary effective 
 pressure of those engines, or to using a coefficient 
 of '5 applied to the theoretic effect of the engine. 
 He thus explains this loss in the effect produced *. 
 Force necessary to bring the steam into the cylinder . '007 
 Force necessary to drive the steam into the atmosphere "007 
 Loss from cooling in the cylinder and in the pipes . . *016 
 
 Friction of the piston, losses, and waste *200 
 
 Force necessary for the opening of the valves and the 
 
 friction of the different parts of the engine . . . *062 
 Loss in consequence of the steam being intercepted 
 
 before the end of the stroke -100 
 
 392 
 
 Reflecting that the numbers here given express 
 fractions of the total gross power of the engine, 
 we shall immediately be convinced of the impossi- 
 bility of admitting such estimates. If, for in- 
 stance, the engine had a useful effect of 100 horse- 
 power, which, from the coefficient, supposes a gross 
 effect of 200, it would require the power of 12 
 horses to move the machinery, of 40 to draw the 
 piston, &c. The exaggeration is self-evident. 
 
 Besides, in applying this estimate of the fric- 
 tions to a locomotive engine, which is also a high- 
 pressure engine, and supposing it to work at 60 
 Ibs. effective pressure, or at 75 Ibs. total pressure 
 per square inch, we perceive that, were the cy- 
 
 * Tredgold, Treatise on Steam-engine, Article 367. 
 
OF THE ORDINARY MODE OF CALCULATION. 13 
 
 linders 12 inches in diameter, or their surface 
 226 square inches, the force here reckoned as 
 representing the friction of the piston would he 
 226 x 75 x -20 = 3390 Ibs. Now, our own experi- 
 ments on the friction of the mechanical organs of 
 the locomotive engine the Atlas, which has those 
 dimensions and which works at that pressure, de- 
 monstrate that the force requisite to move, not only 
 the pistons, hut all the other mechanical organs, 
 including wastes, if it be true that such exist in 
 an engine in good order, is but 48 Ibs. applied to 
 the wheel, or 48x5'9 = 283 Ibs. applied to the 
 piston *. 
 
 It is impossible, therefore, to admit estimates 
 so exaggerated as these ; and what will it be, when 
 it becomes necessary to explain a loss, not merely 
 of half, but of two thirds of the effect produced, as 
 required by the coefficient *33, adopted by many 
 in practice, particularly for locomotive engines? 
 
 2d. This deduction, moreover, of f , considerable 
 as it is, in very many cases does not suffice to 
 harmonize the practical effects with the effects 
 called theoretical. 
 
 In Wood's Treatise on Railroads, second edition, 
 pp. 277 284, appears the calculation of five steam- 
 engines, not locomotive but stationary, two working 
 at low pressure and three at high pressure, in which 
 the real effects are to the theoretic, in the pro- 
 
 Treatise on Locomotive Engines, Second Edition. 
 
14 CHAPTER I. 
 
 portions represented by the following coefficients : 
 
 26 -29 -31 -27 and -30. 
 
 Here then are examples, wherein it would be neces- 
 sary to explain a loss of three fourths of the total 
 power of the engine. 
 
 But, with respect to locomotives, the difficulty 
 becomes still greater, for it often occurs, when the 
 load of the engine is light, that it would be neces- 
 sary to apply a coefficient less even than '25 ; 
 and yet in these engines the steam passages have 
 an area of instead of -$ of the area of the cylin- 
 der ; they are immerged in the steam of the boiler 
 itself, which precludes all possibility of waste ; the 
 cylinders are in contact with the flame issuing from 
 the fire-box, which entirely prevents condensation. 
 Thus there remains only the friction to explain the 
 enormous loss sustained, of more than *75 of the 
 total power ; and this friction measured in our own 
 experiments, as will be seen farther on, never rises 
 above *18 of what is termed the theoretic effect of 
 the engine. 
 
 3d. It has just been said that, in locomotive 
 engines, the coefficient would in certain cases sink 
 below '&5. But there are other cases on the 
 contrary, wherein, for the same engine, it would 
 rise to *80, examples of which may be seen in our 
 Treatise on Locomotives, in all the cases when the 
 engine drew a heavy load. Thus all the loss 
 hitherto so laboriously explained disappears on a 
 sudden. 
 
 4th. The measure of the theoretic effect of the 
 
OF THE ORDINARY MODE OF CALCULATION. 15 
 
 engine results from three elements, to wit : the sur- 
 face of the piston, the pressure of the steam, and the 
 velocity of the motion. The causes which are said to 
 explain the reduction to which this theoretic effect 
 is liable, are : first, the friction of the engine, then 
 the contraction of the passages, their changes of 
 direction, the friction of the steam, its waste and its 
 condensation. Now of the last five causes, the 
 condensation is the only one that can diminish the 
 pressure of the steam during its passage, and that 
 condensation is almost entirely obviated by the 
 precautions used in practice : all the remaining 
 causes of reduction act merely on the velocity. If 
 then these causes produce definitively a reduction 
 in the theoretic effect, it can only be by reason of 
 their action on the velocity. 
 
 But, to calculate, by this method, the theoretic 
 effect of an engine, the area of the piston is multi- 
 plied first by the pressure of steam in the boiler, 
 which gives the theoretic effort ; then this result 
 is multiplied, not by the theoretic velocity of the 
 engine which is unknown, but by its observed or 
 practical velocity. All reduction then applicable 
 to the theoretic velocity is, by the proceeding itself, 
 already made in the calculation, and cannot be in- 
 troduced into it anew. 
 
 Consequently, if notwithstanding the use of the 
 practical velocity in the calculation, it be still ne- 
 cessary to retrench f or from the result obtained, 
 that loss of of the total effect must be due wholly 
 to the friction, which is evidently impossible. 
 
16 CHAPTER I. 
 
 It is clear, then, that these inexplicable differences 
 between the theory and the facts, can arise only 
 from an error in the theory itself; that the results 
 obtained from it are to be considered at most but 
 as approximations, and not as being proper to 
 determine, in an exact and analytical manner, either 
 the effects or the proportions of steam-engines. 
 
 SECT. III. Formula proposed by divers authors to 
 determine the velocity of the piston under a given 
 load ; and proofs of their inaccuracy. 
 
 We have said that, in the above theory, the velo- 
 city of the piston under a given load, had not been 
 made the object of a special research. Some essays 
 had, however, been made to estimate this, but in a 
 different manner. 
 
 1st. Tredgold, in his Treatise on Railways, (p. 
 83,) proposes a formula, without, however, esta- 
 blishing it in any way on reasoning or on fact. 
 This formula is as follows : 
 
 W 
 
 V is the velocity of the piston in feet per minute ; 
 / the stroke of the piston ; P the effective pressure 
 of the steam in the boiler ; and W the resistance 
 of the load. This formula, he says, will give the 
 velocity of the piston. But as no mention is made 
 therein, either of the diameter of the cylinder, or 
 of the quantity of steam furnished by the boiler per 
 minute, it clearly cannot give the velocity required ; 
 
OF THE ORDINARY MODE OF CALCULATION. 17 
 
 for if it could, the velocity of an engine would be 
 the same with a cylinder of 1 foot diameter as with 
 a cylinder of 4 feet, though the latter expends 16 
 times as much steam as the former. The heating 
 surface, or evaporation of the boiler, would be 
 equally indifferent. An engine would not move 
 faster with a boiler evaporating a cubic foot of 
 water per minute, than with another that should 
 evaporate but ^ or ^V. Hence this formula is un- 
 founded. 
 
 2d. Wood, in his Treatise on Railways, (2nd 
 edition, p. 351,) proposes, also without discussing 
 it, the following formula : 
 
 W 
 
 V is the velocity of the piston in feet per minute, 
 / the stroke of the piston, W the resistance of the 
 load, and P the surplus of pressure in the boiler 
 above what is necessary to balance the resistance 
 W. This formula, containing no term to represent 
 either the diameter of the cylinder or the evaporat- 
 ing force of the boiler, is, like the preceding, de- 
 monstrated inexact d priori. 
 
 We know of no other attempt made to attain the 
 solution of this problem, and these are altogether 
 unsatisfactory. 
 
 * In the third edition of his work, which is just published, 
 1838, Mr. Wood abandons this formula to adopt ours, as also 
 our theory in general, developed in the Treatise on Locomotive 
 Engines, 1835. 
 
 C 
 
18 CHAPTER I. 
 
 Consequently, of the three fundamental problems 
 which we have presented, two have received in- 
 accurate solutions hy means of coefficients, and the 
 third remains unsolved. 
 
 SECT. IV. View of the theory proposed. 
 
 We have so far demonstrated that there exists 
 no analytical formula, nor any exact means for cal- 
 culating the effects of steam-engines, and, conse- 
 quently, for determining the proportions proper to 
 he used in their construction, to ohtain desired ef- 
 fects. A great number of engines are constructed 
 and intended to fulfil required conditions ; hut the 
 truth is that, unless they have been designed after 
 others already executed, their precise effects are 
 not known till they are submitted to trial after 
 construction, that is, when it is too late to apply 
 a remedy. In a machine, the most powerful of all 
 known, and which is tending to become almost uni- 
 versal, errors cannot be without importance. Not 
 only such errors have been frequently prejudicial to 
 vast manufacturing enterprises, and the occasion of 
 difficulties between the builders and the purchasers 
 of engines, but they have also compromised the 
 lives of travellers ; for when a steam- vessel has 
 been found incapable of accomplishing its destined 
 task, the only remedy to the evil that has occurred 
 to the engineer, has been that of overloading, or 
 even of making fast, the safety valve, and frightful 
 explosions have often been the result. No doubt 
 
OF THE ORDINARY MODE OF CALCULATION. 19 
 
 then can be entertained as to the usefulness of new 
 researches on the subject. 
 
 After having exposed the state of science in what 
 concerns the theory of the steam-engine, it remains 
 to shew on what principles is grounded the theory 
 we are about to present. 
 
 We shall first explain this theory in all its sim- 
 plicity, supposing the steam to preserve the same 
 temperature throughout its action in the engine, 
 limiting ourselves to rotative engines without ex- 
 pansion, and taking our basis on the consideration 
 merely of the uniform motion which the engine 
 necessarily attains after a very short lapse of time. 
 A great number of theoretical and practical proofs 
 will then corroborate the accuracy of our reasoning; 
 and, finally, in the subsequent chapters, we shall 
 resume the theory in all its generality, so as to ren- 
 der it applicable to all systems of steam-engines, 
 taking into account the circumstances neglected in 
 our first exposition. 
 
 It is well known that in every machine, the effort 
 of the mover first being superior to the resistance, 
 a slow motion is produced, which accelerates gra- 
 dually till the machine attains a certain velocity 
 which it does not surpass, the mover being incapa- 
 ble of sustaining a greater velocity with the mass it 
 has to move ; the machine having once attained 
 this point, which requires but a very little time, the 
 velocity continues the same, and the motion remains 
 uniform during the rest of the work. It is but from 
 this moment, viz., the commencement of uniform 
 
20 CHAPTER I. 
 
 motion, that the effects of machines begin to be 
 calculated, and the few minutes during which the 
 velocity regulates itself, or the transitory effects from 
 the .velocity null to uniform velocity, are always 
 neglected. 
 
 Now in every machine which has attained a uni- 
 form motion, the power is strictly in equilibrio with 
 the resistance ; for were it greater or less, there 
 would be acceleration or retardation of motion, 
 which is not the case. In a steam-engine, the force 
 applied by the mover is no other than the pressure 
 of the steam against the piston or in the cylinder. 
 This pressure then, in the cylinder, is strictly equal 
 to the resistance opposed by the load against the 
 piston. 
 
 Consequently, the steam in passing from the 
 boiler into the cylinder changes its pressure, as- 
 suming that which represents the resistance to the 
 piston. This principle, of itself, explains all the 
 theory of the steam-engine, and in a manner lays its 
 play open. 
 
 It becomes, in fact, easy to render an account of 
 what passes in a steam-engine set in motion. The 
 steam confined in the boiler, at a certain degree of 
 pressure, as soon as the regulator or distributing- 
 cock is open, rushes into the steam-pipes, and from 
 thence into the cylinders. Arriving in the cylinder, 
 whose area is much greater than that of the pipes, 
 the steam dilates at first, losing proportionally a 
 part of its elastic force ; but as the piston is as yet 
 immoveable, and as the steam continues to flow in 
 
OF THE ORDINARY MODE OF CALCULATION. 21 
 
 rapidly, the balance of pressure is soon established 
 between the two vessels ; and the piston, urged by 
 all the force of the steam, begins slowly to move. 
 The fly-wheel of the engine, its entire machinery, 
 and the resistance opposed to it, begin then to 
 acquire a small velocity, which accelerates by in- 
 sensible degrees ; and, if at the end of the stroke 
 of the piston, the coming vapour were suddenly 
 withheld, the piston would not stop instantaneously 
 on that account ; it would itself be impelled for 
 some time by the effect of the velocity previously 
 communicated to the mass. The result of this is, 
 that at the following stroke, the steam finds the 
 piston already slowly receding, at the moment it 
 impresses thereon a new quantity of motion ; 
 which again passes on to the fly-wheel, and to the 
 total mass, where it continues to accumulate. Re- 
 ceiving thus, at every stroke, a new impulse, the 
 piston accelerates its motion by degrees, and, at 
 length, acquires all the velocity the motive power 
 is capable of communicating to it. 
 
 During all this time the steam continues to be 
 generated in the boiler with the same rapidity, and 
 to flow into the cylinder ; but as the piston ac- 
 quires a quicker motion and developes a greater 
 volume before the steam, the latter dilates, assum- 
 ing a lower pressure, till at length, the piston having 
 assumed all the velocity that the steam can impress 
 upon it, with the load that it supports, the pressure 
 of the steam in the cylinder becomes equal to the 
 resistance of the piston, and the motion remains 
 in a state of uniformity, as has been said above. 
 
22 CHAPTER I. 
 
 Thus, from what precedes, we have the pressure 
 which the steam really exercises against the piston ; 
 so that if P' represent that pressure per unit of sur- 
 face, and if R represent the resistance of the load 
 against the piston, divided in like manner per unit 
 of surface, the condition of the uniformity of motion 
 will furnish the first equation of analogy 
 
 F=R. 
 
 This equation establishes the intensity of the 
 effort exerted by the power. Were the case merely 
 one of equilibrium, this determination would suffice ; 
 but in a case of motion, not only the intensity of 
 the force is to be considered, but also the velocity 
 with which it is applied. Now, in the case before 
 us, it is evidently the velocity of production of 
 steam in the boiler, which indicates the velocity with 
 which the above force is renewed or applied. To 
 this latter element then of calculation we must 
 recur, in order to obtain a second relation among 
 the data of the problem, comprising the velocity of 
 the motion. 
 
 This relation will be furnished by the consider- 
 ation, that there is necessarily an equality between 
 the quantity of steam produced and the quantity 
 expended, a proposition which is self-evident. If, 
 then, we continue to express by S the volume of 
 water evaporated in the boiler per unit of time, and 
 effectively transmitted to the cylinder, and by m 
 the ratio of the volume of the steam formed under 
 the pressure P of the boiler, to the volume of water 
 that has produced it, it is clear that 
 
 mS 
 
OF THE ORDINARY MODE OF CALCULATION. 23 
 
 will be the volume of steam formed per unit of 
 time, and under the pressure P, in the boiler. This 
 steam passes into the cylinder and there assumes 
 the pressure P'; but if it be supposed that the 
 steam in this movement preserves its temperature, 
 which, as to the engines under consideration, will 
 make but little change in the^ results, the steam, 
 in passing from the pressure P to the pressure P', 
 will increase its volume in the inverse ratio of the 
 pressures. Thus, transmitted to the cylinder, the 
 volume mS of steam, supplied at each unit of time 
 by the boiler, will become 
 
 On the other hand, v being the velocity of the pis- 
 ton and a the area of the cylinder, av will be the 
 volume of steam expended by the cylinder per unit 
 of time. Therefore, on account of the equality 
 necessarily existing between the production and 
 the expenditure of steam, we shall have the relation 
 
 which is the second relation sought. 
 
 Consequently, by eliminating P' from these two 
 equations, we have as a definitive analytical relation 
 among the various data of the problem, 
 
 mS P 
 '~T' R' 
 
 This relation, which is very simple, suffices for 
 the solution of all questions relative to the deter. 
 mining of the effects or the proportions of steam- 
 
CHAPTER I. 
 
 engines. As we shall develop its terms hereafter, 
 on taking it up in a more general manner, we will 
 leave it for the present under this form, which will 
 render the discussion of it more easy and more 
 clear. 
 
 We have, then, the velocity which the piston of 
 an engine will assume under a given resistance R. 
 If, on the contrary, the velocity of the motion he 
 supposed known, and it be required to calculate 
 what resistance the engine may move at that velo- 
 city, it will suffice to solve the same equation with 
 reference to R, and we shall have 
 
 av 
 
 Finally, supposing the velocity and the load to 
 be given, and that it be required to know the eva- 
 poration proper for the boiler, that the given 
 load may be set in motion at the desired velocity, 
 the value of S must be drawn from the same rela- 
 tion, and it will be 
 
 avR 
 o= . 
 mr 
 
 We limit our deductions here, because, as has 
 been already observed, these three problems are 
 the basis of all problems that can be proposed on 
 steam engines, and that they are sufficient, more- 
 over, to enable us to establish our theory and to 
 compare it with the mode of calculation now in 
 use. But on resuming the same questions with 
 more detail in the following parts of the work, we 
 shall give to the equations their full development, 
 
OF THE ORDINARY MODE OF CALCULATION. 25 
 
 and treat of all the other accessory determina- 
 tions which occur in problems relative to steam 
 engines. 
 
 From what has been stated, it plainly appears 
 that we ground all our theory on these two incon- 
 testable facts ; 1st, that the engine having at- 
 tained uniform motion, there is necessarily equi- 
 librium between the power and the resistance, that 
 is, between the pressure of the steam in the cylin- 
 der, and the resistance against the piston ; which 
 furnishes the first relation 
 
 F = R. 
 
 And 2dly, that there is also a necessary equality 
 between the production of the steam and its ex- 
 penditure, which furnishes the second relation 
 
 mS P 
 
 V = . ,. 
 
 a P 
 
 And these two equations suffice for the solution of 
 all the problems. 
 
 SECT. V. New proofs of the accuracy of the theory 
 proposed, and of the inaccuracy of the ordinary 
 theory. 
 
 As we shall draw from the examination of loco- 
 motive engines, the greater part of the considera- 
 tions we are now about to offer on the two theories, 
 we will first observe, with respect to those engines, 
 that we look upon them as being incontestably 
 more proper than all others to make known the 
 true theory of the motion and action of steam. 
 The reasons of this preference are, 1st, that those 
 
26 CHAPTER I. 
 
 engines are of a remarkable simplicity ; 2d, that 
 the determination of the resistance which they have 
 to move is easy, and susceptible of great exacti- 
 tude, since it consists merely in weighing the train 
 they have to draw ; whereas to estimate the resist- 
 ance opposed to stationary engines often requires 
 calculations both various and uncertain ; 3d, that 
 the friction of locomotive engines is known from 
 our own experiments, and with a degree of pre- 
 cision that seems to be trustworthy, since that fric- 
 tion has been determined by several methods which 
 have served to verify each other ; 4th, that it is 
 easy to observe a locomotive engine under a hun- 
 dred circumstances different from each other, by 
 varying at pleasure the load and the velocity, 
 which may be done in very wide limits ; whereas 
 in stationary engines, it happens most frequently 
 that the resistance to be moved is incapable of 
 variation, whence results that the steam is never 
 seen to act but in one manner, and thus the study 
 of those engines reduces itself nearly to that of one 
 particular case. 
 
 To return to the theory we have exposed, it 
 visibly rests chiefly on this, that though the steam 
 be formed in the boiler at a certain pressure P, 
 yet in passing into the cylinder it assumes a pres- 
 sure R, strictly determined by the resistance 
 against the piston, whatever else may be the 
 pressure in the boiler ; so that, according to the 
 intensity of that resistance, the pressure in the 
 cylinder, far from being always equal to that of 
 the boiler, or from differing always from it in any 
 
OF THE ORDINARY MODE OF CALCULATION. 27 
 
 constant ratio, as is believed, may at times be fully 
 equal to it, and at other times considerably differ- 
 ent. Thus, when in the ordinary theory, the cal- 
 culation is performed under the supposition that 
 the steam acts in the cylinder at the pressure of 
 the boiler, an error often very considerable, and 
 independent of all the real losses to which the 
 engine is liable, is introduced into the calculation ; 
 since a force is considered as applied, which is two 
 or three times greater than the real one. No won- 
 der then it became necessary to use a coefficient 
 ^ or ^, which makes the supposed losses of the 
 engine appear enormous, whereas the real error is 
 in the very basis of the calculation itself. 
 
 We have already proved this mode of action of 
 the steam in the cylinder, from the consideration 
 of uniform motion ; but in examining what takes 
 place in the engine, we shall presently find many 
 other proofs. 
 
 1st. The steam being generated at a certain de- 
 gree of pressure in the boiler, passes into the 
 steam-pipe, and from thence into the cylinder ; 
 there it dilates at first, because the area of the 
 cylinder is ten or twenty times that of the pipe, 
 and it would quickly rise to the same degree of 
 pressure as in the boiler, were the piston immove- 
 able. But as the piston, on the contrary, opposes 
 only a certain resistance determined by the load 
 imposed on the engine, 40 Ibs. for instance, per 
 square inch, it will give way as soon as the elastic 
 force of the steam in the cylinder shall have at- 
 
28 CHAPTER I. 
 
 tained that point. A piston sustaining a resist- 
 ance of 40 Ibs. per square inch is nothing more 
 than a valve loaded with 40 Ihs. per square inch. 
 Were the communication perfectly free between 
 the boiler and the cylinder, without tube or con- 
 traction, so that the two vessels should form but 
 one, the piston would become a real valve to the 
 boiler ; and that valve yielding before the safety 
 valve, which is loaded perhaps at 50 Ibs. per square 
 inch, the steam could never rise in the boiler above 
 the pressure of 40 Ibs. per square inch. Since the 
 communication between those two vessels is not 
 wholly free, the piston is not a valve to the boiler, 
 but it still continues to be one to the cylinder. 
 Wherefore the pressure in the cylinder can never 
 exceed the resistance of the piston. 
 
 2dly. Another consideration will readily prove 
 to us again that the pressure of the steam in the 
 cylinder must necessarily be regulated, not by the 
 pressure in the boiler, but by that of the resist- 
 ance. In fact, were it actually true that the 
 steam be expended in the cylinder, either at the 
 pressure of the boiler, or at any other pressure 
 that were in any fixed ratio whatever to that of 
 the boiler, then, since the quantity of steam raised 
 per minute in the boiler would be expended 
 by the cylinder at one and the same pressure in 
 all cases, and would consequently fill the cylin- 
 der a fixed number of times in a minute, it 
 would follow that the engine, so long as it should 
 work with the same pressure in the boiler and the 
 
OF THE ORDINARY MODE OF CALCULATION. 29 
 
 same apertures or steam passages, would assume 
 the same velocity with all loads. Now, we see 
 that the very contrary takes place ; for, the lighter 
 the load, the greater becomes the velocity of the 
 engine. 
 
 The effect produced is explained easily, in con- 
 sidering what really passes in the engine. If it 
 be supposed that the evaporation, producing, for 
 instance, 200 cubic feet of steam per minute at 
 the pressure of the boiler, be sufficient to fill the 
 cylinder 200 times, when the piston is loaded with 
 the resistance R ; as soon as that resistance R shall 
 be replaced by a resistance ^R, the same mass of 
 steam assuming in the cylinder a pressure of only 
 half of what it was before, will furnish 400 cy- 
 linders full of steam per minute at the new pres- 
 sure. It is then clear that the resistance ^R will 
 be set in motion with a velocity double that of the 
 resistance R ; which does in fact accord with ob- 
 servation, if, in estimating that resistance, account 
 be taken of all the partial resistances and frictions 
 really opposed to the motion of the engine. 
 
 3dly. Applying the same reasoning inversely, 
 we see that; were the pressure of the cylinder in 
 a fixed ratio with that of the boiler, or were- it 
 constant so long as that of the boiler remained the 
 same, then in calculating the effort of which the 
 engine is capable, this would always be found the 
 same, whatever might be the velocity of the pis- 
 ton. Thus, at any velocity whatever the engine 
 would always be capable of drawing the same load. 
 Now, this result again is contrary to experience ; 
 
30 CHAPTER I. 
 
 and the reason of it is that the greater the velocity 
 of the piston, the lower the pressure in the cylin- 
 der ; whence results, that the load the engine is 
 capable of moving diminishes in the same propor- 
 tion. 
 
 4thly. Another proof no less evident is easily 
 adduced. Were it true that the steam be ex- 
 pended by the cylinder at a pressure equal to that 
 of the boiler, or in any fixed ratio to it, indicated 
 by any coefficient whatever, since any one locomo- 
 tive engine requires the same number of turns of 
 the wheel, or the same number of strokes of the 
 piston to traverse the same distance, it would fol- 
 low that so long as these engines work at the same 
 pressure, they ought in all cases to consume the 
 same quantity of water for the same distance. 
 Now, the quantity of water evaporated, far from 
 being constant, decreases, on the contrary, with 
 the load, as may be seen in the experiments we 
 have published on this subject. The Atlas engine, 
 for instance, evaporated 132 cubic feet of water in 
 drawing 195*5 tons, and 95 cubic feet only in 
 drawing 127*6 tons. Since the same number of 
 cylinders-full of steam was expended in each case, 
 the steam of the first must have been of a density 
 different from that of the second ; and here again 
 it is manifest that, notwithstanding the equality of 
 the pressure in the boiler, and of the opening of 
 the regulator in the two cases, the density of the 
 expended steam followed the intensity of the re- 
 sistance, that is to say, the pressure of steam in 
 the cylinder was regulated by the resistance. 
 
OF THE ORDINARY MODE OF CALCULATION. 31 
 
 5thly. From the same cause, since the consumption 
 of fuel must be in proportion to the evaporation 
 effected, it would follow too, were the ordinary 
 theory exact, that the quantity of fuel consumed by 
 a given engine would always be the same for the 
 same distance, whatever might be the load. Now, 
 we find again, by experiment, that the quantity of 
 fuel, on the contrary, diminishes with the load, 
 conformably to the explanation we have given of 
 the effects of the steam in steam-engines. 
 
 6thly. It is clear, moreover, that if the pressure in 
 the cylinder were, as it is thought, constant for a 
 given pressure in the boiler, then after an engine 
 has been found capable of drawing a certain load 
 with a certain pressure, and of communicating to it 
 a uniform motion, it would follow that the same 
 engine could never draw a less load with the same 
 pressure in the boiler, without communicating to it 
 a velocity indefinitely accelerated ; since the power 
 having been found equal to the resistance in the 
 first case, would necessarily be superior to the 
 resistance in the second. Now, experience proves 
 that in the second case the motion is quicker, but no 
 less uniform than in the first ; and the reason is, 
 that though the steam be generated in the boiler at 
 a pressure more or less elevated, which matters little, 
 yet in passing into the cylinder it always assumes 
 the pressure of the resistance ; whence results that 
 the power is no more superior to the resistance in 
 the second case than in the first, and that the 
 motion ought therefore to remain uniform. 
 
3 CHAPTER 1. 
 
 7thly and lastly. On looking over our experiments 
 on locomotives, the same engine will be seen some- 
 times drawing a very light load with a high pressure 
 in the boiler, and sometimes, on the contrary, a 
 very heavy load with a low pressure. It is then 
 impossible to admit, as the ordinary theory would 
 have us, that there is any fixed ratio whatever 
 between the two pressures. This effect, moreover, 
 is most easy to explain ; for it depends simply on 
 this, that in both cases the pressure in the boiler 
 was superior to the resistance against the piston, 
 and no more was needful in order that the steam, 
 generated at that pressure, or at any other fulfilling 
 merely that condition, might, on passing into the 
 cylinder, assume the pressure of the resistance. 
 
 It is moreover to be observed, that these effects 
 cannot take place in a locomotive steam-engine, 
 without equally occurring in a stationary one ; for 
 the steam acts in the same manner in the cylinders 
 of both, and it is unimportant whether during the 
 action of that steam, the engine moves or remains 
 at rest, or whether its own weight do or do not 
 form a part of the load imposed on the piston. 
 
 All these proofs then establish clearly that the 
 pressure of the steam in the cylinder, is strictly 
 regulated by the resistance on the piston and by 
 nothing else ; and that all methods, like that of 
 the coefficients, founded on the principle of its 
 being in any fixed ratio whatever with the pressure 
 in the boiler, are necessarily inaccurate. 
 
 It is, however, essential to observe, that we wish 
 
OF THE ORDINARY MODE OF CALCULATION. 33 
 
 to establish by these reasonings, that, since the 
 pressure in the cylinder is fixed d priori, it cannot 
 depend on the pressure of the boiler ; but we believe, 
 on the contrary, as will be seen, Sect. VI L, that the 
 pressure in the cylinder being once regulated by 
 the resistance on the piston, that of the boiler after- 
 wards depends on it, in proportion to the size of 
 the passages, the volume of steam produced, and 
 the weight of the safety-valves. It would only be 
 for want of making this needful distinction, that 
 we could be thought to admit an entire independ- 
 ence between the two pressures. 
 
 SECT. VI. Comparison of the two theories in their 
 application to particular examples. 
 
 The foregoing already establishes sufficiently, in 
 principle, the accuracy of the theory which we 
 propose, and the inaccuracy of that which has 
 hitherto been made use of. It may, however, be 
 thought by some that the inaccuracy alleged against 
 the latter is of trifling importance, and that in 
 practical examples it gives results very near the 
 truth at least, if not quite correct. We are now, 
 therefore, about to submit that method as well as 
 our own to the scrutiny of practice. When in 
 action together, the difference of the results to 
 which they lead will be apparent, and it will be 
 recognized which of the two is more in harmony 
 with the facts ; and finally, a clear idea will be 
 
34 CHAPTER I. 
 
 formed of the causes from whence the errors of the 
 ordinary theory derive. 
 
 The coefficient of correction for high pressure 
 steam-engines without expansion and without con- 
 densation, not being fixed to the same amount by 
 the authors who have treated on these subjects, sup- 
 pose it be attempted to determine it from the two 
 following facts of which we were eye-witnesses. 
 
 1. The locomotive engine Leeds, which has two 
 cylinders of 11 inches diameter; stroke of the piston, 
 16 inches ; wheels, 5 feet ; weight, 7*07 tons ; drew 
 a load of 88*34 tons, ascending a plane inclined 
 TiVo, at the velocity of 20*34 miles per hour ; the 
 effective pressure in the boiler being 54 Ibs. per 
 square inch, or the total pressure 68*71 Ibs. per 
 square inch. 
 
 2. The same day the same engine drew a load 
 of 38*52 tons, descending a plane inclined ToV*, at 
 the velocity of 29*09 miles per hour ; the pressure 
 in the boiler being precisely the same as in the 
 preceding experiment, and the regulator opened to 
 the same degree. These experiments are given, 
 pages 233, 234, of our Treatise on Locomotives 
 (1st edition). 
 
 Reckoning, on the one part, the theoretical effort 
 applied on the piston according to the ordinary 
 calculation ; and on the other part, the effect 
 really produced, viz., the resistance of the load 
 together with that of the air against the train, we 
 find, referring the area of the pistons and the 
 pressure to the square foot : 
 
OF THE ORDINARY MODE OF CALCULATION. 35 
 
 1st Case. Theoretical effort applied to the piston, ac- 
 cording to the ordinary calculation, ibs. 
 
 1-32 X (6871X144) 13060 
 
 Real effect 8846 
 
 Coefficient of correction *68 
 
 2d Case. Theoretical effort, the same as above . . 13060 
 Real effect 6473 
 
 Coefficient of correction * *50 
 
 The mean coefficient between the two is %59. 
 We thus find three different coefficients. Let the 
 first be chosen, an error will be made in the second 
 case ; let the second be preferred, and an error will 
 be made in the first case. Let the third be ad- 
 
 * The detailed calculation of the effects produced is this : 
 
 Ibs. 
 
 1st Case. Resistance of the 88-34 tons, at 7 Ibs. per ton 
 
 (v. 2d edition of Treatise on Locomotives) . 618 
 
 Gravity of 95-41 tons (train and engine) on a 
 plane rising y^Vs 164 
 
 Resistance of the air against the train, at the 
 velocity of the motion 134 
 
 Resistance overcome at the velocity of the wheel 916 
 
 And as that resistance is here measured at the velocity of the 
 wheel, it produced against the piston, a force augmented in the 
 inverse ratio of the velocities of the piston and of the wheel, 
 
 that is to say a resistance of 916x5-9 5404 
 
 Add for the pressure of the atmosphere against 
 the piston, the engine being a high pressure 
 
 one, 1-32 X (14-71x144) 2796 
 
 And for the pressure caused by the blast-pipe 
 1-32 X (3-4x144) 646 
 
 Total resistance against the piston, exclusive of friction, 8846 
 
 n 2 2d Case 
 
36 CHAPTER I. 
 
 mitted, and the error will only be divided between 
 the two cases. In any way error is inevitable, and 
 that, of itself, suffices to prove that any method, 
 like the ordinary theory, which consists in employ- 
 ing a constant coefficient, is necessarily erroneous, 
 whatever be the coefficient adopted, and to what- 
 ever system of engine the application be made ; for 
 it is evident that the same fact would occur in 
 every species of steam-engine. It might only be 
 less striking, if the velocities in the two instances 
 were less different, and that is what has hitherto 
 prevented the error of that method from being 
 perceived ; for all the engines of the same system 
 being imitated from each other, and working nearly 
 at the same velocity, from a factitious limit that had 
 been imposed on the velocity of the piston, the 
 same coefficient of correction appeared tolerably 
 to suit them all. 
 
 In stationary engines, moreover, it was impos- 
 sible, for want of precise determinations of the 
 
 2d Case. Resistance of the 38-52 tons . . . 270 
 Gravity on the plane descending T ^^ . 93 
 
 177 
 Resistance of the air ..... 282 
 
 Resistance against the wheel . . . 459 
 Or against the piston, 459x5-9 .... 2708 
 Add for the pressure of the atmosphere . . 2796 
 And for the pressure caused by the blast-pipe 
 1 -82 X (5-1x144) ......... 969 
 
 Total resistance against the piston, exclusive of friction, 6473 
 
OF THE ORDINARY MODE OF CALCULATION. 37 
 
 frictions, to separate that part of the result which 
 is really attributable to them, from that part which 
 constitutes a veritable error. But here we easily 
 obtain the conviction that neither of these co- 
 efficients of correction represent, as we are told, 
 the frictions, losses, and divers resistances of the 
 engine ; for direct experiments on the engine under 
 consideration, which are given in our Treatise on 
 locomotives, enable us to estimate separately all 
 those frictions and resistances. Now from these 
 experiments, the friction of that engine, when iso- 
 lated, is equivalent to a force of 82 Ibs. applied to 
 the wheel, and that friction afterwards augments by 
 1 Ib. for every ton of load added to the engine. 
 
 Besides, the losses to which the engine is liable 
 either from condensation or from waste of steam 
 (during its passage from the boiler to the cylinder) 
 are nothing, or at least inconsiderable. 
 
 It becomes easy, then, with these elements, to 
 estimate the amount of the frictions really ex- 
 perienced by the engine. Now in calculating them 
 separately thus, we find : 
 
 1st Case. Friction 1257 Ibs., or '10 of the theoretical result. 
 2cl Case. Friction 873 Ibs., or -07 of the theoretical result *. 
 
 * We have in fact : its. 
 
 1st Case. Friction of the engine without load . 82 
 Additional friction for a resistance 
 
 equivalent to ?15 = 131 tons . . . 131 
 
 213 
 
 Which produces against the piston 213 X 5*9 . 1257 
 
 2d Case. 
 
38 CHAPTER I. 
 
 Thus it appears that in these two cases the 
 frictions omitted in the calculation amounted really 
 to hut 10 and 7 hundredths of the theoretic result ; 
 and if sV or '05 of loss he added for the filling up of 
 the vacant spaces in the cylinder, which we have 
 not heen ahle to estimate in Ibs., they will amount 
 to *15 and *12 ; whereas the coefficients of correc- 
 tion raise them to *32 on the one part, and to 
 50 on the other ; that is, from two to four times 
 what they really are. Deducting, then, from these 
 coefficients, the true value of the frictions and 
 losses, it will be found that the theoretical error 
 which this method brings into the calculation, under 
 the denomination of friction, is 17 per cent, of the 
 total force of the engine in one case, and 38 per 
 cent, in the other. 
 
 It will now be observed that, from the preceding 
 determinations, viz., of the resistances first and then 
 of the frictions, we have for each of the two cases 
 which occupy our attention, the sum of the total 
 effects really produced by the engine, that is : 
 
 Ibs. 
 
 1st Case. Resistances 8846 
 
 Frictions 1257 
 
 10103 
 
 Ibs. 
 
 2(1 Case. Friction of the engine without load . 82 
 Additional friction for a resistance 
 equivalent to = 66 tons ... 66 
 
 148 
 Which produces against the piston 148 x 5-9 . 873 
 
OF THE ORDINARY MODE OF CALCULATION. 39 
 
 Ibs. 
 
 2d Case. Resistances \ 6473 
 
 Frictions 873 
 
 7346 
 
 We are now, therefore, enabled to compare these 
 effects produced, with the results, either of the or- 
 dinary calculation, or of that which we propose to 
 substitute for it. 
 
 1st. In applying, first, the ordinary calculation 
 with the mean coefficient %59 determined above, 
 and comparing its result with the real effect, we 
 find: 
 
 Ibs. 
 
 1st Case. Effort applied on the piston, from the ordinary 
 
 calculation, 1-32 X (68-71 X 144) X -59 . * . 7705 
 Effect produced, including all frictions and re- 
 sistances 10103 
 
 Error, over and above the frictions and resistances, 2398 
 
 2d Case. Effort applied on the piston, from the ordinary 
 
 theory, the same as above 7705 
 
 Effect produced, including, &c 7346 
 
 Error, besides frictions and resistances .... 359 
 Mean error of the two cases 1378 
 
 It is plain, then, that there would be risk of a 
 very great error in attempting to calculate the ef- 
 fects of this engine with the coefficient '59 ; but 
 it is plain, moreover, that in applying any other 
 coefficient whatever, the error would only be trans- 
 ferred from the one case to the other, without ever 
 disappearing; and thus, in fact, the coefficient %59 
 has almost rendered null the error of the second 
 case above, by transferring it to the first. 
 
40 CHAPTER I. 
 
 To apply now our formula relative to the same 
 problem, viz., 
 
 nothing more is requisite than to replace the let- 
 ters m, S, P, and v by their respective values, 
 taking care only to refer all the measures to the 
 same unit. 
 
 Thus, P is the total pressure of the steam in 
 the boiler, viz., 68*71 Ibs. per square inch, or 
 68*71 x 144 Ibs. per square foot. 
 
 m is the ratio of the volume of the steam, at the 
 total pressure of 68*71 Ibs. per square inch, to the 
 volume of the same weight of water ; and, accord- 
 ing to tables which will be given in the following 
 chapter, m = 411. 
 
 S is the volume of water evaporated per minute, 
 and converted to use in the cylinders. Now, during 
 the journey, of which the first of these experiments 
 was a part, the engine evaporated 60*52 cubic feet 
 of water per hour, ( Treatise on Locomotives, 1st 
 edition, page 175) ; which, after a deduction of 
 for the loss of steam by the safety valve, measured, 
 as explained in section VII., and of A on the rest 
 for the filling up of the vacant spaces in the cylin- 
 der, leaves an effective evaporation of *77 cubic 
 feet of water per minute. We here then have 
 
 Finally, v is the velocity of the piston ; and as 
 the engine moved at a velocity of 20*34 miles per 
 hour in the first case, and of 29*09 miles per hour 
 
OF THE ORDINARY MODE OF CALCULATION. 41 
 
 in the second, which correspond respectively to 
 298 and 434 feet per minute for the piston, we 
 shall have successively v = 298 and v = 434. 
 Hence the formula gives : 
 
 1st Case. Effort exerted by the engine at the given ve- 
 locity, from our calculation, 
 411 X -77 X (687 IX 144) 1Q507 
 
 298 
 
 Effect produced, including the frictions and 
 resistances, as above ........ 10103 
 
 Difference ...... 404 
 
 2d Case. Effort exerted by the engine at the given ve- 
 
 434 
 Effect produced, including, &c ...... 7346 
 
 Difference ...... 131 
 
 Mean difference of the two cases ..... 267 
 
 We attain then the effect really produced, within 
 a difference of only 267 Its., a difference which is 
 less than can generally be expected in experiments 
 of this kind, wherein all depends on the manage- 
 ment of the fire ; whereas the preceding theory 
 gives a mean and inevitable error of 1378 Ibs., 
 which is i of the real effect of the first case, and 
 j- of the real effect of the second. 
 
 2d. To continue the same comparison of the 
 two theories, suppose it were required to calculate 
 what quantity of water the boiler should evaporate 
 per minute to produce either the first or the se- 
 cond effect. The mode of calculation followed by 
 the ordinary theory consists, as we have said, in 
 supposing, first, that the volume described by the 
 
42 CHAPTER I. 
 
 piston has been filled with steam at the same pres- 
 sure as in the boiler, and then in applying to it a 
 coefficient of reduction for the losses. 
 
 Now, in the first case, the volume described by 
 the piston at the given velocity, is 
 
 av = l-32 x 298 = 393 cubic feet. 
 If this volume had been filled with steam at the 
 pressure of the boiler, it would have required an 
 evaporation of water of 
 
 3Q3 
 
 - = 96 cubic foot of water. 
 
 411 
 
 But the real evaporation was no more than 
 77. Therefore the theoretic evaporation of the 
 first case requires a coefficient of 
 
 In the second case, the evaporation similarly 
 computed, supposing the steam to have acted in 
 the cylinder at the pressure of the boiler, is 
 
 1-32 x 434 . 
 
 - =1*39 cubic toot ot water. 
 411 
 
 So, for this case the necessary coefficient is '55. 
 In this problem, therefore, as in the preceding, no 
 constant coefficient whatever can be satisfactory. 
 
 If, however, the calculation be performed with 
 the mean coefficient *68, there results : 
 1st Case. Evaporation per minute calculated by the or- 
 dinary theory, with the coefficient '68, 
 1-38X898 X . 68 ......... . 65 
 
 Real evaporation .......... '77 
 
 Error ........ *12 
 
OF THE ORDINARY MODE OF CALCULATION. 43 
 
 2d Case. Evaporation per minute, calculated by the or- 
 dinary theory, with the coefficient -68, 
 1J2X4S4 
 
 411 
 Real evaporation .......... *77 
 
 Error ........ -18 
 
 The mean error committed is then ~ of the eva- 
 poration ; and, for the very reason that it is a 
 medium, it may, in extreme cases, become twice 
 as much. 
 
 This is the error committed, when a coefficient 
 is sought expressly for the evaporation. But 
 when, instead of that, the coefficient *59, deter- 
 mined in the preceding problem from the compari- 
 son of the theoretic and practical effects, is used 
 as a divisor, as by many authors it is, far greater 
 errors are induced ; for we find : 
 
 1st Case. Evaporation per minute, calculated by the or- 
 dinary theory, with the coefficient '59 as a 
 
 -,. . 298X1-32 
 
 dmsor ' 
 
 Error '85 
 
 434x1-32 
 
 Real evaporation -77 
 
 Error 
 
 2d Case. Evaporation per minute 
 
 Real evaporation 77 
 
 Error 1-59 
 
 In our method, on the contrary, the evaporation 
 necessary to put in motion the resistance aR at 
 the velocity v 9 is given by the formula 
 
 R x v 
 mP 
 
44 CHAPTER I. 
 
 Which gives : 
 
 1st Case. Evaporation given by our calculation, 
 10103X298 
 
 41 IX (68-71x144) 
 Real evaporation *77 
 
 Difference -03 
 
 Case. Evaporation given by our calculation, 
 
 7346X434 
 
 411 X (68-71x144) 
 Real evaporation "77 
 
 Difference *01 
 
 3dly and finally, for the case wherein the ve- 
 locity of the piston is sought, supposing the re- 
 sistance given, no method like the ordinary one 
 could do otherwise than lead to error, but on this 
 head comparison is unnecessary, since the problem 
 has never yet been solved. 
 
 We shall merely, therefore, show the verifica- 
 tion of our theory. The formula relative to this 
 problem is : 
 
 mSP 
 
 And we find : 
 
 1st Case. Velocity of the piston, in feet per minute, cal- 
 culated by our theory, 
 411X-77X(68-71X144) 
 
 10103 
 Real velocity 298 
 
 Difference . 12 
 
OF THE ORDINARY MODE OF CALCULATION. 45 
 
 2d Case. Velocity by our calculation, 
 
 411 x -77 X (68-71x144) _ 426 
 
 7346 
 Real velocity 434 
 
 Difference 
 
 It consequently appears that in each of the 
 three problems in question, the theory we propose 
 leads to the true result ; whereas the ordinary 
 theory, besides that it leaves the third problem 
 without solution, may, in the two others, lead to 
 very serious errors. 
 
 Before abandoning this comparison we will re- 
 call attention to an effect, in the calculation of the 
 ordinary theory, of which we have already spoken, 
 but which is here found demonstrated by the facts. 
 It is, that that calculation gives the same force ap- 
 plied by the engine in both the cases considered, 
 notwithstanding their difference of velocity; and 
 such will always be the result, since the calcula- 
 tion consists merely in multiplying the area of the 
 piston by the pressure in the boiler, and reducing the 
 product in a constant proportion. Thus the ordinary 
 theory maintains in principle, that the engine may 
 always draw the same load at all imaginable ve- 
 locities. Again we see that, in the same computa- 
 tion, viz., that of the load or of the force applied, the 
 evaporation of the engine is not mentioned ; which 
 implies that the engine would always draw the same 
 load at all velocities, and whatever might be the 
 evaporation of the boiler ; which is impossible. 
 
 Lastly, we shall remark that, in the calculation 
 
46 CHAPTER I. 
 
 made by the ordinary theory, in order to find the 
 evaporation of the engine, no mention whatever is 
 made of the resistance the engine is supposed to 
 draw ; so that the evaporation necessary to draw a 
 given resistance, is independent of that resistance ; 
 another result equally impossible. 
 
 To these omissions, then, which we regard as 
 errors of principle, and to other causes already 
 noted, are to be attributed the deviations observable 
 in the results of the ordinary theory in the ex- 
 amples proposed. 
 
 SECT. VII. Of the area of the steam-passages. 
 
 There yet remains one point which needs ex- 
 amination, and that is the area of the steam- 
 passages, or the size of the opening of the regu- 
 lator. 
 
 The ordinary theory recognizes in this opening 
 a very important effect on the engine, since it 
 affirms that by increasing or diminishing it, any 
 desired pressure may be produced in the cylinder. 
 Yet no means are afforded us of taking account of 
 this opening in the calculation ; unless obliged, as 
 we are already, to have a coefficient for the useful 
 effects and for every species of engine, and another 
 for the evaporation, modified also for every system 
 of engines, and again a different coefficient for all 
 velocities, we be required to have a new one also 
 for every opening of the regulator. But these co- 
 efficients are not given, and notwithstanding that 
 
OF THE ORDINARY MODE OF CALCULATION. 47 
 
 the action of the engine is considered to change 
 with the opening of the regulator, yet the calcula- 
 tion is always the same, and made with the same 
 coefficient, whatever that opening may he. 
 
 Now, when a stationary engine is at work, its 
 regulator is in constant motion by the effect of the 
 governor, and, as it were, unperceived by the 
 engineer. The calculation then of the ordinary 
 theory will be continually at fault ; it will be in- 
 exact in all cases and at all moments wherein the 
 regulator shall happen to have an opening different 
 from that for which the coefficient employed shall 
 have been determined. 
 
 In the theory which we propose, on the contrary, 
 account is taken of the opening of the regulator, or 
 at least of the effects it produces, though its direct 
 measure does not appear ostensibly in the equations. 
 To set this fact in a perfectly clear light, we will 
 first of all establish what are the real effects of the 
 regulator. 
 
 We will first prove that the degree of opening of 
 the regulator can have no influence on the pressure 
 in the cylinder, but that its reaction, on the con- 
 trary, is upon the pressure in the boiler ; we will 
 then shew that, whatever be the contraction of the 
 regulator, the formula will keep account of it, and 
 will continue to give the true effects produced ; and 
 finally, we will examine, under each circumstance, 
 what changes do take place in those effects, by 
 reason of the contracting of the orifice of the 
 regulator. 
 
48 CHAPTER I. 
 
 1. It is supposed, in the ordinary theory, that 
 the pressure of steam in the boiler being given and 
 fixed, the contracting more or less of the aperture 
 of the regulator may be made to produce at 
 pleasure a certain pressure in the cylinder. But 
 we have proved that the pressure in the cylinder is, 
 on the contrary, always strictly determined, a priori, 
 by the resistance on the piston ; the greater or less 
 opening, then, of the regulator can effect no change 
 in it. Besides, how could the contracting of the 
 passage change the pressure of the steam which 
 issues through it? It may, we agree, change the 
 quantity, because the smallness of the opening will 
 prevent more than a certain portion from passing 
 in a given time, but it certainly never can change 
 its pressure. It will, in fact, always happen, that 
 as soon as the steam, on passing into the cylinder, 
 shall attain there the pressure of the resistance, 
 the piston will recede and not allow the steam to 
 assume a greater pressure. And if it be supposed 
 that by enlarging the passage, the steam be made 
 to flow in 10 times, 20 times, 30 times quicker, 
 the piston will recede 10 times, 20 times, 30 times 
 quicker also, since its motion is the result of the 
 arrival of the steam ; but never will the pressure of 
 the steam exceed the resistance of the piston, since 
 the piston being a valve to the cylinder, that would 
 be supposing a boiler in which the pressure of the 
 steam were greater than that of the valve. 
 
 The regulator, then, can make no change in the 
 pressure in the cylinder, but this is what happens. 
 
OF THE ORDINARY MODE OF CALCULATION. 4<9 
 
 The quantity of steam of a given density, which 
 flows through a determined orifice, being in propor- 
 tion to the area of that orifice, it follows that when 
 the opening of the regulator is contracted, the quan- 
 tity of steam, at the pressure of the boiler, which 
 passes into the cylinder, is thereby diminished; 
 nevertheless the same quantity is still generated in 
 the boiler. The steam which has ceased to find 
 an issue towards the cylinder, will then accumulate 
 in the boiler, and will there rise to a greater and 
 greater density and elastic force, till at length it finds 
 an issue somewhere ; till, for instance, having at- 
 tained the pressure necessary to raise the safety 
 valves, it escape into the atmosphere. Then a 
 balance will be established, according to which the 
 surplus of steam generated above what can reach 
 the cylinder, will find a constant issue by the safety 
 valves ; and the rest will pass through the orifice 
 of the regulator and go into the cylinders to pro- 
 duce the motion of the piston. From this moment 
 all will persevere in the same state, and the 
 pressure in the boiler will continue as high as it 
 must be, to keep the safety-valve open and 
 give egress to the steam, as quickly as it is pro- 
 duced. 
 
 Hence it is plain that the contracting more or 
 less of the regulator can have no action on the 
 pressure in the cylinder, but that it has a very 
 direct action on the pressure in the boiler. 
 
 2. We have just said that, according as the aper- 
 ture of the regulator is contracted, the pressure of 
 
 
 
50 CHAPTER I. 
 
 the steam will rise in the boiler and its density in- 
 crease at the same time ; and so long as the steam 
 shall not find an opening to escape entirely as fast as 
 it is produced, this increase of density and elastic 
 force will continue ; for we suppose that the same 
 mass of steam per minute is still generated in 
 the boiler, and that the fire is maintained in the 
 same state. Now the steam is retained in the 
 boiler by two obstacles : the orifice of the regulator 
 which opposes its passage on account of the density, 
 and the safety-valve which opposes its passage on 
 account of the pressure. Two cases then may now 
 occur, according to which of the two obstacles 
 shall give way first : either the steam becoming 
 more and more dense, will in the end so reduce its 
 volume, as to issue entirely by the orifice of the 
 regulator, notwithstanding the contraction of the 
 latter; or else the safety-valve, opposing less re- 
 sistance to the elastic force than the narrowed ori- 
 fice opposes to the density, the steam will escape 
 by the safety-valve. 
 
 In the first of these two cases, then, the engine 
 will be thus regulated : in the cylinder, as inva- 
 riably, the pressure of the resistance ; and in the 
 boiler the pressure necessary for the corresponding 
 density of the steam, to admit of its issuing entirely 
 by the aperture afforded by the regulator. 
 
 And in the second case, the engine, on the con- 
 trary, will be thus regulated ; in the cylinder still 
 the pressure of the resistance, and in the boiler 
 that of the safety-valve. 
 
OF THE ORDINARY MODE OF CALCULATION. 51 
 
 We must now consider separately each of these 
 cases. Let us suppose that the safety-valve being 
 set at a very high pressure, and the orifice of the 
 regulator, on the contrary, being but moderately 
 contracted, the steam accumulating in the boiler, 
 has acquired the density which allows its issue by 
 the orifice, before it has acquired the pressure 
 which procures its escape by the safety-valve. Then 
 it will happen that the total quantity of steam pro- 
 duced will pass into the cylinder, that it will there 
 assume the pressure of the resistance, dilating it- 
 self in proportion ; and by dividing the volume of 
 the steam thus dilated by the area of the cylinder, 
 we shall always have the velocity of efflux by the 
 cylinder, which is nothing else but the velocity of 
 the piston. Thus all will go on as before in the 
 engine, and consequently the effects produced will 
 always be given by the same formulae, P being 
 made of course to express the new pressure pro- 
 duced in the boiler, and S the new vaporisation, if 
 that vaporisation has changed in consequence of 
 the change of pressure. 
 
 Let us now suppose that the safety-valve is set 
 at a low pressure, and that the regulator, on the 
 contrary, is considerably contracted ; so that the 
 steam raises the valve before it acquires the den- 
 sity that would permit it to issue entirely by the 
 regulator. The valve will then be raised, and a 
 part of the steam which continues to be generated 
 in the boiler, will be lost in the atmosphere ; and 
 necessarily the effects of the engine will be by so 
 
CHAPTER I. 
 
 much diminished. But let it be observed, that, 
 with respect to that part of the steam which is not 
 lost, that is the part which finds an issue towards 
 the cylinder, it may always be truly said, that it 
 will there assume the pressure of the resistance, 
 and act in the same manner as the total mass of 
 the steam did before. 
 
 The only difference will be, then, that the effects 
 produced, instead of being due to the totality of the 
 steam, will now be due to a portion only of that 
 steam. 
 
 Thus, provided our formula take account of this 
 difference, they will thereby take into account the 
 whole change that has taken place. Now this is 
 precisely what they do, for we have said that the 
 quantity S, in those formulae, represents the effective 
 vaporisation of the engine, that, in fact, which is 
 really transmitted to the cylinders ; or, in other 
 words, the total vaporisation, minus that which is 
 lost by the safety-valve. It will, then, suffice to 
 substitute for S the real value proper to the case, 
 and the formulae will continue to represent what 
 passes in the engine. 
 
 As the total quantity of water evaporated in a 
 given time is measured directly in the feeding ap- 
 paratus, all that remains to be sought is the means 
 of estimating that which is lost by the safety-valve, 
 in order to subtract it from the former. This valu- 
 ation is easily made, by noting how much the valve 
 is raised at the moment of the loss, which the 
 length of the valve-levers, and the graduated scale 
 
OF THE ORDINARY MODE OF CALCULATION. 53 
 
 with which they are furnished, render very easy to 
 do ; afterwards the regulator must he completely 
 closed, so as to force the whole of the steam produced 
 to escape by the valve, and note taken again of the 
 degree of elevation which this will cause to the valve. 
 Then the proportion of the first elevation to the 
 second will give the ratio of the steam lost to the 
 whole steam produced. This is the means we 
 have employed for locomotives. Should this va- 
 luation not appear sufficiently precise, the waste 
 steam may he condensed in a separate vessel, and 
 the quantity of water measured. It will always, 
 then, he easy to know the effective vaporisation of 
 the engine, and consequently, by introducing it 
 into the formula, we shall continue to have the 
 true effects produced. 
 
 In the two preceding cases we have supposed 
 that the boiler continues, after the contracting of 
 the regulator, to produce the same quantity of 
 steam. A third case, however, may occur, namely, 
 that wherein the engineer shall lower the damper, 
 the moment he sees the valve blow, and reduce his 
 fire so as to stop the blowing of the valve. Then 
 the mass of steam produced per minute will di- 
 minish ; but since it is clear that the quantity 
 which is produced, however small it may be, will 
 always act in the engine in the same manner, it 
 follows that, provided we substitute this new evapor- 
 ation in the formula}, we shall have also the new 
 corresponding effects. Thus, for this third case as for 
 the other two, the formulae will always satisfy the 
 
54t CHAPTER I. 
 
 exigencies, as soon as the substitutions proper to 
 the supposed case shall have been made. 
 
 3. It will now be proper to examine what 
 changes the effects of the engine will undergo in 
 the three preceding suppositions. We have seen 
 that the proposed formulae will always give those 
 effects, on the proper substitutions being made in 
 them. Let us then examine the results of those 
 substitutions. 
 
 In the first case, to wit, the fire continued at 
 the same degree of intensity, and the orifice nar- 
 rowed, though not sufficiently to make the valve 
 blow, the pressure P in the boiler becomes greater. 
 But in the exposed formulae, the pressure P figures 
 only as multiplied by m, which is the relative vo- 
 lume of the steam. This volume being inversely 
 as the density, and the density itself varying very 
 nearly in the direct ratio of the pressure, it follows 
 that, unless a very great change of pressure take 
 place, the product m P will remain constant. If 
 it be supposed, as is generally admitted, that the 
 evaporation of a given boiler is the same under 
 different pressures of the steam, the quantity S 
 will not vary either. In this case, then, the formu- 
 lae will give the same results ; and consequently 
 the engine will produce the same effects, after the 
 contracting of the regulator as before that con- 
 traction was made. 
 
 In the second case, to wit, contraction of the 
 regulator, attended with blowing of the safety-valve, 
 there is still increase of pressure in the boiler, 
 
OF THE ORDINARY MODE OF CALCULATION. 55 
 
 which, as we have just seen, produces no change 
 in the effects. But moreover there is a certain 
 loss by the safety-valves, and that loss diminishes 
 by so much the effective vaporization S. There will 
 then be a diminution of effect precisely proportional 
 to the quantity of steam lost by the valve, which 
 we have given the means of measuring. 
 
 Finally, in the third case, to wit, contraction 
 of the regulator, accompanied by a reduction of 
 the intensity of the fire, the blowing of the safety- 
 valve will be suppressed only by producing a 
 smaller mass of steam in the boiler. But since 
 this mass of steam, generated and transmitted to 
 the cylinders, is less than before the contraction of 
 the regulator, it follows that the effect produced by 
 the engine, or the result given by the formulae, 
 will be reduced just so much. Thus this third 
 case is similar to the second, and will similarly be 
 attended with a reduction of effect. 
 
 The first of the three cases which we have just 
 presented, takes place without the smallest attention 
 being paid to it, whenever the orifice of the regu- 
 lator is but slightly diminished. 
 
 The second occurs almost continually in locomo- 
 tive engines, because these having to overcome very 
 variable resistances, according to the inclinations 
 of the road they traverse, it is necessary to main- 
 tain an intense fire, and to keep the engine always 
 ready to develope on an emergency an increase of 
 power. 
 
 The third is that which happens generally in 
 
50 .CHAPTER I. 
 
 stationary engines, when the regulator is pretty 
 much contracted, because the regulator, in those 
 engines, being never reduced but when the work of 
 the engine requires less force, the engineman takes 
 advantage of that circumstance to dimmish the in- 
 tensity of the fire, and to produce no more evapora- 
 tion than what is strictly necessary. 
 
 These three cases may then occur in the different 
 engines, but the exposed formula will always adapt 
 themselves to them. 
 
 SECT. VIII. Of the differences which exist between 
 the theory proposed and the ordinary theory. 
 
 In terminating the general exposition of our man- 
 ner of viewing the action of steam in steam-engines, 
 we will resume in a few words the differences exist- 
 ing between the method we propose and that which 
 has been in use hitherto. 
 
 1. The ordinary theory passes from the theo- 
 retic effects to the practical by means of a constant 
 coefficient. 
 
 Ours rejects entirely the use of that coefficient, 
 which we regard as resulting from a fundamental 
 error in the calculation of what are termed the the- 
 oretic effects. 
 
 2. The ordinary theory acknowledges not know- 
 ing the pressure in the cylinder ; it seeks to deduce 
 it from that of the boiler. 
 
 Our theory determines, d priori, the pressure in 
 the cylinder, as being, not equal nor proportional 
 
OF THE ORDINARY MODE OF CALCULATION. 57 
 
 to that of the holler, but equal to that of the re- 
 sistance on the piston. 
 
 3. The ordinary theory determines the load 
 which an engine is capable of drawing, without 
 taking the velocity into the calculation. That is to 
 say, it maintains that the engine will always draw 
 the same load at any velocity that can he imagined. 
 
 Our theory brings the velocity into the calcula- 
 tion in such sort, that the greater the velocity the 
 less will be the load the engine can draw. 
 
 4. The ordinary theory calculates the evapor- 
 ation of the engine for a resistance and a velocity 
 given, exclusively of any consideration of the re- 
 sistance ; that is to say, it maintains again that 
 the evaporation necessary to effect the motion 
 shall be independent of the resistance to be moved. 
 
 Ours, on the contrary, introduces the load and 
 the velocity into the calculation. 
 
 5. The ordinary theory has no means of calculat- 
 ing the velocity that an engine will assume with a 
 given resistance. 
 
 Ours gives this calculation with the same sim- 
 plicity as the preceding. 
 
 6. The ordinary theory regards the regulator 
 as determining the pressure in the cylinder. And 
 yet in that calculation it takes no account of the 
 variations of the regulator. 
 
 Ours regards the regulator as fixing the pressure 
 in the boiler and not in the cylinder. It intro- 
 duces the effects of the regulator into the formulae. 
 
58 CHAPTER I* 
 
 7. The ordinary theory is but an approximation 
 more or less exact. 
 
 Ours, on the contrary, which will be seen still 
 more developed, is a method completely analytic in 
 all its parts. 
 
 Nothing then can be more distinct than these 
 two methods ; and as, not only since the year 1835, 
 when we first laid down these principles in our 
 Treatise on Locomotive Engines, but even as late 
 as December, 1837, the authors who have treated 
 these questions, whether in their writings or in their 
 public lectures, have employed the method of co- 
 efficients, we think that the recapitulation we have 
 just made sufficiently establishes that their con- 
 ception of these questions is altogether different 
 from our own. 
 
 We do not then deem it necessary to insist any 
 more on this subject, and shall now pass on to the 
 complete development of the formula?, of which we 
 have as yet given but a general outline. 
 
.59 
 
 CHAPTER II. 
 
 
 
 OF THE LAWS WHICH REGULATE THE MECHANICAL 
 ACTION OF THE STEAM. 
 
 SECT. I. Relation between the temperature and 
 the pressure of the steam in contact with the 
 liquid. 
 
 BEFORE entering upon considerations which have 
 for their hasis the effects of the steam, it may be 
 necessary to lay down in a few words, some of the 
 laws according to which the mechanical action of 
 the steam is determined or modified. 
 
 In the calculation of steam engines it is requisite 
 to consider four things in the steam. 
 
 Its pressure, which is also called tension or 
 elastic force, and which is the pressure it exercises 
 on every unit of the surface of the vessel that con- 
 tains it. 
 
 Its temperature, which is the number of degrees 
 marked by a thermometer immerged in it. 
 
 Its density, which is the weight of a unit of its 
 volume. 
 
 And its relative volume, which is the volume of 
 a given weight of steam compared to the volume of 
 
60 
 
 CHAPTER II. 
 
 the same weight of water, or, in other words, to 
 the volume of the water that has served to produce 
 it. We deem it necessary to add here the word 
 relative, in order to avoid the confusion which would 
 otherwise arise continually between the absolute 
 volume filled by the steam, which may depend on the 
 capacity of the vessel that contains it, and the 
 relative volume which is the inverse of the density. 
 Thus, for instance, steam generated under the 
 pressure of the atmosphere may fill a vessel of any 
 size, but its relative volume will always be 1700 
 times that of water. 
 
 When the volumes occupied by the same weight 
 of two different steams are compared together, 
 it is evidently a comparison of what we call the 
 relative volumes of those two steams. For, the 
 two steams compared having the same* weight, 
 correspond to the same volume of water evapo- 
 rated. But the relative volume of the steam is 
 the quotient of the absolute volume of the steam 
 by the corresponding volume of water. Therefore, 
 it follows that the ratio of the relative volumes of 
 the two steams is the same as the ratio of their 
 absolute volumes ; and this proposition must be 
 kept in mind for what will follow hereafter. 
 
 The steam may be considered at the moment of 
 its generation in the boiler, when still in contact 
 with the liquid from which it emanates, or else as 
 being separated from that liquid. 
 
 When the steam, after having been formed in 
 
OF THE MECHANICAL ACTION OF THE STEAM. 6l 
 
 a boiler, remains in contact with the generating 
 water, it is observed that the same temperature 
 corresponds invariably to the same pressure, and 
 vice versa. It is impossible then to increase its 
 temperature, without its pressure and density in- 
 creasing spontaneously at the same time. In this 
 state the steam is therefore at its maximum density 
 and pressure for its temperature, and then a constant 
 connexion visibly exists between the temperature 
 and the pressure. 
 
 If on the contrary the steam be separated from 
 the water that generated it, and that the tempe- 
 rature be then augmented, the state of maximum 
 density will cease, since there will be no more 
 water to furnish the surplus of steam, or increase 
 of density, corresponding to the increase of tempe- 
 rature. That invariable connexion above men- 
 tioned, between the temperature and the pressure, 
 will then no longer exist, and, by accessory means, 
 the one may at pleasure be augmented or diminished, 
 without any necessity of a concomitant variation 
 taking place in the other, as it happens in the case 
 of the maximum density. 
 
 It is necessary then to distinguish between these 
 two states of the steam. 
 
 One of the most important laws on the properties 
 of steam, is that which serves to determine the elastic 
 force of the steam in contact with the liquid, when 
 the temperature under which it is generated is 
 known ; or, reciprocally, to determine that tempe- 
 rature when the elastic force is known. Not 
 
2 CHAPTER II. 
 
 only this enquiry is of a direct utility, but we shall 
 see in the sequel, that it serves equally to determine 
 the density or the relative volume of the steam 
 formed under a given pressure, a point of know- 
 ledge indispensable in the calculation of steam- 
 engines. 
 
 Experiments on this subject had long been taken 
 in hand, and they were very numerous for steam 
 formed under pressures less than that of the atmo- 
 sphere ; but for high temperatures, the experiments 
 extended but to pressures of four or five atmo- 
 spheres. Some few only went as far as eight, arid 
 that without completing the scale in the interval. 
 The extreme difficulty of researches of this kind, if 
 made with proper attention, the heavy expenses 
 they occasion, and the danger attending them, 
 had prevented the experiments from being carried 
 farther. But to the Academy of Sciences of the 
 Institute of France we are indebted for a complete 
 table on this subject. The academy confided the 
 conduct of these delicate experiments to two dis- 
 tinguished scientific men, Messrs. Arago and Du- 
 long, who evinced in them every nicety that a per- 
 fect knowledge of the laws of natural philosophy 
 could suggest, to avoid the ordinary causes of er- 
 ror. Never were researches of this kind conducted 
 on so vast a scale, nor with more accuracy. The 
 pressure of the steam was measured by effective 
 columns of mercury contained in tubes of crystal 
 glass, which together extended to the height of 87 
 feet English. The instruments were constructed 
 
OF THE MECHANICAL ACTION OF THE STEAM* 63 
 
 by the most skilful makers, and no expense was 
 spared *. Therefore the greatest degree of confi- 
 dence is to be attached to their results. 
 
 These beautiful experiments furnish a series of 
 observations, from the pressure of 1 atmosphere to 
 that of 24. To form however a table extending 
 beyond this limit, Messrs. Dulong and Arago 
 have sought to deduce from their observations a 
 formula which might represent temperatures for 
 still higher pressures without any noticeable 
 error. They have in fact attained that end, by 
 means of a formula which we shall presently report, 
 and whose accord with experience is such, for all 
 that part of the scale above four atmospheres, as to 
 give room to think that, on being applied to pres- 
 sures up to 50 atmospheres, the error in tempera- 
 ture would not in any case exceed 1 degree of the 
 centigrade thermometer or 1*8 degree of Fahren- 
 heit. They were enabled then, as well from the 
 result of their observations as by means of that for- 
 mula, to compose a table of temperatures of steam 
 up to 50 atmospheres of pressure, with the cer- 
 tainty of committing no error worthy of note. 
 
 Though the formula of Messrs. Arago and Du- 
 long may be applied to pressures comprised be- 
 tween 1 and 4 atmospheres, with an approximation 
 that would suffice for most of the exigencies in the 
 
 * Vide Expose des recherches fartes par ordre de 1'Academie 
 des Sciences, pour determiner les forces elastiques de la vapeur 
 d'eau a de hautes temperatures. Memoires de V Academic des 
 Sciences, Tome X. ; Annales de Chimie et de Physique, Tome 
 XLIII. 1830. 
 
04 CHAPTER II. 
 
 arts, they did not indicate the use of it for that 
 interval, because in that part of the scale, other 
 formulae already known accord more exactly with 
 the results of observation, and ought, in conse- 
 quence, to be preferred. Among those formulae, 
 that originally proposed by Tredgold, and after- 
 wards modified by his translator, Mr. Mellet, gave 
 the most exact results ; and no inconvenience 
 arises from the use of it, when it is required merely 
 to compose a table by intervals of half-atmospheres. 
 But as, for the more commodious use of the formu- 
 lae which we have to propose in this work, we shall 
 want to establish a table by intervals of pounds per 
 square inch ; we deem it better to employ a formula 
 which we shall give with the others presently, and 
 which, approaching as near as that of Tredgold to 
 the results of direct observation, in the points fur- 
 nished by experiment, has moreover the advantage 
 of coinciding exactly at 4 or 4^ atmospheres with 
 the formula of Messrs. Dulong and Arago, which is 
 to form the continuation of it*. 
 
 * In fact, comparing, in French measure, the two formulae 
 with the observation, we find the following results, as it will 
 be easy to verify hereafter. 
 
 Elastic force 
 in atmo- 
 spheres. 
 
 Observed 
 temperature. 
 
 Temperature 
 given by 
 Tredgold's 
 form, modi- 
 fied byMellet. 
 
 Temperature 
 given by the 
 proposed for- 
 mula. 
 
 Temperature 
 given by the 
 form, of 
 Messrs. Arago 
 and Dulong. 
 
 1 
 
 100- 
 
 99-96 
 
 100 
 
 
 
 2-14 
 
 123-7 
 
 123.54 
 
 123-34 
 
 M 
 
 2-8705 
 
 133-3 
 
 133-54 
 
 133-17 
 
 n 
 
 4- 
 
 n 
 
 145-43 
 
 144-88 
 
 5 
 
 4-5735 
 
 149-7 
 
 150-39 
 
 149-79 
 
 149-77 
 
OF THE MECHANICAL ACTION OF THE STEAM. 65 
 
 These formulae, as well as other similar ones, 
 have the inconvenience of suiting only a limited 
 part of the scale of temperatures. That of Tred- 
 gold modified, as well as that which we propose to 
 substitute for it, represent very closely the observa- 
 tions for the interval between 1 and 4 atmospheres; 
 but below that point they are incorrect, and above it 
 they are inferior in point of accuracy to that of 
 Messrs. Dulong and Arago. 
 
 The latter accords remarkably with the facts, 
 from 4f atmospheres to 24. In this interval its 
 greatest difference with observation is "4 degree 
 of the centigrade thermometer or *7 of Fahrenheit, 
 and nearly all the other differences are only *1 de- 
 gree centigrade or *18 Fahrenheit; but, as we have 
 already said, it begins to deviate from the ob- 
 servation below 4 atmospheres. 
 
 Finally, among the formulae proposed by different 
 authors on the same subject, that of Southern is 
 very suitable to steam formed under pressures in- 
 ferior to that of one atmosphere ; it deviates then 
 from the truth only in very low pressures, as ap- 
 pears from the experiments of that engineer. But 
 for pressures superior to 1 atmosphere it ceases to 
 have the same accuracy : from 1 to 4 atmospheres 
 
 It appears that the formula which we propose differs from 
 the observed temperatures nearly as much as that of Tredgold 
 modified ; but as the difference from the observation is on the 
 minus side instead of the plus, there results a coincidence at 
 4| atmospheres with that of Messrs. Arago and Dulong. 
 
 F 
 
66 CHAPTER II. 
 
 it gives more error than that of Tredgold modi- 
 fied, and above 4 atmospheres the error rises ra- 
 pidly to 1 and 1'5 degree of the centigrade ther- 
 mometer, or 1'8 and 2*6 degrees of Fahrenheit; 
 so that the formula of Messrs. Arago and Dulong, 
 which is, besides, of more easy calculation, becomes 
 then far preferable to it. 
 
 No one then of these formulae suits the whole 
 series of the scale of temperatures, and to hold 
 exclusively to any one of them would be knowingly 
 to introduce errors into the tables. As moreover, 
 the true theoretic law which connects the pressures 
 with the temperatures is unknown, and that these 
 formulae are formulae of interpolation, established 
 solely from their coincidence with the facts, and 
 used merely to fill up the intervals of the experi- 
 ments, according to what is wanted for the regular 
 division of the tables, the only means of making use 
 of them is to apply each respectively to that portion 
 of the series which it suits. Then, from the compa- 
 rison of their results with experience, one may rest 
 assured that the error on the temperature will in no 
 point exceed seven-tenths of a degree of Fahrenheit, 
 or four-tenths of a degree of the centigrade ther- 
 mometer. This was the means employed before 
 us, and we shall adopt it in the formation of the 
 tables we are about to present. 
 
 The formulae, which will serve to compose 
 these tables, are then the following, which we 
 report here, not in their original terms but trans- 
 formed, for greater convenience, into the measures 
 
OF THE MECHANICAL ACTION OF THE STEAM. 67 
 
 usual in practice; that is, expressing the pressure 
 p in pounds per square inch or in kilograms per 
 square centimetre, and the temperature t, in degrees 
 of Fahrenheit's or of the centigrade thermometer, 
 reckoned in the ordinary manner. 
 
 Southern's formula, suitable to pressures less 
 than that of the atmosphere (French measures) : 
 
 46- 5 - 13 
 
 t - 145-360 5 /p- -0034542- 46'278. 
 
 Tredgold's formula modified by Mr. Mellet, 
 suitable to pressures of 1 to 4 atmospheres 
 (French measures): 
 
 75 + 1- 6 
 
 Formula suitable, like the preceding, to pres- 
 sures from 1 to 4 atmospheres (French measures) : 
 
 -72-67. 
 
 Formula of Messrs. Dulong and Arago, suitable 
 to pressures from 4 to 50 atmospheres (French 
 measures) : 
 
 p = (-28658 + -0072003 O 5 , 
 * = 138-883 ^/p- 39-802. 
 
 F2 
 
68 
 
 CHAPTER II. 
 
 Southern's formula, suitable to pressures less 
 than that of the atmosphere (English measures): 
 
 13 
 
 t= 155-7256 5 / --04948-51 -3. 
 
 Tredgold's formula modified by Mr. Mellet, 
 suitable to pressures from 1 to 4 atmospheres 
 (English measures) : 
 
 103 
 
 = 201-18 /- 103. 
 
 Formula suitable, like the preceding, to pres- 
 sures from 1 to 4 atmospheres (English measures): 
 
 98-806 
 
 198-562 
 t= 198-562 {/>- 98-806. 
 
 Formula of Messrs. Dulong and Arago, suitable 
 to pressures from 4 to 50 atmospheres (English 
 measures) : 
 
 p = (-26793 + -0067585 0*, 
 *= 147-961 {/p-39'644. 
 
 Besides the formulae which we have just reported, 
 there exists another proposed by Mr. Biot, which, 
 compared by that illustrious natural philosopher to 
 the above-mentioned experiments on high pressures, 
 to those of Taylor on pressures approaching nearer 
 
OF THE MECHANICAL ACTION OF THE STEAM. 69 
 
 to 100 degrees centigrade, and to a numerous series 
 of manuscript observations made by Mr. Gay-Lussac, 
 from 100 to 20 degrees centigrade, reproduces 
 the results observed, with very slight accidental de- 
 viations, such as the experiments themselves are 
 liable to. This formula, which has consequently 
 the advantage over the preceding, of being appli- 
 cable to all points of the scale, is the following : 
 
 log p a a^ b* 0+t a- 2 b 
 
 20+* 
 
 Log p is the tabulary logarithm of the pressure ex- 
 pressed in millimetres of mercury at centigrade ; 
 t is the centesimal temperature counted on the air 
 thermometer, and the quantities a, a l9 a$> b l9 b 2 , are 
 constant quantities which have the following values : 
 
 a- _5-96l31330259, 
 Ioga 1= 1-82340688193, 
 log b l= --01 309734295, 
 Ioga 2 = -74110951837, 
 log J a= _ -00212510583. 
 
 This formula cannot fail to be extremely useful 
 in many delicate researches on the effects of steam j 
 but to establish, by its means, a table of the form 
 we require, the pressure ought first to be deduced 
 from it for each degree of the air thermometer ; 
 then these degrees ought to be afterwards changed 
 into degrees of the mercury thermometer ; and as 
 this would not give the temperatures corresponding 
 to given pressures, by regular intervals, a subsequent 
 interpolation would be still necessary to make the 
 table in the proper disposition. These long opera- 
 
70 CHAPTER II. 
 
 tions induced us to give the preference to the pre- 
 viously cited formulse, for the construction of the 
 tables which we shall shortly present. 
 
 SECT. II. Relation between the relative volumes 
 and the pressures, at equal temperature, or be- 
 tween the relative volumes and the temperatures, 
 at equal pressure, in the steam separated from 
 the liquid. 
 
 We have said that when the steam is in contact 
 with the generating liquid, its pressure is neces- 
 sarily connected with its temperature ; and as the 
 density of an elastic fluid depends only on its tem- 
 perature and its pressure, it follows that the density 
 is then always constant for a given temperature or 
 pressure. But when the steam is separated from 
 the liquid, that connexion between the temperature 
 and the pressure no longer exists. The temperature 
 of the steam may then be varied without changing 
 its pressure, or reciprocally ; and according as the 
 one or the other of these two elements is made to 
 vary, the density of the steam undergoes changes 
 which have been an object of investigation among 
 natural philosophers. 
 
 One very remarkable law in the effects of gas 
 and steam is that which was discovered by Mariotte 
 or Boyle, and has since been confirmed, as far as to 
 pressures of 27 atmospheres, by Messrs. Arago and 
 Dulong. It consists in this, that if the volume of 
 a given weight of gas or of steam be made to vary 
 without changing its temperature, the elastic force 
 
OF THE MECHANICAL ACTION OF THE STEAM. 71 
 
 of the gas will vary in the inverse ratio of the 
 volume it is made to occupy. That is to say, if 
 v and v' express the volumes occupied hy the same 
 weight of steam, and p and p' the pressures which 
 maintain the steam compressed under those respective 
 volumes, the temperature, moreover, being the same 
 in both cases, the following analogy will exist : 
 
 p _v r 
 p'~ v' 
 
 And therefore, /* and yl being the relative volumes 
 of the steam at the pressures p and p', we shall have 
 
 According to this law, if a given weight of an 
 elastic fluid be compressed to half its primitive vo- 
 lume, without changing its temperature, the elastic 
 force of that fluid will become double. But it is 
 plain that this effect cannot take place in the steam 
 in contact with the liquid, because it supposes that 
 during the change of pressure the temperature re- 
 mains constant, whereas we have seen that in such 
 state the pressure always accompanies the tempera- 
 ture, and vice versa. 
 
 Another property equally important in the ap- 
 preciation of the effects of steam has been dis- 
 covered by a celebrated chemist of our times, 
 Mr. Gay-Lussac. It consists in this, that if the 
 temperature of a given weight of an elastic fluid 
 be made to vary, its tension being maintained at 
 the same degree, it will receive augmentations of 
 
CHAPTER II. 
 
 volume exactly proportional to the augmentations 
 of temperature ; and for each degree of the centi- 
 grade thermometer, the increase of volume will be 
 00364 of the volume which the same weight of 
 fluid occupies at the temperature zero. If the 
 temperatures are taken from Fahrenheit's ther- 
 mometer, each augmentation of I degree in the 
 temperature will produce an increase of -00202 of 
 the volume occupied hy the fluid at the temperature 
 of 32. 
 
 If then we call V the volume of the given weight 
 of the elastic fluid, under any pressure, and at the 
 temperature of 32 degrees of Fahrenheit, the vo- 
 lume it will occupy under the same pressure, and 
 at the temperature t of Fahrenheit will be 
 v = V + V x -00202 (t - 32). 
 
 It follows that, between the volumes v and v' oc- 
 cupied by the same weight of steam, at the same 
 pressure and under the respective temperatures t 
 and ff, there will be the following analogy : 
 v __!+ -00202 (-32) 
 v'~ 1+ -00202 (/-32)' 
 
 which will also be true, when we replace the ratio 
 of the two absolute volumes v and v' y by the ratio 
 of the relative volumes /JL and y! of the steam. 
 
 This law, supposing that the temperature of the 
 steam changes, without the pressure undergoing 
 any change, obviously cannot apply to the effects 
 produced in steam in contact with the liquid, since 
 in those the pressure changes necessarily and spon- 
 taneously with the temperature. 
 
OF THE MECHANICAL ACTION OF THE STEAM. 
 
 SECT. III. Relation between the relative volumes, 
 the pressures, and the temperatures, in the steam 
 in contact or not in contact with the liquid. 
 
 As it has just been observed, neither Boyle's 
 law nor that of Gay-Lussac can apply alone to 
 changes which take place in the steam remaining in 
 contact with the liquid. But it is clear that from 
 the two, a third relation may be deduced, whereby 
 to determine the variations of volume which take 
 place in the steam, by virtue of a simultaneous change 
 in the temperature and in the pressure ; and this 
 relation may then comprehend the case of the steam 
 in contact with the liquid, since it will suffice to in- 
 troduce into the formulae the pressures and tempera- 
 tures which, in this state of the steam, correspond 
 to each other. 
 
 Suppose then it be required to know the volume 
 occupied by a given weight of steam, which passes 
 from the pressure p f and temperature tf, to the pres- 
 sure p and temperature t. It may be supposed 
 that the steam passes first from the pressure p' to 
 the pressure p without changing its temperature, 
 which, from Boyle's law, will give between the re- 
 lative volumes of the steam, the analogy 
 
 then supposing this steam to pass from the tem- 
 perature tf to the temperature t, without changing 
 
74> CHAPTER II. 
 
 its pressure, the relative volume of the steam, from 
 the law of Gay-Lussac, will become 
 
 1+ '00202 (-32) 
 ^~^ 1 + -00202 ^- 
 
 ,p' 1+ '00202 (-32) 
 "** p" 1+ -00202 O^ -325* 
 
 This formula will then express the law according 
 to which the relative volume of the steam changes, 
 by virtue of a given combination of pressure and 
 temperature. Consequently, substituting in this 
 equation for p and t, p' and f, the pressures and 
 temperatures only which correspond to each other 
 in the steam in contact with the liquid, we shall have 
 the analogous changes which take place in the 
 relative volume of the steam, when it is not sepa- 
 rated from the water which generated it. 
 
 On the other hand, it is known by experience, 
 that under the atmospheric pressure, or 14 '706 Ibs. 
 per square inch, and at the temperature of 212 of 
 Fahrenheit's thermometer, the relative volume of 
 the steam in contact with the liquid is 1700 times 
 that of the water which has produced it. Hence 
 it is easy to conclude the relative volume of the 
 steam at any given pressure p and at the corre- 
 sponding temperature t. It suffices, in fact, to in- 
 sert the above values for p' 9 if, and //, in the 
 general equation obtained above, and the result 
 will be 
 
OF THE MECHANICAL ACTION OF THE STEAM. 
 
 14706 1 + -00202 (1- 
 =170 * 
 
 1+ -00202 x 180 - 
 
 1 + -00202 (-32) 
 = 18329 - - - -' 
 
 Thus we may, by means of this formula, calcu- 
 late the relative volume of the steam generated 
 under a given pressure, as soon as we know the 
 temperature answering to that pressure in steam 
 at the maximum of density for its temperature. 
 
 It is what we have done in the construction of 
 the following table. The second column has been 
 formed by calculating the temperature of the steam 
 at the maximum density, from the formulae which 
 we have given in the first section of this chap- 
 ter. Then using this series of temperatures in the 
 formula which precedes, we have concluded the 
 third column, or the relative volumes of the steam 
 in contact with the liquid, under all the pressures 
 comprised between 1 and 8 atmospheres. This 
 table will, in consequence, dispense from all calcu- 
 lation with regard either -to the research of the 
 temperatures, or to that of the relative volumes of 
 the steam ; and its extent will suffice for all appli- 
 cations that occur in the working of steam engines. 
 
 When we speak of steam generated under a 
 given pressure, we understand the steam considered 
 at the moment of its generation, and consequently 
 still in contact with the liquid. We have explained 
 elsewhere that the volume of the steam, compared 
 to that of the water which has produced it, is pre- 
 cisely what we call the relative volume of the steam. 
 
76 
 
 CHAPTER II. 
 
 TABLE of the volume of the steam generated under dif- 
 ferent pressures, compared to the volume of the water 
 that has produced it. 
 
 Total pres- 
 sure, in Eng- 
 lish pounds, 
 per square 
 inch. 
 
 Correspond- 
 ing tempera- 
 ture by 
 Fahrenheit's 
 thermometer. 
 
 Volume of the 
 steam compared 
 to the volume 
 of the water that 
 has produced it. 
 
 Total pres- 
 sure, in Eng- 
 lish pounds, 
 per square 
 inch. 
 
 Correspond- 
 ing tempera- 
 ture by 
 Fahrenheit's 
 thermometer. 
 
 Volume of the 
 steam compared 
 to the Volume 
 of the water that 
 has produced it. 
 
 1 
 
 102-9 
 
 20954 
 
 37 
 
 263-7 
 
 727 
 
 2 
 
 126-1 
 
 10907 
 
 38 
 
 265-3 
 
 710 
 
 3 
 
 141-0 
 
 7455 
 
 39 
 
 266-9 
 
 693 
 
 4 
 
 152-3 
 
 5695 
 
 40 
 
 2684 
 
 677 
 
 5 
 
 161-4 
 
 4624 
 
 41 
 
 269-9 
 
 662 
 
 6 
 
 169-2 
 
 3901 
 
 42 
 
 271'4 
 
 647 
 
 7 
 
 176-0 
 
 3380 
 
 43 
 
 272-9 
 
 634 
 
 8 
 
 182-0 
 
 2985 
 
 44 
 
 274-3 
 
 620 
 
 9 
 
 187-4 
 
 2676 
 
 45 
 
 275-7 
 
 608 
 
 10 
 
 192-4 
 
 2427 
 
 46 
 
 277-1 
 
 596 
 
 11 
 
 197-0 
 
 2222 
 
 47 
 
 278-4 
 
 584 
 
 12 
 
 201-3 
 
 2050 
 
 48 
 
 279-7 
 
 573 
 
 13 
 
 205-3 
 
 1903 
 
 49 
 
 281-0 
 
 562 
 
 14 
 
 209-0 
 
 1777 
 
 50 
 
 282-3 
 
 552 
 
 15 
 
 213-0 
 
 1669 
 
 51 
 
 283-6 
 
 542 
 
 16 
 
 216-4 
 
 1572 
 
 52 
 
 284-8 
 
 532 
 
 17 
 
 219-6 
 
 1487 
 
 53 
 
 286-0 
 
 523 
 
 18 
 
 222-6 
 
 1410 
 
 54 
 
 287-2 
 
 514 
 
 19 
 
 225-6 
 
 1342 
 
 55 
 
 288-4 
 
 506 
 
 20 
 
 228-3 
 
 1280 
 
 56 
 
 289-6 
 
 498 
 
 21 
 
 231-0 
 
 1224 
 
 57 
 
 290-7 
 
 490 
 
 22 
 
 233-6 
 
 1172 
 
 58 
 
 291-9 
 
 482 
 
 23 
 
 236-1 
 
 1125 
 
 59 
 
 293-0 
 
 474 
 
 24 
 
 238-4 
 
 1082 
 
 60 
 
 294-1 
 
 467 
 
 25 
 
 240-7 
 
 1042 
 
 61 
 
 294-9 
 
 460 
 
 26 
 
 243-0 
 
 1005 
 
 62 
 
 295-9 
 
 453 
 
 27 
 
 245-1 
 
 971 
 
 63 
 
 297-0 
 
 447 
 
 28 
 
 247'2 
 
 939 
 
 64 
 
 298-1 
 
 440 
 
 29 
 
 249-2 
 
 909 
 
 65 
 
 299-1 
 
 434 
 
 30 
 
 251-2 
 
 882 
 
 66 
 
 300-1 
 
 428 
 
 31 
 
 253-1 
 
 855 
 
 67 
 
 301-2 
 
 422 
 
 32 
 
 255-0 
 
 831 
 
 68 
 
 302-2 
 
 417 
 
 33 
 
 256-8 
 
 808 
 
 69 
 
 303-2 
 
 411 
 
 34 
 
 258-6 
 
 786 
 
 70 
 
 304-2 
 
 406 
 
 35 
 
 260-3 
 
 765 
 
 71 
 
 305-1 
 
 401 
 
 36 
 
 262-0 
 
 746 
 
 72 
 
 306-1 
 
 396 
 
OF THE MECHANICAL ACTION OF THE STEAM. 
 
 77 
 
 Total pres- 
 sure, in Eng- 
 lish pounds, 
 per square 
 inch. 
 
 Correspond- 
 ing tempera- 
 ture by 
 Fahrenheit's 
 thermometer. 
 
 Volume of the 
 steam compared 
 to the volume 
 of the water that 
 has produced it. 
 
 Total pres- 
 sure, in Eng- 
 lish pounds, 
 per square 
 incn. 
 
 Correspond- 
 ing tempera- 
 ture by 
 Fahrenheit's 
 thermometer. 
 
 Volume of the 
 steam compared 
 to the volume 
 of the water that 
 has produced it. 
 
 73 
 
 307-1 
 
 391 
 
 92 
 
 323-5 
 
 317 
 
 74 
 
 308-0 
 
 386 
 
 93 
 
 324-3 
 
 313 
 
 75 
 
 308-9 
 
 381 
 
 94 
 
 325-0 
 
 310 
 
 76 
 
 309-9 
 
 377 
 
 95 
 
 325-8 
 
 307 
 
 77 
 
 310-8 
 
 372 
 
 96 
 
 326-6 
 
 305 
 
 78 
 
 311-7 
 
 368 
 
 97 
 
 327-3 
 
 302 
 
 79 
 
 312-6 
 
 364 
 
 98 
 
 328-1 
 
 299 
 
 80 
 
 313-5 
 
 359 
 
 99 
 
 328-8 
 
 296 
 
 81 
 
 314-3 
 
 355 
 
 100 
 
 329-6 
 
 293 
 
 82 
 
 315-2 
 
 351 
 
 105 
 
 333-2 
 
 281 
 
 83 
 
 316-1 
 
 348 
 
 120 
 
 343-3 
 
 249 
 
 84 
 
 316-9 
 
 344 
 
 135 
 
 352-4 
 
 224 
 
 85 
 
 317-8 
 
 340 
 
 150 
 
 360-8 
 
 203 
 
 86 
 
 318-6 
 
 337 
 
 165 
 
 368-5 
 
 187 
 
 87 
 
 319-4 
 
 333 
 
 180 
 
 375-6 
 
 173 
 
 88 
 
 320-3 
 
 330 
 
 195 
 
 382-3 
 
 161 
 
 89 
 
 321-1 
 
 326 
 
 210 
 
 388-6 
 
 150 
 
 90 
 
 321-9 
 
 323 
 
 225 
 
 394-6 
 
 141 
 
 91 
 
 322-7 
 
 320 
 
 240 
 
 400-2 
 
 133 
 
78 CHAPTER II. 
 
 SECT. IV. Direct relation between the relative 
 volumes and the pressures, in the steam in contact 
 with the liquid. 
 
 It has just been seen, from the formulae given in 
 the preceding section, that the density and the 
 relative volume of the steam, whether separated 
 from the liquid or not, are known in terms of the 
 simultaneous pressure and temperature. It is like- 
 wise known that in the steam in contact with the 
 liquid, the temperature depends immediately on the 
 pressure. It should therefore he possible to find a 
 relation proper to determine directly the relative 
 volume of the steam in contact with the liquid, or, 
 in other words, of the steam at the maximum den- 
 sity and pressure for its temperature, by means of 
 the sole knowledge of the pressure under which it 
 is formed. 
 
 The equation which gives the relative volume of 
 the steam in any state whatever, in terms of its 
 pressure and temperature, has been given above. 
 We have also shewn the formulae which serve to 
 find the temperature in terms of the pressure, in 
 steam in contact with the liquid. Eliminating 
 then the temperature from the equation of the 
 volumes and that of the temperatures, we shall 
 obtain definitively the relation sought, or the rela- 
 tive volume of the steam at the maximum density, 
 in terms of the pressure only. 
 
 But here starts the difficulty. First, Mr. Biot's 
 
OF THE MECHANICAL ACTION OF THE STEAM. J9 
 
 formula not being soluble with reference to the 
 temperature, does not admit the necessary elimina- 
 tion. In the next place, the assemblage of the 
 three formula} reported above, which are made to 
 succeed each other, suit very well in the form- 
 ation of tables of correspondence between the pres- 
 sures and the temperatures, when that is the end 
 proposed. Likewise, in an enquiry relative to the 
 expansion of the steam in an engine, when it is 
 known precisely within what limits of pressure that 
 expansion will take place, it may immediately be 
 discerned which of the three formula} is applicable 
 to the case to be considered, and then t may be 
 eliminated between that formula and the tequation 
 of volumes. But if the question regards, for in- 
 stance, the case wherein the steam generated in 
 the boiler under a pressure of 8 or 10 atmospheres, 
 might, according to the circumstances of the motion, 
 expand during its action in the engine, either to a 
 pressure less than 1 atmosphere, or to a pressure 
 between 1 and 4 atmospheres, or in fine to a pres- 
 sure superior to 4 atmospheres ; then we shall not 
 know which of the three formula} to use in the 
 elimination, and it will be impossible to arrive at a 
 general equation representing the effect of the 
 engine in all cases. 
 
 Besides, were we even to adopt any one of those 
 equations, the radicals they contain would render 
 the calculation so complicated as to make it unfit 
 for practical applications. 
 
 The equations of temperature hitherto known 
 
80 CHAPTER II. 
 
 cannot then solve the question that presents itself, 
 that is to say, satisfy the wants of the calculation of 
 steam-engines in this respect ; and, consequently, 
 the only means left is to seek, in a direct manner, an 
 approximate relation, proper to give immediately 
 the relative volume of the steam at the maximum 
 density in terms of the pressure alone. 
 
 With this view Mr. Navier had proposed the ex- 
 pression : 
 
 1000 
 
 ^~ -09 + '0000484 j> ' 
 
 in which /* is the relative volume, or the ratio of 
 the volume of the steam to that occupied hy the 
 same weight of water, and p the pressure expressed 
 in kilograms per square metre. But this formula, 
 though exact enough in high pressures, deviates 
 considerably from experience in pressures helow 
 that of the atmosphere, which, however, come 
 under consideration in condensing engines. More- 
 over, for non-condensing engines, it is possible to 
 find one much more exact, as will presently be seen. 
 We deem it then proper to propose, on this subject, 
 the following formulae : 
 
 Formula for condensing engines of various sys- 
 tems ; 
 
 10000 
 
 Formula for non-condensing engines ; 
 
 10000 
 + -0023 p. 
 
OF THE MECHANICAL ACTION OF THE STEAM. 81 
 
 In these formulae, the pressure p is expressed in 
 pounds per square foot. 
 
 The former of the two suits equally to pressures 
 superior or inferior to that of the atmosphere, at 
 least within the limits that it may occur to consider 
 in applying them to steam-engines. 
 
 We know that the greatest pressure used in the 
 boiler never surpasses 8 atmospheres, or 120 Ibs. 
 per square inch ; and on the other hand, that it can, 
 in no case, be required to calculate the effects of 
 steam acting as a moving force in an engine, at a 
 pressure inferior to 8 or 10 Ibs. per square inch, 
 or about f of an atmosphere. In a condensing 
 engine, for instance, the steam, after the communi- 
 cation with the condenser has been opened, never 
 descends into the cylinder at a pressure less than 4 
 Ibs. per square inch ; the friction of the engine, 
 besides, may be estimated at 1 Ib. per square inch ; 
 and it is impossible to suppose a load which shall 
 not, of itself and with the additional friction it occa- 
 sions in the engine, produce a resistance against 
 the piston of at least 3 Ibs. per square inch. Thus 
 the resistance to be overcome by force of the steam, 
 cannot in any case be less than 8 Ibs. per square 
 inch ; consequently the steam cannot descend into 
 the cylinder at a pressure less than 8 Ibs. per square 
 inch. A formula which gives the exact volumes 
 down to that pressure, is then all that can be 
 necessary for the calculations that may occur, and 
 we shall presently see that the proposed formula 
 fulfils that condition. 
 
CHAPTER II. 
 
 The first of the formulae might also, without any 
 noticeable error, he applied to non-condensing 
 engines. Since, however, in these the steam can 
 hardly be spent at a total pressure less than two 
 atmospheres, by reason of the atmospheric pressure, 
 the friction of the engine and the resistance of the 
 load, it is needless to require of the formula exact 
 volumes for pressures less than 2 atmospheres. 
 In this case, then, the second formula will be found 
 to have a greater degree of accuracy, and we shall 
 in consequence prefer it. 
 
 It will be remarked that, besides the necessity of 
 these formulae in the general calculation of the 
 effect of steam-engines, they have the advantage 
 moreover, for other purposes in the arts, of dispens- 
 ing entirely with tables of temperature, and of sup- 
 plying the place of tables of the volume of the 
 steam, when these are not at hand. 
 
 Finally, to give a precise idea of the approxima- 
 tion given by the two formula just mentioned, we 
 here subjoin a table of the values they furnish for 
 the principal points of the scale of pressures. 
 
OF THE MECHANICAL ACTION OF THE STEAM. 83 
 
 Relative Volume of the Steam generated under 
 different pressures, calculated by the proposed 
 formulae. 
 
 TOTALPRESSURE 
 
 of the Steam, in 
 Bounds per square 
 'nch. 
 
 VOLUME of the 
 Steam, calculated 
 by the ordinary 
 formula?. 
 
 VOLUME calcu- 
 lated by the pro- 
 posed formula for 
 condensing en- 
 gines. 
 
 VOLUME calcu- 
 lated by the pro- 
 posed formula for 
 non-condensing 
 engines. 
 
 5 
 
 4624 
 
 4386 
 
 
 
 6 
 
 3901 
 
 3771 
 
 
 
 7 
 
 3380 
 
 3307 
 
 
 
 8 
 
 2985 
 
 2946 
 
 
 
 9 
 
 2676 
 
 2655 
 
 
 
 10 
 
 2427 
 
 2417 
 
 
 
 11 
 
 2222 
 
 2218 
 
 j> 
 
 12 
 
 2050 
 
 2049 
 
 99 
 
 13 
 
 1903 
 
 1904 
 
 99 
 
 14 
 
 1777 
 
 1778 
 
 
 
 15 
 
 1669 
 
 1668 
 
 ft 
 
 20 
 
 1280 
 
 1273 
 
 1243 
 
 25 
 
 1042 
 
 1030 
 
 1031 
 
 30 
 
 882 
 
 864 
 
 881 
 
 35 
 
 765 
 
 745 
 
 768 
 
 40 
 
 677 
 
 654 
 
 682 
 
 45 
 
 608 
 
 583 
 
 613 
 
 50 
 
 552 
 
 526 
 
 556 
 
 55 
 
 506 
 
 479 
 
 509 
 
 60 
 
 467 
 
 440 
 
 470 
 
 65 
 
 434 
 
 407 
 
 436 
 
 70 
 
 406 
 
 378 
 
 406 
 
 75 
 
 381 
 
 354 
 
 381 
 
 80 
 
 359 
 
 332 
 
 358 
 
 85 
 
 340 
 
 312 
 
 338 
 
 90 
 
 323 
 
 295 
 
 320 
 
 105 
 
 281 
 
 254 
 
 276 
 
 120 
 
 249 
 
 222 
 
 243 
 
 135 
 
 224 
 
 198 
 
 217 
 
 150 
 
 203 
 
 178 
 
 196 
 
84 CHAPTER II. 
 
 SECT. V. Of the constituent heat of the steam in 
 contact with the liquid. 
 
 There is yet an enquiry, relative to the proper- 
 ties of steam, which has long fixed the attention of 
 natural philosophers : it is that of the quantity of 
 heat, necessary to constitute the steam in the state 
 of an elastic fluid under various degrees of elas- 
 ticity. 
 
 It is well known that when water is evaporated 
 under the atmospheric pressure, in vain new quan- 
 tities of heat may be added by means of the fur- 
 nace, neither the temperature of the water, nor that 
 of the steam ever rise above 100 of the centi- 
 grade thermometer, or 212 of Fahrenheit. All 
 the heat then which is incessantly added to the 
 liquid must pass into the steam, but must subsist 
 there in a certain state which is called latent, be- 
 cause the heat, though really transmitted by the 
 fire, remains nevertheless without any effect upon 
 the thermometer, nor does it afterwards become 
 perceptible till the moment of disengaging itself, 
 on the steam being condensed. 
 
 This latent heat evidently serves to maintain the 
 molecules of water in the degree of separation suit- 
 able to their new state of elastic fluid ; and it is 
 then absorbed by the steam, in a manner similar to 
 that which is absorbed by the water, on passing 
 from the solid state, or state of ice, to the liquid. 
 But it is important to know the quantity of the latent 
 
OF THE MECHANICAL ACTION OF THE STEAM. 85 
 
 heat, in order to appreciate with accuracy the mo- 
 difications the steam may undergo. 
 
 Some essays made hy Watt had already elicited, 
 that the steam, at the moment of its generation, or 
 in contact with the liquid, contains the same quan- 
 tity of total heat, at whatever degree of tension, or, 
 in other words, at whatever degree of density it 
 may be formed. The experiments of Messrs. Sharpe 
 and Clement have since confirmed this result. 
 From them is deduced, that the quantity of latent 
 heat contained in the steam in contact with the 
 liquid, is less and less, in proportion as the tempe- 
 rature is higher; so that the total heat, or the sum 
 of the latent heat plus the heat indicated by the 
 thermometer, form in all cases a constant quantity 
 represented by 650 of the centigrade thermometer, 
 or, 1170 of Fahrenheit's. 
 
 Southern, on the contrary, has concluded from 
 some experiments on the pressure and temperature 
 of steam, that it is the latent heat which is constant ; 
 and that, to have the total quantity of heat actually 
 contained in steam formed at a given temperature, 
 that temperature must be augmented by a constant 
 number, representing the latent heat absorbed by 
 the steam in its change of state. 
 
 Some authors have deemed this opinion more 
 rational, but the observations we are about to 
 report seem to us to set the former beyond all 
 doubt. 
 
 It is known, that when an elastic fluid dilates 
 itself into a larger space, the dilatation is invari- 
 
86 CHAPTER II. 
 
 ably attended with a diminution of temperature. 
 If, then, the former of the two laws is exact, it 
 follows that the steam, once formed at a certain 
 pressure, may be separated from the liquid, and 
 provided it lose no portion of its primitive caloric, 
 by any external agent, it may dilate into greater and 
 greater space, passing at the same time to lower 
 and lower temperatures, without ceasing on that 
 account, to remain at the maximum density for its 
 actual temperature. In effect, since we suppose 
 that the steam has in reality lost no portion of its 
 total heat, the consequence is that it always con- 
 tains precisely as much as suffices to constitute it 
 in the state of maximum density, as well at the new 
 temperature as at the former. 
 
 If, on the contrary, Southern's law be exact, 
 when the steam, once separated from the liquid, 
 will diminish in density as it dilates into a larger 
 space, it will not remain at the maximum density 
 for the new temperature. To admit indeed that it 
 would do so, would be to verify Watt's law, since 
 the new steam would be at the maximum density, 
 although containing precisely the same quantity of 
 total heat as the old. But since we admit, on the 
 contrary, that the primitive steam contained more 
 heat than was necessary to constitute the new at 
 the maximum density, it follows that the surplus 
 heat, now liberated, will diffuse itself in the new 
 steam; and as this is separated from the liquid, 
 the increase of heat cannot have the effect of 
 increasing the density of the steam, but will be 
 
OF THE MECHANICAL ACTION OF THE STEAM. 87 
 
 altogether sensible in the temperature. Thus the 
 result will be, a steam at a certain density, in- 
 dicated by the spaces into which it is dilated, and 
 at a temperature higher than what is suitable to 
 that density, in steams at the maximum of density 
 for their temperature. 
 
 Now, in a numerous series of experiments of 
 which we shall speak hereafter, we have found 
 that in an engine whose steam-pipes were com- 
 pletely protected against all external refrigeration, 
 the steam was generated at a very high pressure in 
 the boiler ; and, after having terminated its action 
 in the engine, escaped into the atmosphere at pres- 
 sures very low and very varied ; and that in every 
 case, the steam issued forth precisely in the state 
 of steam at the maximum of density for its tempe- 
 rature. Southern's law then is inadmissible, un- 
 less any one choose to suppose that in these varied 
 changes of pressure, the steam lose, by contact 
 with the very same external surfaces, always pre- 
 cisely and strictly just that quantity of heat, some- 
 times very considerable, at other times very small, 
 by which its temperature should have increased. 
 Consequently we regard the law of Watt as the 
 only one supported by the facts. 
 
 The total quantity of heat contained in the steam 
 in contact with the liquid, and under any pressure 
 whatever, is then a constant quantity ; and accord- 
 ing as the sensible heat increases, the latent heat 
 diminishes in an equal quantity. 
 
 On the other hand, according to the same law, 
 
88 CHAPTER 11. 
 
 if we conceive water to be enclosed in a vessel 
 capable of sufficient resistance, and submitted to 
 temperatures of greater and greater intensity ; the 
 latent heat of the steam thence arising, will be less 
 and less as the sensible heat or temperature shall 
 become greater ; and as soon as the steam shall be 
 generated at a temperature equal to 650 centigrade 
 or 1170 degrees of Fahrenheit, it will cease to ab- 
 sorb heat in a latent state, and will no longer re- 
 ceive any portion of it, but which will be sensible 
 on the thermometer. We must then conclude that 
 at this point the steam will have a density equal 
 to that of water ; since in passing from one state 
 to another, it requires no farther increase of caloric, 
 as would be necessary if any farther increase of se- 
 verance were to take place between the molecules. 
 Thus the water, though still contained in the ves- 
 sel, will all have passed into the state of steam. 
 From this moment then, new quantities of heat 
 may be applied to the vessel ; but instead of acting 
 on a liquid, it will now act only on an elastic fluid, 
 and therefore all the increase of heat which is 
 added, will, as in all gases, become sensible on the 
 thermometer. 
 
 This observation explains the difficulty, which 
 would otherwise present itself: viz., that beyond 
 650 degrees centigrade or 1170 of Fahrenheit, 
 the preceding law could not subsist without the 
 latent heat becoming a negative quantity, which 
 had been the cause of this law being rejected by 
 some authors. 
 
OF THE MECHANICAL ACTION OF THE STEAM. 89 
 
 SECT. VI. Of the conservation of the maximum 
 density of the steam for its temperature, during 
 its action in the engine. 
 
 When an engine is at work, the steam is gene- 
 rated in the boiler at a certain pressure ; from 
 thence it passes into the cylinder, assuming a dif- 
 ferent pressure, and then, if it be an expansive 
 engine, the steam after its separation from that of 
 the boiler, continues to dilate itself more and more 
 in the cylinder, till the end of the stroke of the 
 piston. It is commonly supposed that, during all 
 the changes of pressure which the steam may un- 
 dergo, its temperature remains the same, and the 
 consequent conclusion is that, during the action of 
 the steam in the engine, its density or relative vo- 
 lume follows the law of Boyle or Mariotte ; that 
 is to say, the relative volume varies in the inverse 
 ratio of the pressure. This supposition simplifies 
 indeed the formulae considerably, but we shall pre- 
 sently see that it is contrary to experience ; and 
 therefore it becomes necessary to seek what is the 
 true law, according to which the steam changes 
 temperature in the engine, at the same time that 
 its pressure changes. And as calculations relative 
 to the effects of steam, depend essentially on the 
 volume it occupies, we must seek also what changes 
 that volume undergoes, by reason of the variations 
 of temperature and pressure, which take place in 
 the steam during its action. 
 
90 CHAPTER II. 
 
 We shall then substitute for the relation pre- 
 cedently indicated, another more real, and, what 
 is essentially necessary to calculate the effects of 
 steam with accuracy, deduced from the facts them- 
 selves. 
 
 We have just said that the calculations relative 
 to steam-engines suppose the steam to preserve in- 
 variably its original temperature, which allows the 
 application of Boyle's or Mariotte's law to all the 
 changes of density or of pressure it may undergo. 
 However, as it is known that elastic fluids never 
 dilate without cooling in some degree, this suppo- 
 sition obviously could not be realized, but on condi- 
 tion that the steam have time to recover from the 
 bodies with which it is in contact, supposed to be 
 sufficiently heated, the quantity of caloric necessary 
 to restore its temperature, after expansion, to the 
 same degree at which it was before. Now, the ra- 
 pidity of the motion of the steam in the cylinders 
 and the pipes will not suffer the admission of such 
 an hypothesis. 
 
 To obtain satisfaction on this head, in a numerous 
 series of experiments which will be found related in 
 the second edition of our TREATISE ON LOCOMOTIVES, 
 we adapted to the boiler of a locomotive engine a 
 thermometer and an air-gauge or manometer ; we 
 applied also two similar instruments to the pipe 
 through which the steam, after having terminated its 
 action in the engine, escaped into the atmosphere ; 
 and observed their simultaneous indications. The 
 steam was generated in the boiler at a total pressure 
 
OF THE MECHANICAL ACTION OF THE STEAM. 91 
 
 varying from 40 Ibs. to 65 Ibs. per square inch, and 
 escaped into the atmosphere at a pressure varying, 
 according to different circumstances, from 20 Ibs. 
 to 15 Ibs. per square inch. Had the steam pre- 
 served its temperature during its action in the 
 engine, it would have issued forth with the pres- 
 sure, for instance, of 15 Ibs. per square inch, but 
 with the temperature proper to the pressure at 
 which it had been formed, that is, 65 Ibs. per 
 square inch. Now, nothing like this took place: 
 during some hundreds of experiments wherein we 
 observed and registered these effects, we found in- 
 variably that the steam escaped precisely with the 
 temperature suitable to its actual pressure ; so that 
 the thermometer, graduated to indicate the pressure 
 in the steam of maximum density, gave identically the 
 same degree of pressure as the air-manometer, and 
 accorded equally with a siphon-manometer, which 
 we had superadded to the apparatus at the point 
 of the outlet of the steam. The steam then was 
 generated in the boiler at a very high pressure and 
 quitted the engine at a very low one ; but, on its 
 leaving the engine, as well as at the moment of its 
 production, the steam was at the same temperature 
 that it would have had, if immediately formed at 
 the pressure which it had at the moment of the 
 observation. 
 
 Consequently, we are to conclude from these 
 experiments, that during its whole action in the 
 engine, the steam remains in the state of steam 
 at the maximum pressure or density for its tern- 
 
92 CHAPTER II. 
 
 perature. Hence it results that, when the pressure 
 of the steam changes in the engine, its temperature 
 changes spontaneously at the same time, and vice 
 versa; so that they always preserve the mutual 
 relation which connects the pressures and tempera- 
 tures in the steam in contact with the generating 
 liquid. 
 
 Now, we have shown in the fourth section of this 
 chapter, that, with regard to steam in contact with 
 the liquid, the relative volume, that is the ratio 
 between the volume of the steam and the volume 
 of an equal weight of water, may be expressed in 
 terms of the pressure by the following very simple 
 formula, 
 
 /* = , .... (a) 
 
 n + qp 
 
 in which //, is the relative volume of the steam, p 
 the pressure expressed in pounds per square foot, 
 and the constant quantities n and q, have, according 
 to the engines considered, the numerical values 
 already indicated, viz. : 
 
 Condensing engines : 
 
 rc = -00004227 .... ^='000000258; 
 Non-condensing engines : 
 
 n- -0001421 .... q- '00000023. 
 
 This relation, then, will be applicable to all the 
 states of the steam during its action in the engine. 
 
 Now, according to equation (), if we suppose 
 that a certain volume of water represented by S, be 
 transformed into steam at the pressure />, and that 
 
OF THE MECHANICAL ACTION OF THE STEAM. 93 
 
 we call M the absolute volume of steam which will 
 be produced by it, we shall have, 
 
 = _ 
 
 " "~ 
 
 S n + qp 
 
 If afterwards the same volume of water be trans- 
 formed into steam at the pressure p, and that we 
 call M' the absolute volume of the resulting steam, 
 we shall have also, 
 
 M!= J_ 
 
 JS n + qp' 
 
 Consequently, between the absolute volumes of 
 steam which correspond to the same weight of 
 water, we shall have the definitive relation, 
 
 that is to say : the volumes of the steam will be, 
 not in the inverse ratio of the pressures, as was 
 supposed in admitting Boyle's or Mariotte's law, but 
 in the inverse ratio of the pressures augmented by 
 a constant quantity. 
 
 The last equation gives also 
 
 n 
 
 And the two equations (V) and (c) will serve to 
 determine, either M, or p, according to the one of 
 these two quantities, which will be unknown. 
 
 These relations then must be substituted for that 
 of Boyle or Mariotte, which is not applicable to the 
 operation of steam in the steam-engine. 
 
94 CHAPTER II. 
 
 As, in all calculations relative to the effects of 
 steam-engines, the volume occupied by a given 
 weight of steam forms the important element of 
 calculation, it is very ohvious that the use of the 
 ahove principle ; that is, of the conservation of the 
 maximum density of the steam for its temperature, 
 during its action in the engine, and the formula by 
 which we have represented it, will tend to the 
 avoiding of many considerable errors in the results. 
 
 If we consider, for instance, an engine in which 
 the steam generated at the pressure of 8 atmo- 
 spheres, or 120 Ibs. per square inch, shall expand 
 to 10 Ibs. per square inch ; then in the usual mode 
 of calculation, it will be supposed that the steam, 
 during its expansion, will preserve its temperature, 
 and that its volume will vary in the inverse ratio of 
 the pressures. The volume of the steam at the 
 pressure of 120 Ibs. per square inch is 249 times 
 that of the water which produced it. If its tempera- 
 ture remained unchanged during its action in the 
 engine, its volume after the expansion would be- 
 come 
 
 12O 
 
 249 x = 2988. 
 
 The supposition, then, amounts to admitting that 
 under the pressure of 10 Ibs. per square inch, the 
 volume of the steam would be 2988 times that of 
 the water. Now, from accurate tables, this volume 
 is 2427. An error, then, is induced of i on the 
 real volume of the steam, that is to say, on the effect 
 of the engine ; and this error will be almost entirely 
 
OF THE. MECHANICAL ACTION OF THE STEAM. 95 
 
 avoided by the use of our formula, since it gives in 
 this case 2417, instead of 2427, that is to say, it 
 differs inconsiderably from the true volume of the 
 steam. 
 
 Let us, however, add, that in slight differences of 
 pressure, such as take place in some engines, the 
 error may become scarcely noticeable. 
 
96 
 
 CHAPTER III. 
 
 GENERAL THEORY OF THE STEAM-ENGINE. 
 
 ARTICLE I. 
 
 OF THE EFFECT OF STEAM-ENGINES, IN THE CASE OF A GIVEN 
 EXPANSION, WITH ANY VELOCITY OR LOAD WHATEVER. 
 
 SECT. I. Of the different problems which present 
 themselves in the calculation of steam-engines. 
 
 AFTER having exposed succinctly, in the first 
 chapter of this work, the manner in which we con- 
 ceive the mode of action of the steam in steam- 
 engines, we now proceed to the full development of 
 the theory of which, as yet, we have given but a 
 very imperfect sketch, and to the solution of the 
 different problems that may occur in the working 
 or in the construction of steam-engines. 
 
 We distinguish three cases in the working of a 
 steam-engine ; that in which it works at a given 
 expansion of the steam, and with any load or ve- 
 locity whatever ; that in which it works at a given 
 expansion, and with the load or velocity proper to 
 produce its maximum useful effect with that ex- 
 
OF THE EFFECT OF STEAM-ENGINES. 9? 
 
 pansion; and lastly, that in which the expansion 
 having been previously regulated for the most fa- 
 vourable working of the steam in that engine, it is 
 loaded, moreover, with the most advantageous load 
 for that expansion ; which consequently produces the 
 absolute maximum useful effect for that engine. 
 
 We have already observed, that the three fun- 
 damental problems of the calculation of steam-en- 
 gines consist in finding successively the velocity, 
 the load, and the evaporation of the engine. After 
 the solution of these three problems, that which 
 first presents itself as a corollary to them, consists 
 in determining the useful effect of the engine, 
 which determination itself may be expressed under 
 eight different forms, viz. by the number of pounds 
 raised one foot by the engine in a minute ; by 
 the force of the engine in horse power ; by the 
 effect of 1 Ib. of coal ; by the effect of one cubic 
 foot of water evaporated ; by the number of pounds 
 of coal or of cubic feet of evaporated water, neces- 
 sary to produce one horse power ; and finally, 
 by the number of horses represented by each pound 
 of fuel consumed or by each cubic foot of water 
 evaporated. We have, then, to give successively 
 the means of solving these different questions. 
 
 For the sake, however, of greater precision, the 
 following are the problems we purpose to solve in a 
 general manner, for each of the three cases above 
 noticed, and for the different kinds of engines. 
 
 1. Given the load of an engine, in other respects 
 
 H 
 
98 CHAPTER III. 
 
 fully known, to determine what velocity the engine 
 will assume with that load. 
 
 2. Knowing, on the contrary, the velocity at 
 which it is intended to work the engine, to deter- 
 mine what load it can set in motion at that velocity. 
 
 3. Given the load to be moved hy the engine 
 and the velocity at which it is to move, to deter- 
 mine what evaporation the engine must he capable 
 of, and consequently the dimensions requisite for 
 the boiler, in order to produce the desired effects. 
 
 4. The evaporation, the pressure, and the di- 
 mensions of an engine being known, to calculate 
 the useful effect it will produce in a given time, at 
 a determined velocity or with a determined load. 
 
 To determine, from the same data, the horse 
 power of the engine. 
 
 Having the same data, and moreover the con- 
 sumption of fuel per hour, to find successively : 
 
 The useful effect the engine will produce per 
 pound of fuel. 
 
 The useful effect the engine will produce per 
 cubic foot of water evaporated. 
 
 The weight of fuel that will produce one horse 
 power. 
 
 The volume of evaporated water that will pro- 
 duce one horse power. 
 
 The horse power which will be produced by the 
 consumption of one pound of fuel. 
 
 The horse power which will be produced by one 
 cubic foot of water evaporated. 
 
OF THE EFFECT OF STEAM-ENGINES. 99 
 
 These various problems will be solved in the three 
 cases mentioned above. Consequently in the two 
 latter, the question will be to calculate the velocity, 
 the load, and the effects, corresponding to the re- 
 lative or absolute maximum useful effect of the 
 engine. 
 
 In the ordinary theory of steam-engines, the 
 solution of three questions only had ever been at- 
 tempted ; namely, to determine the load, the eva- 
 poration, and the useful effect (under different 
 forms) ; and we have seen that their solution was 
 defective. As for determining the velocity of the 
 engine for a given load, no solution had ever been 
 proposed ; and the very nature of the theory em- 
 ployed did not permit of distinguishing in the en- 
 gine, the existence of the three cases which do in 
 reality occur. It is possible, then, that the ques- 
 tions we have just presented may at first appear 
 rather obscure, expressed as they are in general 
 terms, and inferring relations under which it is not 
 usual to consider steam-engines ; but they will be 
 explained as we proceed, and their indispensable 
 necessity will be felt, to calculate with accuracy 
 either the proportions or the effects of sieam en- 
 gines of every kind. 
 
 SECT. II. Of the velocity of the piston under a 
 given load. 
 
 In the 6th section of the preceding chapter, we 
 have demonstrated that during all its action in the 
 
 H 2 
 
100 CHAPTER III. 
 
 engine, the steam constantly remains at the state 
 of maximum density for its temperature ; and we 
 have shewn that, accordingly, when the steam 
 passes, in the engine, from a certain volume M' to 
 another volume M equally known, and that its 
 pressure varies in consequence, and passes from 
 the known pressure p to another unknown pressure 
 p, the pressure p may be determined by the follow- 
 ing equation : 
 
 _ M' /n ,\ n 
 P ~M\q +P )~q' 
 
 This preliminary relation once established, in 
 order to embrace immediately the most complete 
 mode of action of the steam, we will suppose an 
 engine working with expansion and condensation, 
 and with any pressure whatever in the boiler. 
 Then, to pass afterwards to unexpansive engines, or 
 to those without condensation, it will suffice to 
 make the proper suppressions and substitutions in 
 the general equations. 
 
 From what is already known of the proposed 
 theory, the relation we seek between the various 
 data of the problem, will be deduced from two 
 general conditions : the former expressing that the 
 engine has attained an uniform motion, and conse- 
 quently, that the quantity of work applied by the 
 power is equal to the quantity of action developed 
 by the resistance ; the second, that there is neces- 
 sarily, equality between the mass of steam expended 
 by the cylinder, and the mass of steam generated 
 in the boiler. 
 
OF THE EFFECT OF STEAM-ENGINES. 101 
 
 Let P be the total pressure of the steam in the 
 boiler, and P' the pressure the same steam will 
 have on arriving in the cylinder, a pressure which 
 will always be less than P, except in a particular 
 case, which we shall treat of shortly. The steam 
 then will enter the cylinder at the pressure P 7 , 
 and will continue to flow in with that pressure 
 and to produce a corresponding effect, till the 
 communication between the boiler and the cylinder 
 is intercepted. The arrival of any new steam into 
 the cylinder will then be stopped, but that which 
 is already there will begin to dilate during the rest 
 of the stroke of the piston, producing by its ex- 
 pansion a certain quantity of work, which will go 
 to augment that already produced during the 
 period of the admission of the steam. 
 
 P being, as has been said, the pressure of the 
 steam in the boiler, and P' the pressure it will as- 
 sume on reaching the cylinder before the expansion, 
 let TT be the pressure of that steam at any point of 
 the expansion. At the same time let / be the total 
 length of the stroke of the piston, I' the portion 
 traversed at the moment when the expansion be- 
 gins, and X that which corresponds to the point 
 where the steam has acquired the pressure IT. 
 Lastly, let a be the area of the piston, and c the 
 clearance of the cylinder, that is to say, the vacant 
 space which exists at each end of the cylinder, be- 
 yond the portion traversed by the piston, and 
 which necessarily fills with steam at every stroke ; 
 this space, including the adjoining passages, being 
 
102 CHAPTER III. 
 
 represented by an equivalent length of the cylin- 
 der. 
 
 If the piston be taken at the moment when the 
 portion of the stroke traversed is \, and the pres- 
 sure TT, it will appear that if the piston traverse, 
 moreover, an elementary space d\ the elementary 
 work produced in that motion will be irad\. But 
 at the same time, the volume a (T + c), occupied 
 by the steam before the expansion, will have be- 
 come a (\ 4- c). Hence, from the equation (c), in- 
 dicated above, there will exist between the two 
 corresponding pressures P' and TT, the analogy 
 
 f n , T>/V'+ C U 
 7T=I - + ) - 
 
 v q '\ + c q 
 
 Multiplying the two members of this equation by 
 a d X, we shall deduce 
 
 \+c q 
 
 This expression will give then the quantity of ele- 
 mentary work produced by the expansion, while the 
 piston traverses the space d \ ; and if the integral 
 be taken between the limits V and /, we shall have 
 the total effect produced by the expansion of the 
 steam, from the moment of its being intercepted 
 to the end of the stroke : viz. 
 
 an expression in which the logarithm is a hyper- 
 bolic one. 
 
OF THE EFFECT OF STEAM-ENGINES. 103 
 
 This quantity expressing the work performed in 
 that portion of the stroke during which there was 
 expansion, if we add to it the effect P'aF, produced 
 during the anterior part 1! of the stroke, or before 
 the beginning of the expansion, we shall have for 
 the total work developed by the steam during the 
 whole stroke of the piston, 
 
 , it -^ / n T>/\ ( f l + c) n j 
 
 a (I + c) ( - + F ) < r/ - + log - - [ -- al. 
 \q / (I + c ' l f + c) q 
 
 But the engine being supposed to have attained 
 uniform motion, the work developed by the mover 
 must be equal to that developed by the resistance. Re- 
 presenting by R the total pressure exerted on the unit 
 of surface of the piston by virtue of that resistance, 
 or rather by virtue of the divers resistances which 
 take place in the engine, the work it will have de- 
 veloped in one stroke, will have for its expression, 
 
 aRL 
 We must therefore have the analogy 
 
 which is the first general relation between the dif- 
 ferent data of the problem. 
 
 This equation expressing that the work deve- 
 loped by the power, is entirely found in the effect 
 produced, it will be remarked that, for the analogy 
 to take place, it is not necessary that the motion 
 of the engine be strictly uniform. It may be com- 
 posed of equal oscillations, beginning from zero of 
 velocity, and returning to zero again ; provided 
 
104 CHAPTER III. 
 
 the successive oscillations be made in equal times, 
 and that the changes of velocity take place by in- 
 sensible degrees, so as to suffer no loss of vis viva. 
 
 It must be observed also, that, if in this ex- 
 pression we make I' -=1, which amounts to sup- 
 posing that the engine works without expansion, 
 the equation reduces itself to P' = R ; that is to 
 say, the pressure of the steam in the cylinder will, 
 in this case, be equal to the pressure of the resist- 
 ance against the piston, as we have already demon- 
 strated directly for unexpansive engines, of which 
 we spoke in the first chapter. 
 
 We have just obtained the first general relation 
 between the data and the incognita of the problem. 
 Let us now seek a second analogy resulting from 
 the equality between the production and the ex- 
 penditure of the steam. If S be made to express 
 the volume of water evaporated by the boiler in a 
 unit of time, and transmitted to the cylinder, this 
 volume on reaching the cylinder, transformed into 
 steam at the pressure P', will there become, from 
 the relation already given (a), 
 
 S 
 
 n + qP r 
 
 This will then be the volume of steam, at the pres- 
 sure P', supplied by the boiler in a unit of time, in 
 one minute for instance. On the other hand, a (l f + c) 
 being the volume of the steam expended at each 
 stroke of the piston, if there be K strokes per 
 minute, the expense per minute will be 
 
 Ka ;(?+*) 
 
OF THE EFFECT OF STEAM-ENGINES. 105 
 
 But expressing by v the velocity of the piston per 
 minute, we shall have also v = K/; which gives 
 
 K = -. Whence the above expenditure will be 
 
 v a (f + c) 
 
 ~~T 
 
 Since, then, there is an equality between the 
 production and the expenditure of the steam, we 
 shall have the equation 
 
 S V +C x-px 
 
 = va , . . . (15) 
 
 n + q F 
 
 which is the second general relation between the 
 data and the incognita of the problem. 
 
 Consequently, on eliminating P 7 from the two 
 equations (A) and (B), we shall have as the final 
 relation sought, 
 
 |?= S _l_ 
 " a n + qR 
 
 In this equation the logarithm log _ is a hy- 
 
 i ~T~ C 
 
 perbolic logarithm. As it is known that these 
 logarithms are deduced from those of the tables, 
 by multiplying the latter by the constant number 
 2*302585, or approximatively by 2*303, the term 
 
 log 7 '- might, for practical purposes, be replaced 
 
 L ~\~ C 
 
 by 2*303 log - t -, in which log. would then ex- 
 
 press an ordinary logarithm. But as tables of hy- 
 perbolic logarithms are found in several works, and 
 as besides, we shall give in the sequel, a table 
 
106 CHAPTER III. 
 
 which will dispense from all research on this head, 
 we will not here make any change in the formulae. 
 
 This equation is less simple than that which 
 would he obtained in the same enquiry, by suppos- 
 ing the steam to preserve its temperature through 
 the whole of its action in the engine ; but that 
 supposition, though producing often but slight dif- 
 ferences in the definitive results of the calculations, 
 is not really exact, since it is incontestable that the 
 steam changes its pressure during the expansion, 
 and that the experiments quoted above prove that 
 it changes temperature in a manner exactly cor- 
 respondent. The last formula which we have pre- 
 sented, has then the advantage of taking this 
 important circumstance into account, and conse- 
 quently of being more accurate in the applications. 
 Besides, if in equation (1) the effect of the change 
 of temperature be annulled, the formula becomes 
 the same that we have presented in the first chap- 
 ter, supposing the preservation of the temperature 
 of the steam. 
 
 In effect, we have seen, from equation (a), that 
 after the steam has assumed in the engine the 
 pressure R, the absolute volume of that steam, 
 which corresponds to the volume of water S, is 
 given by the relation 
 
 S 
 
 n + qR' 
 
 On the contrary, when the steam is supposed to 
 preserve its temperature, the volume varies in the 
 inverse ratio of the pressure. If, then, we call 
 
OF THE EFFECT OF STEAM-ENGINES. 107 
 
 m the relative volume of the steam generated at 
 the pressure P of the boiler, a relative volume 
 which can be known by the tables already given, it 
 is clear that the absolute volume of the steam cor- 
 respondent to the volume S of water, will first be, 
 under the pressure P, expressed by 
 
 mS; 
 
 and that, in passing afterwards to the pressure R, 
 this volume will change in the inverse ratio of the 
 pressures, that is to say, will become 
 
 m R* 
 
 Therefore, to pass from one law to the other, we 
 must write 
 
 S _. g P t 
 n+qR~ R' 
 
 or, what comes to the same, we must, in the for- 
 mulae already obtained, make 
 
 n = o, and - =mP. 
 
 Then the equation which gives the velocity, be- 
 comes 
 
 r== mPS /_j[_ + lo / + g\. 
 
 which, for the case of unexpansive engines, or for 
 V I, reduces itself to the following : 
 
 And this is precisely the equation we made use of 
 in the first chapter, if only we neglect in it the 
 clearance of the cylinder c. 
 
108 CHAPTER III. 
 
 The quantityR contained in equation (1), is the 
 total resisting pressure which takes place on the 
 unit of surface of the piston in the motion. But 
 this resisting pressure is evidently composed of 
 three parts, namely, the resistance arising from the 
 motion of the load, which we will call r ; that aris- 
 ing from the friction proper to the engine, which 
 we will express hy (f+ r\ calling f the friction 
 of the engine unloaded, and 8 the augmentation of 
 that friction per unit of the load r ; and finally the 
 pressure which may subsist on the face of the piston 
 opposed to the arrival of the steam, which we will 
 represent by p ; the latter quantity p expressing 
 the atmospheric pressure, when the engine is with- 
 out condensation, or only the pressure of condensa- 
 tion in the cylinder, when the engine is a condens- 
 ing one. The quantities r,f>p and 8, are besides, 
 as well as R, referred to the unit of surface of the 
 piston. 
 
 In the calculations relative to locomotive engines 
 we shall introduce three terms more : the first to 
 express the resistance of the air against the train 
 in motion, a force which, increasing in the ratio of 
 the square of the velocity, could not be neglected 
 without error ; the second to represent the resist- 
 ance offered by the engine itself in the transport of 
 its own weight on the rails ; and the third, to take 
 account of the force expended by the engine in 
 animating its fire, according to the method in use 
 in those engines. But as these divers circumstances 
 do not in general occur in stationary engines, we 
 
OF THE EFFECT OF STEAM-ENGINES. 109 
 
 will omit them at present, it being easy to .repro- 
 duce them in the particular cases, as it may become 
 necessary. 
 
 From what has just been said, the resistance R 
 may be replaced by 
 
 We shall then substitute this value in that of v, 
 and at the same time make 
 
 I' I + c 
 
 an expression, which in the case of f == /, that is to 
 say, for unexpansive engines, reduces itself simply 
 to the ratio 
 
 I 
 l + c' 
 
 Then, the value of v will become 
 
 =. k 
 
 a ' n 
 or 
 
 _S k 
 
 ~ 
 
 It will be remarked that the quantity 
 
 S 
 n + qR 
 
 is nothing else but the absolute volume of the steam 
 correspondent to S, in contact with the liquid at 
 the pressure R. Therefore, to have the velocity v, 
 we must calculate the volume of the steam which 
 corresponds to the volume of water S, supposed im- 
 mediately transformed into steam at a pressure 
 
110 CHAPTER III. 
 
 equal to the resistance R, afterwards divide that 
 volume by the area a of the piston, and lastly, 
 multiply the quotient by the quantity k, of which 
 we have a little before given the developed expres- 
 sion. 
 
 The formula (1) contains the general relation 
 between all the data of the problem, and will serve 
 us to solve successively the different questions we 
 have proposed elucidating. It will, however, be 
 observed that the homogeneity of the formula re- 
 quires that the dimensions of the engine a, / and /' 
 be expressed in the same unit as the volume of 
 water evaporated S, and that the pressures per 
 unit of surface P, r, and p, be also referred to the 
 same unit as S. We mention this circumstance 
 because these various quantities are usually referred 
 to different units, according to what may be, in 
 practice, the most convenient manner of expressing 
 each. 
 
 Besides, from the mode of our reasoning itself, 
 it is to be understood that the quantity S, in the 
 equation, is the effective evaporation of the engine ; 
 that is, it represents the volume of water which 
 really enters the cylinder in the state of steam, and 
 there acts upon the piston. If then, from any mode 
 of construction of the engine, it should occur that a 
 portion of the steam generated in the boiler, escape 
 without acting on the piston, that portion is not to 
 be considered as included in the quantity S, and 
 ought, therefore, to be deducted before all calcula- 
 tion. 
 
OF THE EFFECT OF STEAM-ENGINES. Ill 
 
 The formula just obtained will give the velocity 
 of the piston for any load r, when the dimensions 
 and different data of the engine contained in the 
 equation are known. This formula is general, and 
 applies to every kind of rotative steam engine. If 
 the engine he expansive, it will suffice to replace I 1 
 hy the length of the stroke traversed when the 
 steam begins to be intercepted ; if the engine be 
 unexpansive, it will suffice to make l r =. I. If there 
 be condensation, p must be replaced by the pres- 
 sure of condensation ; and if the engine be not a 
 condensing one, p is to be replaced by the atmo- 
 spheric pressure. However, before making these 
 deductions relative to the different systems of en- 
 gines, we shall continue to seek the general for- 
 mulae for all the problems we have undertaken to 
 solve. 
 
 Let it only be observed, that the velocity of the 
 piston in a given engine, is totally independent of 
 the pressure at which the steam is formed in the 
 boiler, and that, on the contrary, it depends essen- 
 tially on the evaporation S of the boiler per unit of 
 time, and on the total resistance [(1 + 8) r +p +/*] 
 opposed to the motion of the piston. 
 
 SECT. III. Of the load of the engine, for a given 
 velocity. 
 
 The analogy we have just obtained will show 
 reciprocally the resistance a known engine can set 
 in motion at a determined velocity. In effect, it 
 suffices to draw from it the value of r ; or rather, as 
 
CHAPTER III. 
 
 r is only the resistance per unit of surface of the 
 piston, it will he preferable to have the whole re- 
 sistance, by taking immediately the value of a x r, 
 that is, 
 
 Sk a 
 
 From the form of this expression, it would ap- 
 pear at a first glance, that on making v = 0, that 
 is, on supposing the velocity null, the result would 
 be an infinite load ; but, on examining the formula 
 more attentively, we soon perceive that the result 
 would by no means be such. 
 
 In effect, if v = 0, it follows also that S = ; for 
 S is the quantity of steam which effectively traverses 
 the cylinders in a unit of time ; and no quantity of 
 steam whatever can traverse the cylinders without 
 moving the piston, and consequently creating some 
 velocity in the engine. If, then, the velocity be 
 supposed equal to zero, we must necessarily have 
 at the same time S = 0. But, making at once v = 
 and S = 0, we find 
 
 
 
 ar =o> 
 
 and not ar= GO , as it at first appeared. 
 
 Thus, in this case, the formula reduces itself to 
 the indeterminate form ; but it is to be observed, 
 that the present formulae give the effects of the 
 engine, only after the uniform motion has taken 
 place. Now we shall presently see that, for a given 
 evaporation S, the uniform velocity can never be 
 less than 
 
OF THE EFFECT OF STEAM-ENGINES. 113 
 
 ,__ 
 
 '~a 
 
 since it is that which corresponds to the passage of 
 the steam into the cylinders, at its state of greatest 
 density, and that at any other density, that steam 
 would form a larger volume, and consequently could 
 not traverse the cylinders in the same time, without 
 producing a greater velocity. All supposition of 
 less velocity than this, is then inadmissible in this 
 problem, as being incompatible with that state of 
 uniformity of motion, for which alone the effects of 
 machines are calculated. 
 
 SECT. IV. Of the evaporation of the boiler, to pro- 
 duce wanted effects. 
 
 To find the evaporation of which an engine 
 ought to be capable, in order to set in motion a 
 certain resistance r at a known velocity v 9 the value 
 of S must be drawn from the same equation, 
 
 rC 
 
 This equation gives the quantity of water the 
 engine ought to be capable of evaporating and 
 transmitting to the cylinder per minute. It will 
 then be easy, according to the mode of construction 
 intended to be used for the boiler, and the prac- 
 tical data proper to estimate the quantity of water 
 evaporated by such form of boiler, to know what 
 extent of heating surface should be given to the 
 boiler of the engine, in order to obtain the pro- 
 posed effects. 
 
114 CHAPTER III. 
 
 As the quantity S represents here the effective 
 evaporation disposable by the engine, it is under- 
 stood that, if the usual construction of the engines 
 under consideration give rise to a certain loss of 
 steam, either by safety-valves or otherwise, account of 
 this must be taken, with as close an approximation 
 as possible, by first adding that loss to the quan- 
 tity S deduced from the preceding equation, then 
 by estimating the heating surface suitable to the 
 production of the useful steam augmented by the 
 lost steam. 
 
 SECT. V. Of the different expressions of the useful 
 effect of the engines. 
 
 1. The useful effect produced by the engine in 
 the unit of time at the velocity v, is evidently a r v, 
 since the velocity v is at the same time the space 
 traversed by the piston in a unit of time. Conse- 
 quently, by multiplying both members of equation 
 
 by v, we shall have the useful effect : 
 
 , Sk av 
 
 == . 
 
 This may be expressed in terms of the load, by 
 multiplying the two members of equation (1) by 
 a r. We have then for the useful effect the engine 
 may produce with a given load, 
 
 S r k 
 
 (4 bis) 
 
 n + q {(l+S)r+p+J} 
 It will be remarked, that in a given engine this 
 
OF THE EFFECT OF STEAM-ENGINES. 115 
 
 useful effect does not depend on the pressure at 
 which the steam is generated in the boiler, since 
 the quantity P does not appear in the above 
 equations ; but that it depends essentially on the 
 evaporation S effected by the boiler in a unit of 
 time. 
 
 2. If it be required to know the horse-power 
 which represents the effect of the engine, when 
 working at the velocity v, or when loaded with 
 the resistance r, it suffices to observe that what 
 is called one-horse power represents an effect 
 of 33,000 Ibs. raised one foot per minute. All 
 consists then in referring the useful effect produced 
 by the engine in the unit of time, to the new mea- 
 sure just chosen, that is, to the power of one horse ; 
 and consequently it will suffice to divide the ex- 
 pression already obtained in equation (4) by 
 33,000. 
 
 Thus the horse-power of the engine, at the ve- 
 locity v, or with the resistance r, will be 
 
 u.HR = JL^: .(5) 
 33000 
 
 We will here observe, that what is designated 
 by horse-power would, with much more propriety, 
 be termed horse-effect, since it is an effect and not 
 a force. It should then be said, that an engine is 
 of so many horse-effect, instead of saying that it is 
 of so many horse-power. 
 
 3. In the two preceding questions, we have ex- 
 pressed the power of the engine from the total ef- 
 fect it is capable of developing, without regard to 
 
116 CHAPTER III. 
 
 its consumption of fuel or water. We are now 
 about to express the same, either from the effect 
 it produces per unit of fuel or of water expended ; 
 or from its consumption while performing a given 
 work. 
 
 The useful effect obtained in equation (4), is that 
 which is produced by the volume S of water trans- 
 formed into steam; and as that volume of water S 
 is evaporated in a unit of time, the result is, as has 
 been said, the useful effect produced by the engine 
 in a unit of time. But if it be supposed that dur- 
 ing the unit of time, there be consumed N pounds 
 of fuel, the useful effect produced by each pound of 
 fuel will plainly be the N th part of the above 
 effect. 
 
 Hence the effect arising from the consumption 
 of 1 Ib. of fuel will be 
 
 u. E. 1 Ib. co.= U ;^' (6) 
 
 To apply this formula, it suffices to know the 
 quantity of fuel consumed in the furnace per 
 minute, that is to say, while the evaporation S is 
 taking place. This datum may be determined by 
 a direct experiment on the boiler itself, or by analogy 
 with other boilers similarly disposed. And the 
 datum once obtained, may be used for every qther 
 case, and for every supposition of velocity of the 
 engine. 
 
 4. We have seen above, that the effect indicated 
 by u. E. is that which is due to the volume of 
 water S transformed into steam. If, then, it be 
 
OF THE EFFECT OF STEAM-ENGINES. 117 
 
 required to know the useful effect arising from each 
 cubic foot of water, or from each unit of the volume 
 S, it will obviously suffice to divide the total effect 
 u. E. by the number of units in S. Thus, for the 
 useful effect due to the evaporation of one cubic 
 foot of water in the engine, we have 
 
 u.E. 1 ft wa.zz^l. . . . (7) 
 
 S 
 
 5. We have obtained above the useful effect 
 produced by one pound of fuel. It consequently 
 becomes easy to know the number of pounds of 
 fuel which represent any given useful effect, as, for 
 instance, one horse-power. A simple proportion 
 is, in fact, enough, and we have for the quantity, 
 in weight, of fuel requisite to produce one horse- 
 power, 
 
 Q.co.forlhp. = 3300 g N (8) 
 
 u. hi. 
 
 6. By a simple proportion will also be found the 
 quantity of water that must be evaporated, in order 
 to produce one horse-power, viz. : 
 
 Q. wa. for 1 hp.rr 380 ^ 8 . ... (9) 
 u. .h. 
 
 7. It may yet be required to know what horse- 
 power will be produced by a pound of coal ; which 
 will evidently be 
 
 u. HP. for lib. 00. = ^. . . .(10) 
 
 8. Finally, the horse-power produced by the 
 
118 CHAPTER III. 
 
 evaporation of 1 cubic foot of water will likewise 
 be 
 
 u. HP. for 1 ft.wa.^-1^. . . . (11) 
 33000 S 
 
 Substituting, then, in these several equations, 
 for u. E. its value determined by the formula, (4,) 
 we immediately deduce the numerical solution of 
 the proposed problems. 
 
 SECT. VI. Table for the numerical solution of the 
 formulce (rotative engines). 
 
 As the formulae we have just obtained, and those 
 which are about to follow, contain hyperbolic 
 logarithms, the use of which is inconvenient, we 
 here subjoin a table which gives, without calcula- 
 tion, the principal elements of the equations, and 
 will greatly simplify the matter. 
 
 In this table we have supposed the clearance of 
 the cylinder c '05 /, as is the case in rotative 
 steam-engines, of which we are now treating. In 
 single-acting engines the clearance of the cylinder, 
 including the adjoining passages, amounts to *1 of 
 the stroke, because the motion of the piston not 
 being limited by a crank, it is more liable to strike 
 the bottom of the cylinder. 
 
 We have not inserted in the table a column to 
 represent the fraction 
 
 
OF THE EFFECT OF STEAM-ENGINES. 119 
 ft 
 
 because it is evident that - being known by the first 
 
 column, the fraction 
 
 I' + c 
 ~T 
 
 s* 
 
 will be equal to the former augmented by -, that is, 
 by '05. 
 
120 
 
 CHAPTER III. 
 
 Table for the numerical solution of the formula 
 (rotative engines). 
 
 PORTION of 
 stroke per- 
 formed before 
 the expan- 
 sion, or value 
 of the fraction 
 V 
 
 CORRESPONDING 
 value of the 
 fraction 
 I 
 f+c 
 
 CORRESPONDING 
 value of k, or 
 of the expression 
 V l+c 
 
 r+- c +los 'i^-c 
 
 PORTION of 
 stroke per- 
 formed befor 
 the expan- 
 sion, or value 
 of the fraction 
 I' 
 
 CORRESPONDING 
 value of the 
 fraction 
 I 
 
 I'+e 
 
 CORRESPONDING 
 value of Jc, or 
 of the expression 
 
 r i+c 
 ,-q^+log jrp c 
 
 I 
 
 
 
 I 
 
 
 
 10 
 
 6-667 
 
 2-613 
 
 51 
 
 1-786 
 
 1-539 
 
 11 
 
 6-250 
 
 2-569 
 
 52 
 
 1-754 
 
 1-523 
 
 12 
 
 5-882 
 
 2-526 
 
 53 
 
 1-724 
 
 1-507 
 
 13 
 
 5-556 
 
 2-485 
 
 54 
 
 1-695 
 
 1-491 
 
 14 
 
 5-263 
 
 2-446 
 
 55 
 
 1-667 
 
 1-476 
 
 15 
 
 5-000 
 
 2-408 
 
 56 
 
 1-639 
 
 1-461 
 
 16 
 
 4-762 
 
 2-371 
 
 57 
 
 1-613 
 
 1-445 
 
 17 
 
 4-546 
 
 2-336 
 
 58 
 
 1-587 
 
 1-431 
 
 18 
 
 4-348 
 
 2-301 
 
 59 
 
 1-563 
 
 1-417 
 
 19 
 
 4-167 
 
 2-268 
 
 60 
 
 1-539 
 
 1-402 
 
 20 
 
 4-000 
 
 2-235 
 
 61 
 
 1-515 
 
 1-388 
 
 21 
 
 3-846 
 
 2-203 
 
 62 
 
 1-493 
 
 1-374 
 
 .22 
 
 3-704 
 
 2-173 
 
 63 
 
 1-471 
 
 1-361 
 
 23 
 
 3-571 
 
 2-142 
 
 64 
 
 1-449 
 
 1-347 
 
 24 
 
 3-448 
 
 2-114 
 
 65 
 
 1-429 
 
 1-334 
 
 25 
 
 3-333 
 
 2-085. 
 
 66 
 
 1-409 
 
 1-321 
 
 26 
 
 3-226 
 
 2-059 
 
 67 
 
 1-389 
 
 1-308 
 
 27 
 
 3-125 
 
 2-032 
 
 68 
 
 1-370 
 
 1-295 
 
 28 
 
 3-030 
 
 2-006 
 
 69 
 
 1-351 
 
 1-282 
 
 29 
 
 2-941 
 
 1-980 
 
 70 
 
 1-333 
 
 1-269 
 
 30 
 
 2-857 
 
 1-955 
 
 71 
 
 1-316 
 
 1-257 
 
 31 
 
 2-778 
 
 1-931 
 
 72 
 
 1-299 
 
 1-240 
 
 32 
 
 2-703 
 
 1-908 
 
 73 
 
 1-282 
 
 1-233 
 
 33 
 
 2-632 
 
 1-884 
 
 74 
 
 1-266 
 
 1-221 
 
 34 
 
 2-564 
 
 1-862 
 
 75 
 
 1-250 
 
 1-210 
 
 35 
 
 2-500 
 
 1-840 
 
 76 
 
 1-235 
 
 1-197 
 
 36 
 
 2-439 
 
 1-818 
 
 77 
 
 1-220 
 
 1-186 
 
 37 
 
 2-381 
 
 1-797 
 
 78 
 
 1-205 
 
 1-175 
 
 38 
 
 2-326 
 
 1-776 
 
 79 
 
 1-191 
 
 1-164 
 
 39 
 
 2-273 
 
 1-755 
 
 80 
 
 1-177 
 
 1-152 
 
 40 
 
 2-222 
 
 1-736 
 
 81 
 
 1-163 
 
 1-141 
 
 41 
 
 2-174 
 
 1-716 
 
 82 
 
 1-149 
 
 1-131 
 
 42 
 
 2-128 
 
 1-697 
 
 83 
 
 1-136 
 
 1-119 
 
 43 
 
 2-083 
 
 1-678 
 
 84 
 
 1-123 
 
 1-109 
 
 44 
 
 2-041 
 
 1-660 
 
 85 
 
 1-111 
 
 1-099 
 
 45 
 
 2-000 
 
 1-642 
 
 86 
 
 1-099 
 
 1-088 
 
 46 
 
 1-961 
 
 1-624 
 
 87 
 
 1-087 
 
 1-078 
 
 47 
 
 1-923 
 
 1-606 
 
 88 
 
 1-075 
 
 1-067 
 
 48 
 
 1-887 
 
 1-589 
 
 89 
 
 1-064 
 
 1-057 
 
 49 
 
 1-852 
 
 1-572 
 
 90 
 
 1-053 
 
 1-047 
 
 50 
 
 1-818 
 
 1-555 
 
 
 
 
OF THE MAXIMUM USEFUL EFFECT. 
 
 We limit ourselves to the preceding problems, 
 because they are those which are most commonly 
 wanted ; but it is obvious that, by means of the 
 same general analogies, any one of the quanti- 
 ties which appear in the problem may be deter- 
 mined, in case that quantity should be unknown, 
 and that it were desired to determine it according 
 to a given condition. Thus, for instance, might be 
 determined the area of the piston, or the pressure 
 in the boiler, or the pressure of condensation, &c., 
 corresponding to given effects of the engine, as we 
 have done for locomotive engines, in a preceding 
 work (TREATISE ON LOCOMOTIVES). But as these 
 questions rarely occur, and as they offer no difficulty, 
 we deem it sufficient to indicate here the manner of 
 obtaining their solution. 
 
 ARTICLE II. 
 
 OF THE MAXIMUM OF USEFUL EFFECT WITH A 
 GIVEN EXPANSION. 
 
 SECT. I. Of the velocity of maximum useful effect. 
 
 The preceding problems have been solved in the 
 most general way, that is to say, supposing the 
 engine to set in motion any load whatever at any 
 velocity whatever, with the single condition that 
 the load and velocity be compatible with the power 
 of the engine. In constructing an engine for a 
 
CHAPTER III. 
 
 determinate object, or to move a certain load with 
 a given velocity, it must not be planned in such 
 manner as to require the greatest effort of which it 
 is capable, to perform that task which is to be its 
 regular work ; for in that case, it would have no 
 power in reserve, to meet whatever emergencies 
 may occur in the service. On the other hand, 
 since the maximum effort of the engine with a 
 given expansion, corresponds, as we shall presently 
 see, to its maximum useful effect, it follows that 
 we are not to expect regularly from the engine its 
 maximum of useful effect, nor can the engine be 
 constructed with such pre-intention. It is neces- 
 sary however, when an engine is constructed, or 
 to be constructed, to know what is the velocity at 
 which it will produce its maximum useful effect, 
 and what this maximum useful effect will be ; 
 for it is evidently that knowledge which must 
 decide the regular working load of the engine, 
 and mark the possible limits of its effects in case 
 of emergency. 
 
 What is that velocity or that load, most advan- 
 tageous for the work, and what are the divers ef- 
 fects which will then be produced by the engine ? 
 This is what now remains to determine, first, 
 in supposing the expansion of the engine fixed 
 a priori, then in making that expansion itself to 
 vary, in order to obtain a further increase of 
 effect. 
 
 To know the velocity corresponding to the 
 greatest useful effect, it suffices to examine the 
 
OF THE MAXIMUM USEFUL EFFECT. 123 
 
 expression of the useful effect produced by the 
 engine under any velocity whatever, namely, 
 (equa. 4) : 
 
 Sk av n 
 
 It is observable here, at the first glance, that 
 since the velocity enters only into the negative 
 terms, the less that velocity is, for a given ex- 
 pansion, the greater will be the useful effect of 
 the engine. On the other hand, referring to the 
 expression of the velocity of the engine under a 
 given load, before having substituted for P' its 
 numerical value, viz. (equa. B) : 
 
 S J_ 
 
 a(n + q P') I' + c 
 
 we perceive that the velocity is the smallest pos- 
 sible, without loss of steam, when P 7 is greatest ; 
 and as P', which is the pressure of the steam in 
 the cylinder, can in no case exceed P, which is the 
 pressure in the boiler, the condition of the minimum 
 velocity, or of the maximum useful effect, will be 
 given by the equation P' = P, or 
 
 S / 
 
 t/= 7 - ^r- 7f - . . . . (12) 
 
 a (n + q P) l f + c 
 
 Expressing by m the volume of the steam under 
 the pressure P, referred to the volume of the same 
 weight of water, this formula may, from equation (a) 
 take the form 
 
 , raS / 
 t/= -- Tf - .... 
 
 a I +c 
 
CHAPTER III. 
 
 In this manner the calculation of the term 
 (n + q P) is avoided, since the quantity m is given 
 by the tables of Chapter II., and may thence 
 be taken with greater accuracy than from its ap- 
 proximative value 
 
 1 
 
 This observation will equally apply to all the fol- 
 lowing formulae, wherein the quantities n and q 
 recur united under the form (n + q P). 
 
 It is to be remarked, with respect to the pre- 
 ceding formula, that, mathematically speaking, the 
 pressure P 7 can never be quite equal to P. In 
 fact, since there exist pipes between the boiler and 
 the cylinder, through which the steam must pass, 
 and that the passages of those pipes form an ob- 
 stacle to the free motion of the steam, there must 
 necessarily be, on the side of the boiler, a small 
 surplus of pressure equivalent to the resistance of 
 the obstacle in question ; otherwise the motion of 
 the steam could not take place. This surplus of 
 pressure, then, on the side of the boiler, prevents 
 P 7 from becoming mathematically equal to P, and 
 thus the real velocity will always be rather greater 
 than v f . The difference between P 7 and P (we 
 mean the difference merely arising from the ob- 
 stacle just mentioned) will be by so much the less, 
 as the area of the passages is larger and their way 
 more direct ; but as, with the dimensions of ordi- 
 nary use in steam-engines, that difference is very 
 trifling, we shall not notice it here. Seeking it, in 
 
OF THE MAXIMUM USEFUL EFFECT. 
 
 fact, by known formulae for the flowing of gases, we 
 find that it is hardly appreciable by the instruments 
 used for measuring the pressure in the boiler ; con- 
 sequently, to introduce them into the calculation 
 would only complicate the formula}, without ren- 
 dering them more exact. 
 
 To return to the enquiry before us, the maxi- 
 mum useful effect will be given by the condition 
 P=zP, or 
 
 S J_ 
 
 l f + c 
 
 This is, then, the velocity at which the engine 
 must work, in order to obtain the greatest effect 
 possible ; and the equation P' = P shows recipro- 
 cally, that, when that velocity takes place, the 
 steam enters the cylinder at full pressure, that is, 
 nearly at the same pressure which it had when in 
 the boiler. 
 
 It is necessary to remark, that this velocity of 
 maximum useful effect, or of full pressure in the 
 cylinder, will not be the same in all engines, but 
 that on the contrary, cfBteris paribus, it will vary 
 in a direct ratio with the evaporation S of the 
 boiler, and in inverse ratio with the area of the 
 cylinder. It may then be found, in one engine, 
 the half or the double of what it would be in an- 
 other ; which shews how erroneous is the belief 
 that, because the velocity of the piston of stationary 
 engines does not in general exceed a certain ve- 
 locity of 150 to 300 feet per minute, the steam of 
 
126 CHAPTER III. 
 
 the boiler necessarily reaches the cylinder without 
 changing its pressure. If it be supposed that, in 
 a certain engine, the maximum of effect be pro- 
 duced at the velocity of 200 feet per minute, it will 
 be easy to plan another, in which the same effect 
 shall take place at a very different velocity. To 
 such end, it would evidently suffice to retain the 
 same boiler, and to diminish or augment the dia- 
 meter of the cylinder ; or else, on the contrary, to 
 preserve the dimensions of the cylinder, and to 
 change those of the boiler. Nay more ; it will suffice 
 to let the fire go down, which will diminish the eva- 
 poration, and the velocity of full pressure will vary 
 at the same time. It cannot, then, be decided a 
 priori, as it is in the theory usually applied to steam- 
 engines, that the velocity of 150 to 250 feet per 
 minute is, for all engines, the velocity of full pressure. 
 The fact is, that there are no other means of 
 knowing the velocity of maximum useful effect or 
 of full pressure for an engine, than by calculating 
 it directly for that engine, which is the object of 
 the formula we have given above. That formula, 
 moreover, is of remarkable simplicity, and requires 
 no other experimental knowledge than that of the 
 evaporation of which the boiler is capable. As to 
 that evaporation, it may be determined by a special 
 experiment, or deduced from the heating surface of 
 the boiler and the quality of the fuel, taking as a 
 basis some special experiment on the subject, made, 
 not on the boiler of the engine itself, but on some 
 
OF THE MAXIMUM USEFUL EFFECT. 127 
 
 other boiler of similar construction. Thus the ve- 
 locity of the maximum useful effect of an engine 
 may always he calculated a priori. 
 
 SECT. II. Of the load of maximum useful effect. 
 
 To know what useful resistance an engine is 
 capable of setting in motion at the above velocity 
 of maximum useful effect, it suffices to introduce 
 for v, in the general expression of the resistance, 
 (equation 2,) the value we have just found; and 
 we obtain for the corresponding load /, 
 
 q 
 
 Examining the same equation (2), which gives 
 the resistance in the general case, we perceive that 
 resistance to be the greater as the velocity of the 
 motion is less. The resistance which we have just 
 obtained, as corresponding to the velocity of maxi- 
 mum useful effect, or to the minimum velocity of 
 the engine without loss of steam, is then, at the 
 same time, the greatest resistance of which the en- 
 gine is capable with the given expansion. Thus, 
 the greatest useful effect will be obtained by mak- 
 ing the engine work at its smallest velocity and 
 with its maximum load, which moreover is evident 
 a priori. 
 
 This load suitable to the production of the 
 greatest useful effect, may also be found directly. 
 It suffices for that, to consider the expression of the 
 
128 CHAPTER III. 
 
 useful effect in terms of the load, that is, to examine 
 the equation (4 bis). There we see that the maxi- 
 mum of effect, for a given expansion, corresponds 
 to the maximum of the fraction 
 
 in which r alone is variable ; and whose maximum 
 will evidently be given by the maximum of the 
 resistance r. We are therefore led to the same 
 solution as above. 
 
 SECT. III. Mode of determining the friction of un- 
 loaded engines, and the additional friction per 
 unit of the load ; derived from the preceding 
 enquiry. 
 
 The analogy which we have obtained above, may 
 serve to determine the friction of the engine work- 
 ing without load, and its additional friction per 
 unit of the load. It will, in fact, be remarked 
 that, since there is a maximum load corresponding 
 to every pressure in the boiler, every load whatever 
 may be made a maximum load for the engine, by 
 lowering sufficiently the pressure. Suppose, then, 
 that no load be imposed on the engine, but that 
 the pressure in the boiler be lowered, by means of 
 the safety-valve, till the engine can do no more 
 than just overcome its own friction and keep itself 
 in motion. With this reduced pressure, the fric- 
 tion of the engine alone becomes a maximum load 
 for it. Let P" be that pressure, ' determined by 
 
OF THE MAXIMUM USEFUL EFFECT. 
 
 experiment ; the preceding equation, then, will be 
 true in changing P for P" and making r = 0. 
 Therefore we shall have 
 
 and this equation will determine the quantity f, or 
 the friction of the engine without load. 
 
 To obtain afterwards the quantity 8, resort will 
 be had to a similar means. Without making any 
 change in the ordinary pressure of the engine, the 
 load must be augmented more and more, till it ap- 
 pear that the engine would stop if any further re- 
 sistance were imposed on it. Then we shall have ob- 
 tained the maximum load corresponding to the 
 pressure of the boiler. If, therefore, the load thus 
 determined by experiment be denoted by /', P 
 being still the pressure of the boiler, equation (13) 
 will give 
 
 which determines the quantity 8, or the additional 
 friction accruing to the engine per unit of the load 
 r imposed on it. 
 
 If the load of the engine be not capable of aug- 
 mentation, instead of increasing the load as has 
 just been said, the pressure in the boiler must be 
 lowered so as to bring it in equilibrio with the or- 
 dinary load of the engine ; and the value of P thus 
 found, must be introduced into the equation, as in 
 the case in which the engine was made to work with- 
 out a load. 
 
 K 
 
130 CHAPTER III. 
 
 The quantities f and 8 being thus determined, 
 the friction of the engine, with any given load r, 
 will be 
 
 These are the means we used to determine the 
 friction of the locomotive engines, either isolated 
 or followed by their trains, and we propose them 
 equally with respect to steam-engines of every kind. 
 
 It is to be remarked that if, having omitted pre- 
 viously to certify that the engine is working with 
 its maximum load, we should happen erroneously 
 to take a case of general velocity for a case of 
 minimum velocity, and pretend to deduce therefrom 
 the friction of the engine, the value thus found 
 would necessarily be too great, and by so much the 
 more exaggerated, as the load with which the en- 
 gine is working may be more remote from the 
 maximum. In effect, the effort of which the en- 
 gine is capable diminishing as its velocity increases, 
 it is plain that, in the above calculation, there 
 would be introduced for /' a quantity smaller than 
 it ought to be ; whence results that the quantity 8, 
 and consequently the definitive friction of the en- 
 gine, would appear too great. This explains how 
 the ordinary theory, in comparing its theoretical 
 results to those of experiment, is led to coeffi- 
 cients of reduction, which make the friction of 
 the engine appear much more considerable than it 
 really is. 
 
OF THE MAXIMUM USEFUL EFFECT. 131 
 
 SECT. IV. Of the evaporation of the engine. 
 
 The evaporation necessary to an engine, in order 
 to exert a certain maximum effort / at the mini- 
 mum velocity v', will be given by equation (3), on 
 substituting in it / and v' instead of the general 
 values r and v, or more simply, it may be derived 
 immediately from equation (12), which amounts to 
 the same, and gives 
 
 It will be observed that on substituting / and v' 
 for r and v in equation (3), we should have 
 
 but on introducing for / its value given in equa- 
 tion (13), we fall into the preceding formula 
 
 SECT. V. Of the maximum useful effect of the 
 engine. 
 
 The maximum of useful effect the engine can 
 produce in a unit of time, with a given expansion, 
 will be known by the general formula (4), on in- 
 troducing into it the velocity proper to produce that 
 effect ; or it may be deduced from the product of 
 the two equations (12), (13), viz. 
 
 max.u.E.=arV= _ _* -- . 1 L3P. .(15) 
 
 n + qP ) 
 
 K 2 
 
132 CHAPTER III. 
 
 This maximum useful effect, it will be observed, 
 in nowise depends either on the area of the cylin- 
 der, or on the velocity, that is the number of strokes 
 of the piston per minute. If we suppose an engine 
 working without expansion, which amounts to mak- 
 ing I' I and reduces the expression k to the value 
 
 , we find even that the maximum useful effect 
 
 l + c 
 
 does not depend on the length of stroke of the 
 piston. In effect the quantity - , which still 
 
 V ~\~ C 
 
 subsists in the equation, expresses merely a ratio, 
 that of the volume traversed by the piston to the 
 whole volume of the cylinder. On the other hand, 
 the quantities f and 8 are constant in a given en- 
 gine, and vary but little in engines of the same 
 system \ the quantities n and q are constant co- 
 efficients, as has been seen ; in fine, the pressure of 
 condensation p depends on the mode of condensa- 
 tion used, and especially on the quantity and tem- 
 perature of the water applied to produce the con- 
 densation ; consequently it is a constant quantity 
 under given conditions. We may then say, that 
 the maximum useful effect of an engine depends 
 essentially on but two things : the evaporating 
 power S of the boiler, and the pressure P at which 
 the steam is generated. This result, indeed, must 
 appear evident a priori, for they are the only real 
 causes of power. As to the dimensions of the cy- 
 linder, and of the stroke, they are no more than 
 means of transmitting that power under one form 
 
OF THE MAXIMUM USEFUL EFFECT. 133 
 
 or another, but are utterly unable to create it, or 
 fundamentally to change it ; and as to the velocity 
 of the piston, that cannot in any way influence the 
 maximum useful effect, since, for a given produc- 
 tion of steam, that velocity is susceptible of all 
 values according to the diameter given to the 
 cylinder. 
 
 Here we see what error we commit, when we 
 pretend to calculate the useful effect of engines, 
 from the diameter of the cylinder, taking no ac- 
 count of the evaporation produced, which not only 
 enters not into the calculation, but forms no part 
 even of the observations. 
 
 The horse-power of the engine when exerting its 
 greatest effort, or when it produces its greatest use- 
 ful effect, will be as before, 
 
 max. u. E. 
 
 33000 
 
 The other modifications already given of the 
 expression of the useful effect, will also be as fol- 
 lows : 
 
 (17) m. u. E. 1 lb. co. = ^^ L - ......... Maximum useful 
 
 effect of 1 lb. 
 of fuel. 
 
 /i o\ T i c max. u. E. 
 
 (18) m. u. E. 1 ft. wa. = - _ - ......... Maximum useful 
 
 effect of 1 cu- 
 bic foot of 
 water evapo- 
 rated. 
 
 (19) mi.Q. co. for 1 hp. = - ......... Minimum quan- 
 
 max. u. E. ^ 
 
 tity of fuel 
 
 consumed per 
 horse-power. 
 
134 
 
 CHAPTER III, 
 
 (20) mi. Q. wa. for 1 hp._ g300oS , 
 max. u.E. 
 
 (21) m.u. HP.l Ib. co. = 
 
 (22) m.u. HP. 1ft. wa. = 
 
 max. u. E. 
 
 33000 N 
 
 max. u. E. 
 
 33000 S 
 
 Minimum quan- 
 tity of water 
 evaporated per 
 horse-power. 
 
 Maximum horse- 
 power produc- 
 ed by 1 Ib. of 
 fuel. 
 
 Maximum horse- 
 power produc- 
 ed by 1 cubic 
 foot of water 
 evaporated. 
 
 ARTICLE III. 
 
 OF THE ABSOLUTE MAXIMUM OF USEFUL EFFECT. 
 
 The preceding enquiries contain all that is ne- 
 cessary for engines in which the expansion is fixed 
 a priori. They suffice also for unexpansive en- 
 gines, because these are naturally included in the 
 case of engines with a fixed expansion ; and we 
 have only to make in the formula f /, which gives 
 
 at the same time k = 
 
 But there still remains 
 
 a problem to occupy our attention, for engines in 
 which the expansion is susceptible of variation. 
 
 It has been shewn that, for a given expansion, 
 the most advantageous mode of working the engine 
 is to give it its maximum load, which we have made 
 known above, and which may be calculated a priori 
 by equation (13). Hence may be known what load 
 
OF THE ABSOLUTE MAXIMUM USEFUL EFFECT. 135 
 
 to prefer for every expansion. But the question is 
 now to determine, among the divers degrees of ex- 
 pansion that may be given to the engine, each accom- 
 panied hy its maximum corresponding load, which 
 of them will produce the greatest definitive useful 
 effect. 
 
 To this end, recourse must be had to equation 
 (15), which gives the maximum useful effect of the 
 engine with the expansion 1! ; viz., in putting for k 
 its value, 
 
 max. u. E. = __ ?_J JL+ log '+ _ _L 5ffj+_(/>l ; 
 (l+i)g\f+c+ >g /'+c l'+c n+qP /' 
 
 and we must seek, among the values that may be 
 given to /', that which will render this useful effect 
 a maximum. Now, making equal to zero the dif- 
 ferential coefficient of this equation, taken with 
 reference to /', we find for the condition of the 
 maximum sought 
 n 
 
 9 
 
 This equation may be written as follows : 
 
 1 
 
 n 
 
 and, under this form, it is easy to perceive that the 
 second part is the ratio of the relative volumes of 
 the steam generated under the respective pressures 
 Pand(p+/). 
 
136 CHAPTER III. 
 
 f 
 
 Therefore, to find the value of the ratio -, which 
 
 i 
 
 corresponds to the production of the absolute maxi- 
 mum of useful effect, the relative volume of the 
 steam generated under the pressure P of the boiler 
 must he sought first, either hy formula (a), or by 
 the table given in Chapter II. ; then, afterwards, 
 the relative volume of the steam suppose_d imme- 
 diately generated under a pressure indicated by the 
 sum (p +/"), must be sought in the same manner ; 
 
 and the first volume divided by the second will give 
 
 f 
 the wanted value of -, that is the ratio which ought 
 
 i 
 
 to exist between the portion of the stroke traversed 
 before the beginning of the expansion, and the total 
 stroke of the piston. 
 
 We also see that, in supposing n 0, this re- 
 lation reduces itself to 
 
 f-p+ 
 
 r P 
 
 If, then, in the calculation of the effects of the 
 engine, we neglected the change of temperature of 
 
 f 
 the steam, the value of -, proper to the production 
 
 if 
 
 of the absolute maximum of useful effect, would 
 be equal to the ratio between the two quan- 
 ties (p +/) and P. We must consequently con- 
 sider this last equation as an approximation to the 
 preceding one. 
 
 f 
 
 Introducing the value of - given by equation (34), 
 
 i 
 
OF THE ABSOLUTE MAXIMUM USEFUL EFFECT. 137 
 
 in the formulae of Article II., we shall have all the 
 determinations relative to the maximum useful 
 effect produceable by the engine with that expan- 
 sion ; and since that expansion is the most favour- 
 able for the engine, it follows that those determina- 
 tions will correspond to the absolute maximum of 
 
 useful effect of which that engine is capable. 
 
 f 
 The formulae in which the value of thus found 
 
 L 
 
 is to be substituted, are those already given, and in 
 the calculation of which, account is taken of the 
 change of temperature of the steam. Since, how- 
 ever, the supposition of the preservation of the tem- 
 perature of the steam, that is to say, the hypothesis 
 
 of n = 0, q = , in which m represents the rela- 
 mr 
 
 tive volume of the steam at the pressure P, greatly 
 simplifies the equations, and gives an approxima- 
 tion near enough for a great number of cases in 
 which the expansion is not carried very far, we 
 present here the corresponding results of all the 
 formulae. They make known very nearly to the 
 truth, the absolute maximum of useful effect that 
 it is possible to obtain from a known engine, taking 
 at the same time the most advantageous expansion 
 and the most advantageous load. 
 
 absolute maxi- 
 mum of useful 
 effect. 
 
138 CHAPTER III. 
 
 ( o 4 ) ar - a .'O+/) + Pc lo~ ('-HOP 
 
 Load of the pis- 
 ton,correspond- 
 ing to the abso- 
 lute maximum 
 of useful effect. 
 
 Effective evapo- 
 ration per unit 
 of time. 
 
 Absolute maxi- 
 mum of useful 
 effect. 
 
 Absolute I maxi- 
 mum of useful 
 horse-power. 
 
 Absolute maxi- 
 mum of useful 
 effect arising 
 from 1 Ib. of 
 fuel. 
 
 Absolute maxi- 
 mum of useful 
 effect due to the 
 evaporation of 1 
 cubic foot of 
 water. 
 
 Minimum quan- 
 tity of fuel per 
 horse-power. 
 
 Minimum quan- 
 tity of water 
 evaporated per 
 horse-power. 
 
 
 (~ 5) S m IP 
 ( 9 6) ab. max u E wSP 1 (^+ c ) P 
 
 (7) ab m u HP - aba max * u ' E ' 
 
 33000 
 (S) ab m u E lib co - ab ' maX> U< E< 
 
 (-0) ab m u E 1 ft iva -_ ab - max - u - E - 
 
 S 
 33000 N 
 
 ab.max.u. E. 
 
 ab. max. u. E. 
 
OF THE ABSOLUTE MAXIMUM USEFUL EFFECT. 139 
 
 (32) ab.m.HP.llb.co.zz 
 
 ab. max. u. E. 
 
 33000 N 
 
 (33) 
 
 33000 S 
 
 (34 bis) 
 
 .... Absolute maxi- 
 mum of horse- 
 power produced 
 by 1 Ib. of fuel. 
 
 .... Absolute maxi- 
 mum of horse- 
 power produced 
 by 1 cubic foot 
 of water eva- 
 porated. 
 
 .... Expansion which 
 produces ap- 
 proximatively 
 the absolute 
 maximum of 
 useful effect. 
 
 The only remark we have to make on the sub- 
 ject of these formulae is, that the load proper to the 
 production of the absolute maximum of useful effect, 
 in expansive engines, is not the maximum load of 
 which the engine is capable. If, in effect, we refer 
 to equation (13), which represents the maximum 
 load with a given expansion, that is, developing in 
 it the quantity k, 
 
 a /n \ /I' Z'-f-c J+c\ a /n \ 
 
 and seek what value of l r will make it a maximum, 
 we shall find that condition to be expressed by 
 /' = /, and not by 
 
 n 
 
140 CHAPTER III. 
 
 Consequently, if the engine be required to move 
 the greatest load of which it is capable, it must 
 work without expansion ; but that load is not the 
 load which produces the absolute maximum of use- 
 ful effect. This will be given by the solution of 
 equation (34) introduced into the formula (13), or 
 approximatively by the formula (24) as has just 
 been seen above. 
 
141 
 
 CHAPTER IV. 
 
 OF HIGH-PRESSURE ENGINES. 
 
 ARTICLE I. 
 
 THEORY OF HIGH-PRESSURE ENGINES, AND OF UNEXPANSIVE 
 ENGINES IN GENERAL. 
 
 SECT. I. Of the effects of the engine with any 
 given load or velocity. 
 
 IN the preceding chapter we have given, in a 
 general manner, the theory of the effects of steam 
 acting in the cylinder of an engine, as well while it 
 is coming directly from the boiler, as during its 
 expansion in the cylinder, after its separation from 
 the hoiler. This theory, and the formula we have 
 deduced from it, form the basis of all the calcula- 
 tions that the particular applications of steam in 
 steam-engines can require ; and nothing now re- 
 mains but to show the modifications of it according 
 to each system of engines, or according to each 
 mode of applying the motive power of steam. 
 
 Such is the object we have now in view in the 
 present and following chapters. We divide steam- 
 engines into three kinds, and each kind into 
 several classes, according to the differences which 
 exist in the various systems of engines. 
 
CHAPTER IV. 
 
 In the first kind we place rotative unexpansive 
 steam-engines, comprising as subdivisions, high- 
 pressure stationary engines, locomotives, and Watt's 
 rotative or double-acting steam-engines. 
 
 In the second kind we place rotative expansive 
 engines, which form likewise three classes : con- 
 densing engines, with expansion in a single cylin- 
 der, or Cornish engines ; condensing engines, with 
 expansion in two cylinders, or Woolf's and Ed- 
 wards' s engines ; and, lastly, expansive engines 
 without condensation, or Evans's engines. 
 
 In the third kind we place single-acting engines, 
 which again form three classes, namely : Watt's 
 single-acting engines, Cornish single-acting engines, 
 and atmospheric engines. 
 
 We shall, in the present chapter, treat specially 
 of high-pressure engines, and shall first give the 
 theory of those engines, under a form so general as 
 to embrace at the same time all unexpansive engines. 
 
 We have already said, in developing the general 
 theory of the action of steam, that the formula 
 suitable to the calculation of unexpansive engines, 
 may be deduced from the general formula?, by sup- 
 posing in them the expansion null, that is to say, 
 by making 1 1. Moreover, as in those engines, 
 the expansion is capable of no variation, since there 
 is no expansion, the third case considered in the 
 general theory, cannot occur. Thus, there will 
 be but two circumstances to notice in their work, 
 namely : the case in which they work with any 
 given load whatever, and that in which they work 
 
HIGH-PRESSURE ENGINES. 143 
 
 with their maximum load, or load of greatest useful 
 effect. 
 
 We might, then, immediately conclude, from the 
 general formula already developed, the particular 
 formulae suitable to the engines of which we are 
 treating. But, as in the general theory which 
 we have developed, there occurs more complica- 
 tion than is necessary for the case of unexpansive 
 engines, and as we have been obliged to make some 
 use of the differential and integral calculus in it, 
 the knowledge of which is not familiar to the greater 
 number of readers, we deem it better to leave aside 
 the general formula already found, and to seek in 
 a direct manner, those which are proper to the 
 action of the steam, considered in unexpansive en- 
 gines only. 
 
 We have already said, in the first chapter, that 
 since the effects of steam-engines are never calcu- 
 lated till after the moment when they have attained 
 uniform motion, it follows that there is necessarily 
 equilibrium between the power and the resistance ; 
 that is, between the force applied by the mover, and 
 the force resulting from the friction of the engine, 
 the resistance of the load, and all the other divers 
 resistances which oppose the motion of the engine. 
 Consequently, if we call P the pressure of the steam 
 per unit of surface in the boiler, P' the unknown 
 pressure that steam will have in the cylinder, and 
 R the total resistance opposed to the motion of the 
 piston, we shall have as a first equation of analogy 
 F = R . . . . (A) 
 
144 CHAPTER IV. 
 
 We have said, besides, that there is necessarily 
 equality also between the expenditure and the pro- 
 duction of steam. Now, if we express by S the vo- 
 lume of water evaporated per unit of time in the 
 boiler, this volume of water will first, in the boiler, 
 be converted into steam at the pressure P ; then 
 in the cylinder it will change into steam at the 
 pressure P 7 . But we have seen in Chapter II., 
 that during the change of pressure of the steam, 
 which, from our experiments, is always found at- 
 tended with a corresponding change of tempera- 
 ture, the steam always remains at the maximum 
 density for its temperature. Again, in the steam of 
 a maximum density for its temperature, the re- 
 lative volume of the steam, that is, the ratio of the 
 volume occupied by the steam to the volume of the 
 water that produced it, may be expressed by the 
 very simple equation (a) 
 
 1 
 n + qp 
 
 in which /* is the relative volume sought, p the 
 pressure, and n and q two constant numbers, of 
 which we have given the values. The volume of 
 steam at the pressure P', transmitted per minute 
 to the cylinder of the engine, will then be /* S, or 
 
 S 
 
 On the other hand, if by v be expressed the velocity 
 of the piston, and by a the area of the cylinder, 
 v a will be the volume described by the piston in 
 
HIGH-PRESSURE ENGINES. 145 
 
 a unit of time. It is however to be remarked that 
 the steam admitted in the cylinder, fills not only 
 the space actually traversed by the piston, but also 
 the clearance of the cylinder and the adjoining pas- 
 sages, wherein a portion of it is lost at every stroke 
 of the piston. Calling I the stroke of the piston, 
 and c the clearance of the cylinder expressed by 
 an equivalent portion of the useful length of the 
 cylinder, then a I will be the volume described by 
 the piston at each stroke, but the real volume 
 expended in the same time will be a (/ + c). 
 The volume of steam expended at every stroke, 
 will be then, to the volume described by the piston, 
 in the ratio 
 
 and consequently there will be the same ratio 
 between the volumes expended and the volumes 
 described by the piston, in the unit of time. 
 
 The volume described by the piston in the unit 
 of time is, as has been said, a v ; the volume of 
 steam, then, expended in the unit of time will be 
 
 l+c 
 
 av j- 
 I 
 
 And thus the equality between the production 
 of the steam and its expenditure, will give for a 
 second general analogy 
 
 S l + c 
 
 .... (B) 
 
146 CHAPTER IV. 
 
 Eliminating therefore P' from the two equations 
 (A) and (B), we obtain 
 
 J_ S _ 1_ 
 ~ l + c ' a ' n + qR, 
 
 On the other hand, expressing by r the resistance 
 of the load, by f the friction of the engine when 
 unloaded, by 8 the increase accruing to that 
 friction, per unit of the resistance r imposed on 
 the engine, lastly, by p the pressure against the 
 opposite face of the piston, proceeding from the 
 atmosphere in non-condensing engines or from im- 
 perfect condensation in the cylinder in condensing 
 ones, these four forces being referred besides to 
 the unit of surface of the piston ; we shall 
 plainly have, for the total resistance R, the value 
 
 And, therefore, the velocity v of the engine with 
 a known load, in terms of all the data of the pro- 
 blem, will be 
 
 I S _ l _ 
 
 f) . . - (1 ) 
 
 ~~l + c a n + q[(l +&)r+p+f] 
 
 We derive reciprocally, for the value of the 
 load ar 9 in terms of the velocity, 
 
 I S a sn -x 
 
 ar=j -- -7- - rr -- = - r I - +J0 +/ 1 . . (2) 
 
 l + c tl+8)gv l+S \q J J 
 Similarly the evaporation capable of producing 
 the velocity v, with the load a r y will be 
 
 
 .(3) 
 
HIGH-PRESSURE ENGINES. 
 
 Finally, knowing, by means of these equations, 
 the load and the velocity of the engine, their 
 product will give, without further calculation, the 
 useful effect of the engine, viz. 
 
 u.E. = arv; (4) 
 
 and successively, as in Section 5, Article I. of the 
 preceding chapter, we shall have the horse-power 
 and the other modes in which the useful effect of 
 the engine may be represented. 
 
 SECT. II. Of the maximum useful effect of the 
 engine. 
 
 The preceding calculation refers to the general 
 case in which the load or the velocity are given 
 a priori, and without any particular condition ; 
 but if it be required to know what is the velocity 
 or load suitable to the production of the maximum 
 useful effect of the engine, we must examine the 
 value of the useful effect arv, which, from equa- 
 tion (2) is 
 
 _, I S av /n 
 
 'l + c' (l+S)q~lT&~\q 
 
 Now, this expression containing the velocity 
 only in the negative terms, will evidently attain 
 its maximum when the velocity shall be the 
 smallest possible ; and on the other hand, referring 
 to equation (B), we perceive that the velocity 
 will be a minimum, when the pressure P' shall, 
 
 L 2 
 
148 CHAPTER IV. 
 
 on the contrary, have attained its highest value, 
 that is, when it shall be equal to the pressure P, 
 in the hoiler. The velocity then of maximum 
 useful effect will he furnished by equation (B), 
 on making P' P ; or that velocity will be 
 
 -- 
 a n + qP 
 
 Consequently, substituting this value in equa- 
 tion (2), we have for the load of maximum useful 
 effect, 
 
 ... (6) 
 
 and, substituting it in equation (3), we have for 
 the evaporation of the engine, in terms of the ve- 
 locity of maximum useful effect, 
 
 S=Lfat/(n + ? P) ... (7) 
 
 I 
 
 Finally, the product a / v r will give the measure 
 of the maximum useful effect of the engine, and it 
 may be expressed, as above, under the different 
 forms indicated in Section 5, Article 1, Chapter 
 III. 
 
 Thus, therefore, we have all the formulae neces- 
 sary to the solution of the different problems, which 
 may occur in the calculation of these engines. 
 
 ARTICLE II. 
 
 PRACTICAL FORMULA FOR THE CALCULATION OF HIGH- 
 PRESSURE ENGINES ; AND EXAMPLE OF THEIR APPLICATION. 
 
 In high-pressure unexpansive engines, the steam 
 is generated in the boiler at a very high pressure. 
 
HIGH-PRESSURE ENGINES. 149 
 
 It passes thence into the cylinder, where it acts 
 successively above and below the piston, to impress 
 on it an alternate motion, which being communi- 
 cated to the great beam, is transmitted to a crank 
 and changed into a continued rotatory motion ap- 
 plied to the shaft of the engine ; and this after- 
 wards determines the particular action of all the 
 parts necessary to the operation of the engine. 
 The steam then penetrates into the cylinder alter- 
 nately by each extremity ; but after it has driven 
 the piston to the end of the stroke, it finds an aper- 
 ture by which it escapes into the atmosphere, 
 without being condensed. Consequently, instead 
 of having, on that face of the piston opposed to the 
 action of the steam, a vacuum more or less perfect 
 resulting from the condensation, that is to say, a 
 pressure null or nearly so, there always subsists, on 
 the contrary, the atmospheric pressure, which im- 
 mediately succeeds to that previously exerted by 
 the steam. In these engines, then, the atmospheric 
 pressure is to be counted one of the forces which 
 oppose the motion of the piston ; and, therefore, 
 the quantity represented by p, should in this case 
 have a value equal to 14*71 Ibs. per square inch. 
 
 To be able to apply the formulae which we have 
 just developed, it is necessary to know the two 
 quantities f and S ; that is, the friction of the 
 engine when unloaded, and its additional friction, 
 per unit of the load r. Special and circumstantial 
 experiments would be requisite for this purpose, 
 but till more precise determinations on the subject 
 
150 CHAPTER IV. 
 
 be obtained, we may take an evaluation of those 
 two quantities from our own experiments on loco- 
 motives, which are also high-pressure unexpansive 
 engines. In these the friction of the engine when 
 unloaded, deduction being made of the force neces- 
 sary to move its own weight along the rails, 
 amounts to about 1 Ib. per square inch of the sur- 
 face of the piston, and the additional friction 
 caused by any additional resistance is *14 of that 
 resistance. We shall take then 
 
 /= 1 x 144 Ibs. and B = -14 *. 
 
 Finally, in these as well as in all rotative engines, 
 where the motion of the piston is checked and 
 
 * A locomotive engine with two cylinders, 11 inches in 
 diameter, and with wheels uncoupled, has, when unloaded, a 
 mean friction of 101 Ibs. ; and deducting 64 Ibs. for the trans- 
 port of its own weight, and the additional friction which that 
 draught causes in the engine, there remain 37 Ibs. for the re- 
 sistance occasioned by the machinery. This resistance being 
 measured at the velocity of the wheel, produces against the 
 piston, a force increasing in the inverse proportion of the re- 
 spective velocities ; that is to say, a force of 
 
 37X5-9 =218 Ibs. 
 
 which amounts to 1-15 Ib. per square inch of the surface of the 
 pistons. 
 
 Again, a load of 1 ton causes on a railway a resistance of 
 7 Ibs., and occasions in the engine a surplus friction of 1 Ib., 
 which is an additional friction equal to ith of the resistance im- 
 posed on the engine. This evaluation may be found too high, 
 with regard to a stationary engine, in which there is no fric- 
 tion of carrying wheels ; but we let it remain, because in the 
 absence of positive determinations, it is better to avoid the 
 risk of rating the passive resistances too low. 
 
HIGH-PRESSURE ENGINES. 151 
 
 regulated by a crank, the space left between the 
 end of the stroke of the piston and the bottom of 
 the cylinder, amounts, including the adjoining 
 passages, to but -^ of the useful length of the cy- 
 linder. Therefore weJiave 
 
 * + C - 21 = l.Qff 
 
 / 20 
 
 Introducing these values into the formulae, ex- 
 cept those of /and 8, which are but approximative ; 
 and replacing the constant quantities n and q by 
 the values suitable to them in non-condensing 
 engines, that is, in English measure, and when the 
 pressure is expressed per square foot, 
 
 rc = -0001421, 
 
 ? = -00000023, 
 
 we obtain, for the numerical formulae suitable to 
 the calculation of the engines under consideration : 
 
 Practical formula for high-pressure engines. 
 
 GENERAL CASE. 
 
 of the 
 
 _ 
 6-6075 +.00.415 [ 
 
 per minute. 
 arr:4140750- r -? _ -_?_ (273G+/) Useful load of 
 
 v 
 
 thc piston>in 
 
 pounds. 
 
 [(! + >) r+f]} Effective evapo- 
 ration, in cubic 
 feet of water 
 per minute. 
 
CHAPTER IV. 
 
 u. E. arv Useful effect, in 
 
 pounds raised 
 1 foot per mi- 
 nute. 
 
 u.HP ~ U - E> Useful force, in 
 
 33000 
 
 horse-power. 
 
 u. E. 1 Ib. co.... -~ U<E - Useful effect of 
 
 1 Ib. of coal, in 
 pounds raised 
 1 foot. 
 
 u. E. 1 ft. wa... U ' Et Useful effect of 
 
 1 cubic foot 
 of water, in 
 pounds raised 
 1 foot. 
 
 Q.. fnrl lip . - 3300QN , , Quantity of coal, 
 
 u. E. j 
 
 in pounds, 
 
 which pro- 
 duces 1 horse- 
 power. 
 
 Q. wa. for 1 hp ~~ir~E~ Quantity of wa- 
 ter, in cubic 
 feet, which 
 produces one 
 horse-power. 
 
 u. HP. 1 Ib. co = _ U 1 E ', T Horse - 
 
 produced by 1 
 Ib. of coal. 
 
 u.HP. 1 ft.wa = 
 
 produced by 
 1 cubic foot of 
 water evapor- 
 ated. 
 
HIGH-PRESSURE ENGINES. 153 
 
 CASE OF MAXIMUM USEFUL EFFECT. 
 
 S 10000 -r i > i 
 
 v zi . Velocity 01 the 
 
 a 1-492 + -002415 P * ' 
 
 piston, in feet 
 
 per minute. 
 ar'= (P-/-2118) Useful load of 
 
 1 "f" i 
 
 the piston, m 
 pounds. 
 
 / 
 
 S=-^- (1-492 4- '002415 P) Effective evapor- 
 
 10000 
 
 ation, in cubic 
 
 feet of water 
 per minute. 
 
 max. u. E.=arV Useful effect, in 
 
 pounds raised 
 to 1 foot per 
 minute. 
 
 We do not add to the formulae of the case of 
 maximum useful effect, the different expressions of 
 the useful effect, in horse-power, in weight of fuel, 
 &c., because the formulae which give them are the 
 same as those of the general case. 
 
 To shew an application of these formulae, let us 
 suppose it be required to determine the effects to 
 be expected from an engine of this system already 
 constructed, and of which the dimensions and other 
 data of the calculation are known, viz. : 
 
 Cylinder, 17 inches diameter ; or a=z 1*57 sq. feet. 
 
 Stroke of the piston, 16 inches ; or /= 1-33 feet. 
 
 Effective evaporation of which the boiler is capa- 
 ble, '67 cubic foot of water per minute ; or S = '67 
 cubic foot. 
 
 Consumption of coke in the same time, 8 Ibs. ; 
 orN = 8lbs. 
 
154 CHAPTER IV. 
 
 Total pressure in the boiler, 65 Ibs. per square 
 inch ; or P = 65 x 144 Ibs. per square foot. 
 
 Performing the calculation with these data, we 
 obtain the following results, as the effects which 
 this engine is capable of producing, at the velocity 
 of maximum useful effect, and also at the respec- 
 tive velocities of 250 and 300 feet per minute : 
 
 Maximum 
 of useful effect. 
 v =300 250 176 
 
 ar =4146 5769 9777 
 
 -^7 = 18-34 25-52 43-25 
 
 144 
 
 S =-67 -67 -67 
 
 u. E = 1243800 1442250 1724580 
 
 u. HP =38 44 52 
 
 u. E. 1 Ib. co =155475 180280 271390 
 
 u. E. 1 ft. wa =1856450 2152640 2574000 
 
 Q. co. for 1 hp = -212 -183 -153 
 
 Q. wa. for 1 hp =-018 -015 -013 
 
 u. HP. 1 Ib. co =4-71 5-46 6-53 
 
 u. HP. 1 ft. wa =56 65 78 
 
 Such will be the effects produced. With respect, 
 however, to those presented in the Table, as result- 
 ing from the combustion of one pound of fuel, it is 
 to be observed that they refer to the use of coke ; 
 that is to say, it is from experiments made with 
 that kind of fuel that we have concluded the quan- 
 tity employed in the evaporation of -67 cubic foot 
 of water per minute. From Smeaton's experiments, 
 it would appear that coke produces but i of the 
 effect produced by the same weight of coal. There- 
 fore, it would follow, that if the fire, in these ex- 
 periments, had been fed with coal instead of coke, 
 
HIGH-PRESSURE ENGINES. 155 
 
 the fuel necessary for the above evaporation, would 
 have been but 6*66 Ibs. ; and the effect due to the 
 consumption of 1 Ib. of fuel, in the engine in ques- 
 tion and in the case of maximum useful effect, would 
 have become, 
 
 u. E. lib. co. = 58950. 
 
 However, as the experiments of Smeaton on that 
 head have been regarded as not being conclusive, 
 we do not lay a stress upon the result, but make 
 the observation, only to call attention upon the 
 difference that may arise from the use of the two 
 sorts of fuel. 
 
156 
 
 CHAPTER V. 
 
 LOCOMOTIVE ENGINES. 
 
 ARTICLE I. 
 
 THEORY OF LOCOMOTIVE ENGINES. 
 
 THE locomotive engines in use on railways are con- 
 structed on the same principles as the preceding, 
 with regard to the application of the steam as a 
 motive force. The steam is generated at a very 
 high pressure in the boiler. It then passes into 
 the cylinders, where it is admitted without inter- 
 ruption during the whole stroke of the piston ; and 
 finally it escapes into the atmosphere without hav- 
 ing been condensed. These engines, therefore, act 
 without expansion or condensation. 
 
 The steam having entered the cylinder, acts 
 successively on either side of the piston, and thus 
 communicates to it a rectilineal alternate motion, 
 which, by means of a crank, changes to a rota- 
 tory motion applied to the wheels that support 
 the engine ; and the effect of this rotation is to 
 carry forward the engine itself, followed by all its 
 train. 
 
LOCOMOTIVE ENGINES. 157 
 
 As these engines are no more than a particular 
 application of the preceding, the formulae proper 
 to calculate their effects will he similar to those 
 just given. There, however, occur in them some 
 accessory circumstances which must be introduced ; 
 and this renders it necessary to treat of them 
 apart. 
 
 These circumstances are : 1st, that the engine is 
 obliged to draw its own weight, which by so much 
 diminishes its useful effect ; 2d, that the waste 
 steam being driven into the chimney through the 
 orifice of the blast-pipe, to cause an artificial cur- 
 rent of air intended to excite the fire, and thus to 
 compensate for the smallness of the boiler, it re- 
 sults that a certain force is expended by the engine 
 in driving this steam with the necessary velocity; 
 3d, that the train led by the engine, having in its 
 motion to contend against the resistance of the air, 
 and this force increasing as the square of the ve- 
 locity, there results a variable resistance to be add- 
 ed to those already considered ; 4th, and lastly, 
 that some of these engines are liable to a consider- 
 able loss of steam by the safety-valves, and till this 
 defect is entirely corrected, it is necessary to no- 
 tice it. 
 
 In order to take account of these different cir- 
 cumstances, we will express by p the force requisite 
 to move the engine itself; and by p'v the pressure 
 on the opposite face of the piston, resulting from 
 the action of the blast-pipe, a pressure which, from 
 our own experiments on this subject, we will sup- 
 
158 CHAPTER V. 
 
 pose to be proportional to the velocity of the mo- 
 tion ; we will express by g v 2 the resistance of the 
 air against the train, and will introduce these three 
 forces into the calculation, supposing them to be 
 referred to the unit of surface of the piston and to 
 its velocity. With regard to the loss of steam by 
 the valves, it will be taken account of, as may be 
 required, in the manner we have already indicated, 
 and then subtracted from the total evaporation, in 
 order to conclude from it the effective evaporation, 
 or the true value of S. 
 
 This being premised, the resistance of the load 
 against the piston, which we have expressed by r, 
 will now become 
 
 r + p+gv 2 
 
 and the pressure j9, on the opposite face of the piston, 
 will become 
 
 p +p'v. 
 
 Consequently, the formulae proper for the calcula- 
 tion of these engines, in the general case and in 
 that of maximum useful effect, will be 
 
 GENERAL CASE. 
 / S 1 
 
 viz.. 
 
 l+c a 
 
 ar=. 1 
 
 u.E.zz 
 
 arv. 
 
LOCOMOTIVE ENGINES. 159 
 
 CASE OF MAXIMUM USEFUL EFFECT. 
 
 I S 
 
 = rrv ~TT 
 
 l-\-c a(n 
 
 a r'= 
 
 max.u.E.iz ar' v. 
 
 The other different expressions of the useful 
 effect will be expressed by the formulae developed 
 in Sect. V. Article I. Chapter III., which are suit- 
 able to all engines. 
 
 The velocity v, in these equations, is that of the 
 piston. It is practically more convenient to use 
 the velocity of the engine itself, as we have done in 
 our TREATISE ON LOCOMOTIVES ; but to retain the 
 same form in the application of our formulae to all 
 steam-engines, we shall make no change here in 
 that respect. Howbeit, as the piston performs two 
 strokes while the wheel makes one revolution, the 
 velocity of the engine is easily deduced from that 
 of the piston, by multiplying the latter by the ratio 
 
 g or 1*708 
 
 in which TT is the ratio of the circumference to the 
 diameter, D the diameter of the wheel, and I the 
 stroke of the piston. 
 
 In the application of these formulae, it will be 
 remarked that the formula which gives the velocity 
 in the general case, contains still in the denominator 
 
160 CHAPTER V. 
 
 one term p'v and another gv z ; but to avoid the so- 
 lution of an equation of the third degree, an evalua- 
 tion may be made of the resulting velocity, and 
 thus approximate values of each of those terms cal- 
 culated, which being introduced into the equation 
 as constant quantities, a value of v may be derived. 
 With a little experience of these engines, the first 
 trial thus made, leads to a value of v sufficiently 
 near in many cases. If, however, it should appear 
 from the result, that too great an error has been 
 made in the original evaluation of v, for this first 
 trial to be depended on, the result of it must be 
 used as an approximation for a second, and the 
 result of this for a third, if necessary. The diffi- 
 culty, however, occurs only in the equation of the 
 velocity, and not in any of the other formula. 
 
 ARTICLE II. 
 
 PRACTICAL FORMULAE FOR THE CALCULATION OF LOCOMO- 
 TIVE ENGINES, AND EXAMPLE OF THEIR APPLICATION. 
 
 Now, to form the numerical equations proper for 
 the calculation of these engines, the constant quan- 
 tities in the algebraic formulae just given, must be 
 replaced by the values which experience has as- 
 signed to them. 
 
 The friction of a locomotive engine, with un- 
 coupled wheels, is, as we have said, about 1 Ib. per 
 square inch of the surface of the piston ; but in the 
 engines with coupled wheels, that friction is rather 
 
LOCOMOTIVE ENGINES. l6l 
 
 greater, and is equivalent to 1*25 Ib. per square 
 inch of the piston. In order therefore to include 
 those engines, we will adopt the latter determina- 
 tion, and take 
 
 It has already been said, that the surplus accru- 
 ing to that friction, when the engine draws a given 
 resistance, amounts to yth of that resistance ; we 
 have then 
 
 S = -14. * 
 
 The pressure arising from the blast-pipe varies 
 not only with the velocity of the engine but also 
 with the evaporation of the boiler per minute, and 
 with the size of the orifice of the blast-pipe, as 
 will be seen by the experiments developed in the 
 TREATISE ON LOCOMOTIVES ; but to simplify the 
 formulae, we will here refer that effect to the mean 
 evaporation of the engines and to the size of orifice 
 commonly adopted. With these mean dimensions 
 and data, it is found that, when the velocity is 
 10 miles per hour for the engine, or 150 feet 
 per minute for the piston, the pressure arising 
 from the blast-pipe is 1'75 Ib. per square inch of 
 the surface of the piston, and that it varies in 
 the direct ratio of the velocity of the motion ; 
 whence is deduced 
 
 p f v 1*75 x 14*4, when vzzl50; 
 and, consequently, 
 
 p' = LZl 5 x 144 = -0117 x 144 Ibs. 
 
 M 
 
162 CHAPTER V. 
 
 The resistance of the air against a train of mean 
 surface is 33 Ibs. at the velocity of 10 miles per hour 
 for the engine. This resistance in transmitting itself 
 to the piston, increases in the inverse ratio of the 
 respective velocities of the piston and of the engine, 
 that is to say, of the velocities of the piston and of 
 the circumference of the wheel. Consequently, if 
 D represent the diameter of the wheel, that force 
 becomes 
 
 and supposing it to he divided per unit of the 
 surface of the piston, it produces a resistance ex- 
 pressed by 
 
 a Z 
 
 Admitting then the mean proportions most in use 
 in these engines, that is, taking the diameter of the 
 wheel at 5 feet, the stroke of the piston at 16 inches 
 or 1'33 foot, the diameter of the piston at 12 inches 
 or 1 foot, the above resistance, divided per square 
 foot of the surface of the piston, amounts to 
 
 8615 x 144. 
 
 And as we have said that it is the intensity of that 
 force, when measured at the velocity of 150 feet per 
 minute for the piston, and that it increases as the 
 square of the velocity, we have 
 
 g V 2 = -8615 x 144, when v = l50, 
 which gives 
 
 #=00003829 x 144 Ibs. 
 
LOCOMOTIVE ENGINES. 163 
 
 Finally, as locomotive engines are high pressure 
 engines, without condensation, the values of n and 
 q which are suitable to them, are 
 
 n- -0001421 
 ? = -00000023. 
 
 Introducing these values then in the algebraic 
 formulae above, they become as follows : 
 
 Practical formula for locomotive engines. 
 
 GENERAL CASE. 
 
 __ S. 10000 VI' 
 
 V "~ a 7-042 +'00275 (r+g) +-0040688 W+-00001518 v* Velocitv * tne 
 
 piston, in feet 
 
 , per minute. 
 
 a 
 
 ar=3632300-~ a[2558+ f + l-4779v + '0055t; 2 ]Useful load of 
 
 the piston, in 
 
 pounds. 
 av 
 S = io<Jo "[7'042 + -00275 (r+ e ; + -0040688 t> + -00001518 v a ] Effective 
 
 evaporation, in 
 cubic feet of 
 water per mi- 
 nute. 
 
 u. E zz arv Useful effect, in 
 
 pounds raised 
 1 foot per mi- 
 nute. 
 
 U ' HP = So Useful force ' in 
 
 horse-power. 
 
 u. E. 1 Ib. co. = Useful effect of 1 
 
 Ib. of coke, in 
 pounds raised 
 1 foot. 
 M 2 
 
164 
 
 CHAPTER V, 
 
 u. E. 1 ft. wa. ... =-^' Useful effect of 
 
 ij^ 
 
 1 cubic foot of 
 water evapor- 
 ated, in pounds 
 raised 1 foot. 
 
 33000 N 
 Q. co. for 1 hp...=: ^ Quantity of coke, 
 
 in pounds, 
 which pro- 
 duces 1 horse- 
 power. 
 
 Q. wa. for 1 hp. = ^r Quantity of water 
 
 in cubic feet, 
 which produces 
 1 horse-power. 
 
 u.HP.llb.co...= 3^0^- Horse - power 
 
 produced per 
 pound of coke. 
 
 u. HP. 1 ft. wa. =33Q 00 'g Horse - P ower 
 
 produced per 
 cubic foot of 
 water evapor- 
 ated. 
 
 CASE OF THE MAXIMUM USEFUL EFFECT. 
 
 S 
 
 10000 
 
 ' 1-492 + -002415 P 
 
 , Velocity of the 
 
 piston, in feet 
 per minute. 
 
 ar'=a(-8772P-2016-p-l-47r9t/--005$t/ 2 )Useful load of 
 
 the piston, in 
 pounds. 
 
LOCOMOTIVE ENGINES. 165 
 
 S= j^ (1-492+ '002415 P) Effective eva- 
 poration, in 
 cubic feet of 
 water per mi- 
 nute. 
 
 max. u. E -=.arv Useful effect, in 
 
 pounds raised 
 1 foot per mi- 
 nute. 
 
 We omit, in the case of maximum useful effect, 
 the horse-power and the other modes of expressing 
 the effect of the engine, because the formulae proper 
 to those determinations are the same as those of 
 the general case, which dispenses us from repeating 
 them here. 
 
 To give now an example of the application of 
 these formulae, we will suppose a locomotive engine 
 similar to the Atlas, which has the following dimen- 
 sions and capabilities : 
 
 Two cylinders of 12 inches diameter; or = 
 1-57. 
 
 Stroke of the piston 16 inches ; or /=1'33. 
 Clearance of the cylinder ^V of the stroke ; or 
 c=-05l. 
 
 Wheels 5 feet in diameter, coupled. 
 Total pressure in the boiler, 65 Ibs. per square 
 inch ; or P = 65 x 144 Ibs. per square foot. 
 
 Effective evaporation, '67 cubic feet of water per 
 minute ; or S = '67. 
 
 Consumption of fuel in the same time 8 Ibs. ; or 
 N = 8. 
 
166 CHAPTER V. 
 
 Resistance caused by the transport of the engine 
 itself, 3^-5- of its own weight, or 80 Ibs. ; which 
 makes against the piston a resistance of 80 x 5.9 = 
 472 Ibs. or 2*09 Ibs. per square inch of its surface. 
 Thus p- 2-09x144 Ibs. 
 
 Performing, then, the calculation with these data, 
 we find the following results, at the velocity of 
 maximum useful effect, and at the velocities of 250 
 and 300 feet per minute for the piston. 
 
 Maximum useful effect. 
 v ........................ = 300 ............ 250 ............ 176 
 
 ar ..................... = 2151 ......... 4124 ......... 8576 
 
 = 9-52 ......... 18-25 ......... 37-95 
 
 S ...................... = -67 ............ -67 ............ -67 
 
 u. E ................... 645280 ...... 1031010 ...... 1512690 
 
 u. HP ................ = 20 ............ 31 ............ 46 
 
 u.E.l Ib.co .......... = 80663 ......... 128880 ...... 189090 
 
 u. E. 1 ft. wa ....... = 963150 ...... 1538850 ...... 2257800 
 
 Q. co. for 1 hp ...... = -409 ......... -256 ......... -175 
 
 Q. wa. forlhp ...... = '034 ......... -021 ......... -015 
 
 u. HP. lib. co ...... zz 2-44 ......... 3-91 ......... 5-73 
 
 u. HP. 1ft. wa ...... = 29 ............ 47 ........... 68 
 
 Were it desired to refer these results to the 
 measures usual on railways, that is, to count the 
 velocity in miles per hour for the engine, and the 
 load in tons drawn at the same velocity, then, from 
 what has been said, it would suffice to multiply the 
 above velocities by the factor 
 
 5-9 x 60 
 5280" ' 
 
LOCOMOTIVE ENGINES. 
 
 because, in the ordinary proportions, the velocity 
 of the engine is 5*9 times that of the piston, a mile 
 contains 5280 feet, and an hour 60 minutes. The 
 result of this first multiplication would convert the 
 velocity of the piston in feet per minute, to that of 
 the engine in miles per hour. 
 
 To convert afterwards the loads in pounds on the 
 piston, which have just been calculated above, into 
 loads of the engine in tons, the former must be 
 multiplied by the factor 
 
 1 
 
 5^7' 
 
 on account of the proportion of the velocities, and 
 because the draught of 1 ton requires a force of 
 7lbs. 
 
 Thus the three cases reported above would then 
 be expressed as follows : 
 
 Velocities 20-11. . .16-76... 11-83 miles per hour. 
 
 Loads 52 100 207 tons. 
 
 The other results would not be changed. 
 
 We will, however, observe, that these effects 
 will be those of the engine, for the data intro- 
 duced above ; but by forcing the fire, in order to 
 increase the evaporation S, or by augmenting the 
 pressure P of the steam in the boiler, the engine 
 may be made to produce greater effects ; and, on 
 the contrary, the effects will be less if the evapo- 
 ration of the boiler is less than '67 square foot per 
 minute, or if a part of the steam is lost by the 
 safety valves. 
 
 We have designedly chosen a locomotive and a 
 
168 CHAPTER V. 
 
 stationary steam-engine of the same dimensions, 
 in order to render it easy to judge of the disad- 
 vantages the locomotive engines lie under, by 
 reason of the circumstances we have just men- 
 tioned, viz. : the necessity of transporting their 
 own weight, the resistance of the air against the 
 train in great velocities, and the force consumed in 
 blowing the fire in a furnace of small dimen- 
 sions. 
 
169 
 
 CHAPTER VI. 
 
 WATT'S ROTATIVE OR DOUBLE-ACTING ENGINES. 
 
 SECT. I. Practical formula suitable to the calcu- 
 lation of these engines ; and Example of their 
 application. 
 
 IN Watt's rotative or double-acting steam-engines, 
 condensation is used, but not the expansion of the 
 steam. The steam is generated in the boiler at a 
 pressure which exceeds that of the atmosphere by 
 about 1-5 to 3lbs. per square inch. It then passes 
 into the cylinder, flowing in without interruption 
 during the whole stroke of the piston ; but when 
 the piston has terminated its descending stroke, a 
 communication opens between the top of the cy- 
 linder and the condenser. The steam which filled 
 the cylinder passes immediately into the condenser, 
 leaving on the upper surface of the piston only a 
 very slight pressure, owing to imperfect condens- 
 ation. At this moment the communication from 
 the boiler to the cylinder is changed. The steam, 
 instead of continuing to enter above the piston, 
 now comes in below it, driving the piston back 
 to the top of the cylinder, and then is condensed 
 
170 CHAPTER VI. 
 
 again, while a new quantity of steam is admitted 
 above the piston, and so on. The alternate motion 
 thus communicated to the piston is transmitted to 
 a crank, and therefore changed to one of rotation, 
 imparted to the divers pieces of machinery used for 
 various purposes in the arts. 
 
 These engines heing unexpansive, the formulae 
 suitable to them are the same as those of stationary 
 high-pressure engines, which have been treated of 
 in Chapter IV. ; only the quantity P, which ex- 
 presses the pressure in the boiler, will represent a 
 much lower pressure, and the quantity p 9 instead 
 of representing the atmospheric pressure, will now 
 stand for the pressure which still subsists in the 
 cylinder, after imperfect condensation of the steam. 
 
 We shall not, then, repeat those formula here 
 in their algebraic shape, but transform them at 
 once into the corresponding numerical formula ; 
 to which end we will first seek the constant quan- 
 tities that figure in them, in order to make the 
 proper substitutions. 
 
 The quantity p represents the pressure of con- 
 densation under the piston ; but it is necessary 
 here, to make an observation relative to that 
 pressure. In good engines, well provided with 
 injection water, at a degree of temperature not 
 exceeding 50 of Fahrenheit, the pressure in the 
 condenser, taken by means of a manometer, is 
 usually reduced to 1*5 Ib. per square inch ; but 
 that is clearly not the true value of p, or the pres- 
 sure of condensation in the steam cylinder itself, 
 
WATT'S ROTATIVE OR DOUBLE-ACTING ENGINES. 17! 
 
 and under the piston. In effect, since the con- 
 densation of the steam takes place only by degrees, 
 as it passes from the cylinder to the condenser, and 
 as the velocity of its transit depends only on the 
 difference of pressure between the uncondensed 
 steam remaining in the cylinder and the imperfect 
 vacuum in the condenser, it follows that an equi- 
 librium of pressure between the emptying cylinder 
 and the condenser can only be established gra- 
 dually. The mean pressure of condensation, then, 
 under the piston, must be superior to that of the 
 condenser. Direct experiments on this subject, 
 made with Watt's indicator, prove that in ordinary 
 velocities, and with the usual dimensions of the 
 passages, the mean pressure under the piston is 
 ordinarily 2 '5 Ibs. higher than that of the con- 
 denser. The latter, then, being 1*5 lb., we per- 
 ceive that the value of p will generally be 
 
 p 4 x 144 Ibs. ; 
 
 it ought, however, to be measured specially for each 
 occasion. 
 
 As to the friction of these engines, some notions 
 tolerably precise have been acquired by expe- 
 rience. It is a datum admitted among practical 
 men, and founded on numerous trials made on 
 Watt's engines, that their friction when working 
 under a moderate load, varies from 2*5 Ibs. per 
 square inch of the piston, in the smaller and less 
 elaborate engines ; to 1 '5 Ib. in those of larger 
 dimensions and better made ; and this includes 
 
CHAPTER VI. 
 
 the friction of the different parts of the engine 
 and the force necessary to work the feeding and 
 discharging pumps, &c. 
 
 By a moderate load in these engines is under- 
 stood a load of about 8 Ibs. per square inch of the 
 piston ; and our own experiments on locomotives 
 give room to think that the additional friction 
 created in the engine by reason of that load, will 
 be |th of the load, or 1 Ib. per square inch. The 
 above datum goes then to establish that Watt's 
 engines, working without load, have a friction of 
 1 -5 Ib. to '5 Ib. per square inch, according to 
 their dimensions : namely, l\5lb. for the smaller 
 ones, or those termed 10 horse power, having a 
 cylinder IJ'5 inches in diameter ; and '5 Ib. for 
 those termed 100 horse power, or having a cy- 
 linder 48*5 inches in diameter ; which makes 1 Ib. 
 for engines of mean power. This result agreeing 
 with that which we have deduced from our re- 
 searches on locomotives, the data precedently in- 
 dicated may be admitted for engines of mean di- 
 mensions of this system ; but it is to be recollected 
 that these data can only serve in default of special 
 and circumstantial experiments on the subject. 
 
 In an engine of this system of mean dimensions, 
 we may then take 
 
 /= 1 x 144, B = -14. 
 
 The engine, too, working by condensation, we 
 
 have, 
 
 n = -00004227 
 q = -000000258. 
 
WATT'S ROTATIVE OR DOUBLE-ACTING ENGINES. 173 
 
 Finally, the clearance of the cylinder in these 
 engines is ordinarily ^th of the stroke, which gives 
 c = -05 /. 
 
 Substituting in the algebraic formulae, those 
 among these values which are constant, we have 
 the following numerical formulae : 
 
 Practical formulae for Watt's rotative or double- 
 acting engines. 
 
 GENERAL CASE. 
 
 S 10000 
 
 of the 
 piston, in feet 
 
 per minute. 
 
 ar=369HOO ^_-^L (l64+p+/) Useful load of 
 
 the piston, in 
 pounds. 
 
 S = -^ {4438 + '00271[(l+5)r4-p+/]}Effective eva- 
 
 poration, in 
 cubic feet of 
 water per 
 minute. 
 
 u.E.zz arv ........................ . ................. Useful effect, in 
 
 pounds raised 
 1 foot per mi- 
 nute. 
 
 u ' HP .......... = jjjjfr ........................... _ Useful force, in 
 
 horse-power. 
 
 u. E. 1 Ib. co. = ^7^ ........................... Useful effect of 
 
 1 Ib. of coal, 
 in pounds 
 raised 1 foot. 
 
174 
 
 CHAPTER VI. 
 
 u. E. 1 ft. wa. = 
 
 u.E. 
 
 Q. co. for 1 hp. 
 
 Q. wa. for 1 hp. zi 
 
 33000 N 
 u.E. 
 
 33000 S 
 u. E. 
 
 u. HP. lib. co. = 
 
 u.HP.l.ft.wa. = 
 
 u. E. 
 
 33000 N 
 
 u.E. 
 
 33000 S 
 
 Useful effect of 
 1 cubic foot 
 of water, in 
 pounds raised 
 1 foot. 
 
 Quantity of 
 coal in Ibs., 
 which pro- 
 duces 1 horse 
 power. 
 
 Quantity of 
 water in cu- 
 bic feet, which 
 produces 1 
 horse-power. 
 
 Horse - power 
 produced per 
 pound of coal. 
 
 Horse - power 
 produced by 
 1 cubic foot 
 of water eva- 
 porated. 
 
 CASE OF THE MAXIMUM USEFUL EFFECT. 
 
 __S^ _ 10000 
 ~~ a 
 
 4438 + -00271P 
 
 ar' = 
 
 8= 
 
 av 
 
 10000 
 
 Velocity of the 
 
 piston, in feet 
 per minute. 
 
 (P-p-/) Useful load of 
 
 the piston, in 
 pounds. 
 
 ( -4438 + -00271 P) Effective eva- 
 poration, in 
 cubic feet of 
 water per 
 minute. 
 
WATT'S ROTATIVE OR DOUBLE-ACTING ENGINES. 175 
 
 max. u. E.zzarV Maximum use- 
 ful effect, in 
 pounds raised 
 1 foot per 
 minute. 
 
 As an application of these formulae, we will per- 
 form the calculation on an engine constructed by 
 Watt at the Albion Mills, and put to trial soon 
 after it was built by Watt himself. The following 
 were the dimensions of the engine : 
 
 Diameter of the cylinder, 34 inches ; or a = 
 6-28? sq. feet. 
 
 Stroke of the piston, 8 feet ; or I 8 feet. 
 
 Clearance of the cylinder, aVth of the stroke ; or 
 c = -05 /. 
 
 Pressure in the boiler, 16'5 Ibs. per square inch ; 
 or P = 16'5 x 144 Ibs. per square foot. 
 
 Effective evaporation, '927 cubic feet of water 
 per minute ; or S = '927* 
 
 Consumption of coal in the same time, 6*71 Ibs. ; 
 or N = 6-71 Ibs. 
 
 The engine had been constructed to work at the 
 velocity of 256 feet per minute, which was con- 
 sidered as its proper velocity ; but when put to 
 trial by Watt, accompanied by Sir J. Rennie, it 
 assumed, in performing its regular work, which 
 was esteemed 50 horse-power, the velocity of 286 
 feet per minute for the piston, consuming per 
 minute the quantity of water and coal which we 
 have just stated. 
 
 On seeking the effects which this engine ought 
 
176 CHAPTER VI. 
 
 to produce, at its velocity of maximum useful effect, 
 and then at those of 256 and 286 feet per minute, 
 they are found to be : 
 
 Maximum useful effect. 
 
 v zz 286 256 214 
 
 ar = 5621 6850 9133 
 
 . , = 6-21 7-57 10-09 
 
 144 
 
 S zz -927 -927 -927 
 
 u. E zz 1607610 1753600 1957180 
 
 u. HP ... zz 49 53 59 
 
 u. E. lib. co zz 239585 261340 291680 
 
 u. E. 1ft. wa zz 1734200 1891700 2111300 
 
 Q. co. for 1 hp = -138 -126 -113 
 
 Q.wa.forlhp zz -019 '017 -016 
 
 u.HP. lib. co = 7-26 7-92 8-84 
 
 u. HP. lft.wa zz 53 57 64 
 
 Such are the effects that might have been ex- 
 pected from this engine ; and we see in consequence 
 that, performing a work estimated at 50 horses, it 
 was to be expected to assume the velocity which 
 in fact it did ; namely, 286 feet per minute. 
 
 SECT. II. Considerations on the application of 
 the ordinary mode of calculation to Watt's 
 ' steam-engines. 
 
 THE above calculations and the formulae which 
 precede them, testify our free admission that Watt's 
 engines may work at a full pressure of steam in the 
 cylinder, that is to say, at a pressure in the cylinder 
 sensibly equal to that of the boiler, since the case 
 of maximum effect here mentioned is nothing else 
 
WATT'S ROTATIVE OR DOUBLE-ACTING ENGINES. 177 
 
 but that. But we have proved that this effect can- 
 not be produced at all velocities, as the calculation 
 commonly applied to steam-engines would have it. 
 It can be produced only at one velocity, that of the 
 maximum of effect, and this will vary in every 
 engine, so that it becomes necessary to calculate it 
 separately for each. 
 
 Some persons think that stationary engines, and 
 those of Watt in particular, working always at a 
 very moderate velocity, and having, as they allege, 
 very large passages for the circulation of the steam, 
 must necessarily work at full pressure in the cy- 
 linder ; and they conclude that, however true the 
 theory we have explained may be, it is nevertheless 
 unnecessary for calculating the effects of these 
 engines, because they are found to realize the sup- 
 positions of the ordinary calculation. But, besides 
 that the steam-passages are much narrower in Watt's 
 engines than in the locomotives, since they are but 
 T ^th instead of ^th of the area of the cylinder, it 
 is easy to obtain conviction, that Watt's engines, as 
 well as those we have already treated, may work at 
 pressures very much reduced in the cylinder, which 
 fact sets all the suppositions and calculations of the 
 ordinary theory at fault. 
 
 Not to return to the general demonstration of 
 this fact, which results from principles we have al- 
 ready explained, the proof of it will be found in three 
 authentic circumstances. 
 
 1st. Watt's indicator applied to the cylinder of 
 several of his engines, while working at their usual 
 
 N 
 
178 CHAPTER VI. 
 
 velocity and in their regular state, made known that 
 the pressure in the cylinder was several pounds per 
 square inch below that of the hoiler. Whatever 
 reasons, then, be adduced to the contrary, there is 
 not an equality between the two pressures. 
 
 2d. Experience has demonstrated to Watt, that in 
 his engines, every cubic inch of water evaporated 
 produced a cubic foot of steam of an elasticity suf- 
 ficient to effect the motion. This practical datum 
 he has registered in his notes to the article on 
 steam-engines in Robinson's Encyclopaedia. From 
 this observation the steam is expended in the cy- 
 linder at a volume equal to 17^8 times that of 
 water. But the steam is formed in the boiler at the 
 pressure of 16%5 Ibs. per square inch, which gives 
 but 1530 times the volume of the water. There- 
 fore, the volume of the steam, in passing into the 
 cylinder, becomes 1728 instead of 1,530 ; that is to 
 say, the steam undergoes a considerable diminution 
 of pressure, and at the same time a corresponding 
 augmentation of volume. 
 
 o 
 
 3d. It is known that Watt's engines, when put to 
 the proof, will produce about half as much more than 
 their nominal power, without any change being made 
 in the engine. Hence, in their daily work, they 
 do not work at full pressure ; for if they did, it 
 would be impossible to make them execute more. 
 
 Thus we see that these engines work habitually 
 at a reduced pressure, as well as those of which we 
 have treated hitherto. However, we do not give 
 this observation as new in itself. A diminution of 
 
WATT'S ROTATIVE OH DOUBLE-ACTING ENGINES. 179 
 
 pressure in the cylinder, and even an augmentation 
 of the volume of the steam, had already been re- 
 cognized in these engines ; but the observation of 
 such a change, instead of leading to the theory of 
 the steam-engine, such as we have developed it, was 
 nothing but a material fact used to explain the co- 
 efficient^ which we cannot admit. 
 
 As to that coefficient itself, it would be wrong 
 to deem that, notwithstanding its inexactitude in 
 principle, it might yet, without error, be used in 
 practice to calculate Watt's steam-engines, for the 
 reason alleged, that these engines, undergoing but 
 slight variations of velocity, the application of a 
 constant coefficient would suit them in all cases. 
 This is certainly an error ; for the smallest velocity 
 of the piston in those engines being 150 feet per 
 minute, and the greatest 300 feet per minute, it 
 will be remarked that these are also the velocities 
 of the piston in a locomotive engine, moving at 
 from 10 to 20 miles an hour, as may be seen in 
 the example of the ATLAS given in Chapter V. of 
 this work. Consequently, there must always be, 
 between the loads corresponding to these extreme 
 velocities, or in other words, between the useful 
 effects produced by the stationary engines, differ- 
 ences analogous to those which are observed for 
 similar velocities in locomotives. This, besides, is 
 made completely manifest by the calculation which 
 we have given above. 
 
 If in fact we turn back to the useful effects ob- 
 tained for the respective velocities of 286, 2.56, and 
 
180 CHAPTER VI. 
 
 214 feet English per minute, and seek to represent 
 them hy a coefficient applied to what is called the 
 theoretic effect, we shall have the following results : 
 
 Velocity. Coefficient. 
 
 286 Theoretic effect 32365001 Krk 
 
 } ........ *o(J 
 
 Real 1607610) 
 
 256 Theoretic effect 2897000} , gl 
 
 Real 1753600} ' 
 
 214 Theoretic effect 2013300 1 , gl 
 
 Real 1957180) 
 
 Thus, for these engines, as well as for those pre- 
 cedently treated of, no constant coefficient what- 
 ever can supply the place of the analytical calcula- 
 tion which we offer in its stead. And considering 
 the great variation that takes place in that coeffi- 
 cient, from the very ordinary differences of the ve- 
 locities 286, 256, and 214 feet per minute, for 
 which the coefficient assumes the three values given 
 ahove, we cannot but he convinced of the fallacy of 
 the mode of calculation hy coefficients ; for how- 
 ever little the velocity of the engine to which the 
 calculation is applied, differ from the velocity at 
 which the coefficient has heen originally deter- 
 mined, which is altogether unknown, no reliance 
 whatever is to be placed on the effect indicated by 
 that coefficient. 
 
 We thus see that Watt's engines make no ex- 
 ception to the general rule in this respect ; and it 
 may further be observed, that the experiment of 
 Watt reported above, would give rise, in the or- 
 dinary theory, to precisely the same contradictions 
 
WATT'S ROTATIVE OR DOUBLE-ACTING ENGINES. 181 
 
 and inaccuracies as the experiments which we have 
 discussed in Chapter I. on locomotives. 
 
 In effect, in this experiment the engine evapor- 
 ating *927 cubic foot of water per minute, and ex- 
 ercising a force of 50 horses, assumed a velocity of 
 286 feet per minute. 
 
 We find, then, that since the engine had a use- 
 ful effect of only 50 horse-power, and that its 
 theoretic force calculated according to that me- 
 thod, from the area of the cylinder, the effective 
 pressure in the hoiler and the velocity of the pis- 
 ton, was 
 
 6-287 x (16*5-4) x 144x286 = g8 hors , power , 
 
 33000 
 
 it followed that, to pass from the theoretic effects 
 to the practical required the coefficient of '51. 
 Consequently following the reasonings of that 
 theory, we are led to these conclusions : 
 
 1st. The observed velocity having been 286 feet 
 per minute, the evaporation, calculated from the 
 quantity of water which, being reduced to steam at 
 the pressure of the boiler, might occupy the volume 
 described by the piston, and afterwards divided, 
 as indicated by several authors, by the coefficient, 
 to account for the losses, would have been 
 
 1 
 
 1530 x 6-287 x 286 ,. 
 
 ' - = 2*305 cubic feet of water per 
 
 *O L 
 
 minute, instead of '927. 
 
 2d. The engine having evaporated only -927 cu- 
 bic foot of water per minute, the velocity of the 
 
182 CHAPTER VI. 
 
 piston, calculated from the volume of steam formed 
 at the pressure of the boiler, and afterwards re- 
 duced by the coefficient, not as indicated, since this 
 problem has never been solved, but as one must 
 naturally conclude from the signification attributed 
 to that coefficient, could be no other than 
 
 1580 x -927 X . dl -u5 f eet per m i nu te, instead 
 6-287 
 
 of 286. 
 
 3d. The coefficient found by the comparison of 
 the theoretical to the practical effects being %51, 
 the frictions, losses and divers resistances of the 
 engine amounted to '49 of the effective power ; 
 whereas these frictions, losses and resistances con- 
 sisting merely in the friction of the engine and the 
 clearance of the cylinder, cannot be estimated at a 
 higher rate than the following : 
 Total friction, (additional friction included,) 
 
 2 Ibs. per sq. inch, or as a fraction of the 
 
 effective pressure, -* .... '17 
 
 Clearance of the cylinder, TO th of the effective 
 
 stroke, or '05 
 
 22 
 
 Hence we see that Watt's experiment, as well as 
 those made on locomotives, would be altogether in- 
 explicable by that theory, and it would be the 
 same with any experiment wherein should be noted 
 the quantity of water evaporated. 
 
 Before quitting this example we must recall to 
 notice that some authors employ constant coeffi- 
 
WATT'S ROTATIVE OR DOUBLE-ACTING ENGINES. 183 
 
 eients also, but without using the same coefficient 
 to determine the evaporation as to determine the 
 useful effect. This mode of calculating originated 
 in those authors having recognized hy experience 
 that the steam assumed in the cylinder a pressure 
 and density less than in the boiler. But as they 
 have not been able to fix a priori what was that 
 pressure in the cylinder, and as they always seek 
 to deduce it from that of the boiler, instead of 
 concluding it immediately and in principle, as we 
 do, from the resistance against the piston, the ob- 
 served diminution of pressure could not be defined 
 as to its limits, and it remained merely a practical 
 fact of which they availed themselves to explain 
 their coefficient. The change they make in the co- 
 efficient makes them avoid the first and second of 
 the contradictions we have just signalized, but the 
 third as well as all the objections which we have 
 advanced elsewhere against the use of any constant 
 coefficient whatever, remain in full force. That is 
 to say, in this method, the power of the engine is 
 always calculated independently of the evaporating 
 force of the boiler, and the evaporation independ- 
 ently of the resistance t'o be moved ; the effort of 
 which the engine is capable is always found the 
 same at all velocities ; no account can be taken of 
 the degree of opening of the regulator, except by 
 introducing for that purpose a new series of co- 
 efficients, and the same for all the changes of ve- 
 locity. In a word, it is the method which we have 
 applied, in the first chapter, to the experiments on 
 
184 CHAPTER VI. 
 
 the locomotive engine the Leeds, and consequently 
 it is directly liable to all the objections we have 
 there made. 
 
 To return to the results of our formulae, it is 
 understood that the effects of steam-engines as we 
 have shewn them, are producible only in as much 
 as the various conditions of the calculation are 
 fulfilled. But, as it often occurs that several im- 
 portant circumstances suffer change without any 
 notice being taken of their alteration, we must 
 here add : 1st, that as all effect produced depends 
 directly and absolutely on the evaporation effected, 
 it is utterly impossible to know the power or to 
 compute the effects of a given steam-engine, with- 
 out having previously measured or estimated its 
 evaporation ; 2d, that the pressure in the boiler, 
 as well as that in the condenser, ought, for every 
 case, to be observed by the manometer, because a 
 change of pressure in the boiler is a circumstance 
 occurring continually, and it is plain that if the 
 engine works at a higher pressure, some of the ef- 
 fects produced will change in consequence ; 3d, 
 with respect especially to the effects attributable 
 to the consumption of a given weight of fuel, we 
 must observe that nothing can be more uncertain. 
 Smeaton's experiments demonstrate that the differ- 
 ent qualities of coal in use in England may pro- 
 duce effects varying in divers proportions between 
 the numbers 86 and 133. He found, moreover, 
 that refuse coal, such as is often used in some 
 mines, compared to the same quality of coal in 
 
WATT'S ROTATIVE OR DOUBLE-ACTING ENGINES. 185 
 
 pieces of the size of an egg, produced effects in 
 the proportion of 80 to 100 ; and finally that if the 
 fire is ill managed, loaded in layers too thick and 
 seldom stirred, the effects produced may he but i ths 
 of what they would he with a clear well-stoked 
 fire. Thus we see that all these causes combined 
 may produce a difference of effect, between the 
 least favourable and the most advantageous circum- 
 stances, in the proportions of 
 
 _80 5 86 _. 43 . 
 
 Too x 6 x i33~ioo ; 
 
 that is, the difference between the two may exceed 
 the half. 
 
 It is clear, besides, that the effect due to a given 
 weight of fuel will depend also on the more or less 
 judicious construction of the boiler, as well as on 
 the incrustations that may form upon it during the 
 work, For these various reasons, then, it is im- 
 possible to found a mathematical comparison of the 
 effects of steam-engines on the quantity of fuel 
 reported to have been consumed by them in pro- 
 ducing a given effect. This datum must evidently 
 be accompanied by many others in order to be de- 
 cisive. 
 
 In Watt's experiment, for instance, which has 
 just been mentioned, the consumption of fuel was 
 in the ratio of 7*24 Ibs. of coal per cubic foot of 
 water evaporated ; whereas, from the medium ob- 
 servation of Watt himself, that evaporation is not 
 generally produced at a consumption of less than 
 8 '4 Ibs. of the best coal. If, then, in the instance 
 
186 CHAPTER VI. 
 
 in question, the ordinary data had not been ex- 
 ceeded, the evaporation of '9^7 cubic foot of water 
 per minute, would have required a consumption of 
 7*79 Iks. of c al ; and thus the effect arising from 
 1 Ib. of fuel, would have been, in the case of max- 
 imum useful effect, only 
 
 max.u. E. 1957180 
 U.E. llb.CO.rz: ; = 251240 Ibs.; 
 
 a result' which, as we see, is much inferior to that 
 deduced from the experiment itself. 
 
187 
 
 CHAPTER VII. 
 
 CORNISH DOUBLE-ACTING STEAM-ENGINES. 
 
 Practical formulce for calculating those engines. 
 
 IN the county of Cornwall both double-acting and 
 single-acting engines are in use. The latter are 
 a modification of Watt's single-acting engines. 
 We shall not notice these at present, but direct our 
 attention wholly to the former. 
 
 The Cornish double-acting engines, are the rota- 
 tive or double-acting engines of Watt, with con- 
 densation, working at a higher pressure in the boiler, 
 that is, at a total pressure of about 3%5 atmospheres, 
 or more, and constructed to admit the use of the ex- 
 pansion of the steam, a force of which Watt availed 
 himself only in his single-acting engines. These 
 engines having but one cylinder, which serves at once 
 for the direct admission of the steam and for its ex- 
 pansion, are no other than those we have treated of 
 in Chapter III., in developing the general formula? 
 of the action of the steam on the piston, by direct 
 admission, expansion and condensation. We shall 
 not, then, here reproduce those formulae under their 
 
188 CHAPTER VII. 
 
 algebraic form, but shall replace them by the cor- 
 responding numerical formulae, to which end it 
 will be necessary, first, to settle the value of the 
 constant quantities which figure in them. 
 
 The friction of Watt's steam-engines, as has 
 been said above, may be estimated at 1*5 Ib. per 
 square inch of the surface of the piston, when the 
 piston is of the smallest dimensions in use in those 
 engines, that is, about 17^ inches in diameter, and 
 at '5 Ib. per square inch, when, on the contrary, 
 the diameter of the piston is as much as about 48^ 
 inches. It is for this reason that we have consi- 
 dered 1 Ib. per square inch as representing, nearly 
 enough for practical purposes, the friction of 
 Watt's engines of mean dimensions. The Cor- 
 nish engines being, as we have said, merely a mo- 
 dification of those of Watt, their friction may be 
 estimated by the same rule, at least till special 
 experiments be made on the subject. But as, in 
 these engines, the expansion requires a cylinder of 
 much greater capacity, it follows that a diameter 
 of 48 inches, instead of being an extreme size for 
 the piston, is, on the contrary, no more than a 
 medium dimension. Consequently the friction pro- 
 per to the mean dimensions of these engines, may 
 be valued at '5 Ib. per square inch of the surface of 
 the piston ; which gives per square foot 
 f= -5 x 144 Ibs. 
 
 In this valuation, however, we comprehend only 
 the friction proper to the engine itself, and not 
 that of any connecting rods, and other pieces of 
 
CORNISH DOUBLE-ACTING ENGINES. 189 
 
 machinery more or less complex, which may serve 
 to transmit the motion to points often very distant 
 from the engine. The friction, or the resistance 
 of these different parts, when they exist, ought to 
 be valued separately, according to the circum- 
 stances of their construction, and then deducted 
 from the result which we obtain for the useful 
 effect ; this representing, in our calculation, only 
 the quantity of power as disposable on the very 
 shaft of the engine. This observation, however, 
 refers more particularly to pumping or single-acting 
 engines. 
 
 Furthermore, we may, as in Watt's steam-en- 
 gines, estimate also the additional friction caused 
 by a given resistance, at ^th of that resistance, or 
 make 
 
 8 = -14 : 
 
 and admit that the pressure subsisting in the steam 
 cylinder, after communicating with the condenser, 
 will be about 4 Ibs. per square inch, that is to say, 
 we shall have 
 
 p =L 4 x 144 Ibs. 
 
 Thus, from what precedes, we might in the 
 algebraic equations, replace the term (^ + f} by 
 its mean value 
 
 4-5 x 144. 
 
 But as our value of the friction is only an estima- 
 tion ; as it is to be taken differently, according to 
 the dimensions of the cylinder ; and as, moreover, 
 
190 CHAPTER VII. 
 
 the pressure of condensation, p 9 varies also accord- 
 ing to the construction of the engine, and likewise 
 according to the quantity and temperature of the 
 water used for condensation, we prefer to let the 
 term (jo 4-jO remain in the equations, leaving 
 its precise value to be given suitably to each 
 case. 
 
 As to the coefficients of the relative volume of 
 the steam, since the engine is a condensing one, 
 they will have the values already given : viz. 
 
 n - -00004227" 
 q - -000000258. 
 
 Finally, the motion of the piston being regu- 
 lated by a crank, the clearance of the cylinder is 
 still, at a medium, -~th of the stroke, as in all ro- 
 tative engines, which gives 
 
 Admitting, then, these valuations, and making 
 the proper substitutions in the algebraic formulae 
 already developed, we shall have the following re- 
 sults, in which the value of k is 
 
 t I + c 
 
CORNISH DOUBLE-ACTING ENGINES. 191 
 
 Practical formula for Cornish double-acting 
 engines. 
 
 CASE OF AN INDEFINITE LOAD OR VELOCITY, 
 WITH A GIVEN EXPANSION. 
 
 10000 k 
 
 Velocity of the 
 
 (1+J) r + -00*58 
 
 piston, in feet 
 
 per minute. 
 ar- 3875970 ^- --- (164+p+/) ... Useful load of 
 
 (1 -\-o) V 1 -}-o 
 
 the piston, in 
 pounds. 
 
 S = , *V 1-4227 + -00258 (1+3) r+-002580>+/)] Effective eva- 
 10000 Ic 
 
 poration, in 
 
 cubic feet of 
 water per mi- 
 nute. 
 
 u. E ................... arv ..................... Useful effect, 
 
 in Ibs. raised 
 1 foot per mi- 
 nute. 
 
 u. HP ................ = ^^- ..................... Useful force, in 
 
 33000 
 
 horse-power. 
 
 u. E. lib. co ....... zz Ut E< ..................... Useful effect of 
 
 1 Ib. of coal, 
 in Ibs. raised 
 1 foot. 
 
 u. E. 1 ft. wa ....... = U<E - ..................... Useful effect 
 
 of one cubic 
 foot of water, 
 in Ibs. raised 
 1 foot. v 
 
192 
 
 CHAPTER VII, 
 
 Q. eo. for 1 hp.... = - Quantity of 
 
 coal, in Ibs., 
 producing 1 
 horse-power. 
 
 Q. wa. for 1 hp.... = ^_ S Quantityofwa- 
 
 ter, in cubic 
 feet, produc- 
 ing 1 horse- 
 power. 
 
 . HP. for 1 Ib. co. = JL|_ Useful horse . 
 
 power pro- 
 duced by 1 
 Ib. of coal. 
 
 u. HP. for 1 ft. wa. - U ' E ' Useful horse- 
 
 33000 S 
 
 power pro- 
 duced by 1 
 foot of water 
 evaporated. 
 
 CASE OF MAXIMUM USEFUL EFFECT, WITH A 
 GIVEN EXPANSION. 
 
 , I S 10000 
 
 V ~- 
 
 a -4227 + '00258 P 
 
 Velocity of the 
 piston, in feet 
 per minute. 
 
 
 S = . ('4227 + '00258 P) 
 
 ]oad of 
 
 the piston, in 
 pounds. 
 
 Effective eva- 
 poration, in 
 cubic feet of 
 water per mi- 
 nute. 
 
CORNISH DOUBLE-ACTING ENGINES. 193 
 
 max.u. E, =arV Useful effect, 
 
 in Ibs. raised 
 1 foot per mi- 
 nute. 
 
 CASE OF ABSOLUTE MAXIMUM USEFUL EFFECT. 
 
 I' 164 +p+f 
 
 1= 164 + P .................................... Expansion 
 
 which pro- 
 
 duces the ab- 
 solute max- 
 imum of use- 
 ful effect. 
 
 To make use of these formulae, the first thing to 
 be done is to determine the quantity k. But as 
 the expansion at which the engine works is known, 
 or rather that point of the stroke at which the ex- 
 pansion begins, or the ratio 
 
 e 
 
 7' 
 
 the corresponding value of k will be immediately 
 concluded, by means of the table given in Chapter 
 III. Article I., as well as that of the fraction 
 
 I' + c 
 
 If the table above mentioned is not at hand, it 
 will be necessary to seek directly the value of the 
 term 
 
 o 
 
194 CHAPTER VII. 
 
 Then, to obtain the term 
 
 which is an hyperbolic logarithm, the common 
 logarithm must he taken of 
 
 and this being multiplied by 
 
 will give the hyperbolic logarithm sought. So 
 the value of k will be known as before, and then 
 the various formulae will offer no other difficulty, 
 since they contain no terms beyond the first de- 
 gree. 
 
 We have said that the Cornish engines generally 
 work at the total pressure in the boiler of 50lbs. 
 per square inch. It has been seen also that their 
 mean friction may be estimated at '5 Ib. per square 
 inch of the surface of the piston, and the pressure 
 of condensation under the piston at 4 Ibs. per square 
 inch. Now, the table of Chapter III. shows that 
 the relative volume of the steam, under the pressure 
 of 4 '5 Ibs. per square inch, is very nearly 51 60 times 
 that of water; and that under the pressure of 50 Ibs. 
 per square inch, the relative volume is expressed 
 by 552. We see then, bearing in mind at the same 
 time, that the formula of the expansion of absolute 
 maximum useful effect expresses the ratio of the 
 relative volumes of the steam, under the respective 
 pressures P and (p +/), that in the Cornish engines 
 
CORNISH DOUBLE-ACTING ENGINES. 1Q5 
 
 the absolute maximum of useful effect will be given 
 in general by the equation 
 
 V 552 
 
 It is, however, to be remarked that, if the en- 
 gine should condense less perfectly, or its friction 
 be somewhat greater than we have supposed, the 
 preceding fraction will tend rather to increase. 
 On the other hand, if the pressure in the boiler 
 be above 50 Ibs. per square inch, the fraction will 
 tend to diminish. Thus it may happen in some of 
 these engines, that the maximum of useful effect 
 shall be produced by cutting the steam when the 
 piston has performed -11 or -12 of the stroke; and 
 in others, when it has performed *10 of it, or even 
 less. 
 
 The expansion which produces the absolute 
 maximum of useful effect having been determined 
 by the last formula given above, substituting the 
 value of l r thus found, in the formulae of the case of 
 maximum effect with a given expansion, we obtain 
 all the determinations relative to the maximum 
 useful effect which that expansion can produce ; 
 and since that expansion is the most favourable for 
 the engine, these determinations will be found to 
 be precisely those of the absolute maximum useful 
 effect of which the engine is capable. 
 
 As an example of the application of these 
 formulae, we will suppose that the Watt engine, 
 whose dimensions and capacities have been given 
 
 o 2 
 
196 CHAPTER VII. 
 
 in Article III. of the preceding chapter, is ad- 
 justed to generate the steam at the total pressure 
 of 50 Ibs. per square inch in the boiler, and to use 
 the expansion according to the Cornish system ; 
 and we will then calculate its effects. This sup- 
 position will furnish a natural comparison of the 
 two systems, and will enable us to recognise the 
 considerable advantages to be derived from the use 
 of the expansion of the steam. 
 
 We will suppose then that the same boiler above- 
 mentioned be retained, but that the cylinder be 
 changed for another of greater capacity, in which 
 the admission of the steam shall be intercepted 
 when the piston has performed only one quarter of 
 the stroke. Therefore we will suppose the follow- 
 ing dimensions : 
 
 Diameter of the cylinder, 48 inches ; or surface 
 of the piston, a=. 12*566 square feet. 
 
 Stroke of the piston, 10 feet ; or I 10. 
 
 Space traversed by the piston before the expan- 
 sion begins, ^ of the stroke ; or I' = '25 I. 
 
 From these dimensions, we may estimate the 
 friction of the engine, when unloaded, at *5 Ib. per 
 square inch of the surface of the piston ; and if, 
 moreover, we suppose the condensation to be 
 operated to the same degree as in the Watt original 
 engine, we shall have for f and p the values given 
 above, that is : 
 
 /= -5 x 144 Ibs., p =4 x 144 Ibs. 
 
 We have seen, besides, that it requires the same 
 quantity of total heat, and therefore nearly the same 
 
CORNISH DOUBLE-ACTING ENGINES. 19? 
 
 expenditure of fuel, to produce the same weight of 
 steam under different degrees of tension. Whence 
 it is to be concluded, that at the pressure of 50 Ibs. 
 per square inch, at which we purpose to work the 
 engine, the boiler will retain the same evaporation 
 of water per minute as before, consuming nearly 
 the same quantity of fuel in the same time. We 
 will admit then for the engine, in this respect, the 
 same data already reported, and shall have 
 
 Total pressure in the boiler, 50 Ibs. per square 
 inch ; or P = 50 x 144 Ibs. per square foot. 
 
 Effective evaporation, "9^7 cubic foot of water 
 per minute ; or S = '9^7. 
 
 Consumption of fuel in the same time, 6*71 Ibs. ; 
 or N = 6-71. 
 
 Applying the formulae to this engine, we shall 
 find, for the effects produced at the velocity of 
 maximum useful effect and at the respective veloci- 
 ties of 200 and 250 feet per minute, the following 
 results : 
 
 Effects of the engine with the given expansion. 
 
 Maximum of useful effect. 
 
 r 
 
 *25 
 
 25 .... ... ' 
 
 25 
 
 F 
 
 = 250 
 17337 
 
 200 
 
 23909 , . 
 
 136 
 39465 
 
 r 
 
 9*58 . ... 
 
 13-21 
 
 91. Q1 
 
 144 
 
 s 
 
 927 
 
 927 
 
 .097 
 
 u. E 
 
 4334210 
 
 4781800 
 
 5356860 
 
 u. HP 
 
 131-34 
 
 144-90 
 
 162-33 
 
 u. E. 1 Ib. co 
 u. E. 1 ft. wa 
 
 = 645940 
 = 4675520 
 
 712640 
 5158350 
 
 798350 
 5778700 
 
198 
 
 CHAPTER VII. 
 
 Q. CO. for 1 hp 
 
 Q. wa. for 1 hp 
 
 u. HP. for 1 Ib. co. 
 u. HP. for 1 ft. wa. 
 
 = -051 -046 -041 
 
 = -007 -006 -0057 
 
 = 19-57 21-60 ......... 24-19 
 
 = 141-7.. . 156-3.. . 175-1 
 
 Maxima useful effects of the engine, with 
 expansions. 
 
 various 
 
 
 
 Absolute max. of 
 
 useful effect. 
 
 I 
 
 
 
 
 T 
 
 - -50 
 
 25 
 
 11 
 
 v ' 
 
 74 
 
 136 
 
 255 
 
 ar' 
 
 ~ 57247 
 
 39465 
 
 22865 
 
 r 
 
 31-64 
 
 21-81 
 
 12-64 
 
 144 
 
 
 
 
 S 
 
 -927 
 
 927 
 
 927 
 
 max. u. E 
 
 = 4238480 
 
 5356860 
 
 5819300 
 
 u. HP 
 
 = 128-44 
 
 162-33 
 
 176-34 
 
 
 = 631670 
 
 798350 
 
 867260 
 
 u. E. 1 ft. wa 
 
 == 4572250 
 
 5778700 
 
 6277560 
 
 Q. co. for 1 hp 
 
 zz -052 
 
 041 
 
 038 
 
 Q. wa. for 1 hp 
 
 = -007 
 
 0057 
 
 005 
 
 u. HP. for 1 Ib. co. 
 
 - 19-14 
 
 24-19 
 
 26-28 
 
 u. HP. for 1 ft. wa. 
 
 138-6 
 
 175-1 
 
 190-2 
 
 These two tables shew what diversity of effect 
 may be produced by the same engine, according to 
 the degree of expansion at which it works and the 
 load imposed on it. In calculating the maxima 
 useful effects for different rates of expansion, they 
 are found continually to increase, on diminishing 
 the portion t of the stroke described before the 
 expansion, till we have l' = 'll Z; and beyond that 
 point, they are on the contrary found to diminish ; 
 which demonstrates by the facts, that the absolute 
 
CORNISH DOUBLE-ACTING ENGINES. 199 
 
 maximum useful effect of the engine is really at- 
 tained. We find in effect : 
 
 7 = '10 max. u. E. = 5806300 
 
 I 
 
 11 5819300 ab. max. 
 
 12 5817820 
 
 From the results we have just obtained, it is 
 observable, that by using a sufficient degree of 
 pressure in the boiler of a Cornish engine, and by 
 carrying the principle of the expansion far enough, 
 a useful effect may be obtained nearly treble of 
 that which would be produced in the Watt engine, 
 by the same quantity of water evaporated. It is 
 without reason, then, that the effects of the Cornish 
 engines have been considered so extraordinary ; 
 and that they have sometimes even been reputed 
 incredible. 
 
 The effect, however, of every engine must ob- 
 viously depend: on the pressure at which the 
 steam is generated in the boiler ; on the improve- 
 ments introduced into the construction of the boiler 
 and the furnace, whence will result that the same 
 quantity of fuel more judiciously applied, will 
 evaporate a greater quantity of water ; on the 
 quality of the fuel employed ; on the condensation 
 more or less complete in the condenser ; on the 
 care taken to clear the boiler of the incrustations 
 which form on it and obstruct its evaporating 
 power ; on the more perfect workmanship in the 
 execution of the engine, which may diminish its 
 friction ; and, in fine, on the nature and number 
 
200 CHAPTER VII. 
 
 of the connecting rods, which, in transmitting the 
 action of the engine to distant points, absorb in 
 their motion a portion, more or less considerable, 
 of the useful effect really produced by the engine. 
 
 For this reason, if it be required to compute 
 with great accuracy, what would be, in certain cir- 
 cumstances, the effects of an engine already con- 
 structed, it will not suffice to take an approximate 
 valuation of the friction, nor of the pressure of 
 condensation in the cylinder. But the friction of 
 the engine must be determined by the method 
 developed in Sect. III. of Article II., Chapter III. 
 The pressure subsisting in the steam cylinder, after 
 communication with the condenser, must also be 
 measured by Watt's indicator ; and the two quan- 
 tities f and p, thus determined, must be intro- 
 duced into the formulae, with the other data of the 
 problem equally deduced from immediate admea- 
 surement or observation. 
 
201 
 
 CHAPTER VIII. 
 
 WOOLF'S OR EDWARDS' STEAM-ENGINES, 
 
 ARTICLE I. 
 
 THEORY OF WOOLF'S STEAM-ENGINE. 
 
 WOOLF'S or Edwards' engines are of the same kind 
 as the preceding: that is to say, the expansion of 
 the steam is used also, but instead of that expan- 
 sion being effected in a single cylinder, it is per- 
 formed in two unequal ones, which the steam 
 traverses, successively dilating itself more and 
 more. 
 
 The steam coming directly from the boiler is 
 first admitted into the upper part of the small 
 cylinder, and there acts at full pressure during a 
 portion of the stroke. Then the communication 
 with the boiler is intercepted, and the steam al- 
 ready received into the small cylinder expands till 
 it has driven the piston to the bottom of its stroke. 
 At this moment, a communication opens between 
 the top of the small cylinder and the bottom of 
 the large one. The steam which filled the upper 
 part of the small cylinder passes then into the 
 larerer, and coming under the large piston and ex- 
 
202 CHAPTER VIII. 
 
 panding, makes it perform a stroke upwards. But 
 at the moment the communication opens from the 
 small cylinder to the large one, the passage from 
 the boiler to the small cylinder opens also, to let 
 in a new supply of steam, which now penetrates 
 under the smaller piston, so as to make it perform 
 an ascending stroke. 
 
 The two pistons then rise together : the smaller, 
 by the direct action of the steam from the boiler 
 at first, and then fyy that steam expanded ; the 
 larger, by the expansion of the steam which has 
 produced the preceding motion in the small cy- 
 linder. Thus, the two pistons are brought simul- 
 taneously to the top of their cylinders, and the 
 same effect is renewed. Finally, after having ter- 
 minated its action in the greater cylinder, the 
 steam is received into a separate vessel and there 
 condensed. 
 
 The two piston rods articulate on the beam of 
 the engine, and both tend to communicate to it a 
 motion of oscillation, which is afterwards con- 
 verted, by means of a crank, into a circular motion ; 
 but as these articulations may be placed at different 
 distances from the centre, the stroke of the larger 
 piston may be made longer than that of the smaller, 
 and thus the large cylinder may offer a capacity 
 by so much the more considerable for the expan- 
 sion of the steam. 
 
 It is plain that the mode of action of the steam 
 in these engines is precisely the same as in the 
 preceding ; but as the difference in the surface of 
 
WOOLF'S ENGINES. 203 
 
 the two pistons, on which the steam acts during the 
 expansion, must give rise to some modifications in 
 the formulae, we will briefly introduce those cir- 
 cumstances into the general theory. 
 
 Let P he the pressure 6f the steam in the boiler, 
 and P' the pressure that steam assumes on enter- 
 ing the small cylinder, before the expansion. Let 
 A and a be the surfaces of the two pistons, L and 
 I their respective strokes, and C and c the clear- 
 ances of the cylinders. Let 1' 9 in fine, be the 
 portion of the stroke performed by the small piston 
 before the expansion. 
 
 If we take the engine after it has attained 
 uniform motion, the quantity of labour applied 
 by the power will be equal to the quantity of 
 action developed in the same time by the resist- 
 ance. Now the effort applied by the power consists 
 of the pressure exerted against the small piston, 
 by the steam coming from the boiler, and of the 
 pressure exerted against the large piston, by the 
 steam coming from the small cylinder. The resist- 
 ance, on the contrary, consists of the pressure pro- 
 duced against the small piston, by the reaction of 
 the steam which serves as a motive force to the 
 larger ; of the pressure subsisting behind the large 
 piston, after imperfect condensation of the steam 
 which effected the preceding stroke ; and, finally, 
 of the load and of the friction of the engine. 
 
 We will then estimate successively the quantity 
 of action developed by each of these five forces, 
 during one oscillation of the beam, and will after- 
 
204 CHAPTER VIII. 
 
 wards form the equation of the dynamical equili- 
 brium of the engine. 
 
 1st. Referring to the calculation which has al- 
 ready been made in Chapter III., Article I., Sect. 
 II., of the action of the expansion of the steam, it 
 will be found,,that the quantity of labour developed 
 by the direct entrance of the steam into the small 
 cylinder, and by its expansion there, has for its 
 expression 
 
 a (r + c ) + P' +log - al 
 
 '\q / Lf + c /' + cj q 
 
 2d. To obtain the quantity of motive labour de- 
 veloped also in the large cylinder during one oscil- 
 lation of the engine, it must be observed that it is 
 the same steam which, after having first occupied 
 the length I' of the small cylinder, with the pressure 
 P r , is now diffused partly under the small piston 
 and partly above the large one, with a pressure 
 corresponding to the space it fills. 
 
 If, then, we consider the point of the oscillation 
 of the engine, at which the small piston has per- 
 formed a portion \ of its stroke, and at which the 
 pressure of the expanded steam, above the large 
 piston or below the small one, is become TT ; it is 
 plain that we shall have, between the two pres- 
 sures TT and P' and the spaces respectively occupied 
 by the steam, the general relation (c), which we 
 have originally written under the form 
 
 n 
 
 and which expresses that, during its change of 
 
WOOLF'S ENGINES. 20,5 
 
 volume, the steam remains at the maximum density 
 for its temperature. 
 
 But, since the strokes L and I of the two pistons 
 are performed in the same time, it follows that 
 when the smaller one shall have performed the 
 length X of its stroke, the larger one will have per- 
 formed of its stroke 
 
 Consequently, the space occupied by the expanded 
 steam, as well ahove the large piston as below the 
 small one, will be 
 
 A 
 
 We will for a moment write it under the form 
 
 Bx + Q, 
 
 B representing the coefficient of X, and Q the con- 
 stant quantity. 
 
 This expression gives, then, the value of the 
 present volume of the steam, under the pressure ?r. 
 Now the same steam under the pressure P', occu- 
 pied the volume a (I 1 + c). We have then, from the 
 general analogy indicated a little above, 
 
 w = <rtO/^ + pV-- 
 
 " Bx + Q V? J q 
 
 Consequently, proceeding as before, that is to say, 
 multiplying both members of the equation by 
 
 - d\, and then taking the integral between the 
 
 i 
 
 limits -X=Oand - X^L, or X=O and X=/, we 
 
206 CHAPTER VIII. 
 
 shall have the value of the total labour pro- 
 duced by the expansion of the steam on the large 
 piston, from the beginning of the stroke to the end, 
 viz : 
 
 3d. To obtain the expression of the quantity of 
 action developed by the resistance of the steam 
 under the small piston, it is to be observed that 
 that steam being at the pressure TT, like the steam 
 in the large cylinder, it will suffice to multiply the 
 value of TT obtained above, by ad\ and to take 
 the integral between the limits X = O and \ l. 
 That operation will give us the quantity of action 
 developed by the resistance of the steam against 
 the motion of the small piston, from the beginning 
 of the stroke to the end. The result will obviously 
 be the same as that just obtained, except that 
 
 AL 
 
 - will be replaced by a It will then be 
 
 4th. Expressing still by p the pressure subsist- 
 ing in the cylinder which communicates with the 
 condenser, as this cylinder is here the great one, 
 the quantity of resisting action developed by the 
 force p, during one stroke, will be 
 
 5th. If, in fine, we represent by R the resistance 
 set in motion by the engine, measured not per unit 
 of surface, but in absolute magnitude, and by h 
 
WOOLF'S ENGINES. 207 
 
 the space which that resistance traverses at each 
 stroke of the piston, the quantity of action pro- 
 duced by that force during one stroke will evi- 
 dently be 
 
 R^. 
 
 Again, supposing the friction of the engine to 
 be represented by two forces : one f exerted upon 
 each unit of the surface of the small piston, and the 
 other F exerted upon each unit of the surface of 
 the large piston, the labour produced by these two 
 forces together, during one stroke, will be 
 
 Besides, if we call B the surplus friction of the en- 
 gine, accruing from each unit of the resistance r, 
 and measured at the same velocity as that resistance, 
 the quantity of action produced by the latter force 
 during one stroke, will be 
 
 BRh. 
 
 Thus, the quantity of action produced by these 
 three resistances together, during one oscillation 
 of the engine, will be 
 
 We have, then, all the elements of the labour 
 developed by the power and by the resistance. 
 Consequently, forming the equation of their dyna- 
 mical equilibrium, and transposing into the first 
 member, the term which expresses the quantity of 
 action developed by the third of the forces esti- 
 mated, that is to say, by the reaction of the steam 
 against the small piston, we shall have : 
 
208 CHAPTER VIII. 
 
 Finally, replacing the quantities B and Q by 
 their values, we obtain the relation 
 
 .............. (A) 
 
 This equation, then, is the first general relation 
 that we shall establish between the data and the 
 incognita of the problem. The second will, as in 
 the preceding calculations, be deduced from the 
 consideration of the equality between the produc- 
 tion and the expenditure of the steam. 
 
 S being the volume of water evaporated per unit 
 of time in the boiler, 
 
 S 
 n + q' 
 
 will be the volume of the resulting steam, measured 
 at the pressure P', that is, at the pressure of the 
 small cylinder, before the expansion. Again, v 
 being the velocity of the small piston, 
 
 will be the expenditure of steam per unit of time, 
 measured at the moment of its passage into the 
 small cylinder, before expansion, that is, at the 
 same pressure P'. Thus the second relation will 
 be as before, 
 
WOOLF'S ENGINES. 209 
 
 Consequently, eliminating P' from these two 
 equations and making, in order to simplify the 
 formulae, 
 
 we shall obtain for the required value of v 9 
 
 ST / 
 k 
 
 This velocity will be that of the small piston ; 
 and from this velocity we conclude that of the 
 
 large piston which will be r v, and that of the 
 
 I 
 
 point of application of the resistance R, which will 
 
 evidently be - v, or 
 I/ 
 
 since the resistance R traverses the distance A, and 
 the large piston the distance L, in the same time 
 that the small piston performs its stroke /. 
 
 It is to be observed that the value of V, whence 
 all the other formulae are afterwards deduced, is 
 precisely similar to that which we have obtained in 
 general, in Chapter III., with the exception only 
 
 h * 
 
 that the quantity =- k' replaces k, and that the 
 
 JU 
 
 factor 
 
210 CHAPTER VIII. 
 
 _L 
 
 replaces the similar factor 
 
 It will be easy then to derive the formulae of the 
 divers effects of the engine, from the general for- 
 mulae given in Chapter III. We shall, moreover, 
 remark that if, in the two expressions just ob- 
 tained for the values of k' and V, we were to make 
 A a, L =r / m A, and C = c, these values would be 
 reduced precisely to those of Chapter III. ; and 
 this in effect must be so, since the supposition of 
 A a and L = I amounts to reducing the two cy- 
 linders to one. 
 
 Performing, then, the calculations as before, we- 
 obtain the following formulae : 
 
 Case of any load or velocity, with a given 
 expansion. 
 
 jfe'S n AL /al+FAL + y AL 
 
 S = . w + 
 
 u. E. = RV. 
 
WOOLFS ENGINES. 
 
 Case of maximum useful effect, with a given ex- 
 pansion. 
 
 / __ l'+c al j, fn * n AL /g/4-FAL + pAL 
 
 ~~* + p ~~"' 
 
 max. u. E. = R'V 7 . 
 
 Case of absolute maximum useful effect. 
 
 This latter formula makes known, as we have 
 explained elsewhere, the expansion which produces 
 the absolute maximum of useful effect ; and by in- 
 troducing the value thus found into the formulae 
 which give the maximum effect for a given expan- 
 sion, we have the maximum effect which that ex- 
 pansion can produce, and consequently the absolute 
 maximum effect of the engine. 
 
 ARTICLE II. 
 
 PRACTICAL FORMULAE FOR CALCULATING WOOLF'S ENGINES. 
 
 The formulae which we have obtained above, 
 offer no difficulty in their application. To resolve 
 them numerically, recourse may be had to the table 
 
212 CHAPTER VIII. 
 
 given in Article I. Chapter III., which furnishes, 
 without calculation, the value of the terms 
 
 / I' l + c 
 
 - and - - + log - -- 
 
 i f +c r+c ^y + o 
 
 Then, having calculated the expression 
 
 its hyperbolic logarithm must be taken directly in 
 tables of that system ; or not having such tables, its 
 ordinary logarithm will be taken in a table of 
 common logarithms, and then multiplied by the 
 number 
 
 2-302585. 
 
 The product will be the hyperbolic logarithm re- 
 quired. In case the table of Chapter III. should 
 not be at hand, the hyperbolic logarithm of the 
 quantity 
 
 + c 
 
 would be found in the same way. The value, 
 therefore, of the expression which we have repre- 
 sented by k' will be easily obtained, and thus the 
 solution of the formula will offer no difficulty, for 
 the quantities contained in them are connected in 
 the first degree only. 
 
 To transform the equations which we have just 
 obtained into accurate numerical formula, it would 
 be necessary to know the precise value of the 
 constant quantities, for which purpose special ex- 
 periments would be requisite. However, till such 
 
WOOLF'S ENGINES. 
 
 experiments are made, we will calculate these for- 
 mulae numerically, contenting ourselves with such 
 approximate value of the constant quantities, as 
 may be deduced from the analogy of those engines 
 with those already treated of. 
 
 In Watt's engines, which have but one cylinder, 
 the friction of the engine, without load, amounts to 
 l%5lb. per square inch of the surface of the piston, 
 for a cylinder of small dimensions, and to '5 Ib. per 
 square inch of the piston, for a cylinder of large 
 dimensions, as has been explained above. In 
 Woolf s engines there are two cylinders instead 
 of one, and it will be observed that each of them 
 requires nearly the same machinery and produces 
 nearly the same friction as if it were alone in the 
 engine. The safest way then of estimating the 
 friction of these engines, is to attribute to each of 
 the pistons the friction suitable to its dimensions in 
 the Watt engine. 
 
 We will take, moreover, at the same rate as in 
 the other engines, the additional friction per unit 
 of the load, that is, we will make 
 
 8 = -14. 
 
 The pressure of condensation in the cylinder 
 communicating with the condenser, will also be, 
 in good engines, 
 
 p=4f x 144 Ibs. 
 
 The clearance of the cylinder will be -^ of the 
 stroke, which gives 
 
 c = -05/, and C = '05 L 
 
214 CHAPTER vm. 
 
 And in fine, since the engine is a condensing one, 
 we must take 
 
 n = -00004227, 
 ? = -000000258. 
 
 Admitting then these approximate valuations, 
 and bearing in mind that the value of k r is 
 
 a j 
 
 ^=___+ log,, _ + log 
 
 we obtain the following formulae. 
 
 PRACTICAL FORMULA FOR WOOLF's ENGINES. 
 
 Case of any load or velocity, with a given 
 expansion. 
 
 V-*. 8 __ _ 10000*' __ 
 
 " L A -4227+-00258 [(!+*) Rh+fal+F A L+pA L] 
 A L 
 
 Velocity of the 
 load, in feet 
 per minute. 
 
 Useful load of 
 the engine, 
 in Ibs. 
 
 
 Effective eva- 
 poration, in 
 cubic feet of 
 water per mi- 
 nute. 
 
WOOLF'S ENGINES. 
 
 u . E =1 R V Useful effect, 
 
 in Ibs. raised 
 one foot per 
 minute. 
 
 u HP . .. - "' E> Useful horse- 
 
 " 33000 
 
 power. 
 
 u. E. 1 Ib. co. ... = Hl5l Useful effect < 
 
 of 1 Ib. of 
 coal, in Ibs. 
 raised 1 foot. 
 
 u. E. 1 ft. wa. ... = ^A* Useful effect 
 
 of 1 cubic 
 foot of wa- 
 ter, in Ibs. 
 raised 1 foot. 
 
 Q. co. for 1 hp.... = 83000N Quantity of 
 
 u. E. . . ,, 
 
 coal, in Ibs., 
 
 which pro- 
 duces one 
 horse-power. 
 
 Q. wa. for 1 hp.... = 3800 S Quantityofwa- 
 
 u. E. ... 
 
 ter, in cubic 
 
 feet, which 
 produces one 
 horse-power. 
 
 u. HP. for 1 Ib. co.= 3 " 0( *|' N Horse -power 
 
 produced by 
 1 Ib. of coal. 
 
 u. HP. for 1 ft. wa.= JlJL Horse - power 
 
 OO\)(J\J O 
 
 produced by 
 1 cubic foot 
 of water eva- 
 porated. 
 
CHAPTER VIII. 
 
 Case of maximum useful effect, with a given 
 
 expansion, 
 k I S 10000 
 
 V = 7* lc' a' -4227 + -00258P Velocity of the 
 
 load, in feet 
 
 per minute. 
 
 Useful load of 
 
 the engine, 
 
 in Ibs. 
 c / I' + c aV 
 
 = k' ' 10000 <' 4 ** 7 +'* 68p > Effective eva " 
 
 poration, in 
 cubic feet of 
 water per 
 minute. 
 
 max. u. E.=R'V Useful effect, 
 
 in Ibs. raised 
 1 foot per 
 minute. 
 
 Case of absolute maximum of useful effect. 
 
 I' AL 164 +^(/^+ FAL +P AL ) Expansion 
 
 7=TT' : 16ITF- which P r - 
 
 duces the ab- 
 solute maxi- 
 mum of use- 
 ful effect. 
 
 The expansion of the absolute maximum of useful 
 effect being determined by this formula, then in- 
 troducing it into the formulae of the case of maxi- 
 mum effect for a given expansion, we have the 
 maximum of useful effect producible by that ex- 
 pansion, and consequently the absolute maximum 
 useful effect of the engine. 
 
CHAPTER IX. 
 
 EVANS'S ENGINES. 
 
 PRACTICAL FORMULAE FOR CALCULATING THESE ENGINES. 
 
 IN Evans's engines, the steam generated at the 
 pressure of about 8 atmospheres, or 120 Ibs. per 
 square inch, is admitted into the cylinder during 
 about a third part of the stroke of the piston ; 
 the communication with the boiler is then inter- 
 cepted, and the piston continues its motion by 
 means of the expansion of the steam ; after which 
 the steam generally escapes into the atmosphere 
 without any use being made of condensation. 
 
 The steam, therefore, in these engines acts by 
 expansion, as in the Cornish engines ; and thus 
 the formula? proper to calculate them, are the 
 same as those which have been given in Chapter 
 VII. The only difference is that the quantity P 
 will represent a higher pressure of formation of the 
 steam, and that the quantity p, instead of express- 
 ing the pressure due to the imperfect condensation 
 of the steam, will express the atmospheric pres- 
 sure. 
 
218 CHAPTER IX. 
 
 We shall not give these formulae under their 
 algebraic form, since it would be a mere repetition 
 of those of Chapter III. ; but we will present them 
 under their numerical form, taking the value of 
 the constant quantities by approximation, from 
 analogy with the Watt engines and the locomo- 
 tives, of which we have given the determinations 
 above. 
 
 To this end we will notice that the high pressure 
 at which the steam is employed in this system, 
 admits of a cylinder of very small capacity. The 
 result is, that in a middle-sized engine of Evans, 
 the diameter of the cylinder is scarcely half what 
 it is in the smallest of Watt's engines that it has 
 occurred to us to mention, and in which the friction 
 is about 1*5 Ib. per square inch. Supposing then 
 the total friction the same in both, it is visible that 
 considering it divided per square inch of the sur- 
 face of the piston, it must, in Evans's engine, exert 
 four times the resistance that it would in Watt's 
 engine; and consequently in Evans's engines of 
 mean force, the friction of the engine unloaded 
 should be estimated at 1-5x4 = 6 Ibs. per square 
 inch, which gives 
 
 /= 6x144 Ibs. 
 
 Moreover, as before, we may admit the following 
 determinations : 
 
 Additional friction of the engine, per unit of 
 the resistance imposed on it, |th of that resist- 
 ance ; or 8 = '14. 
 
EVANS'S ENGINES. 219 
 
 Atmospheric pressure, per square foot, p = 
 14-71 x 144 Ibs. 
 
 Clearance of the cylinder, c = *05 1. 
 Finally, the coefficients of the relative volume 
 of the steam will be, as in non-condensing en- 
 gines 
 
 n = -0001421, 
 ? = -00000023. 
 
 Admitting then these data, and bearing in mind 
 that the value of k is 
 
 we obtain the following equations : 
 
 PRACTICAL FORMULA FOR EVANS'S ENGINES. 
 
 Case of any load or velocity, with a given 
 expansion. 
 
 10000 * _ ...... Velocity ofthe 
 
 feet per mi- 
 nute. 
 ar = 4347830,-M- --(876+/) ...... Useful load 
 
 of the piston, 
 in Ibs. 
 
 S = av [6-29 + '0023 (!+*> + *0023/] Effective eva- 
 10000 k poration, in 
 
 cubic feet of 
 water per mi- 
 nute. 
 
220 
 u.E.n: arv 
 
 u. HP. = 
 
 CHAPTER IX, 
 
 u. E. 
 
 33000 
 
 u.E.llb. co.zz 
 
 u. E.lft.wa. = 
 
 u.E. 
 
 N 
 
 u, E. 
 
 S. 
 
 Q.co.forlhp.rz 
 
 38000 N 
 u. E. 
 
 Q.wa.forlhp.=: 
 
 83000 S 
 u. E. 
 
 u. HP. for lib. co. = 
 
 u.E. 
 
 33000 N 
 
 u. HP, for 1 ft. wa. = 
 
 u.E. 
 33000 S~ 
 
 Useful effect, 
 in Ibs. raised 
 1 foot per mi- 
 nute. 
 
 Useful horse- 
 power. 
 
 Useful effect 
 ofl pound of 
 coal, in Ibs. 
 raised 1 foot. 
 
 Useful effect 
 of 1 cubic 
 foot of water, 
 in Ibs. raised 
 1 foot. 
 
 Quantity of 
 coal, in Ibs., 
 producing 1 
 horse-power. 
 
 Quantity of 
 water, in cu- 
 bic feet, pro- 
 ducing 1 
 horse-power. 
 
 Horse - power 
 produced by 
 1 pound of 
 coal. 
 
 Horse power 
 produced by 
 1 cubic foot 
 of water eva- 
 porated. 
 
EVANS'S ENGINES. 
 
 Case of maximum useful effect with a given 
 expansion. 
 
 '=-!.. _ 1000 Velocity of 
 
 m 1-1+-00 M P the piston, in 
 
 feet per mi- 
 nute. 
 
 ar'= -f-. F -+k (618 + P)- (2736 +/) Useful load 
 l +* l of the piston, 
 
 in Ibs. 
 
 av ' 
 
 -0023 P) ........ Effective eva- 
 
 1 1000 poration, in 
 
 cubic feet of 
 water per mi- 
 nute. 
 
 max. u. E. = a r' t>' ................................. Useful effect, 
 
 in Ibs. raised 
 1 foot per mi- 
 nute. 
 
 Case of absolute maximum useful effect. 
 
 Expansion, 
 
 1 618+P which pro- 
 
 duces the ab- 
 solute maxi- 
 mum of use- 
 ful effect. 
 
 This latter relation having made known the ex- 
 pansion which produces the ahsolute maximum of 
 useful effect, if that expansion he introduced into 
 
222 CHAPTER IX. 
 
 the formulae of maximum effect for a given expan- 
 sion, all the determinations relative to the abso- 
 lute maximum useful effect of the engine will be 
 derived. 
 
 As, in this system, the steam is usually em- 
 ployed at the pressure of 120 Ibs. per square inch, 
 we see from the last formula, and adopting also 
 the valuation just made of the friction, that the 
 expansion of absolute maximum useful effect will 
 generally be given by the relation 
 
 *'_3600 
 
 7~17898~ 
 
 that is to say, the greatest possible effects are to be 
 obtained by intercepting the action of the steam at 
 about a fifth part of the stroke. 
 
CHAPTER X. 
 
 WATT'S SINGLE-ACTING ENGINES. 
 
 ARTICLE I. 
 
 THEORY OF WATT'S SINGLE-ACTING ENGINES. 
 
 SECT. I. Of the regulation of the engine. 
 
 IN Watt's single-acting engines, the steam is ap- 
 plied only on the upper side of the piston, and the 
 action of the engine, to raise the water in the 
 pumps, or to produce the useful effect, is exercised 
 only during the descending stroke of the piston ; 
 and thence these engines are termed single-acting. 
 The communication being first opened between 
 the boiler and the upper part of the cylinder on 
 the one hand, and between the lower part of the 
 cylinder and the condenser on the other, the steam 
 passes into the cylinder, and presses with all its 
 force above the piston, whereas below it, the steam 
 which has effected the preceding stroke is con- 
 densed. The piston then begins its descending 
 stroke. When it has performed a certain portion 
 of the stroke, the communication from the boiler 
 to the cylinder is intercepted. The piston, how- 
 ever, still continues its motion, by the effect of the 
 expansion of the steam already admitted in the 
 
CHAPTER X. 
 
 cylinder. But on its attaining nearly the end of 
 the stroke, a valve, called the equilibrium valve, 
 opens and establishes a free communication between 
 the upper and lower portions of the cylinder. 
 Then the steam, diffusing itself on both sides of 
 the piston, puts the latter in a state of equilibrium, 
 that is, presses equally on both its faces. Conse- 
 quently there ensues a total cessation in the action 
 of the motive power, and as the resistance opposed 
 by the load remains the same, the piston, whose 
 motion has already slackened during the decreas- 
 ing action of the expansion, is quickly brought to 
 a state of rest. 
 
 The piston then has reached the bottom of the 
 cylinder, and as the equilibrium valve still con- 
 tinues open, the piston remains also in equilibrio in 
 the steam. On the other hand, during the stroke 
 just performed, a counterweight, suspended at the 
 opposite end of the beam, has been raised at the 
 same time with the load. It now, therefore, tends 
 to descend again by its own weight, and conse- 
 quently to raise the piston in the cylinder. The 
 piston, too, no longer bears any load to oppose this 
 motion, since the action of the pump to raise water, 
 or of the load of the engine, is exercised only when 
 the piston tends to descend, whereas at this moment 
 it tends to move upwards. Nothing remains, then, 
 but the friction of the engine, to oppose a resist- 
 ance to the effort of the counterweight ; but as the 
 weight given to this is sufficient to overcome the 
 friction of the engine, it follows that the piston 
 
WATT'S SINGLE-ACTING ENGINES. 
 
 is raised and brought back to the top of the 
 cylinder. 
 
 Thus all has returned to the same state as at 
 first, and the action of the steam recommences, to 
 make the piston perform a new descending stroke. 
 
 Comparing these engines with those which we 
 have hitherto treated of, they are found to differ 
 in three points : 1st. The counterweight acts al- 
 ternately as a resistance and as a motive force. 
 2d. During its action as a motive force, it effects 
 the motion by means of its fall. 3d. There is nei- 
 ther continuity nor uniformity in the motion of the 
 resistance. It will be proper, then, to examine 
 whether these circumstances ought to change the 
 bases whereon we have established the theory of 
 the other engines. 
 
 1st. With respect to the alternate action of the 
 counterweight as a resistance and as a motive force, 
 no change in the mode of reasoning need result 
 from that ; for the counterweight acts precisely in 
 the manner of an ordinary fly-wheel, such as are 
 met with in the engines of which we have hitherto 
 treated. In fact, the motion of the single-acting 
 engine is composed of two distinct parts : 1st, the 
 descending stroke, during which the water is raised 
 in the pumps, that is to say, during which the 
 useful effect is produced ; and 2dly, the ascending 
 stroke, effected by the counterweight, and unat- 
 tended with any useful effect, that stroke serving 
 merely to replace matters as they were originally, 
 and to enable the steam to renew its action. Now, 
 
CHAPTER X. 
 
 during the first period of this motion, a certain 
 quantity of work is communicated to the counter- 
 weight, by raising it to a certain height ; and 
 during the second period, the counterweight does 
 no more than restore this quantity of work to the 
 engine, hy its descent from the same height to 
 which it was raised. Its office then is limited to 
 receiving a certain quantity of action, in order to 
 restore it at a proper time, during the suspension 
 of the effort of the motive force ; and consequently 
 its action is of the same nature as that of an or- 
 dinary fly-wheel. 
 
 2dly. During the ascending stroke of the piston, 
 the weight of the counterweight becomes the mo- 
 tive power, and the velocity produced in that stroke 
 depends incontestably on the superiority of the 
 counter-weight over the resistance then opposed by 
 the engine. If then, we were to make a dis- 
 tinction between the velocity of the descending 
 and of the ascending stroke of the piston, we 
 ought to calculate the latter from the circum- 
 stances of the fall of the counterweight. But 
 it will be remarked that a complete oscillation 
 of the engine consists of an ascending and a de- 
 scending stroke of the piston. It matters little 
 that one of these be rapid and the other slow ; a 
 new oscillation will never take place till the time 
 requisite for both shall have expired. If the 
 respective velocities of the two strokes are equal, 
 the motion of the piston will be regular throughout 
 the oscillation ; if they are unequal, that motion 
 
WATT'S SINGLE-ACTING ENGINES. 
 
 on the contrary will be accelerated at one point and 
 retarded at another. But in either case, the true 
 velocity of the engine, that which we want to know 
 in order to calculate the useful effects, will always 
 be the mean velocity of the two strokes. 
 
 In fact, as soon as we know the number of com- 
 plete oscillations made by the engine per minute, 
 we necessarily know its useful effect ; since at every 
 complete oscillation, we know that the load advances 
 the length of one stroke. 
 
 Thus we see, first, that the velocity wanted, in 
 order to calculate the useful effect, is neither the 
 velocity of the rising stroke of the piston, nor that 
 of the descending, but the mean velocity of the 
 two strokes taken together. Now, this is neces- 
 sarily regulated, d priori, by the production of 
 steam in the boiler, and may be calculated directly 
 from that datum. For, if the boiler produce per 
 minute, and transmit to the cylinder, a certain 
 quantity of steam, whose volume be known, it is 
 plain that, since that volume of steam issues by 
 the cylinder in one minute, dividing it by the con- 
 tent of the cylinder, we have the number of com- 
 plete oscillations it will cause in the engine. This, 
 therefore, will be the mean velocity of the piston, 
 however rapid the motion of the piston may have 
 been at one point of its oscillation, and however 
 slow, on the contrary, its motion may have been at 
 another point of the same oscillation. 
 
 As to the greater or less velocity of the piston 
 during its ascending stroke, it_will in no wise alter 
 
 Q 2 
 
CHAPTER X. 
 
 the mean velocity of the engine, as long as the pro- 
 duction of steam be supposed^ the same in the 
 boiler. If, in fact, the rising stroke of the piston 
 be performed in a longer time, since^the issue of 
 the steam by the cylinder takes place only during 
 the descending strokes, it will happen that the suc- 
 cessive interruptions thus made in the efflux of the 
 steam, will be so much the longer. But, since the 
 production of steam per minute in the boiler is a 
 fixed and determinate quantity, and that its issue 
 by the cylinder will now be found suspended 
 during a longer time, the steam during that interval 
 will accumulate in the boiler. At the beginning 
 then of the descending stroke, there will be a greater 
 quantity ready to be applied to the effecting of the 
 motion ; and it will consequently supply a velocity 
 in that stroke by so much the greater. The velo- 
 city, therefore, of the descending stroke will aug- 
 ment as that of the rising stroke diminishes. If, 
 on the contrary, the rising stroke is performed 
 more rapidly, the steam will accumulate in a smaller 
 quantity during the interval of the rising strokes, 
 and will in consequence afford a smaller supply of 
 steam, that is to say, a less velocity of the piston, 
 during the descending stroke. Thus, in all cases, 
 the velocity of the descending stroke will vary in 
 the inverse manner to that of the rising stroke, and 
 the mean or definitive velocity will always be regu- 
 lated by the number of cylinders-full of steam sup- 
 plied by the boiler per minute. One case only can 
 occur wherein this rule would fail, namely, that 
 
WATT'S SINGLE-ACTING ENGINES. 
 
 in which the motion of the ascending stroke were 
 so slow'that the time left for the issue of the steam 
 during the descending stroke, should be insufficient 
 for the total efflux of the steam generated per 
 minute in the boiler, and that a part of the steam 
 should escape by the safety-valves. Then the mean 
 velocity could no longer be calculated, as above, on 
 the total efflux of the steam by the cylinder, since 
 that total efflux would not take place. But as, in 
 single-acting engines, care is always taken to set a 
 counterweight capable of executing the ineffective 
 stroke, with a velocity nearly equal to that which 
 it is purposed to give to the effective stroke, this 
 excessive slowness of one of the two strokes never 
 occurs in practice, nor is it necessary to consider 
 such supposition. 
 
 Hence we see that the use of the counterweight 
 is no hindrance to the velocity of the engine being 
 always calculated from the production of steam in 
 the boiler. 
 
 3dly. These engines have no fly-wheel, nor is the 
 motion of the resistance either continued or uni- 
 form. But the preservation of the engine requires 
 that, at the end of each alternate motion, the piston 
 be brought to rest without shock and by insensible 
 degrees. No loss therefore of vis viva is occasion- 
 ed ; and consequently, in these engines, as well as 
 in the double-acting ones, there is always equality 
 between the quantity of work applied by the power 
 and that which is performed by the resistance. It 
 is necessary, however, before we proceed, to explain 
 
230 CHAPTER X. 
 
 how such a regulation of the engine may be at- 
 tained in practice, both in the descending and in the 
 ascending stroke of the piston. 
 
 For this purpose, care is taken, during the de- 
 scending stroke, to admit no more steam into the 
 cylinder, than what is just requisite to drive the 
 piston by the expansion of that steam, to the point 
 where the stroke should end ; so that it may there 
 stop of itself, without either falling short or going 
 too far. This effect is easily obtained in a few 
 trials, on setting the engine to work. A small 
 quantity of steam is first admitted with caution into 
 the cylinder, and this is augmented by degrees, till 
 the piston is found to attain precisely the point pro- 
 posed as the end of the stroke. Should the piston 
 overpass that point, and strike against the springs 
 which protect the bottom of the cylinder, it is a 
 proof that the quantity of steam admitted is too 
 great, and it must be reduced in consequence till a 
 proper adjustment be obtained. 
 
 During the ascending stroke, on the contrary, as 
 the piston rises in the cylinder only because the 
 equilibrium valve allows the steam then in the up- 
 per part of the cylinder, to pass into the lower 
 part and make room for the piston, this equili- 
 brium valve is closed somewhat before the piston 
 has reached the point where it is intended to stop. 
 The piston then continues its motion for some time 
 by virtue of its acquired velocity and by the effect 
 of the counterweight ; but the steam situated in 
 the lower part of the cylinder dilates as the pro- 
 
WATT'S SINGLE-ACTING ENGINES. 
 
 gress of the piston opens a greater space for it, and 
 the steam which is intercepted in the upper part, 
 is on the contrary gradually compressed, and con- 
 sequently acquires more and more elastic force. 
 Thus the difference between the two pressures in- 
 creasing continually, and creating a greater and 
 greater resistance against the motion of the piston, 
 at last brings it gently to rest. In practice, this 
 adjustment is also obtained by trials, in closing the 
 valve a little sooner or a little later, according as 
 the piston is seen to come too near or not near 
 enough the top of the cylinder. Moreover, in this 
 way of adjusting the equilibrium valve, it will be 
 observed that the steam compressed in the upper 
 part of the cylinder retains all the work developed 
 by the piston, while coming to rest, and that in 
 this state of compression, it immediately contri- 
 butes to produce the new descending stroke of the 
 piston ; whence results that no action is lost, and 
 that the expenditure of steam consists only in that 
 portion which has been intercepted below the pis- 
 ton. 
 
 This previous regulation of the engine must 
 necessarily change every time that the load varies, 
 but when it is once effected, that is to say, as soon 
 as the engine has attained its regular working 
 state, the piston, whether rising or falling, is always 
 brought to rest by insensible degrees, and without 
 loss of vis viva. Thus, in the single-acting en- 
 gines, as well as in those of the divers systems of 
 which we have already treated, there is always 
 
CHAPTER X. 
 
 equality between the quantity of work developed 
 by the power and that which is performed by the 
 resistance ; and consequently we may apply to these 
 engines, the same mode of reasoning that we have 
 hitherto made use of. It is, however, to be ob- 
 served, that a distinction must be made between 
 the descending and the rising stroke of the piston, 
 in as much as the equality of the work produced 
 by the power and the resistance, takes place in 
 each separately. This double condition will fur- 
 nish two equations proper to calculate the expan- 
 sion of the steam in the descending stroke, and the 
 adjustment of the equilibrium valve in the ascend- 
 ing one. Afterwards, to determine the velocity of the 
 engine, there will remain the condition that the 
 expenditure of the steam be equal to its produc- 
 tion. We shall first establish these three funda- 
 mental equations, and then seek to deduce from 
 them the formula suitable to the solution of the 
 divers questions that may occur in these engines. 
 
 SECT. II. Of the effects of the engine in the case 
 of a given counterweight, with an indefinite 
 load or velocity. 
 
 We distinguish three cases in the working of 
 single-acting engines : that in which they work 
 with a given counterweight, and with the load or 
 the velocity indefinite; that in which they work 
 with a given counterweight, and the load or the 
 velocity which produces the maximum of useful 
 effect with that counterweight ; and finally, that 
 
WATT'S SINGLE-ACTING ENGINES. 9233 
 
 in which the counterweight having been first re- 
 gulated according to its most advantageous weight 
 for the engine, the load imposed is also the most 
 advantageous for that counterweight, which conse- 
 quently produces the absolute maximum of useful 
 effect that it is possible to obtain from the engine. 
 
 In the rotative or double-acting engines, which 
 we have treated of farther back, it was seen that 
 at the moment of the starting of the engine, the 
 steam penetrated into the cylinder with a pressure 
 equal to that of the boiler ; but that, by the effect 
 of the continuity of motion and the intervention of 
 a fly-wheel, the impulse given to the piston during 
 one stroke, was continued in the stroke following. 
 The velocity of the piston then accelerated by de- 
 grees from the accession of new quantities of motion, 
 and the pressure of the steam in the cylinder lowered 
 at the same time, till at last the piston attained the 
 greatest velocity capable of being impressed on it 
 by the motive power, with the load imposed on 
 the engine. Then the motion ceased to accelerate. 
 It remained in the state of uniformity, or rather in 
 that state which produced uniformity in the motion 
 of the resistance; and that condition regulated the 
 pressure which the steam definitively acquired in 
 the cylinder, during all the duration of uniform 
 motion. The pressure then, in the cylinder, might 
 be found, in certain cases, much inferior to that of 
 the steam in the boiler. 
 
 But this circumstance does not occur in the en- 
 gines we are now considering, because the motion 
 
CHAPTER X. 
 
 in them not being continued or brought to conti- 
 nuity by the use of a regular fly-wheel, the impulse 
 given to the piston in one stroke, cannot transmit 
 itself to the following stroke. The piston then is 
 placed perpetually under such conditions as in ro- 
 tative engines, it is placed only at the moment of 
 departure ; and consequently, the cause no longer 
 existing of the diminution of pressure of the steam 
 in the cylinder, the only difference of pressure occa- 
 sioned between the two vessels is that resulting from 
 the force requisite to traverse the narrow passages 
 which separate the cylinder from the boiler. But 
 as, with the dimensions usually adopted in steam- 
 engines, the diminution of pressure attributable to 
 this cause alone is found to be unimportant, we 
 shall omit it here, as we have done in calcu- 
 lating the maximum load of rotative engines. On 
 this head, however, it will readily be seen that, 
 if it be deemed necessary to take account of this 
 difference, it will suffice to attribute to the force 
 which will appear in our calculation, a value not 
 precisely equal to the pressure in the boiler, but 
 somewhat inferior, according to its intensity given 
 by the formulae on the flowing of gas. 
 
 This being premised, and first considering the 
 descending stroke of the piston, the quantity of 
 work applied by the steam, as well during its di- 
 rect action as during its expansion, must be found 
 in toto in the work performed by the resistance. 
 Now, adopting all the notations hitherto employed, 
 that is, indicating by P the pressure of the steam in 
 
WATT'S SINGLE-ACTING ENGINES. 235 
 
 the boiler, by a the area of the cylinder, by / the 
 stroke of the piston, and by V the portion of that 
 stroke performed before the expansion begins, and 
 referring besides to the calculation which has been 
 developed in Sect. II., Article L, Chapter III., we 
 shall have for the work performed by the direct 
 action and by the expansion of the steam in the 
 cylinder, the following expression : 
 
 ) 
 / 
 
 log 
 
 On the other hand, the resistance then consists of 
 the load r measured per unit of the surface of the 
 piston ; the friction (f + S r) of the engine loaded 
 with the resistance r, terming f the friction of the 
 engine unloaded, and 8 the increase of that friction 
 per unit of the load ; the counterweight which we 
 will express by n, supposing it divided per unit of 
 the surface of the piston ; and finally, the pressure 
 p subsisting under the piston, by reason of the 
 imperfect condensation of the steam. The work 
 performed during the stroke by these different 
 resistances united, will therefore be 
 
 Consequently, we have the analogy 
 
 and making, for simplification, 
 
 t l+c 
 
236 CHAPTER X. 
 
 the expression obtained above will become 
 
 ? 
 
 or 
 
 +p+f + n 
 
 This will then be the first of the relations sought 
 between the data and incognita of the problem. 
 
 To obtain a similar relation for the ascending 
 stroke, we must express that' the same equality 
 exists between the quantities of action developed 
 by the power and by the resistance. 
 
 Now the power is here the counterweight; and the 
 resistance consists of the opposition exerted by the 
 steam after the closing of the equilibrium valve, and 
 of the friction of the engine. By friction of the 
 engine, is here to be understood the sum of all the 
 resistances overcome by the engine in effecting its 
 motion ; that is to say, not merely the friction of 
 the various moving parts of the machinery, but 
 also, when the engine is used to raise water, the 
 resistance resulting from the pump piston pene- 
 trating into the water to be exhausted, a resistance 
 which does not occur in the ascending stroke. The 
 friction of the unloaded engine in this stroke, is 
 not then exactly the same as the friction of the 
 engine supposed unloaded in the descending stroke, 
 and consequently we will express it by f" instead 
 off. 
 
WATT'S SINGLE-ACTING ENGINES. 237 
 
 This premised, the quantity of work developed 
 by the counterweight during the stroke, is 
 
 n a I, 
 and that developed by the friction is 
 
 f'al. 
 
 As to the work developed by the resistance of 
 the steam, it requires a short calculation, analogous 
 to that which has been performed for the expan- 
 sion. 
 
 During the ascending motion of the piston, and 
 before the closing of the equilibrium valve, the 
 pressure of the steam above the piston is neces- 
 sarily rather higher than that of the steam below, 
 as we have just seen, with respect to the pressure 
 of the boiler compared with that of the cylinder. 
 As, however, the difference between those two 
 pressures may be left out without error, we will 
 here suppose that, till the moment of the closing 
 of the equilibrium valve, the piston is in equilibrio 
 in the steam, that is, pressed equally on both 
 its faces. 
 
 Now, during all the time that the equilibrium 
 valve remains open, the two parts of the cylinder 
 are filled with the steam which has undergone ex- 
 pansion in the preceding downward stroke. This 
 steam, at the moment when it entered the cylinder, 
 was at the pressure P, and filled the length (/' + c) 
 of the cylinder. It now finds itself distended 
 through all the capacity of the cylinder, including 
 the two vacant spaces not traversed by the piston. 
 Therefore, from the relation (c), between the vo- 
 
238 CHAPTER X. 
 
 lumes and the pressures of the same weight of 
 steam, during its action in the engine, demon- 
 strated in Chapter III., Article I., Sect. II., the 
 pressure of the steam, after being dilated in the 
 two parts of the cylinder, has become 
 
 + c q 
 
 This premised, let I" be the length already tra- 
 versed by the piston in its rising stroke, at the mo- 
 ment when the equilibrium valve closes. From 
 this point, the piston still continuing its motion, 
 by virtue of its acquired velocity and of the effort 
 of the counterweight, the steam confined above, 
 and which can no longer escape, acquires by com- 
 pression still more and more elastic force ; while 
 that which subsists below assumes, on the con- 
 trary, a pressure still less and less. 
 
 Suppose, then, the piston arrived at a distance 
 X, from the beginning of its stroke, and at that 
 point let TT' be the pressure of the steam beneath 
 the piston, and IT" that of the steam above it. Let 
 the piston then traverse a further elementary space 
 d \, the corresponding labour, produced by the re- 
 sistance of the steam, will be 
 
 But, as the pressures TT" and TT', which take place in 
 the two parts of the cylinder, result from the 
 spaces respectively occupied by the steam, whether 
 above or below the piston, and that at the origin 
 of this compression and dilatation of the steam, 
 both the portions of the cylinder were filled with 
 
WATT'S SINGLE-ACTING ENGINES. 239 
 
 steam at the pressure TT, we shall again have, be- 
 tween the pressures and the volumes respectively 
 occupied by the steam, the relation 
 
 // /^ , \ l l" + c n 
 
 7T = (- + TT\- - - , 
 
 \q / l \ + c q 
 and 
 
 , (n \l" + c n 
 
 7T IZ I + 7T j - 
 
 q ' X+c q 
 
 Whence the elementary labour above, will be 
 
 Or, substituting for TT its value given above, this 
 expression will become 
 
 Now, as this effect of compression on the one 
 hand, and dilatation on the other, takes place from 
 the length l f of the cylinder to the end of the stroke, 
 taking the integral of that expression between the 
 limits t f and /, we have the quantity of total labour 
 developed by the steam in its resisting action against 
 the piston, namely : 
 
 But if, in order to abridge, we make 
 
 , , l-l"+c 
 
 __.log- 
 
 the expression just obtained may be written under 
 the form 
 
240 CHAPTER X. 
 
 Consequently, referring to what has been said above, 
 of the labour performed during the same stroke by 
 the counterweight and the friction of the engine, 
 the equality between the action developed by the 
 power and that developed by the resistance, during 
 the rising stroke of the piston, will be found to 
 produce the equation 
 
 k"a 
 or 
 
 K'J^^. 1 ^', . . . (B) 
 
 5 
 
 which is the second relation sought between the 
 data and the incognita of the problem. 
 
 Finally, to obtain the third relation, expressing 
 the equality between the expenditure and the pro- 
 duction of the steam, it must be observed that at 
 each stroke of the piston, there is condensed and 
 consequently expended, only the steam which has 
 passed under the piston, during the rising stroke. 
 As to that which is intercepted above the piston, 
 it is driven back towards the boiler and serves to 
 produce the following stroke. 
 
 Now, the volume of the steam to be condensed at 
 each stroke of the piston, taken at the moment of 
 its separation, is 
 
 and its pressure is then 
 
 Cn 
 - + 
 
WATT'S SINGLE-ACTING ENGINES. 
 
 If M be the number of strokes of the piston per 
 minute, the volume of steam expended per minute 
 will then be 
 
 But V being the mean velocity of the piston, or 
 the space it traverses, rising and descending, in a 
 minute, we have V = 2 M I ; or if, conformably to 
 the usage very properly adopted with these engines, 
 we reckon only the space v traversed by the piston 
 in producing the useful effect, a space which is the 
 half of that traversed in the two strokes, we shall 
 have 
 
 s\* 
 
 v = M /, or M rz - . 
 
 v 
 
 Whence the above volume, expended by the cylinder, 
 may be expressed by 
 
 t'l+c 
 av ~T~ 
 
 On the other hand, if we represent by S the vo- 
 lume of water evaporated per minute in the boiler, 
 the volume of steam that will result from it, under 
 the pressure P, at which it is generated, will be, 
 according to what has been seen (equ. a) 
 
 S 
 
 n + qP 
 
 This will be the volume occupied by the steam 
 under the pressure P ; but in passing to the pres- 
 sure TT, at which we measure the steam expended, 
 that volume will become 
 
 S n + qP __ S_ 
 n + qP n + qtr n + qw 
 
 R 
 
CHAPTER X. 
 
 Therefore, since the expenditure of the cylinder is 
 equal to the production of the hoiler, we shall have 
 
 S 
 
 av 
 
 I n + qir 
 
 or, substituting for TT its value, and resolving the 
 equation with reference to v 9 
 
 S n 
 
 7T7* a' ' 
 
 which is the third relation sought. 
 
 Consequently adding to this, the two other rela- 
 tions already obtained, and resolving these equa- 
 tions successively with reference to the quantities 
 r and S, we obtain definitively the following 
 formulae : 
 
 - + (l+8)r+^+/ / + n 
 y = I - ....... (A) 
 
 n-f" 
 
 /'+ c n 
 
 v = 
 
 1 1 + c f + c a n +qP 
 ar = I 
 
 SI . . (1) 
 
WATT'S SINGLE-ACTING ENGINES. 
 
 In these equations, kf and k" represent the ex 
 pressions indicated a little above, viz. : 
 
 SECT. III. Of the velocity of the engine with a 
 given load. 
 
 The equations (A), (B), (1), (2), (3), and (4) 
 resolve all the problems relative to the calculation 
 of Watt's single-acting engines. As, however, the 
 nature of the expressions represented by k' and k" 
 in the equations (A) and (B), do not admit of 
 eliminating between these equations and equation 
 (1); and that we cannot, therefore, attain a direct 
 expression of the velocity in terms of the load, nor 
 of the load in terms of the velocity, as we were 
 enabled to do in the formulae of the rotative en- 
 gines, it will be proper to dwell a moment on the 
 mode of calculation to be adopted, in seeking the 
 solution of the various problems that may occur. 
 
 Suppose it were required to determine at what 
 velocity an engine, fully known in other respects, 
 would set in motion a given load r. 
 
 We shall first introduce the given value of r in 
 equation (A), 
 
 / If s -* f^N. 
 
44 CHAPTER X. 
 
 Thus will be obtained the value of V. Then in 
 the first of the two tables which we shall give pre- 
 
 sently, will be found immediately and without cal- 
 
 jf 
 culation the corresponding value of -. 
 
 V 
 
 I' 
 Introducing the value of - into the equation (B), 
 
 L 
 
 ^ 
 
 I' + c ' n_ p ' 
 
 ? 
 
 the result will give the value of k" 9 and conse- 
 
 quently, seeking that value in the second of the 
 two tables mentioned, we shall there find the cor- 
 
 F 
 
 responding value of . 
 
 If I" 
 
 Finally, the two values of - and thus found 
 
 i L 
 
 must be substituted in the equation (1) 
 
 S 1 
 
 ~ I" + c' I' + c'a'n + qP' 
 which operation will make known the velocity 
 sought. 
 
 SECT. IV. Of the load of the engine with a given 
 velocity. 
 
 Suppose the velocity of the engine be given, and 
 that it be required to find what load it can set in 
 motion at that velocity. 
 
 It must be observed that drawing from equation 
 
 _ 
 
 (B), the value of - 7 - ; and substituting it in 
 I { c 
 
WATT'S SINGLE-ACTING ENGINES. 
 equation (1), the latter becomes 
 
 I l V ' 
 
 "V r + c a n-/" ; 
 
 or 
 
 This is then the analogy proper to give the value 
 of k". Substituting for v the velocity given, we 
 shall have the number which represents the ex- 
 pression 
 
 ^ TJ1 . 
 
 v+- c 
 
 and seeking this number in the Table No. II., we 
 shall immediately find the corresponding value of 
 
 I" 
 
 j- and that of k". 
 
 V 
 
 I" I' 
 
 The value of being known, that of is next 
 
 I/ L 
 
 to be calculated. To this end, the equation (B) 
 furnishes the analogy, 
 
 t' + c _ i+2c n -/" 
 
 ' ?+?)*"* 
 
 The value of V being known by the preceding 
 
 research, substituting it then in the equation just 
 
 f. 
 given, we derive from it -, and afterwards k f by 
 
 L 
 
 means of the Table No. I. Consequently, intro- 
 ducing the value of k ( into equation 
 
246 
 
 CHAPTER X. 
 
 we obtain definitively the load a r corresponding to 
 the velocity v. 
 
 SECT. V. Of the evaporation necessary to produce 
 desired effects. 
 
 Suppose it be required to find what ought to be 
 the evaporation of the engine, to set in motion a 
 given load at a given velocity. 
 
 I' I" 
 The values of - and -- must be calculated by 
 
 V V 
 
 means of the two formula (A) and (B), and sub- 
 
 stituting these, with the given velocity v, in equa- 
 tion (3), 
 
 o r + c i + c 
 
 we shall thence conclude the evaporation sought. 
 
 SECT. VI. Of the useful effects of the engine. 
 
 Let it be required to find what useful effect the 
 engine may produce with a given load. 
 
 We shall seek first the velocity corresponding to 
 that load, by the proceeding indicated above ; then 
 multiplying that velocity by the given load, we shall 
 obtain the corresponding useful effect, viz. : 
 u. E. = arv. 
 
 Were the velocity given instead of the load, we 
 should seek the load corresponding to that velocity, 
 and the product arv of that load by the given 
 velocity would be the useful effect required. 
 
WATT'S SINGLE-ACTING ENGINES. 247 
 
 All the divers expressions of the useful effect are 
 to be derived from the knowledge of the product 
 ar v, by the formulae already given in Sect. V., Ar- 
 ticle L, Chapter III. 
 
 SECT. VII. Of the load and the velocity which cor- 
 respond to a given expansion. 
 
 Finally, as in these engines the expansion is 
 essentially connected with the load, in such sort 
 that the one being known the other follows neces- 
 sarily, it may happen that the expansion at which 
 the engine is to work, be determined beforehand, 
 and that it be required to find what load it can set 
 in motion with that expansion, and what velocity it 
 
 will assume with that load. 
 
 f 
 
 In this case the quantity - is given a priori, and 
 
 L 
 
 consequently we have also the corresponding value 
 of A/, which is found simply by inspection in Table 
 No. I. Therefore the equation (2), viz. 
 
 1+8 
 
 will immediately make known the load r ; and this 
 being found, the corresponding velocity will be 
 deduced, as in the first problem, which we have 
 treated above, Sect. III. 
 
 Thus it appears that the various questions which 
 may occur in the calculation of single-acting en- 
 
248 CHAPTER X. 
 
 gines, are to be resolved less directly, but yet in a 
 manner nearly as simple as those which concern 
 the rotative, or double-acting engines. 
 
 SECT. VIII. Determination of the friction of the 
 unloaded engine, and of its additional friction 
 per unit of the load. 
 
 The making use of the formula? which we have 
 just presented, suppose to be known, or at least 
 that means are had of estimating the friction f of 
 the engine unloaded in the descending stroke, the 
 surplus S accruing to that friction per unit of the 
 load imposed on the engine, and finally the fric- 
 tion f" of the engine in the ascending stroke. It 
 remains then to shew the means of determining 
 these three quantities. 
 
 In order to do this, it will be observed that equa- 
 tion (A) is general, or that we have always the 
 analogy 
 
 This analogy will consequently subsist when the 
 load of the engine shall be null, that is when r = 0. 
 So that, in this case, we have 
 
 q 
 which gives 
 
WATT'S SINGLE-ACTING ENGINES. 249 
 
 This equation then will determine the quantity f' y 
 as soon as the quantities P, p, II and k' become 
 known. 
 
 Now, it is easy to know these quantities by di- 
 rect experiment. Suppose the engine put in mo- 
 tion without any load, and that by admitting cau- 
 tiously a very small quantity only of steam into 
 the cylinder at each stroke of the piston, we have 
 succeeded in discovering, by repeated trials, the 
 
 jr 
 
 portion - of the stroke of the piston, at which the 
 
 V 
 
 steam flowing into the cylinder should be inter- 
 cepted, in order that the piston may be driven just 
 to the end of its stroke by the expansion of the 
 
 steam. We have thus, by experiment, the quan- 
 
 y 
 tity - ; and consequently recurring to the Table 
 
 V 
 
 No. I., or to the developed expression of k' 9 viz. 
 
 it will be easy to find the value of k', correspond- 
 
 f 
 ing to the observed value of -. 
 
 ' 
 
 Again, the counterweight II is known, and by 
 means of a manometer and Watt's indicator^ may 
 be measured the pressures P and p, subsisting in 
 the boiler, and in the steam cylinder after im- 
 perfect condensation. All the elements then of 
 
CHAPTER X. 
 
 the calculation will be had, and consequently, sub- 
 stituting them in the preceding equation 
 
 we easily deduce the quantity^ 7 , or the friction of 
 the engine in the descending stroke. 
 
 To find the quantity 8, it will suffice to set the 
 engine to work with a known load, and to regulate 
 by trials, as is usually done in practice, the portion 
 of the stroke which the piston must be allowed to 
 perform, before intercepting the flow of the steam 
 into the cylinder. Then by experiment we have 
 
 the quantity - suitable to the load r. Examining 
 / 
 
 afterwards the counterweight and measuring the 
 pressure in the boiler, as well as that which sub- 
 sists in the cylinder after imperfect condensation 
 of the steam, we shall likewise have by direct ob- 
 servation the quantities P, p and II. 
 
 Now we have always the relation (A), namely 
 n ,. ., 
 V 
 which gives 
 
 l f 
 
 It will be easy then to deduce 8 ; for - being 
 
 i 
 
 known, as has been said above, the quantity k f 
 
WATT'S SINGLE-ACTING ENGINES. 
 
 may be immediately concluded from the table 
 already indicated, and as the friction f is equally 
 known by the preceding research, we shall have all 
 the elements of the value of 8. 
 
 Finally, to obtain the friction f" of the rising 
 stroke, the equation (B) gives 
 
 /" = n~/ + c " (- + PY 
 
 l+2c \q 
 
 The engine then being supposed regulated by 
 
 I 1 I" 
 
 trials with any load, the two quantities - and and 
 
 I I 
 
 the pressure P must be measured simultaneously 
 on the engine. Then, recurring to the Table No. 
 II., or to the developed expression of k" viz. : 
 
 log {=!_+ l og 
 
 _ 
 
 ' C I 
 
 we shall know the value of k" which corresponds 
 
 I" 
 to the observed value of -- . We have, moreover, 
 
 V 
 
 f 
 
 by direct observation, that of - ; introducing there- 
 
 V 
 
 fore these values in the equation just given, we 
 conclude from it the value of/"". 
 
 It is however to be remarked, with respect to the 
 determination off, that in the engines employed 
 for raising water, this friction is not strictly a con- 
 stant quantity. It includes, in fact, the resistance 
 opposed by the water to the pistons of the pumps, 
 and this resistance varies as the square of the velo- 
 city of the motion. But as the velocity of the pis- 
 
CHAPTER X. 
 
 ton varies but little in these engines, it may suffice 
 to determine the value of f" for the ordinary ve- 
 locity, and this value may afterwards he considered 
 as a mean value applicable to all cases. 
 
 From what has been said then, the friction of 
 the engine in each stroke, and the additional fric- 
 tion per unit of the load imposed on the engine in 
 the descending stroke, may always be determined. 
 After having measured these frictions in several 
 engines, a mean friction may be deduced, which 
 will serve to estimate the friction of engines, before 
 their construction. 
 
 SECT. IX. Tables for the numerical solution of 
 the formula, for single-acting engines. 
 
 The formulae (A) and (B), which we have given 
 in the preceding section, not admitting of a direct 
 solution, and the use of them occurring nevertheless 
 in all the problems relative to the eifects of the en- 
 gines, we here subjoin two tables which give their 
 solution without calculation. 
 
 The first of these tables refers to the descending 
 stroke of the piston, the second to the rising stroke. 
 The correspondence between the numbers of the 
 different columns will immediately show, either the 
 
 F 
 
 value of k r when - is given, or on the contrary the 
 {/ 
 
 l r 
 value of - when k' is known. The same will be 
 
 L 
 
WATT'S SINGLE-ACTING ENGINES. 
 
 I" 
 found with reference to - and k" in the second 
 
 v 
 
 table. 
 
 if 
 When the ratio is determined, that of 
 
 + c 
 
 I 
 
 will be immediately concluded ; since the latter ratio 
 is nothing more than 
 
 f c 
 
 7 + r 
 
 and that in every engine the clearance c of the 
 cylinder is known. 
 
 When the equations contain the term 
 
 T+C"' 
 
 its value will be concluded from the ratio between 
 the two following, 
 
 l+2c , F + c 
 and _, 
 
 -jt 
 
 which are both known from the values of - and - . 
 
 I I 
 
 Thus no difficulty can, in any case, occur in the 
 numerical solution of the formulae. 
 
 In the tables we are about to offer, we have sup- 
 posed c = !/; that is to say, we have taken the 
 clearance of the cylinder, including the adjoining 
 steam passages, as equal to one tenth of the useful 
 stroke of the piston. This proportion is in effect 
 adopted in single-acting engines, in order to avoid 
 the accidents which might happen if the piston, 
 
CHAPTER X. 
 
 carrying its stroke rather too far, should strike 
 against the bottom of the cylinder. 
 
 We must here recall to mind that the quantities 
 l r and I" are the distances traversed by the piston, 
 at the moment when the communication of the 
 steam is intercepted ; and, consequently, that these 
 distances are supposed to be measured, not from 
 the bottom of the cylinder, but from that point 
 whence the piston effectively departs. 
 
WATT S SINGLE-ACTING ENGINES. 
 
 255 
 
 (No. 1.) TABLE for the numerical solution of the formula. 
 {Single-acting engines.) 
 
 Portion of the 
 descending stroke 
 >erformed before 
 expansion, or value 
 of the fraction 
 f 
 
 7* 
 
 Corresponding value of fc', 
 or of the expression 
 + c f V , Z+CN 
 
 (vTc + log rT- c > 
 
 Portion of the 
 descending stroke 
 performed before 
 xpansion, or value 
 of the fraction 
 /' 
 
 7" 
 
 orresponding value of Ac', 
 or of the expression 
 + c f V l+c\ 
 
 (T+-C + log F+-J- 
 
 01 
 
 263 
 
 49 
 
 857 
 
 02 
 
 286 
 
 50 
 
 863 
 
 03 
 
 308 
 
 51 
 
 869 
 
 04 
 
 329 
 
 52 
 
 875 
 
 05 
 
 349 
 
 53 
 
 881 
 
 06 
 
 368 
 
 54 
 
 887 
 
 07 
 
 387 
 
 55 
 
 892 
 
 08 
 
 406 
 
 56 
 
 897 
 
 09 
 
 424 
 
 57 
 
 902 
 
 10 
 
 441 
 
 58 
 
 907 
 
 11 
 
 458 
 
 59 
 
 912 
 
 12 
 
 474 
 
 60 
 
 917 
 
 13 
 
 490 
 
 61 
 
 921 
 
 14 
 
 505 
 
 62 
 
 925 
 
 15 
 
 520 
 
 63 
 
 929 
 
 16 
 
 535 
 
 64 
 
 933 
 
 17 
 
 549 
 
 65 
 
 937 
 
 18 
 
 563 
 
 66 
 
 941 
 
 19 
 
 577 
 
 67 
 
 945 
 
 20 
 
 590 
 
 68 
 
 949 
 
 21 
 
 603 
 
 69 
 
 952 
 
 22 
 
 615 
 
 70 
 
 955 
 
 23 
 
 627 
 
 71 
 
 958 
 
 24 
 
 639 
 
 72 
 
 961 
 
 25 
 
 651 
 
 73 
 
 964 
 
 26 
 
 662 
 
 74 
 
 967 
 
 27 
 
 673 
 
 75 
 
 970 
 
 28 
 
 684 
 
 76 
 
 973 
 
 29 
 
 694 
 
 77 
 
 975 
 
 30 
 
 704 
 
 78 
 
 977 
 
 31 
 
 714 
 
 79 
 
 979 
 
 32 
 
 724 
 
 80 
 
 981 
 
 33 
 
 734 
 
 81 
 
 983 
 
 34 
 
 743 
 
 82 
 
 985 
 
 35 
 
 752 
 
 83 
 
 987 
 
 36 
 
 761 
 
 84 
 
 989 
 
 37 
 
 770 
 
 85 
 
 990 
 
 38 
 
 778 
 
 86 
 
 991 
 
 39 
 
 786 
 
 87 
 
 992 
 
 40 
 
 794 
 
 88 
 
 993 
 
 41 
 
 802 
 
 89 
 
 994 
 
 42 
 
 810 
 
 90 
 
 995 
 
 43 
 
 817 
 
 91 
 
 996 
 
 44 
 
 824 
 
 92 
 
 997 
 
 45 
 
 831 
 
 93 
 
 998 
 
 46 
 
 838 
 
 94 
 
 999 
 
 47 
 
 845 
 
 95 
 
 999 
 
 48 
 
 851 
 
 
 
256 
 
 CHAPTER X. 
 
 (No. II.) TABLE for the numerical solution of the formulae. 
 (Single-acting engines.) 
 
 Portion of the ascending stroke 
 >erformed before the closing of 
 he equilibrium valve, or value 
 of the fraction 
 I" 
 I. 
 
 Corresponding value 
 of k", 
 or of the expression 
 l-l"+c . l-f'+c 
 
 Corresponding value of the ex- 
 pression 
 
 FTC*"' 
 
 1 lo ~ c 
 l"+c , l+c 
 
 - toBF+c 
 
 50 
 
 711 
 
 1-186 
 
 51 
 
 687 
 
 1-127 
 
 52 
 
 664 
 
 1-072 
 
 53 
 
 641 
 
 1-017 
 
 54 
 
 618 
 
 966 
 
 55 
 
 595 
 
 916 
 
 56 
 
 573 
 
 869 
 
 57 
 
 551 
 
 823 
 
 58 
 
 530 
 
 780 
 
 59 
 
 509 
 
 738 
 
 60 
 
 488 
 
 698 
 
 61 
 
 468 
 
 659 
 
 62 
 
 448 
 
 622 
 
 63 
 
 428 
 
 586 
 
 64 
 
 409 
 
 552 
 
 65 
 
 390 
 
 519 
 
 66 
 
 371 
 
 488 
 
 67 
 
 352 
 
 458 
 
 68 
 
 334 
 
 429 
 
 69 
 
 317 
 
 401 
 
 70 
 
 300 
 
 375 
 
 71 
 
 283 
 
 349 
 
 72 
 
 266 
 
 325 
 
 73 
 
 250 
 
 301 
 
 74 
 
 234 
 
 279 
 
 75 
 
 219 
 
 258 
 
 76 
 
 205 
 
 238 
 
 77 
 
 190 
 
 218 
 
 78 
 
 176 
 
 200 
 
 79 
 
 162 
 
 182 
 
 80 
 
 149 
 
 165 
 
 81 
 
 136 
 
 149 
 
 82 
 
 123 
 
 134 
 
 83 
 
 112 
 
 120 
 
 84 
 
 101 
 
 107 
 
 85 
 
 089 
 
 094 
 
 86 
 
 079 
 
 083 
 
 87 
 
 069 
 
 072 
 
 88 
 
 060 
 
 061 
 
 89 
 
 052 
 
 052 
 
 90 
 
 044 
 
 044 
 
 91 
 
 036 
 
 036 
 
 92 
 
 029 
 
 029 
 
 93 
 
 023 
 
 023 
 
 94 
 
 017 
 
 017 
 
 95 
 
 012 
 
 012 
 
WATT'S SINGLE-ACTING ENGINES. 257 
 
 SECT. X. Of the maximum useful effect with a 
 given counterweight^ and of the absolute maxi- 
 mum useful effect of the engine. 
 
 The effects of engines with a given counter- 
 weight, and with indefinite load or velocity, have 
 just been considered. But if different loads be 
 imposed on the engine without any change being 
 made in the counterweight, a certain velocity will 
 be generated for each load, and, consequently, a 
 corresponding useful effect. The useful effects 
 thus produced will necessarily differ. It will then 
 be necessary to seek, which, among these divers 
 loads, will be most advantageous for the engine 
 with the fixed counterweight, or which will produce 
 the maximum of useful effect with that counter- 
 weight, 
 
 To this end, the direct method would be first to 
 form the expression of the useful effect produced 
 with an indefinite load, in terms of that load; then 
 to seek, among all the possible values of the load, 
 that which would make the useful effect a maxi- 
 mum. This is the mode we have followed in the 
 similar research for rotative engines. Here the 
 velocity of the engine with an indefinite load r has 
 for its expression 
 
 I" + c ' I' + c'a'n + qP' 
 and, consequently, the corresponding useful effect is 
 
 Sr 
 arv 
 
 I" + c I' -f c n + q P 
 
258 CHAPTER X. 
 
 To arrive at the expression of this useful effect 
 in direct terms of the load, it would be necessary 
 to substitute the values of l r and I" for those quan- 
 tities, in the second member of the equation. 
 These values are given by the two equations (A) 
 and (B), namely : 
 
 9 
 
 in which k' and k" are logarithmic functions of f 
 
 and I", which we have expressed above. But these 
 two functions not being of a nature to furnish 
 a direct solution for 1! and /", it follows that those 
 quantities cannot, in the expression arv, be re- 
 placed by their value in terms of r, and conse- 
 quently, that this mode cannot lead to the desired 
 result. 
 
 The only method then to follow, is to proceed by 
 successive trials and approximations. A suppo- 
 sition must be made on the load r, or on the ex- 
 
 f 
 pansion - of the steam, which amounts to the 
 
 L 
 
 same, since, in these engines, those two quantities 
 are essentially connected with each other. Then 
 must be sought, by the means indicated above, the 
 corresponding velocity of the engine, and on form- 
 ing the product of the supposed load by the velocity 
 
WATT'S SINGLE-ACTING ENGINES. 259 
 
 found, we shall have the corresponding useful 
 effect. This proceeding must be renewed for a 
 second load, then for a third, and so on ; and ob- 
 serving always to vary the load on that side on which 
 the useful effect is seen to increase, we shall finally 
 attain that load which produces the maximum 
 useful effect sought. 
 
 By the foregoing may be known the most ad- 
 vantageous load for the engine, with a given counter- 
 weight, which is the first problem we had proposed 
 to ourselves in this section. But now it is plain 
 that on varying the counterweight itself, there will 
 exist for each of its values, a load the most ad- 
 vantageous and a corresponding maximum effect. 
 It remains then to seek, among these different 
 maxima effects produced by divers values of the 
 counterweight, that which is the most considerable ; 
 in order to discover, among all the values that may 
 be given to the counterweight, that which is the 
 most advantageous for the engine. If, in effect, 
 the solution of this question be attained, it is plain 
 that by first giving to the counterweight the value 
 thus found, and then laying on the engine the most 
 advantageous load for that counterweight, we shall 
 obtain the absolute maximum useful effect whereof 
 the engine is capable. 
 
 To obtain the required solution, the analytic 
 mode would again be similar to that which we 
 have followed with respect to the double-acting 
 engines, and which we have this instant recalled to 
 mind ; but this direct method is again inapplicable 
 
260 CHAPTER X. 
 
 here, for the reasons already stated. Consequently, 
 it is only by making successive suppositions on the 
 value of the counterweight, then seeking hy trials 
 the corresponding maximum of useful effect, and 
 finally, comparing with each other the divers 
 maxima useful effects, that it will be possible to 
 attain the numerical solution of the question pro- 
 posed. 
 
 This proceeding, at the first glance, appears 
 rather long ; but on observing that the equations 
 employed are very simple, and that in the successive 
 essays the same numbers continually recur, it will 
 soon be recognized that the calculation offers in 
 reality but little difficulty ; a difficulty, moreover, 
 which occurs but once for all, and is nothing in 
 comparison of the importance of the question, 
 namely, to find the means of making an engine work 
 ever afterwards with the greatest possible advantage. 
 
 ARTICLE II. 
 
 PRACTICAL FORMULAE FOR WATT's SINGLE-ACTING ENGINES, 
 AND EXAMPLE OF THEIR APPLICATION. 
 
 To form the numerical equations suitable to 
 the calculation of these engines, it must be ob- 
 served that, since they are condensing engines, 
 the value of the coefficients n and q of the relative 
 volume of the steam, will be 
 
 rc = -00004227 
 ^ = -000000258. 
 As to the pressure P of the steam in the boiler, 
 
WATT'S SINGLE-ACTING ENGINES. 
 
 it is generally from 16%5 to 18 Ibs. per square inch, 
 as in Watt's double-acting engines. The pressure 
 p subsisting, not in the condenser, but in the steam 
 cylinder itself, after imperfect condensation of the 
 steam, is again about 4 Ibs. per square inch, when 
 the engine is well supplied with injection water. 
 
 Referring the pressures to the square foot, we 
 have then generally 
 
 P-16'5 x 144 Ibs., p = 4 x 144 Ibs. 
 
 Finally, to have all the elements of the calculation, 
 it would be necessary to know the precise values 
 of the friction sf>f" and S, in order to substitute 
 them in the algebraic equations already obtained. 
 We want special experiments on this head ; but to 
 furnish at least an example of calculation which 
 may show the process, if not the precise results, 
 we will make, from analogy with Watt's rotative 
 engines, an approximate valuation of those fric- 
 tions. 
 
 These two kinds of engines differing but little 
 from each other, we will take the friction f" of the 
 unloaded engine in the rising stroke, at the same 
 value as in Watt's double-acting engines ; that is, 
 at *5 Ib. per square inch of the piston, for a cylinder 
 of about 48 inches diameter. As this dimension 
 is nearly the mean dimension of Watt's single- 
 acting engines, because the expansion of the steam 
 requires a capacious cylinder, we will consider the 
 above valuation as suitable to middle-sized engines 
 of this system. 
 
CHAPTER X. 
 
 We will estimate at the same value the friction 
 f of the unloaded engine, in the descending stroke ; 
 and, in fine, with respect to the additional friction 
 8, accruing to the engine per unit of the load im- 
 posed on it, we will admit, till an especial deter- 
 mination be obtained, the datum deduced from the 
 observation of locomotive engines. Consequently 
 we take 
 
 /'=/" = '5x144, 8 --14. 
 
 The numerical formula will be obtained by sub- 
 stituting the values of these constant quantities in 
 the algebraic equations. But as the pressure in 
 the boiler varies in different engines, as well as the 
 pressure p of condensation, and as the valuations 
 we have just given of the frictions are but tem- 
 porary, we will introduce no other values into the 
 numerical equations than those of n and q. The 
 following will then be the formula obtained : 
 
 Practical formulce for Watts single-acting engines. 
 
 164 + p 1 ^ 115 .-.Regulation of 
 
 the descending 
 stroke of the 
 piston. 
 
 Regulation of 
 
 the ascending 
 stroke of the 
 piston. 
 
 ar=JL-A:'(164.H-P) (164+p+/+n) Useful load 
 
 of the piston, in 
 pounds. 
 
WATT'S SINGLE-ACTING ENGINES. 263 
 
 1000 . Velocity of the 
 00258 P y 
 
 . . 
 
 piston, in feet 
 
 per minute. 
 
 av (-4227 + *00258P)Effectiveeva- 
 1000 poration, in 
 
 cubic feet of 
 water per mi- 
 nute. 
 
 u. E.= arv ....................................... Useful effect, in 
 
 pounds raised 
 1 foot per mi- 
 nute. 
 
 :-- .................................... Useful horse 
 
 33000 
 
 power. 
 
 u. E. lib. co. = ^-' ......................... Useful effect of 1 
 
 py 
 
 lb. of coal, in 
 pounds raised 
 1 foot. 
 
 u. E, 1 ft. wa. =^-' ............................ Useful effect due 
 
 to the evapora- 
 tion of 1 cubic 
 foot of water, 
 in Ibs. raised 1 
 foot. 
 
 Q. co. for 1 hp.= - =j ........................ Quantity of coal, 
 
 in pounds, 
 which pro- 
 duces 1 horse- 
 
 power. 
 
 33000 8 
 Q. wa. for 1 hp. =z - ^ ..................... Quantity of water, 
 
 in cubic feet, 
 which produces 
 1 horse-power. 
 
264 CHAPTER X. 
 
 "C* 
 
 u. HP. for 1 lb.co....= - ^- Horse - power 
 
 33000 N 
 
 produced per 
 
 pound of coal. 
 
 u. HP. for 1 ft. wa. ' ' Horse - power 
 
 ooOOO o 
 
 produced per 
 
 cubic foot of 
 water evapor- 
 ated. 
 
 Now, to show an application of the preceding 
 formulae, we will suppose an engine of this system 
 offering the following data : 
 
 Diameter of the cylinder, 48 inches ; or surface 
 
 of the piston, a 12*566 square feet. 
 Stroke of the piston, 1=8 feet. 
 Clearance of the cylinder, ^V of the useful stroke 
 
 of the piston ; or c= *1 /. 
 Pressure in the boiler, 16*5 Ibs. per square inch; 
 
 or P:nl6%5 x 144 Ibs. per square foot. 
 Pressure of condensation in the steam-cylinder, 
 
 4 Ibs. per square inch ; or p = 4 x 144 Ibs. per 
 
 square foot. 
 Effective evaporation, S = *506 cubic foot of water 
 
 per minute. 
 
 Consumption of coal in the same time,N = 4'25lbs. 
 Counterweight, 1*25 Ib. per square inch of the 
 
 surface of the piston ; or 11=1 '25 x 144 Ibs. 
 
 per square foot of the surface of the piston. 
 
WATT'S SINGLE-ACTING ENGINES. 265 
 
 With these divers data, it is proposed to deter- 
 mine the effects which the engine may produce. 
 Leaving then at first the counterweight invariable, 
 but making the engine work with different loads, 
 or, which amounts to the same, with different 
 expansions of the steam ; varying afterwards the 
 counterweight, and proceeding in the calculation, 
 according to the mode indicated in Article I. 
 of the present chapter, we obtain the following 
 results : 
 
 Effects of the engine, with the counterweight 
 given. 
 
 144 
 
 Maximum 
 useful effect. 
 
 == -66.. .......... -625 ......... -50 
 
 . ..... = 8-52 ......... 8-30 ......... 7'33 
 
 144 
 
 ar ..................... = 15411 ......... 15009 ........ , 13251 
 
 v ........................ = 100 ............ 105 ............ 130 
 
 S ..................... = -506 ......... -506 ........ -506 
 
 u . E ................... = 1543050 ...... 1580270 ...... 1728580 
 
 u . HP ................ = 47 ............ 48 ............ 52 
 
 u. E. lib. co ....... = 363070 ...... 371830 ...... 406730 
 
 u.E. 1ft. wa ....... = 3049510 ...... 3123060 ...... 3416180 
 
 Q. co. for 1 hp ...... = -091 ......... -089 ......... -081 
 
 Q.wa.forlhp ...... = -0108 ......... '0106 ......... -0097 
 
 u. HP. for 1 Ib. co. = 11-00 ......... 11-27 ......... 12-33 
 
 u. HP. for 1 ft. wa. = 92-41 ......... 94*64 ......... 103-52 
 
266 
 
 CHAPTER X. 
 
 Maxima effects of the engine, with different 
 counterweights. 
 
 II 
 
 Absolute maximum 
 of useful effect. 
 
 1*00 . . i*9/ 
 
 1-50 
 50 
 
 7-10 
 
 12849 
 133 
 506 
 1705030 
 51-7 
 401180 
 3369610 
 082 
 0098 
 12-16 
 102-11 
 
 144 
 t 
 
 -50 
 
 50 
 
 1 
 r 
 
 7-54 
 
 733 
 
 144 
 
 13640 
 
 13251 
 
 s 
 
 125 
 *506 
 
 130 
 506 
 
 u . E 
 
 1708840 
 
 1728580 
 52-4 
 
 u . HP 
 
 51-8 
 
 u.E.l Ib. co 
 
 = 402080 
 = 3377150 
 -082 
 
 406730 
 3416180 
 081 
 
 u. E. 1 ft. wa 
 Q. co. for 1 hp 
 Q. wa. for 1 hp 
 u. HP. for 1 Ib. co. 
 u. HP. for 1 ft. wa. 
 
 -0098 
 
 0097 
 
 12*18 
 
 12'33 
 
 = 102-34 
 
 103-52 
 
 The first of these tables shows that with the 
 constant counterweight of 1*25 Ib. per square inch 
 of the piston, the maximum of useful effect is pro- 
 duced by making the engine work with the expan- 
 
 l 
 sion indicated by - = "50, or with the corresponding 
 
 if 
 
 load, which is here found to be 7*33 Ibs. per square 
 inch of the surface of the piston. An expansion, 
 indicated by a number less than *<50, would, indeed, 
 produce still greater useful effects ; but as, with the 
 pressure at which these engines work, the motion 
 
WATT'S SINGLE-ACTING ENGINES. 267 
 
 becomes too irregular, when the steam is inter- 
 cepted before the half of the stroke, we deem it 
 proper to stop at this practical limit. 
 
 The second table shows moreover, that among 
 the divers counterweights that may be used to 
 work the engine, that of 1*2,5 Ib. per square inch 
 of the surface of the piston, is the most advan- 
 tageous - 9 and that employing this counterweight 
 
 I' 
 
 with the expansion corresponding to - "50, or, 
 
 I 
 
 in other words, with the load of J'33 Ibs. per 
 square inch of the piston, the engine will produce 
 the absolute maximum of useful effect that can 
 possibly be expected from it. 
 
 In performing the calculation, it is also to be 
 remarked, that for each different counterweight 
 that may be supposed for the engine, it is always 
 
 7f 
 
 the same expansion, corresponding to - = *50, 
 
 I 
 
 which is the most advantageous. This observa- 
 tion will greatly simplify the research of the ab- 
 solute maximum of useful effect ; for it will ob- 
 viously suffice, first to make different essays on 
 the expansion with the given counterweight, which 
 will make known the expansion of greatest useful 
 effect ; and then to apply that expansion to divers 
 hypotheses on the counterweight, which will lead 
 very promptly to the counterweight of the absolute 
 maximum of useful effect sought. Nevertheless, 
 as this remark is established only by the fact, and 
 not by a general reasoning, it will be proper to 
 
268 CHAPTER X. 
 
 verify as to the question, whether effectively the 
 expansion suitable to the maximum useful effect 
 for the first counterweight chosen, is equally the 
 expansion of maximum useful effect for the 
 counterweight determined hy the calculation. 
 
 For the above engine, it is easy to be convinced 
 
 p 
 
 that on changing either the expansion = *50, or 
 
 I/ 
 
 the counterweight 1*25 Ib. per square inch, there 
 would be a diminution of useful effect ; for on per- 
 forming the calculation we find the following cor- 
 responding results : 
 
 = 1-25 7 = -50 u. E. = 1728580 
 
 144 l 
 
 625 1580270 
 
 66 1543050 
 
 /' n 
 
 T -50 = 1 u. E. = 1708840 
 
 / 144 
 
 1-25 1728580 
 
 1*50 1705030. 
 
 We do not suppose an expansion represented by 
 
 f 
 
 a number less than - = '50 for the reasons given 
 i 
 
 above ; but were it chosen to step out of this limit 
 fixed by practical experience, the calculation must 
 be continued by trying other values of the expan- 
 sion and of the counterweight, as we shall do in the 
 following chapter with respect to the Cornish en- 
 gines. 
 
 The engine we have just calculated is the pattern 
 single-acting engine, constructed by Watt at his 
 factory in Soho. Its counterweight was 1*25 Ib. 
 
WATT'S SINGLE-ACTING ENGINES. 269 
 
 per square inch of the surface of the piston, and 
 when the steam was intercepted after the piston 
 had traversed 5 feet of the stroke, that is, in the 
 
 l r 5 
 case of - = - =z '625, the engine assumed a velocity 
 
 of 96 feet per minute. This is, then, a verification 
 of the preceding theory ; for the smallest diminu- 
 tion in the quality of the fuel, that is to say, in the 
 intensity of the fire, and, consequently, in the eva- 
 poration of the engine, may produce the slight dif- 
 ference which here appears between the velocity of 
 105 feet per minute, resulting from our calculation, 
 and that of 96 feet per minute, the result of direct 
 observation. 
 
270 
 
 CHAPTER XL 
 
 CORNISH SINGLE-ACTING ENGINES. 
 
 SECT. I. Practical formula for calculating these 
 engines ; and example of their application. 
 
 THE Cornish single-acting engines are nothing 
 more than a modification of Watt's single-acting 
 engines. All the. difference consists in the steam 
 being employed at a total pressure of 50 or 55 Ibs., 
 and sometimes at J5 or 80 Ibs., instead of 1 6 or 18 
 Ibs. per square inch ; and in the expansion of the 
 steam being carried much farther, since the steam 
 is often intercepted, when the piston has performed 
 but one-tenth of the stroke. 
 
 As this modification in no way affects the 
 principle itself of the application of the steam as a 
 motive power, but concerns merely the limits of 
 two quantities which figure in the formulae, it is 
 plain that the theory we have developed with re- 
 gard to Watt's single-acting engines, will equally 
 apply to the Cornish engines, and that we shall 
 only have to substitute for the pressure in the 
 boiler and the expansion of the steam, that is, for 
 
CORNISH SINGLE-ACTING ENGINES. 271 
 
 the quantities P and -, numbers different from those 
 I/ 
 
 which relate to Watt's engines. 
 
 The Cornish engines, therefore, will be calculated 
 by the following numerical formulae : 
 
 Practical formula for the Cornish single-acting 
 engines. 
 
 . R e gu l ation of 
 
 164+ P 
 
 the descend- 
 
 ing stroke of 
 the piston. 
 Regulation of 
 the ascend- 
 ing stroke of 
 the piston. 
 
 ar z:-^-rA:'(164 + P) -^-(164.+^+/+n) Load of the 
 
 1 -j-o 1 -f-o . 
 
 piston, m 
 pounds. 
 
 , - J_ . il c . ? . _ i _ Velocity of the 
 /" + c I' + c a -4227 + ' 00258P 
 
 piston, in feet 
 
 per minute. 
 
 poraton, in 
 cubic feet of 
 water per 
 minute. 
 u.E. zz arv ........................................ Useful effect, in 
 
 pounds raised 
 1 foot per mi- 
 nute. 
 
 Useful horse - 
 
 power. 
 
272 
 
 CHAPTER XL 
 
 u. E. 1 Ib. CO. = 
 
 u. E. 1 ft. wa. = 
 
 u. E. 
 
 "IT 
 
 u. E. 
 
 ~ f , , 
 
 Q. co. for 1 hp. 
 
 33000 N 
 
 Q. wa. for 1 hp. = 
 
 33000 S 
 u. E. 
 
 Useful effect of 
 1 Ib. of coal, 
 in pounds 
 raised 1 foot. 
 
 Useful effect 
 of 1 cubic 
 foot of water, 
 in pounds 
 raised 1 foot. 
 Quantity of 
 coal, in Ibs., 
 which pro- 
 duces 1 horse 
 power. 
 
 Quantity of 
 water, in cu- 
 bic feet, which 
 produces 1 
 horse-power. 
 Horse - power 
 produced per 
 pound of coal. 
 
 Horse - power 
 produced per 
 cubic foot of 
 evaporated 
 water. 
 
 For the mode of solution and for the perfect 
 understanding of these formulsB, we refer to what 
 has been said in the preceding chapter, on Watt's 
 single-acting engines. 
 
 As an example of the calculation, we will suppose 
 the single-acting engine of Watt, the effects of which 
 
 u. HP. for lib. co. = 
 
 u.HP. forl.ft.wa.= 
 
 u. E. 
 
 33000 N* 
 
 u.E. 
 
 33000 S 
 
CORNISH SINGLE-ACTING ENGINES. 
 
 were calculated in the last chapter, to he modified 
 according to the Cornish system ; that is to say, 
 the steam to he at a total pressure of 50 Ibs. per 
 square inch ; the cylinder to be replaced by ano- 
 ther of greater diameter ; and in fine, the steam 
 to be intercepted in the cylinder, when a small 
 portion only of the stroke is performed. We shall 
 then take for the different data of the problem : 
 
 Diameter of the cylinder 80 inches ; or surface 
 
 of the piston, a = 34*910 square feet. 
 Stroke of the piston, Z= 10 feet. 
 Clearance of the cylinder, ^ of the useful stroke 
 
 of the piston, or c = *l /. 
 Total pressure in the boiler, 50 Ibs. per square 
 
 inch, or Pr=50 x 144 Ibs. per square foot. 
 Pressure of condensation in the steam cylinder, 
 
 4 Ibs. per square inch, or j = 4 x 144 Ibs. per 
 
 square foot. 
 Friction of the unloaded engine in each stroke, 
 
 on account of the size of the cylinder, *25 Ib. 
 
 per square inch of the surface of the piston ; 
 
 orf=f" = -25x144. 
 Additional friction of the engine, per unit 
 
 of the resistance imposed on the piston, ^ of 
 
 that resistance; orSzz'14. 
 Effective evaporation, Sm'506 cubic foot of 
 
 water per minute. 
 Consumption of coal in the same time, 
 
 N = 4-25 Ibs. 
 
 T 
 
274 CHAPTER XI. 
 
 Substituting then these values in the formulae, 
 and supposing that, after having adopted a coun- 
 terweight, several degrees of expansion be succes- 
 sively tried, or in other words, several loads for 
 the engine, we shall first obtain the results con- 
 tained in the first of the following tables ; then 
 varying the counterweight itself, and essaying again 
 for each value of the counterweight, different de- 
 grees of expansion, we shall obtain the results of 
 the second table. 
 
 Effects of the engine with the counterweight given, 
 11 = 1-50x144 Ibs. 
 
 Maximum 
 of useful effect. 
 
 j = '25 '12 -10 
 
 =23-16 15-22 13-74 
 
 144 
 
 ar ,.=116426 76523 69053 
 
 v =28-72 47-85 53-28 
 
 S =-506 -506 -506 
 
 u. E =3344260 3661850 3679170 
 
 u. HP = 101 .....Ill 111-5 
 
 u. E. 1 Ib. co. =786880 861610 865690 
 
 u. E. 1 ft. wa = 6609220 7236860 7271100 
 
 Q. co. for 1 hp = -042 -038 -038 
 
 Q. wa. for 1 hp =-00499 -00456 -00454 
 
 u. HP. for 1 Ib. co =23-85 26-11 26-23 
 
 u. HP. for 1 ft. wa = 200... ...219 220 
 
CORNISH SINGLE-ACTING ENGINES. #75 
 
 Maxima effects of the engine with different 
 counterweights. 
 
 Absolute maximum 
 of useful effect. 
 
 , = 1-50 5-00 6-00 
 
 144 
 
 -- =-10 -11 -II 
 
 ., =13-74 11-42 10-54 
 
 144 
 
 ar =69053 57428 52993 
 
 v =53-28 67-28 72-65 
 
 S =-506 -506 -506 
 
 u.E =3679170 3863940 3850000 
 
 u.HP = 111-5 117-1 116-7 
 
 u.E. 1 Ib. co = 865690 909160 905880 
 
 u.E. 1 ft. wa =7271100 7636250. ...7608700 
 
 Q. co. for Ihp = '038 -036 -036 
 
 Q. wa. for 1 hp = -00454 -00432 -00434 
 
 u. HP. for 1 Ib. co =26-23 27'55 27-45 
 
 u. HP. for 1 ft. wa =220 231 230-6 
 
 The first of these tables shows that, with the 
 counterweight 1*50 Ibs. per square inch of the 
 surface of the piston, the most advantageous way 
 
 of working the engine is with the expansion cor- 
 l' 
 
 responding to -~ = *10, or, in other words, with 
 I 
 
 the load of 13*74 Ibs. per square inch of the piston. 
 Any other expansion, greater or less than that in- 
 
 I' 
 
 dicated by-, = *10, tends to diminish the useful 
 I 
 
276 CHAPTER XI. v 
 
 effect ; for the calculation gives the following re- 
 sults : 
 
 j --11 ..u. =3672490 
 
 10 3679170 
 
 09 3675790 
 
 Thus, this first research establishes the expansion 
 or the load which produces the maximum of useful 
 effect with the counterweight given. 
 
 Applying, moreover, the same calculation to dif- 
 ferent values of the counterweight, we find that, 
 for those different values, it is always the same 
 expansion, very nearly, which produces the maxi- 
 mum of useful effect. This will therefore be a 
 guide in the following research, namely, that of 
 the counterweight which produces the absolute 
 maximum of useful effect. For it will suffice to 
 calculate at first the effects of divers counter- 
 weights with the expansion corresponding to 
 
 I' 
 
 ~ n:'10; and no more will be needed afterwards 
 
 // 
 
 than a very slight correction of the expansion, 
 in order to attain the absolute maximum of useful 
 effect sought. 
 
 Proceeding thus, we find, as appears in the se- 
 cond table, that the most advantageous counter- 
 weight is 5 Ibs. per square inch ; and that, to pro- 
 duce the absolute maximum of useful effect, that 
 counterweight is to be used, and at the same time 
 
 F 
 
 the expansion -11. If, in effect, we increase 
 
CORNISH SINGLE-ACTING ENGINES. 277 
 
 or diminish, either the value of the counterweight, 
 or that of the expansion, we immediately find a 
 diminution of the useful effect ; for the calculation 
 gives 
 
 i= 4-75 -7= -10 u. E.zz 3856200 
 
 144 I 
 
 11 3860020 max. 
 
 12 3854250 
 
 = 5-00 -.- zz-10 u. E.= 3859220 
 
 144 I 
 
 11 3863940 ab. max. 
 
 12 3858430 
 
 ^-- 5-25 t-= -10 u.E.zz 3853990 
 
 144 / 
 
 11 3862600 max. 
 
 12... ,. 3858160 
 
 The two preceding tables therefore show, either 
 the load which produces the maximum of useful 
 effect with a given counterweight ; or the counter- 
 weight and load, which produce together the 
 absolute maximum of useful effect. It is to be 
 observed, moreover, that this research, though dif- 
 ficult in appearance, is not so in reality, first, 
 because the same numbers always recur in the dif- 
 ferent cases, and again, because the first calculation 
 having already made known the most favourable 
 expansion, that expansion, as we have said before, 
 varies but little in the second calculation. 
 
 It must be observed also that, on having de- 
 rived from the formula, a table similar to the 
 preceding, it is understood that the effects indi- 
 
#78 CHAPTER XI. 
 
 cated can be produced by the engine, only on con- 
 dition of there being no practical hindrance to the 
 regulation supposed in the calculation. Thus, 
 certain effects could not be attained, unless the 
 motion of the piston were much more rapid, or 
 much slower in one stroke than in the other ; and 
 though some difference may be admitted on this 
 point, yet it would not be possible to carry it be- 
 yond certain limits. In this case then, a regula- 
 tion must be adopted, as near as possible to the one 
 which would produce the maximum useful effect. 
 We shall so have the most advantageous practical 
 regulation, and calculating with it the effects of 
 the engine, we shall have the maximum of the 
 effects which it is capable of producing. 
 
 We will finally remark that it is customary in 
 these engines, to make use of the cataract. Under 
 this circumstance the engine does not evaporate the 
 full quantity of water, that its boiler would other- 
 wise be capable of evaporating per minute ; but 
 on introducing into the formula the evaporation 
 really effected, the formulae will always give the 
 corresponding effects of the engine. 
 
 The comparison which we have made between an 
 engine of Watt and a Cornish engine, that would 
 consume the same quantity of fuel per hour, proves 
 the superiority of the latter engine to the former. 
 It is evident also from the same comparison, that 
 the effects attributed to their engines by the Cornish 
 engineers, have been erroneously called in question ; 
 
CORNISH SINGLE-ACTING ENGINES. 279 
 
 since the example we have chosen shows that the 
 useful effect arising from the combustion of 1 Ib. 
 of fuel is 
 
 909160 Ibs. raised 1 foot, 
 
 which gives per imperial bushel, or 84 Ibs. of fuel, 
 76369440 Ibs. raised 1 foot. 
 
 If then we consider engines, wherein the steam 
 should be employed at a total pressure of 80 Ibs* 
 for instance, instead of 50 Ibs. per square inch, and 
 more especially, whose boiler should be constructed 
 in a more improved system for the application of 
 the heat, still greater useful effects may be ex- 
 pected from them. 
 
280 
 
 CHAPTER XII. 
 
 ATMOSPHERIC ENGINES. 
 
 ARTICLE I. 
 
 ATMOSPHERIC ENGINE WITH A CONDENSER. 
 
 SECT. I. Regulation of the Engine. 
 
 IN the atmospheric engines, the steam is first in- 
 troduced under the piston, in order to raise it hy 
 the help of a counterweight attached to the oppo- 
 site extremity of the heam ; this steam is then con- 
 densed, and the piston pressed on the upper face 
 hy the weight of the atmosphere, re-descends to 
 the bottom of the cylinder, causing the load, that 
 is the water in the pumps, to rise to a correspond- 
 ing height, and raising the counterweight at the 
 same time. Then a fresh quantity of steam is ad- 
 mitted into the cylinder, the piston is raised again, 
 and the action continues as before. 
 
 In this system then the motive power is succes- 
 sively, the pressure of the steam, aided by the 
 counterweight, and afterwards the atmospheric 
 pressure ; and the useful effect, instead of being 
 
ATMOSPHERIC ENGINES. 281 
 
 produced during the first of these periods, that is, 
 at the moment of the application of the steam, is 
 produced on the contrary during the action of the 
 atmospheric pressure. Consequently these two 
 forces act by turns ; however, as the atmospheric 
 pressure is here but an inert force, incapable of 
 producing any effect, unless the means were first 
 afforded to it by the production and application of 
 the steam, it follows that the steam, after all, is 
 the true force creative of the motion. 
 
 The mode of action of this engine may easily 
 be reduced to that of a single-acting engine of 
 Watt ; for, since the atmospheric pressure is 
 equivalent to a weight of 14*71 Ibs. per square 
 inch, we may suppose actual weights of 14*71 Ibs. 
 per square inch of the surface, to be placed on 
 the upper part of the piston, and these weights 
 will produce the same effect as the atmospheric 
 pressure. We may then suppress the atmospheric 
 pressure, and consider things as taking place in a 
 vacuum ; so, nothing in the system will be changed. 
 In this manner the engine reduces itself to a 
 material weight placed on the piston on the one 
 hand, and on the other hand to the pressure of the 
 steam introduced and suppressed under the piston 
 by turns, to raise it first by the help of the counter- 
 weight, and to let it afterwards descend by the 
 effect of the weight which has replaced the atmo- 
 spheric pressure. We are thus led precisely to the 
 case of a single-acting engine of Watt, in which 
 the useful effect, instead of being produced imme- 
 
CHAPTER XII. 
 
 diately, during the action of the steam, would be 
 produced during the return of the piston ; and 
 consequently, in these engines, the steam performs 
 the same office as in Watt's single-acting engines. 
 
 The atmospheric engines, having in general 
 neither fly-wheel nor crank, that is, not being 
 rotative, require like Watt's single-acting engines, 
 a particular regulation ; in order that the piston 
 may stop of itself in the cylinder, after having 
 performed the whole stroke assigned to it, without 
 either going beyond or falling short. To this 
 end, a moment before the piston reaches the top 
 of the cylinder in the rising stroke, the flow of 
 steam from the boiler is intercepted. So, through 
 the space which yet remains to be traversed, the 
 piston moves only by virtue of its acquired velocity, 
 of the effort of the counterweight, and of the 
 decreasing pressure of the steam during its ex- 
 pansion. But as soon as the communication from 
 the boiler is intercepted, the resistance of the at- 
 mosphere above the piston presently becomes supe- 
 rior to the motiye force ; consequently the piston 
 is brought to rest without shock and by insensible 
 degrees. 
 
 In the descending stroke, on the contrary, the 
 condensation of the steam is suspended somewhat 
 before the end of the stroke, either by stopping 
 the injection water when the steam is condensed 
 in the cylinder itself, or by shutting in time the 
 communication from the cylinder to the condenser, 
 when the engine is furnished with a separate con- 
 
ATMOSPHERIC ENGINES. 283 
 
 denser. The steam thus confined under the piston 
 without being condensed, opposes to it therefore, 
 on being compressed, a greater and greater resist- 
 ance, and at last brings it gently to rest. But it 
 is to be remarked that, as this steam thus acquires 
 all the labour developed by the piston in coming 
 to rest, and as itself is to contribute to the next 
 stroke of the piston, no loss of action is incurred. 
 
 By means of this management, the practical 
 object of which is to preserve the bottom of the 
 cylinder from the shocks of the piston, the atmo- 
 spheric engine, like the single-acting engine of 
 Watt, is regulated in its motion and brought to 
 rest at each stroke without loss of vis viva. Con- 
 sequently, referring to what has been said on this 
 head, in treating of Watt's single-acting engines, it 
 will be recognized that the formulae suitable to the 
 calculation of atmospheric engines may still be 
 grounded on the same principles as the preceding, 
 viz. : equality between the labour developed by the 
 power and that performed by the resistance in the 
 two strokes of the piston, and equality between the 
 expenditure of steam by the cylinder and the use- 
 ful evaporation of the boiler. 
 
 SECT. II. Of the effects of the engine with a 
 given counterweight and an indefinite load or 
 velocity. 
 
 Three cases are to be distinguished in the work- 
 ing of the atmospheric engines : that wherein they 
 work with a given counterweight, and with a load 
 
284 CHAPTER XII. 
 
 or velocity indefinite ; that in which they work 
 with a given counterweight, and the load or 
 velocity which produces the maximum of useful 
 effect for that counterweight ; and lastly, that 
 wherein the counterweight having been previously 
 regulated at the degree most advantageous for the 
 working of the engine, the most favourable load for 
 that counterweight be also adopted, which conse- 
 quently produces the absolute maximum of useful 
 effect of which the engine is capable. 
 
 We will first suppose the first case, and call as 
 before : 
 
 IT The weight of the counterweight, supposed 
 
 to be divided per unit of the surface of the 
 
 piston ; 
 
 P the total pressure of the steam in the boiler ; 
 a the area of the cylinder ; 
 / the stroke of the piston ; 
 /' the portion of the ascending stroke, performed 
 
 before the steam from the boiler is intercepted; 
 /" the portion of the descending stroke, per- 
 
 formedbefore the shutting of the injection cock, 
 
 or of the communication from the cylinder to 
 
 the condenser ; 
 r the useful load of the piston, divided per unit 
 
 of the surface of the piston ; 
 f the friction of the engine, then unloaded, 
 
 during the ascending stroke of the piston ; 
 f" the friction of the unloaded engine, during 
 
 the descending stroke of the piston ; 
 
ATMOSPHERIC ENGINES. 285 
 
 the surplus accruing to the latter friction, 
 per unit of the load r imposed on the engine 
 in the descending stroke ; 
 
 <j) the atmospheric pressure ; 
 
 p the mean pressure, subsisting under the pis- 
 ton, by reason of the imperfect condensation 
 of the steam. 
 
 Finally, we will suppose, the engine furnished 
 with a separate condenser. 
 
 This premised, the steam, during the ascending 
 stroke of the piston, penetrates into the cylinder, 
 as in Watt's single-acting engine, with the same 
 pressure as in the boiler, and the quantity of action 
 developed by its full pressure while coming directly 
 from the boiler, and afterwards by its decreasing 
 pressure during the expansion, has likewise for its 
 expression 
 
 n 
 
 Again, during the same stroke, the work de- 
 veloped by the counterweight, in descending from 
 the height /, is II a I ; that performed by the fric- 
 tion of the engine, when unloaded, is fa I ; and 
 finally, that performed by the atmospheric pres- 
 sure, is < a I. Consequently, the equality between 
 the work developed by the power and by the re- 
 sistance will furnish the equation 
 
 !? + p\( + log ^)- n -al+nal=<pal+fal. 
 q^ J \/'+c / +c' q 
 
286 CHAPTER XII. 
 
 And making, in order to simplify, 
 
 the above equation may be written under the form 
 
 %4>+/-n 
 u = l (i). 
 
 %P 
 
 9 
 
 This will then be the first of the three general 
 relations sought. 
 
 To pass on to the descending stroke of the 
 piston, the quantity of work applied by the at- 
 mospheric pressure, which is then the motive 
 power, has for its value <j> a I. On the other hand, 
 the quantity of action developed by the counter- 
 weight, is n a I ; that of the load is r a /, and that 
 of the friction of the loaded engine is (f r + S r) a I. 
 
 As to that which is developed by the pressure p, 
 subsisting under the piston by reason of the imper- 
 feet condensation of the steam, it must be decom- 
 posed into two parts. The pressure p is at first 
 exerted without any additional effect, while the 
 piston is traversing the portion ?' of its stroke; 
 that is to say, till the communication from the 
 cylinder to the condenser is shut, or till the con- 
 densation of the steam is suspended, if the engine 
 has no condenser. This pressure produces then, 
 in this first interval, a quantity of action ex- 
 pressed by 
 
 pal f/ . 
 
ATMOSPHERIC ENGINES. 287 
 
 But beyond this point, the condensation ceases; 
 the steam which still subsists under the piston be- 
 gins to be more and more compressed, and this 
 effect prevails through the length (I I"}, which 
 the piston has yet to traverse to terminate its 
 stroke. It remains then to determine the quan- 
 tity of action developed by the steam during this 
 compression. 
 
 Now, at the moment when this steam begins to 
 be compressed, it is at the pressure p 9 and the 
 volume it occupies below the piston is 
 
 If then we call TT the pressure, measured per unit 
 of surface, that it will have when the piston shall 
 have performed the length X of its stroke, and if 
 we suppose that the piston perform besides an ele- 
 mentary space d\ the corresponding elementary 
 work, produced by the compression of the steam, 
 will be 
 
 But as it is the same steam which, after having 
 occupied, in contact with the condensing water 
 and under the pressure p, the space 
 
 now occupies under the pressure TT, the space 
 
 a (/ \ + c), 
 
 without having lost, in the interval, any portion 
 of its total heat, there exists between the volumes 
 and the corresponding pressures of the steam, the 
 
288 CHAPTER XII. 
 
 relation (c), which we have demonstrated generally 
 (Chap. III., Art. I., Sect. II.), viz.: 
 
 (n \ a (I r + c) n 
 - + P i 71 - ( --- 
 q *Va(/-\+c) q 
 
 Consequently, we thence conclude, 
 
 7^ /n \ fT in N d\ n 7 ^ 
 irad\ a ( h p) (l l+c) --- ad\. 
 \q J J-X + c q 
 
 Thus, proceeding as before, that is to say, taking 
 the integral between the limits I" and /, we have 
 for the quantity of action developed by the gradual 
 compression of the steam on the portion (I I"} of 
 the stroke, the following expression : 
 
 (fl \ SI 7// N 1 I - / + C 71 ,1 j//^ 
 
 - + p) ('"? + c ) ** - " - a (* r)i 
 
 wherein the term 
 
 ! 
 
 log 
 
 expresses an hyperbolic logarithm. 
 
 Adding to this the work p a /", produced during 
 the portion of the stroke anterior to the compres- 
 sion of the steam, we shall have for the total quan- 
 tity of action developed by the resistance of the 
 uncondensed steam 
 
 If, to abridge, we make 
 
 1" 
 
 7 
 
ATMOSPHERIC ENGINES. 289 
 
 the above expression may be written under the 
 form 
 
 Q + p) k"al- n q al. 
 
 Consequently, referring to what has been found for 
 the quantity of action developed by the atmospheric 
 pressure, the load, the counterweight, and the fric- 
 tion of the engine, we will express the equality 
 between the work applied by the power and the 
 work performed by the resistance, in the stroke 
 now under consideration, by the following equation : 
 
 <pal=ral + Hal + (f" + $r) al+( - + p) k"al - al, 
 
 which gives 
 
 -+0_(1 +3) r-/"-n 
 
 W=9 __ . .(B) 
 
 n 
 
 i** 
 
 This is, in consequence, the second relation between 
 the data and the incognita of the problem. 
 
 Finally, the third relation sought will be obtained 
 by expressing the equality between the production 
 and the expenditure of the steam. 
 
 S being still the volume of water evaporated per 
 minute in the boiler, and effectually transmitted to 
 the cylinder, that volume of water, once converted 
 into steam at the pressure P of the boiler, will be- 
 come, as we have seen, 
 
 S 
 
 n + qP 
 
 Moreover, the capacity of the cylinder which is 
 
 u 
 
290 CHAPTER XII. 
 
 filled with steam at every ascending stroke of the 
 piston, is expressed by 
 
 But we have seen that at every descending stroke, 
 a certain quantity of steam remains compressed 
 under the piston, and serves for the next stroke ; 
 that is to say, is returned to the hoiler, instead of 
 being condensed. The pressure of this steam, at 
 the moment of its separation in the cylinder, is p, 
 and the volume it occupies under that pressure is 
 ($_/" + <.). 
 
 If it returned to the pressure P, without losing any 
 of its total heat, its volume would change in the 
 ratio 
 
 n + qp m 
 
 n + qP* 
 that is to say, it would become 
 
 n + qP 
 
 This expression gives then the volume of steam, 
 measured at the pressure of the boiler, which is 
 given back at each descending stroke. So that the 
 real expenditure of steam per double stroke of the 
 piston, is but 
 
 n + qP 
 
 Consequently, representing by M the number 
 of double strokes of the piston given by the engine 
 in a minute, the expenditure of steam per minute 
 will be 
 
ATMOSPHERIC ENGINES. 
 
 M 
 
 But, calling V the mean velocity of the piston, or 
 the space it traverses per minute both in rising and 
 falling, we shall have V = 2 M / ; or, counting for 
 the velocity, only the space traversed by the piston 
 in producing the useful effect, that is, in the de- 
 scending stroke alone, and calling that velocity 
 thus measured v 9 we have 
 
 ?? 
 
 
 
 Thus, the volume of steam expended by the cy- 
 linder in one minute, will be 
 
 The equation expressing the equality between. 
 the production and the expenditure of the steam, 
 will be then 
 
 - 
 
 n + qPJ n + qP 
 which gives 
 
 S 1 
 
 ~ ' ' 
 
 , 
 
 (n + qp} 
 
 This is the third relation sought. 
 
 Consequently, bringing forward the two equa- 
 tions (A) and (B) precedently obtained, and re- 
 solving them with reference to the quantities r and 
 8, we shall have the following formulae : 
 
292 CHAPTER XII. 
 
 (A) 
 
 n - + P 
 
 *" = __ . . .(B) 
 ^ 
 
 J + > 
 
 S 1 
 
 8 =a v ~ (n + q P) ~ 
 
 u.E. arv . . ............ ..... (4) 
 
 The expressions k' and W contained in these 
 equations, have the following values : 
 
 SECT. III. Of the velocity of the engine with a 
 given load. 
 
 The formulae which we have just given, suffice 
 for the solution of all the problems that can 
 occur on the atmospheric engine. As, however, it 
 is impossible to draw from the two equations (A) 
 
ATMOSPHERIC ENGINES. 293 
 
 T 1" 
 and (B) the direct values of -.and--, in terms of 
 
 V V 
 
 the data of the problem, in order to substitute 
 them in the formulae (1), (2), (3), (4), as would 
 be requisite for these formulae to give immediately 
 the solution sought, it will be necessary to use a 
 mode of calculation analogous to that which we 
 have already indicated for Watt's single-acting 
 engine, and to which we will recall attention here. 
 
 Suppose the load of the engine known, and that 
 it be required to find what velocity the engine 
 will assume with that load. 
 
 The given value of r must be substituted in the 
 equation (B), 
 
 ^ + _(i+s) r _/-"_n 
 
 n 
 
 which will consequently give the value of k n '. 
 Then, recurring to the Table which we are about 
 to give, we will find the corresponding value of 
 
 I" 
 
 . On the other hand, by calculating the second 
 
 i 
 
 member of the equation (A), 
 
 the value of k f will be obtained ; and afterwards, by 
 
 I! 
 means of the same Table, the value of -. Thus, 
 
 t 
 
CHAPTER XII. 
 
 F I" 
 
 therefore, will be had - an d - ; and consequently 
 
 t i 
 
 by substituting the value of these two quantities in 
 the equation (1) 
 
 the required velocity will readily be deduced. 
 
 With respect to the latter calculation, we will 
 remark that, if m and m f be put to express the 
 relative volumes of the steam under the respective 
 pressures P and p, there will result 
 
 m - - and m f =. - . 
 n + qr n+qp 
 
 The value of v, then, may also be written 
 
 S 1 
 
 " 
 
 a I' + c 1 l-F' 
 
 I'm I mf 
 
 and under this form it will obviously be more 
 easily calculated and more exactly obtained, for 
 the quantities m and m' will be known with strict 
 accuracy, and yet without calculation, by the Tables 
 which we have given in Sect. III. of Chapter II. 
 
 SECT. IV. Of the load of the engine with a given 
 velocity. 
 
 Suppose the velocity of the engine be known, 
 and that it be required to find what load it can put 
 in motion at that velocity. 
 
ATMOSPHERIC ENGINES. 
 
 The equation (A) 
 
 9 
 
 will first furnish the value of k f , and consequently 
 
 "it 
 that of by means of the table we have men- 
 
 jf 
 
 tioned. Then the value of - thus known must 
 
 I 
 
 be substituted in equation (1), which will give the 
 
 F 
 
 value of -, namely 
 
 /-T + c _ f + c n + qP S_ J__ 
 / / n + qp av ' n + qp 
 
 Consequently, by recurring to the same table, 
 the corresponding value of k ff may be known, and 
 this substituted in equation (2), which is 
 
 will give definitively the value of a r. 
 
 SECT. V. Of the evaporation of the engine, in 
 order to produce desired effects. 
 
 Suppose the load and the velocity of the engine 
 to be fixed a priori, and that it be required to 
 find the evaporation necessary to produce the de- 
 sired velocity with the given load. 
 
296 CHAPTER XII. 
 
 The equations (A) and (B) will first make 
 
 /' I" 
 
 known kf and V and subsequently -; and -j ; sub- 
 
 stituting these then, as well as the given velocity, 
 in equation (3) : 
 
 we will thence conclude the value of S. 
 
 SECT. VI. Of the useful effect of the engine. 
 
 Finally, the load being given, let it be required 
 to find what useful effect the engine will produce 
 with that load. 
 
 The velocity corresponding to the load must 
 first be calculated, as has just been said ; then 
 multiplying that velocity by the load, the product 
 a r v will be the corresponding useful effect. 
 
 If the velocity is known, instead of the load, we 
 will calculate the load corresponding to that velo- 
 city, and their product arv will again give the 
 corresponding useful effect. 
 
 Finally, the divers expressions of the useful 
 effect, whether in horse-power, or measured by the 
 force arising from a given expenditure of water or 
 of fuel, will be determined, in terms of the product 
 arv, by the formulae of Sect. V. Art I. Chap. III., 
 which suit all engines. 
 
 Thus, means are afforded of resolving all ques- 
 tions that can arise with respect to atmospheric 
 engines. 
 
ATMOSPHERIC ENGINES. 
 
 SECT. VII. Determination of the friction of the 
 engine unloaded, and of its additional friction 
 per unit of the load. 
 
 As the use of the preceding formulae supposes that 
 there are means of determining the friction of the 
 engines working without load, and the surplus 
 accruing to that friction for each unit of the load 
 imposed on the engine, it becomes necessary to 
 dwell a moment on that determination. 
 
 The equation (A) viz. : 
 
 gves 
 
 f= (!? + 
 
 9 
 
 To know the friction f of the unloaded engine in 
 the ascending stroke, it suffices then to determine 
 by experiment or by direct observation, the quan- 
 tities </>, II, P, and Id. 
 
 Now, this is an easy matter : for <f> is the atmo- 
 spheric pressure, which is given by the barometer ; 
 II is the counterweight of the engine, which is 
 known ; and P the total pressure of steam in the 
 boiler, which is measured by means of a manometer. 
 Thus, k f alone requires a particular determination. 
 Suppose then an experiment to be made on an engine 
 with any load whatever, and that the rising stroke 
 
298 CHAPTER XII. 
 
 be regulated by trials, as is always done in prac- 
 tice. Then the quantity , that is to say, the 
 
 i 
 
 point at which the steam from the boiler should be 
 intercepted, becomes given by the experiment itself. 
 Consequently, recurring to the table which we shall 
 give presently, or to the developed expression of 
 k f , namely : 
 
 F + c / 1! I + 
 
 it will be easy to know the value of A/ correspond- 
 
 f 
 ing to that of _. Thus, by substituting this value 
 
 V 
 
 of k r in the preceding equation, with the observed 
 values of $, II and P, we shall conclude without 
 difficulty the value of f, or the friction of the un- 
 loaded engine, in the ascending stroke. 
 
 To find afterwards the similar friction /", of the 
 descending stroke, recourse must be had to equa- 
 tion (B) 
 
 n 
 
 l*>. 
 
 which gives 
 
 /"= - +0-(l+8)r-n- (- + p) k". 
 q ^ q 
 
 This relation is general, and consequently must 
 still subsist when the load of the engine is null, 
 that is, when v = 0. , But then it reduces itself to 
 
ATMOSPHERIC ENGINES. 299 
 
 in order, then, to have the value of f", it will 
 suffice to be able to find directly what values the 
 quantities p and k f/ assume, with no load ; for as 
 to <f> and II, they are constant quantities which may 
 always be measured, as has been said above. 
 
 To have the values of p and k" which correspond 
 to the load zero, recourse must be had to experi- 
 .ment. The engine must be set to work without 
 applying any load on it, but effecting the con- 
 densation very imperfectly, that is to say, retaining 
 under the piston a high pressure for the uncon- 
 densed steam, and shutting with precaution and in 
 time the cock of the condenser, or the injection 
 cock ; so that the engine shall be regulated in its 
 descending stroke, by its friction alone and the 
 resistance of the steam compressed under the pis- 
 ton. Then must be measured directly the portion 
 
 I" 
 
 - of the stroke, which is performed before suspend. 
 
 I 
 
 ing the condensation. Recurring afterwards to 
 the developed expression of k 1 ' ', viz. : 
 
 II c 
 
 it will be easy to know the value of k" correspond- 
 
 l" 
 ing to the measured value of At the same mo- 
 
 // 
 
 ment the quantities </>, II and p, that is, the at- 
 mospheric pressure, the counterweight and the 
 pressure of the uncondensed steam, will be taken 
 
300 CHAPTER XII. 
 
 by direct observation. Consequently, substituting 
 these different values in the equation 
 
 we shall conclude the friction of the unloaded 
 engine, in its descending stroke. 
 
 With respect to the direct measure of the quan- 
 tity p, it must be effected with Watt's indicator 
 of the pressure, if the engine has a condenser ; 
 because we have seen, in speaking of Watt's 
 double-acting engines, that the pressure in the 
 cylinder is always superior to that in the con- 
 denser. But if the engine has no condenser, it will 
 suffice to take with a thermometer the temperature 
 of the water coming out of the eduction pipe, that 
 is, of the water which issues from the cylinder 
 after the condensation. This temperature being 
 also that of the steam with which the water was 
 in contact, the corresponding pressure will be 
 found in the tables which we have given in Chapter 
 II. of this work, on the pressure and temperature 
 of steam in contact with the liquid. 
 
 Knowing the quantity f" , or friction of the un- 
 loaded engine, it will be easy to obtain the surplus 
 8 accruing to that friction per unit of the load 
 imposed on the engine. To this end, the engine 
 must be set to work with a known load r, and the 
 stroke regulated by trials in the usual manner. 
 Then, taking as above, the direct measure of the 
 
 I" 
 quantities </>, II, and p, and substituting them 
 
ATMOSPHERIC ENGINES. 301 
 
 with the value. of r, in the equation (B), which 
 gives 
 
 r 
 the value of 8 will be concluded immediately. 
 
 Thus the three constant quantities f' 9 f" and B 
 may be determined on several engines ; and from 
 the aggregate of these determinations, may be 
 concluded a mean valuation of those quantities, 
 which will afterwards be used in general calcu- 
 lations and applied to engines not yet constructed. 
 
 We will here, however, make an observation 
 analogous to that which we have already made 
 relative to Watt's single-acting engines : viz., that in 
 the engines used for raising water, the friction of 
 the engine in the ascending stroke is not strictly a 
 constant quantity. That friction, in fact, comprises 
 the resistance offered by the water to the pump 
 piston or plunger, which resistance changes with the 
 velocity of the motion ; but as the velocity of atmo- 
 spheric engines varies within very narrow limits, it 
 will suffice to determine f for the average usual 
 velocity, and that determination may be considered 
 as a mean applicable to the different cases. 
 
 SECT. VIII. Tables for the numerical solution of 
 the formula. 
 
 As the use of the preceding formulae requires a 
 table, furnishing immediately the values of kf and 
 
302 CHAPTER XII. 
 
 V I" 
 
 k" corresponding to given values of - or , and 
 
 ' i 
 
 vice versa, we will here give that table calculated 
 
 from hundredth to hundredth, for the values of 
 it iff 
 
 r-*r 
 
 When the result of a formula shall have made 
 
 known k' or k ! ', the table will, on inspection, give 
 
 // /// 
 
 - or - ; whence will immediately be concluded 
 
 and l - l " +c 
 
 since these fractions are nothing more than 
 f c A l + c r 
 
 i*i and -p 
 
 and that the quantity c, which represents the 
 clearance of the cylinder, is always known in terms 
 of the useful length of the stroke, or of the quan- 
 tity /. 
 
 We have, in the following table, taken the 
 clearance of the cylinder equal to a tenth of the 
 useful stroke of the piston, that is to say, we have 
 made c = ! /; because that is the proportion 
 generally adopted in practice. However, as in 
 atmospheric engines without a condenser, it often 
 happens that the clearance of the cylinder amounts 
 to 2 or 3 tenths of the stroke, we will add to the 
 table No. II., which alone is required for calcu- 
 lating those engines, two other columns for the 
 
 cases of f = % and - = *3. 
 l I 
 
ATMOSPHERIC ENGINES. 
 
 303 
 
 No. I. TABLE for the numerical solution of the formula. 
 (Atmospheric engines.) 
 
 Portion of the ascending stroke 
 performed during the admission of 
 the steam under the piston, or value 
 of the fraction 
 
 r 
 
 t ' 
 
 Corresponding value of k', or of the 
 expression 
 
 (*)' 
 
 50 
 
 863 
 
 
 
 52 
 
 -375 
 
 53 
 
 -881 
 
 54 
 
 .. -887 
 
 55 
 
 . -892 
 
 56 
 
 -897 
 
 .57 
 
 .902 
 
 .53 
 
 9O7 
 
 .59 
 
 .. 912 
 
 60 
 
 Q17 
 
 61 . 
 
 
 62 
 
 .005 
 
 63 .. 
 
 Q9Q 
 
 .(34 
 
 .. -Q^^ 
 
 65 .. 
 
 
 66 
 
 . 'Q4.1 
 
 67 .. 
 
 . 'Ckd^ 
 
 68 
 
 .. -Q4Q 
 
 69 .. 
 
 
 70 .. 
 
 .. Q'i'i 
 
 71 . 
 
 
 72 .. 
 
 
 73 ., 
 
 
 74 l 
 
 
 75 .. 
 
 
 76 . 
 
 
 77 
 
 
 7ft .. 
 
 
 7Q .. 
 
 
 80 
 
 
 
 
 82 
 
 
 83 .. 
 
 
 
 
 R5 ., 
 
 
 
 
 87 
 
 
 
 
 
 
 90 ... 
 
 
 01 
 
 
 92 
 
 'QQ7 
 
 93 
 
 
 94 
 
 'QQQ 
 
 95 
 
 
 
 
304 
 
 CHAPTER XII. 
 
 No. II. TABLE for the numerical solution of the 
 formulae. (Atmospheric engines.) 
 
 Portion of the descending stroke 
 performed before the shutting of 
 the injection cock, or value of the 
 fraction 
 
 r 
 T 
 
 Corresponding value of Jc", or of the 
 expression 
 
 I" . t-f'+e l-r+c 
 
 I ' I 10 c 
 
 ' T=- 
 
 ?= 
 
 ;-=, | 
 
 5 
 
 1-575 
 1-557 
 1-539 
 1-522 
 1-505 
 1-488 
 1-471 
 1-454 
 1-437 
 1-421 
 1-405 
 1-389 
 1-373 
 1-357 
 1-342 
 1-327 
 1-312 
 1-297 
 1-283 
 1-269 
 1-255 
 1-241 
 1-227 
 1-214 
 1-201 
 1-188 
 1-176 
 1-164 
 1-152 
 1-141 
 1-130 
 1-119 
 1-108 
 1-098 
 1-088 
 1-079 
 1-070 
 1-061 
 1-053 
 1-046 
 1-039 
 1-032 
 1-026 
 1-020 
 1-015 
 1-011 
 
 1-377 
 1-364 
 1-352 
 1-340 
 1-328 
 1-316 
 1-304 
 1-293 
 1-282 
 1-271 
 1-260 
 1-249 
 1-238 
 1-227 
 1-216 
 1-206 
 1-196 
 1-186 
 1-176 
 1-167 
 1-158 
 1-149 
 1-140 
 1-131 
 1-123 
 1-115 
 1-107 
 1-099 
 1-091 
 1-084 
 1-077 
 1-070 
 1-064 
 1-058 
 1-052 
 1-046 
 1-040 
 1-035 
 1-030 
 1-026 
 1-022 
 1-018 
 1-014 
 1-01 L 
 1-008 
 1-006 
 
 1-286 
 1-275 
 1-265 
 1-255 
 1-246 
 1-237 
 1-228 
 1-219 
 1-210 
 1-201 
 1-193 
 1-185 
 1-177 
 1-169 
 1-161 
 1-153 
 1-145 
 1-137 
 1-130 
 1-123 
 1-116 
 1-109 
 1-102 
 1-096 
 1-090 
 1-084 
 1-078 
 1-072 
 1-066 
 1-060 
 1-055 
 1-050 
 1-045 
 1-041 
 1-037 
 1-033 
 1-029 
 1-025 
 1-021 
 1-018 
 1-015 
 1-012 
 1-010 
 1-008 
 1-006 
 1-004 
 
 51 
 
 52 
 
 53 
 
 .54 
 
 .55 
 
 56 
 
 .57 
 
 .58 
 
 .59 
 
 60 
 
 61 .... 
 
 .62 
 
 63 
 
 .64 
 
 .65 
 
 .66 
 
 .67 
 
 .68 
 
 .69 
 
 .70 
 
 .71 
 
 .72 
 
 .73 
 
 .74 
 
 75 .. 
 
 76 
 
 77 
 
 78 
 
 79 
 
 80 
 
 81 
 
 82 
 
 83 
 
 84 
 
 85 
 
 86 
 
 87 
 
 88 
 
 89 
 
 90 
 
 "V)l 
 
 92 
 
 93 
 
 94 
 
 95 
 
 
ATMOSPHERIC ENGINES. 305 
 
 SECT. IX. Of the maximum useful effect with 
 a given counterweight^ and of the absolute 
 maximum of useful effect. 
 
 To arrive at the determination of the maximum 
 useful effect with a given counterweight, and after- 
 wards at that of the absolute maximum of useful 
 effect of the engine, it would be necessary, first to 
 form the general expression of the useful effect 
 under a given load, in terms of the load only. 
 Then putting equal to zero the differential of that 
 expression, taken with reference to the load con- 
 sidered as variable, we should have the conditional 
 equation, proper to determine the load suitable 
 to the production of the maximum of useful effect 
 with a given counterweight. Afterwards would be 
 also concluded, in the same way, an equation for 
 determining the absolute maximum of useful effect, 
 as we have done with respect to the rotative engines, 
 in Articles II. and III. of Chapter III. of this 
 work. 
 
 Referring to equation (1), 
 
 a 
 
 I I 
 
 it is plain that on multiplying both members by ar, 
 we obtain the general expression of the useful effect, 
 viz. : 
 
 Sr 
 
306 CHAPTER XII. 
 
 1" 
 
 As however the quantity - varies with the load of 
 
 ' 
 
 the engine, it would he necessary, in order to con- 
 tinue the calculation, to be able to replace that 
 quantity, in the expression of arv, before or after 
 differentiating, by its value in terms of the load r. 
 This value would have to be concluded from the 
 developed equation (B), viz. : 
 
 r. i-i" + c } 
 T + -7 log 
 
 but as the nature of this equation does not admit of 
 resolving it directly with reference to r, we must 
 here, as in Watt's and the Cornish single-acting 
 engines, have recourse to the method of successive 
 trials and approximations. 
 
 To this end, when the counterweight is given, 
 the calculation must be made of the useful effect 
 with various loads successively, taking care to vary 
 the load on that side on which the useful effect shall 
 be found to increase ; and after a few trials, the 
 load will be attained, which produces the maximum 
 of useful effect with the given counterweight. 
 
 Afterwards the counterweight must be varied 
 in the same manner, and the corresponding maxi- 
 mum useful effect determined for each of its values. 
 Then, on comparing with each other^ the divers 
 maxima useful effects, due to the different values of 
 the counterweight, we shall determine that coun- 
 terweight which produces the absolute maximum 
 of useful effect in the engine. 
 
ATMOSPHERIC ENGINES. 307 
 
 This research, apparently complicated, simplifies 
 itself considerably from the circumstance of the 
 same numbers recurring constantly in the calcula- 
 tion. Besides, when the most advantageous load 
 for a given counterweight is found, and that the 
 counterweight is afterwards made to vary, it is 
 observable that on the counterweight being in- 
 creased a certain quantity, the corresponding most 
 advantageous load will be lowered by a nearly equal 
 quantity, relatively to the first load ; and that if, on 
 the contrary, the counterweight be diminished a 
 certain quantity, the most advantageous load will 
 augment nearly as much. By this observation, the 
 research of the absolute maximum of useful effect is 
 reduced to a very few trials, nor does it present any 
 difficulty at all comparable to the importance of the 
 object proposed, namely, to find the means of 
 making an engine work in the most advantageous 
 manner possible. 
 
 ARTICLE II. 
 
 ATMOSPHERIC ENGINE WITHOUT CONDENSER. 
 
 SECT. I. Modifications to be made in the preceding 
 formula, for the case of the engine not furnished 
 with a separate condenser. 
 
 In the theory of the atmospheric engine, which 
 we have just explained, we have admitted that, till 
 the moment of the communication from the boiler 
 
308 CHAPTER XII. 
 
 to the cylinder being intercepted, the steam acts in 
 the cylinder at the same pressure as in the boiler. 
 This does in fact take place, at least without no- 
 ticeable error, in engines furnished with a separate 
 condenser, and whose cylinder is properly protected 
 by a double casing against all external refrigeration. 
 But in very many atmospheric engines, the con- 
 densation after every stroke of the piston, takes 
 place in the steam cylinder itself; so that the latter 
 is found cooled to the temperature of condensation, 
 at the moment when the steam arrives to cause a 
 new ascending stroke. Immediately then on enter- 
 ing the cylinder, the steam must tend to establish 
 an equilibrium of temperature with it, the effect 
 of which tendency is to raise the temperature 
 of the cylinder by lowering that of the steam. 
 Hence, instead of retaining its original tempera- 
 ture, the steam necessarily acquires in the cylinder 
 a certain temperature, and consequently, a certain 
 pressure, intermediate between that of its forma- 
 tion in the boiler and that of condensation in the 
 cylinder. 
 
 What is then this intermediate pressure which 
 the steam assumes? That is the point which it 
 will be proper to examine first. 
 
 To obtain this determination, it must be observed 
 that on entering the cylinder, the steam comes im- 
 mediately into contact with a cooled surface ; that by 
 degrees, as the piston is driven back in the cylinder, 
 it lays open a new portion of surface to be heated ; 
 and finally, that the more the velocity of the pis- 
 
ATMOSPHERIC ENGINES. 309 
 
 ton increases, the more surface, in a unit of time, 
 is exposed to the heating action of the steam. 
 Now, the quantity of steam produced by the boiler 
 in a unit of time is a fixed and determinate quan- 
 tity. As that steam gradually penetrates into the 
 cylinder, a portion of it will be condensed to effect 
 the heating of the cylinder, and the rest will lower 
 both in temperature and pressure ; so that it can 
 only be with a reduced pressure, that the steam will 
 act on the piston to effect its motion in the cylinder. 
 There may then occur three cases. If the pressure 
 of the steam after refrigeration be found still superior 
 to the resistance of the piston, its effect will be to 
 create the velocity of the piston, or to augment it, 
 if it be already produced. If the pressure of the 
 refrigerated steam be merely equal to the resist- 
 ance of the piston, its effect will be to maintain 
 the motion of the piston in a state of uniformity, 
 without increasing or diminishing its velocity. 
 And finally, if the pressure of the steam after 
 cooling be found inferior to the resistance of the 
 piston, the motion of the latter will slacken and 
 come at length completely to a stand. 
 
 This premised, at the first moment the steam 
 is introduced into the cylinder, to produce the 
 ascending stroke, there is already a considerable 
 portion of surface to be heated, for it consists of 
 the clearance of the cylinder, the bottom of the 
 cylinder, and the lower face of the piston. The 
 first effect of the contact of the steam with this 
 large extent of surface will therefore be sensibly to 
 
310 CHAPTER XII. 
 
 lower the pressure of the steam. But the piston is 
 loaded with a considerable resistance, amounting 
 indeed very nearly to the pressure of the atmo- 
 sphere, for the friction of the engine and the 
 counterweight, which act contrariwise to each 
 other, tend to produce an equilibrium. Thus 
 the refrigerated steam will be found at first too 
 weak for the resistance to be moved, and a moment 
 will elapse before the starting of the piston can 
 be effected. 
 
 However, by degrees, as the steam condensed by 
 the contact of the cylinder is replaced by a fresh 
 quantity of steam supplied from the boiler, and 
 the temperature of the exposed portion of the cylin- 
 der rises, the steam in the cylinder will acquire 
 a greater pressure. It will soon then attain the 
 degree requisite to determine the starting of the 
 piston ; and the latter will assume a certain velocity, 
 which will still increase as long as the pressure of 
 the steam in the cylinder, that is to say of the re- 
 frigerated steam, shall exceed the resistance of the 
 piston. 
 
 But since we have seen that, as the velocity of 
 the piston increases, the surface to be heated in 
 a given time increases also : it follows that the 
 greater the velocity of the piston becomes, the more 
 refrigeration the steam will undergo, and the more 
 the pressure in the cylinder will be diminished. 
 Now the original pressure of the steam in the 
 boiler exceeds but little the atmospheric pressure, 
 which represents very nearly the resistance on the 
 
ATMOSPHERIC ENGINES. 311 
 
 piston. A velocity then will soon occur wherein 
 the pressure of the steam, after its refrigeration in 
 the cylinder, shall not exceed the resistance of the 
 piston ; and consequently, from that moment, the 
 motion of the piston will become uniform, till the 
 state of things be changed by entirely suppressing 
 the admission of the steam from the boiler. 
 
 Finally, as soon as the communication from the 
 boiler to the cylinder shall be intercepted, the 
 steam contained in the cylinder will begin to ex- 
 pand, and consequently to diminish its pressure. 
 But as the receding of the piston continues still to 
 expose a new portion of the cylinder to be heated, 
 the expansion of the steam will be attended with a 
 continual condensation, which will itself contribute 
 to lower the pressure more rapidly. Thus the re- 
 sistance will promptly obtain a preponderance over 
 the motive force, and the piston be brought to rest 
 in a very short time. 
 
 Consequently, there obviously exists in the en- 
 gine a rapid tendency to produce uniformity in the 
 motion of the piston, that is to say, an equilibrium 
 between the pressure of the steam after refrigeration 
 in the cylinder, and the resistance of the piston ; and 
 save in a very short interval at the beginning and at 
 the end of the stroke, that equilibrium must subsist 
 during the whole ascending motion of the piston. 
 
 Now, on examining the motion of an atmospheric 
 engine without condenser, it is observable, that at 
 the moment when the steam penetrates into the 
 cylinder to produce the rising stroke, the piston 
 
CHAPTER XII. 
 
 remains motionless an instant at first, before effect- 
 ing its starting. Then, as soon as it begins to 
 move, it acquires, in a very short time, a remark- 
 ably uniform motion ; and this lasts without inter- 
 ruption till the moment when the steam from the 
 boiler is intercepted, after which the velocity very 
 rapidly decreases. We must then conclude from 
 this fact, which all observers have remarked, that 
 during nearly the whole of the ascending motion, 
 the equilibrium which we have mentioned above, 
 does effectually establish itself between the pressure 
 of the steam in the cylinder and the resistance of 
 the piston. 
 
 Now the resistance of the piston, in this stroke, 
 consists of the atmospheric pressure, augmented 
 by the friction of the engine and diminished by the 
 action of the counterweight. Consequently, calling 
 P 7 the unknown pressure assumed by the steam in 
 the cylinder, we shall have 
 
 F=*+/-n (C). 
 
 This equation makes known the pressure as- 
 sumed by the steam, during the continuance of the 
 uniform motion of the piston. And neglecting the 
 beginning and the end of the stroke, this pressure 
 may be taken as the mean pressure of the steam 
 during the whole stroke of the piston. But if, 
 moreover, it be observed that, at the beginning of 
 the stroke, the pressure of the steam must exceed 
 the resistance by a certain quantity, in order to 
 produce the motion, and that at the end of the 
 stroke, or during the expansion, the resistance 
 
ATMOSPHERIC ENGINES. 313 
 
 must on the contrary exceed the pressure by an 
 equal quantity, in order to destroy the velocity 
 already acquired, it will be recognized that the va- 
 riations which the pressure undergoes in these two 
 extreme points compensate each other. Thus the 
 value of P' given by the equality between the two 
 forces, that is to say, by the preceding equation, 
 is at once the real pressure of the steam during the 
 uniform motion of the piston, and the mean value 
 of the pressure of the steam taken in the whole 
 duration of the stroke ; that is to say, as well 
 before as during the expansion of the steam. 
 
 The circumstance of the cooling of the steam in 
 the cylinder, first gives us then the preceding 
 equation, relative to the ascending stroke of the 
 piston. It will now be proper to examine whether 
 the other analogies already obtained for the en- 
 gines with condensers, ought still to subsist or 
 undergo some modification. 
 
 Among those analogies, equation (A) is intended 
 to make known the quantity 
 
 r 
 
 T 
 
 that is, the point of the stroke at which the flowing 
 of the steam to the cylinder should be intercepted, 
 for the piston to stop of itself and without shock, 
 after having performed the whole of the stroke 
 assigned to it. The considerations which led to 
 the establishing of that equation, suppose that the 
 steam enters the cylinder with a pressure constant 
 and equal to th? t of the boiler ; that the velocity 
 
314 CHAPTER XII. 
 
 communicated to the piston by means of that force, 
 continually increases till the moment of intercept- 
 ing the steam ; and that, from that moment, the 
 steam expands in the cylinder without any loss of 
 the total heat which it contains. Now these dif- 
 ferent circumstances no longer occur in the engines 
 under consideration, and therefore equation (A) is 
 not applicable to them. 
 
 To calculate now the point at which the flowing 
 of the steam into the cylinder is to be intercepted, 
 it would be necessary to establish, between the 
 forces which act on the piston in the ascending 
 stroke, a relation analogous to equation (A), in- 
 troducing into it the variations incident to the pres- 
 sure of the steam during the different periods of 
 the stroke, and the partial condensation attendant 
 on the expansion of the steam, after the communi- 
 cation from the boiler to the cylinder is intercepted. 
 This relation would make known the value of the ratio 
 
 r 
 
 r 
 
 But as the introduction of these circumstances 
 render the research extremely complicate, and as 
 it will presently be shewn to be unnecessary for at- 
 taining the knowledge either of the velocity or of 
 the useful effects of the engine, we deem it needless 
 to pursue the enquiry. 
 As to equation (B), viz. 
 
 -(!+ a) r-/"-n 
 
 n 
 
ATMOSPHERIC ENGINES. 315 
 
 since it expresses the circumstances of the motion 
 during the descending stroke of the piston, and 
 since these circumstances suffer no change from 
 the fact of the cooling and reheating of the cylin- 
 der during the rising stroke, this relation evidently 
 will undergo no modification. 
 
 Finally, to obtain the equation of the velocity of 
 the piston, we will deduce it as before, from the 
 equality which necessarily subsists between the 
 expenditure of steam by the cylinder and the ef- 
 fective or useful evaporation in the boiler. Calling 
 S' the total volume of water evaporated per minute 
 in the boiler, this water is first converted, in the 
 boiler, into steam at the pressure P. This steam 
 then passes into the cylinder. There a certain 
 portion of it is condensed by the contact of the 
 cylinder, and the rest, which answers to what we 
 call the effective evaporation of the engine, passes, 
 by cooling, to the pressure P', indicated above. 
 But, as the water formed in the cylinder by the 
 partial condensation of the steam, subsists there till 
 the end of the stroke, it follows that the steam, on 
 assuming the pressure P', will find itself in contact 
 with a certain quantity of liquid, and that it will 
 in consequence be at the maximum density for 
 its pressure and temperature. Hence, expressing 
 still by S the effective evaporation of the engine 
 per minute, or the volume of water actually em- 
 ployed in working the machinery, that volume of 
 water once passed to the state of steam at the 
 
316 
 
 CHAPTER XII. 
 
 pressure P', will occupy (Chapter II., Sect. IV., 
 V., VI.) a space expressed by 
 
 S 
 
 This will, therefore, be the volume of steam re- 
 sulting from the effective evaporation of the en- 
 gine. 
 
 On the other hand, since P' is the mean pres- 
 sure of the steam in the cylinder, during the entire 
 stroke, it follows that the capacity of the cylinder 
 which, at every ascending stroke, is filled with 
 steam at that pressure, is 
 
 a(7 + c). 
 
 But at every descending stroke, a certain quan- 
 tity of steam is found compressed under the piston 
 and restored to the boiler. The pressure of this 
 steam, at the moment of its separation, is j9, and 
 the volume it occupies under that pressure is 
 
 On repassing to the pressure P', it loses no portion 
 of its total heat, since that action takes place 
 during the descending stroke ; that is to say, while 
 the cylinder undergoes no refrigeration. In this 
 mutation, therefore, its volume will vary in the ratio 
 
 n + qp . 
 
 n + qP'' 
 
 thus, under the pressure P'> this steam, restored to 
 the boiler, will represent a volume expressed by 
 
ATMOSPHERIC ENGINES. 317 
 
 Therefore the real expenditure of steam per double 
 stroke of the piston, will be 
 
 Consequently, if the engine gives M double strokes 
 of the piston per minute, the corresponding expen- 
 diture of steam will be 
 
 But, expressing by v the velocity of the piston, or 
 the space it describes in producing the useful 
 effect, we shall have 
 
 v 
 v = M /, or M = T . 
 
 The volume, therefore, of steam expended by the 
 cylinder in one minute, will be 
 c l 
 
 Consequently, making it equal to the volume of 
 steam resulting from the effective evaporation of 
 the engine, and which has been shown above, the 
 following equation will be deduced : 
 
 _S _ 1 _ 
 
 "a ' l + c , -p,, l-l" + c , -, . (1) 
 
 C*>.ff) --- y--(+?iO 
 
 And replacing P' by its value drawn from equation 
 (C), it may equally be written under the form 
 
 S 1 
 
 11 , 
 
 a l + c- . -, ,_ ll" + c 
 ' 
 
318 CHAPTER XII. , 
 
 This equation, therefore, must now replace that 
 which we have obtained for the case of engines 
 with a condenser ; and it will be remarked that it 
 differs from that equation, only in the substitution 
 of / and P' for V and P. Thus the only modifica- 
 tions to be introduced into the equations obtained 
 for the case of engines with a condenser, will be : 
 first, the substitution just mentioned, and secondly, 
 the suppression of equation (A), which is not appli- 
 cable to engines without a condenser. 
 
 SECT. II. Of the difference between the total and 
 the effective evaporation of the engine, in con- 
 sequence of the refrigeration of the cylinder at 
 every stroke of the piston. 
 
 To enable us to pass on to the application of 
 the formulae just given, there is yet a point which 
 requires some development. We have made, in 
 the engines under consideration, a distinction be- 
 tween the total evaporation and the effective or use- 
 ful evaporation. The latter is that which figures 
 in our formulae, whereas the total evaporation is 
 that alone which can be directly observed, or which 
 can be deduced from the measure of the heating 
 surface of the boilers. It is necessary then to 
 furnish means of passing from the knowledge of 
 one of these quantities to the knowledge of the 
 other. 
 
 When the condensation of the steam is not 
 effected in a separate condenser, but in the steam 
 
ATMOSPHERIC ENGINES. 319 
 
 cylinder itself, the metal of the cylinder is, at each 
 stroke of the piston, first cooled to the temperature 
 of condensation, and then reheated by the steam of 
 the boiler to the temperature corresponding to the 
 pressure P', which the steam definitively assumes 
 in the cylinder. At each stroke then of the piston, 
 there is occasioned, in mere loss, a condensation of 
 steam depending on the quantity of metal to be 
 heated, and on the difference of temperature be- 
 tween the two successive states of the cylinder ; 
 and it is this incidental loss which causes the 
 difference between the total evaporation and the 
 effective evaporation of the engine. 
 
 Now, in the same engine, the quantity of steam 
 thus condensed at every stroke, is proportional to 
 the quantity of heat absorbed by the metal of the 
 cylinder, that is to say, to the difference between 
 its two successive temperatures. Expressing these 
 two temperatures by T' and t, the quantity of 
 steam condensed will be proportional to the dif- 
 ference 
 
 Moreover, between two engines wherein these 
 two temperatures were the same, but which should 
 differ from each other as to the extent of surface 
 exposed to the contact of the steam, the quantity of 
 steam condensed will be in the ratio of the extent 
 of surface and of the thickness of the metal ; or, 
 since the thickness of the metal may be taken as 
 proportional to the diameter of the cylinder, the 
 
320 CHAPTER XII. 
 
 condensation in question would be proportional to 
 the product of the surface exposed to refrigeration, 
 by the diameter of the cylinder. 
 
 If we term d the diameter of the cylinder, and 
 TT the ratio of the circumference to the dia- 
 meter, / and c still expressing the length of the 
 stroke and the clearance of the cylinder, the sur- 
 face exposed to refrigeration, including the lower 
 surface of the piston and the bottom of the cylin- 
 der, will evidently be 
 
 7T d (I + C) + -L 7T d 2 . 
 
 But as, in these engines, the diameter of the cylin- 
 der is usually made equal to two thirds of the 
 stroke, or 
 
 d=il, 
 
 the refrigerated surface will have for its measure, 
 more simply 
 
 Thus, considering at once the difference of tem- 
 perature, the extent of surface to be heated, and 
 the thickness of the metal, the quantity of steam 
 condensed at every stroke will be proportional to 
 the product of those three quantities. If then we 
 measure the steam under the pressure P', which it 
 assumes definitively in the cylinder, and express 
 by H a constant coefficient, the volume of steam 
 condensed at each stroke will be of the form 
 
 But the steam turned to use at each stroke in the 
 
ATMOSPHERIC ENGINES. 321 
 
 cylinder, occupies there, under the pressure P', 
 
 a volume expressed by 
 
 I i 
 
 4 
 
 Therefore the steam condensed will be to the 
 steam turned to use, in the ratio indicated by the 
 number 
 
 and as the effective steam, plus the steam con- 
 densed, represents the total steam produced, the 
 steam effectively turned to use in the cylinder will 
 be definitively to the total production of steam, in 
 the ratio 
 
 S 1 
 
 3 
 
 The factor H, contained in this expression, 
 depends on the capacity of the steam and metal 
 for heat, and ought necessarily to be determined, 
 once for all, by direct experiment. 
 
 For this purpose, must be measured accurately 
 in an engine, and during a sufficient length of 
 time : 1st, the temperature of the water coming out 
 of the eduction pipe, that is, issuing from the 
 cylinder, after the definitive condensation of the 
 steam which has produced its effect ; 2dly, the mean 
 temperature of the cylinder, taken by placing a 
 thermometer in contact with it, for experience 
 proves that a thermometer so placed will readily 
 stop at a constant degree ; 3dly, the quantity of water 
 
CHAPTER XII. 
 
 coming out of the cylinder after the definitive con- 
 densation of the steam ; 4thly, the quantity of water 
 injected into the cylinder to produce that condens- 
 ation ; and, finally, a reckoning must be kept of 
 the number of strokes of the piston given by the 
 engine during the continuance of the experiment. 
 
 This premised, 1st, the temperature of the water 
 coming out of the cylinder, after the definitive 
 condensation of the steam, will also be that of 
 the steam in contact with that water ; that is to 
 say, it will be the temperature t of the steam 
 imperfectly condensed in the cylinder. 
 
 2dly, The observed temperature of the cylinder 
 will be a mean between the temperature of con- 
 densation and the temperature of the steam while 
 it exerts its action in the cylinder ; since, during 
 every descending stroke of the piston, the cylinder 
 tends to be cooled to the temperature of condens- 
 ation, and during every rising stroke, on the 
 contrary, it tends to be heated to the temperature 
 of the steam during its effective action. Putting, 
 therefore, T for the observed temperature of the 
 cylinder, we have 
 
 and consequently 
 
 T' = 2T-. 
 
 Thus will readily be known T', or the mean tem- 
 perature of the steam during its action in the 
 cylinder, that is, while its pressure there is P'. 
 
ATMOSPHERIC ENGINES. 323 
 
 3dly, The useful expenditure of steam at the 
 pressure F, made by the cylinder at each stroke, 
 is expressed by 
 
 a (I + c). 
 
 Therefore, multiplying this quantity by the num- 
 ber of strokes of the piston during the experiment, 
 we shall have the volume of steam effectively turned 
 to use in the cylinder. 
 
 To conclude from this the corresponding volume 
 of water, it suffices to observe that, since the water 
 of condensation, which forms in the cylinder, is not 
 withdrawn by degrees as it is formed, the steam 
 effectively used in the cylinder is always found in 
 contact with the liquid ; that is to say, it is there, 
 as has already been said, at the maximum density 
 for its temperature. Now we have given (Sect. III. 
 Chapter II.) a table of the relative volumes of the 
 steam in contact with the liquid, under divers tem- 
 peratures. Taking then in that table the relative 
 volume corresponding to the temperature T'; since 
 that relative volume is nothing more than the ratio 
 of the volume of the steam to that of the same 
 weight of water, it will suffice to divide by that 
 ratio, the volume of the steam effectively turned to 
 use, which has just been obtained, in order to con- 
 clude the corresponding volume of water, or the 
 effective evaporation S of the engine. 
 
 4thly, The total quantity of water which comes out 
 of the cylinder after condensation, being diminished 
 by the injection water and the water turned to effec- 
 
324 CHAPTER XII. 
 
 tive use, will make known the volume of water 
 condensed by the contact of the cylinder. 
 
 Thus will readily be obtained the ratio between 
 the volume of water condensed in consequence of 
 the refrigeration of the cylinder, and the volume 
 of water turned to useful account. Let M then be 
 that ratio. Since we have seen that it has equally 
 for its general expression the quantity 
 
 we shall have 
 
 3 
 
 and since the temperatures T' and t are known, as 
 well as the quantities / and c, it follows that this 
 equation will definitively make known the value of 
 the constant quantity H, which will be 
 
 4 TT_ M 
 
 a 
 
 Experiments, such as those of which we have 
 just pointed out the process, are as yet wanting ; 
 but till such experiments shall have been made 
 with all proper care, we may make use of some 
 practical observations, which have not indeed the 
 degree of precision necessary to lead to the exact 
 determination of the quantity H, but which, never- 
 theless, will authorize an approximate value being 
 provisionally assigned to it. 
 
 Watt has recognised from a great number of 
 
ATMOSPHERIC ENGINES. 325 
 
 direct observations, ( Watt on the Steam JZngine, 
 pages 66 and 9-5,) that in the atmospheric engines 
 least liable to the loss under consideration, the 
 condensation which arises in consequence of the 
 refrigeration of the cylinder amounts to '75 of the 
 efficient steam ; and that in those which, on the 
 contrary, are most liable to it, that condensation 
 amounts to twice the efficient steam ; and he 
 found at the same time, that the temperature of 
 condensation in these engines, varies between 174 
 and 142 degrees of Fahrenheit's thermometer. 
 
 The engines least liable to loss of steam by the 
 cooling of the cylinder are evidently those in which 
 the condensation takes place at the highest tem- 
 perature, and in which the clearance of the cylin- 
 der is the smallest in use, since that clearance aug- 
 ments by so much the mass of metal to be heated. 
 They are, therefore, the engines in which the con- 
 densation takes place at 174 degrees of tempera- 
 ture, and in which the clearance of the cylinder is 
 but one tenth of the effective stroke of the piston. 
 
 The engines most liable to the same loss are, 
 for the opposite reason, those in which the con- 
 densation is effected at the temperature of 142 de- 
 grees, and in which the clearance of the cylinder 
 amounts to three tenths of the stroke. 
 
 On the other hand, a tolerably accurate valua- 
 tion may be made of the pressure, and consequently 
 of the temperature assumed by the steam in the 
 cylinder of these engines, during the ascending stroke 
 of the piston. It has, in effect, been seen that 
 there is equilibrium between that pressure and the 
 
326 CHAPTER XII. 
 
 resistance of the piston. Now the resistance of the 
 piston consists of the atmospheric pressure, aug- 
 mented by the friction of the engine and diminished 
 by the counterweight. If then the counterweight 
 be considered as equi-balancing the friction of the 
 engine, which experience indicates very nearly in 
 Watt's engines, we see that the pressure P' of the 
 steam in the cylinder will be equal to the atmo- 
 spheric pressure ; and consequently, the tempera- 
 ture T' corresponding to that pressure will gene- 
 rally be 212 degrees of Fahrenheit, or 100 degrees 
 of the centigrade thermometer. 
 
 Making use then of this as a mean valuation, 
 applicable without great error to the observations 
 of Watt, we see that those observations will furnish, 
 for the approximate value of the factor H, the two 
 following equations : 
 
 |H= - -7* ='0050, 
 
 - H= - i-, - =-0076. 
 (212 -142) 3 + - 
 
 Between these two values the mean would be*0063; 
 but as the second case, that is, a loss so considerable 
 as twice the efficient steam, is of much more rare 
 occurrence than the first ; and as it is an extreme 
 case which may depend on an ill-contrived arrange- 
 ment of the parts of the engine, or on some acci- 
 dental cause of which no account can be taken in 
 the calculation, we deem it more correct to hold to 
 the first of the above determinations, viz. : 
 
-ATMOSPHERIC ENGINES. 327 
 
 - Hzz'0091. 
 
 3 
 
 Thus, till more precise researches be made on 
 this subject, we will value, in these engines, the 
 effective evaporation in terms of the total evapo- 
 ration of the boiler, from the following analogy, 
 in which the temperatures are taken from 
 Fahrenheit's thermometer. 
 
 S: 
 
 In this equation, t expresses the temperature of 
 the steam imperfectly condensed in the cylinder 
 during the descending stroke of the piston, and T' 
 the mean temperature of the steam in the cylinder 
 during the, rising stroke of the piston. The latter 
 temperature being, besides, that which corresponds 
 to the pressure P' determined by equation (C), or 
 
 F=4>+/-n, 
 
 we see that, without any direct observation, it will 
 readily be known from the tables given in Sect. III. 
 Chapter II. In practice, however, in the greater 
 number of cases, we may still more simply, take 
 for T' the temperature which answers to the 
 atmospheric pressure, namely, 212 degrees of 
 Fahrenheit's thermometer. 
 
 This estimate of the effective evaporation is, 
 without doubt, but a very imperfect approximation ; 
 but till more precise researches be made, it may 
 perhaps be found sufficient for engines in which so 
 
328 CHAPTER XII. 
 
 little regard is had to the exact employment of the 
 motive force, that no scruple is made at losing from 
 one half to two thirds of the evaporation effected in 
 the boiler, that is, of the motive force produced by 
 the engine. 
 
 The quantity S, determined by the above equa- 
 tion, is that which must be used to determine the 
 effects of the engine ; and referring to the modi- 
 fications denoted in the preceding section of this 
 article, we see that the formulae proper to calcu- 
 late the proportions or the effects of atmospheric 
 engines without condenser, will be the following : 
 
 p '=<K/ / - n (C) 
 
 S 1 
 
 - + <f>-a+SV-/ v/ -IJ 
 
 n 
 
 (B) 
 
 v = -~ rrr- r~Jn C 1 ) 
 
 v I ~T L s 
 
 (3) 
 
 u.E. = arv ................... (4) 
 
ATMOSPHERIC ENGINES. 329 
 
 To make use of these formulae, it is obvious that 
 the first thing to be done is to determine P', or the 
 pressure in the cylinder, by means of equation (C). 
 Then, knowing P', the corresponding tempera- 
 ture T' will *be found immediately, by the help 
 of the tables given in Sect. III., Chapter II. 
 Finally, that value of T', substituted in equa- 
 tion (D), will make known the effective evapora- 
 tion S. This done, in the different problems that 
 may occur to resolve, we have only to proceed 
 according to the mode explained in Sect. III., 
 Article I., of the present chapter. 
 
 ARTICLE III. 
 
 PRACTICAL FORMULAE FOR ATMOSPHERIC ENGINES, AND 
 EXAMPLE OF THEIR APPLICATION. 
 
 To obtain the practical formulae suitable to the 
 calculation of atmospheric engines, the constant 
 quantities in the algebraic equations developed 
 above, must be replaced by their value deduced 
 from experiment or from observation. 
 
 In these engines, the pressure P of the steam in 
 the boiler, is usually from one pound and a half to 
 two pounds per square inch above the atmospheric 
 pressure ; that is to say, we have in general 
 
 P = l6*5 x 144 Ibs. per square foot. 
 The atmospheric pressure varies according to the 
 state of the atmosphere, and in delicate experiments, 
 it is necessary to measure that pressure accurately 
 
330 CHAPTER XII. 
 
 by the barometer ; but in the general calculations, 
 it may be taken at its mean value, which is 14 -7! Ibs. 
 per square inch. Referring it then to the square 
 foot, we have 
 
 = 14-71x144 Ibs. 
 
 The pressure p of condensation in the cylinder, 
 ought to be observed directly in every case wherein 
 the thing may be possible. For this purpose Watt's 
 indicator of the pressure is to be used, if the engine 
 has a condenser ; and if not, it will suffice to take 
 the temperature of the water issuing from the cy- 
 linder after condensation. This temperature being 
 at the same time that of the steam with which the 
 water was in contact, by consulting the tables of 
 correspondence between the pressure and the tem- 
 perature of the steam in contact with the liquid, 
 (Chapter II. Sect. III.,) we shall have the pressure 
 of condensation under the piston. The tempera- 
 ture of condensation thus observed in a great num- 
 ber of atmospheric engines, has been found to vary 
 between 142 and 174 degrees of Fahrenheit's ther- 
 mometer ; which corresponds to the pressures of 
 condensation comprised between 3 and 7 Ibs. per 
 square inch. Thus, referring this pressure to the 
 square foot, we shall have most commonly 
 jt? = 4'7 x 144 Ibs., and t- 158. 
 
 As to the value of the frictions f, f" and S, we 
 want special experiments to give us sure notions on 
 the subject ; but in order to shew the process of 
 the calculation in the applications, we will approxi- 
 
ATMOSPHERIC ENGINES. 331 
 
 matively take these quantities, at the same valua- 
 tion as in Watt's engines. The construction of these 
 two kinds of engines being little dissimilar, gives 
 us in fact room to think, that this valuation may be 
 near enough to the truth, for the results obtained 
 by this means to be of some utility in practice. 
 
 We will take then the friction of the unloaded 
 engine, in either stroke, to be '5 Ib. per square inch 
 of the surface of the piston, in engines which have 
 a cylinder of about 48 to 50 inches diameter, and 
 as varying with the proportions of the cylinder, 
 according to what practice has indicated in Watt's 
 engines; that is to say, amounting to 1*5 Ib. in 
 engines having a cylinder of only 17 inches diameter, 
 or diminishing, on the contrary, in a similar way, 
 when the cylinder has larger dimensions. We will 
 also, till a special determination be obtained, take 
 the additional friction B, at the same rate as in the 
 locomotive engines. 
 
 For the engines then which have a cylinder 
 differing little from 50 inches diameter, we shall 
 have 
 
 f -f" - 72 Ibs. per sq. foot of the surface of the 
 piston ; and 3 = -14. 
 
 Finally, these engines being condensing engines, 
 the value of the coefficients n and ^, of the relative 
 volume of the steam, will be 
 
 n = -00004227, 
 ? = -000000258. 
 
 To obtain the definitive practical formulae, suit- 
 
332 CHAPTER XII. 
 
 able to the calculation of these engines, it would be 
 necessary to substitute in the algebraic equations 
 developed above, the value of each of the constant 
 quantities which figure in them ; but as the pres- 
 sures P and p vary in different engines, and as the 
 valuation which we have given of the frictions, 
 varies likewise according to the diameter of the 
 cylinder, we shall merely substitute the values of 
 <, n and q. The algebraic equations will then be 
 replaced by the following : 
 
 Practical formula for atmospheric engines with a 
 condenser. 
 
 n .................................... Regulation of 
 
 164 + P theascending 
 
 stroke of the 
 piston. 
 
 Regulation of 
 the descend- 
 ing stroke of 
 the piston. 
 
 ar = [2282-/'-n-.(164+j>)ft*] ......... Useful load of 
 
 the piston, in 
 Ibs. 
 
 10000 
 
 a ?(-4227+-00258P) - l ^l (-4^227 +'Q0258p).... 
 
 Velocity of the 
 
 piston, in feet 
 per minute. 
 
ATMOSPHERIC ENGINES. 
 
 333 
 
 
 
 u.E. zz 
 
 u.HP. = 
 
 u.E. 
 
 33000' 
 
 u. E. 1 Ib. co. zz 
 
 u.E. 
 
 N 
 
 u. E. 1 ft. wa. zz 
 
 u.E. 
 
 Q. co. for 1 hp.zz 
 
 33000 N 
 u.E. 
 
 Q. wa. for I hp. zz 
 
 33000 S 
 u. E. ' 
 
 , Effective eva- 
 poration, in 
 cubic feet of 
 water per mi- 
 nute. 
 
 Useful effect, in 
 Ibs. raised 1 
 foot per 
 nute. 
 
 mi- 
 
 . Useful horse- 
 power. 
 
 . Useful effect of 
 1 Ib. of coal, in 
 pounds raised 
 1 foot. 
 
 . Useful effect of 
 1 cubic foot 
 of water eva- 
 porated, in 
 pounds raised 
 1 foot. 
 
 , Quantity of coal, 
 in pounds,pro- 
 ducing 1 horse- 
 power. 
 
 Quantity of wa- 
 ter, in cubic 
 feet, producing 
 one horse- 
 power. 
 
334 CHAPTER XII. 
 
 Horse - power 
 produced per 
 Ib. of coal. 
 
 u. HP. 1ft. wa. = Ut Et . ........ Horse - power 
 
 33000 S _ f 
 
 produced per 
 
 cubic foot of 
 water evapor- 
 ated. 
 
 Those formulae suit the case in which the engine 
 is furnished with a separate condenser ; but for 
 engines without condenser, the following must be 
 used : 
 
 Practical formula for atmospheric engines without 
 a condenser. 
 
 '=p+/ n ...................... . ................... Total pressure 
 
 of the steam 
 in the cylin- 
 der, during 
 the rising 
 stroke. 
 
 CJ 1 
 
 _ _ _ ....... . ..... .. Ratio between 
 
 '-) ( 8 +7Z~c) 
 
 _ 
 
 effetive 
 
 evaporaton 
 of the engine, 
 and the total 
 evaporation 
 of the boiler. 
 
 ///= ,2282--(l+3)r-./''-n ..................... Regulation of 
 
 the descend- 
 ing stroke of 
 the piston. 
 
ATMOSPHERIC ENGINES. 335 
 
 . -fL[2282 /" n (164-fjo)^"] Useful load of 
 
 the piston, in 
 Ibs. 
 
 10000 
 
 / + c f-4^ 
 
 ;7-j-'00 9 58P') . (*4??7 4-*00?58ij) 
 
 / 
 
 I 
 Velocity of the 
 
 
 piston, in feet 
 per minute. 
 
 .................. Effective eva- 
 
 poration, in 
 cubic feet of 
 water per 
 minute. 
 
 u. E ................ arv ............................ Useful effect, 
 
 in Ibs. raised 
 one foot per 
 minute. 
 
 " HP ............... = 3l 
 
 power. 
 
 u. E. 1 Ib. co. ... = HlJ!l ........................ Useful effect 
 
 of 1 Ib. of 
 coal, in Ibs. 
 raised 1 foot. 
 
 u. E. 1 ft. wa. ... = ^~ ........................ Useful effect 
 
 of 1 cubic 
 foot of wa- 
 ter evapor- 
 ated, in Ibs. 
 raised 1 foot. 
 
336 CHAPTER XII. 
 
 Q. co. for 1 hp.... = 330 Q N Quantity of 
 
 u. E. , . ,. 
 
 coal, in Ibs., 
 
 which pro- 
 duces one 
 horse-power. 
 
 Q. wa. for 1 hp.... z= 8300 ^ S/ Quantityofwa- 
 
 u. E. . , . 
 
 ter, in cubic 
 
 .feet, which 
 produces one 
 horse-power. 
 
 .HP.forllb.co.= JL|_ Horse -power 
 
 produced per 
 Ib. of coal. 
 
 u. HP. for 1 ft. wa.= JL|_, Horse - power 
 
 produced per 
 cubic foot of 
 water eva- 
 porated. 
 
 In the first of these formulae, we leave the at- 
 mospheric pressure expressed by </>, in order that, 
 whenever it may he more convenient, the pressure 
 P' may he calculated according to the ordinary 
 measure, that is, in pounds per square inch, instead 
 of pounds per square foot ; which will he done hy 
 assigning to <f>,f and II their value referred to the 
 same unit. 
 
 In all engines wherein there exists a difference 
 between the total and the effective evaporation, it 
 is plain that when the effects due to a determined 
 evaporation are sought, the effects in question are 
 always those of a total determined evaporation. 
 
ATMOSPHERIC ENGINES. 337 
 
 It is for this reason that in the formulae relative to 
 these determinations, we have introduced S' instead 
 of S. If, however, it were required to know the 
 effects arising from a given effective evaporation, 
 it would suffice to restore in the formulae the 
 quantity S, and they would then give the effects 
 sought. 
 
 Now to shew a numerical application of these 
 formulae, we will suppose an engine without con- 
 denser, offering the following dimensions and data : 
 
 Diameter of the cylinder, 52 inches ; or surface 
 of the piston a 14*75 square feet. 
 
 Stroke of the piston, l=J feet. 
 
 Clearance of the cylinder, -29 of the useful 
 stroke of th piston ; or c = '29 / 
 
 Total pressure of the steam in the boiler, 16*5 Ibs. 
 per square inch ; or P = l6'5 x 144 Ibs. per square 
 foot. 
 
 Temperature of condensation, =152 degrees of 
 Fahrenheit; which gives for the pressure of con- 
 densation, 4 Ibs. per square inch, or p = 4< x 144 Ibs. 
 per square foot. 
 
 Total evaporation of the boiler, 1 '50 cubic feet 
 of water per minute ; or S' = 1*50. 
 
 Consumption of coal in the same time, 11*9 Ibs.; 
 or N= 11-9- 
 
 Counterweight, 1*25 Ibs. per square inch of the 
 surface of the piston ; or II 1*25 x 144 Ibs. 
 
 With these data, it is required to determine the 
 effects to be expected from the engine. Adopting 
 then first the counterweight such as is indicated, 
 
338 
 
 CHAPTER XII. 
 
 and supposing that different loads be given to the 
 engine ; trying afterwards divers values of the 
 counterweight, to find that which will produce the 
 most advantageous useful effect, and performing 
 the calculation according to the mode indicated, 
 Sect. III., IV., V., VI., and VIII., Art. I., and 
 Sect. II., Art. II. of this Chapter, the following 
 results will be obtained : 
 
 Effects of the engine with the counterweight given. 
 
 144 
 
 Maximum of 
 useful effect. 
 
 r-. .................... = 7 ............. 7-65 ......... 7-80 
 
 144 
 
 ar ..................... = 14866 ......... 16246 ......... 16565 
 
 ........................ = 81-03 ......... 76-29 ......... 74-48 
 
 S' ..................... = 1-50 ......... 1-50 ......... 1-50 
 
 u. E ................... rr 1204510 ...... 1239370 ...... 1233670 
 
 u. HP ................ 36-50 ......... 37'56 ......... 37-38 
 
 u. E. 1 Ib. co ....... = 101220 ...... 104150 ...... 103670 
 
 u. E. 1 ft. wa ....... = 803010 ...... 826250 ...... 822450 
 
 Q. co. for 1 hp ...... =r -326 ......... -317 ......... -318 
 
 Q. wa. forlhp ...... = -0411 ......... -0399 ......... -0401 
 
 u. HP. for 1 Ib. co. =r3-07 .......... 3-16 ......... 3-14 
 
 n. HP. for 1 ft. wa. = 24-33 ......... 25-04 ......... 24-92 
 
 Maxima effects of the engine with divers 
 counterweights. 
 
 Absolute max. of 
 useful effect. 
 
 ......... ............ ~ 1-25 ......... 1-50 ......... 1-75 
 
 ^ ................... =7-65 ......... 7-40 ......... 7-20 
 
 144 
 
 ar .............. 16246... . 15715... . 15290 
 
ATMOSPHERIC ENGINES. 339 
 
 Absolute max. of 
 useful effect. 
 
 v =z 76-29 79-09 80-92 
 
 S' = 1-50 1-50 1-50 
 
 u. E .' = 1239370 1242880 1237230 
 
 u.H.P = 37-56 37-66 37-49 
 
 u. E. lib. co = 104150 104440 103970 
 
 u. E. 1ft. wa = 826250 828590 824820 
 
 Q. co. for 1 hp = -317 '316 -317 
 
 Q. wa. for 1 hp = -0399 -0398 -0400 
 
 u. HP. for lib. co. =: 3-16 3-165 3-15 
 
 u. HP. for 1 ft. wa. = 25-04 25-11 25-00 
 
 The engine we have just submitted to calcula- 
 tion is that which Smeaton constructed at Long 
 Benton, and which is well known. In evaporating 
 the quantity of water which we have stated, its 
 ordinary velocity, with a moderate load, was 84 
 feet per minute ; and this result comes near enough 
 to those which we have obtained, to serve as a 
 practical verification of them. It will also be re- 
 cognized, on examining the effects produced with 
 different counterweights, that the counterweight 
 attributed to that engine, was not the most advan- 
 tageous possible, and that with a counterweight of 
 1%50 instead of 1*25 Ib. per square inch of the 
 surface of the piston, and an appropriate load, the 
 engine would have produced a greater useful effect. 
 
 It is easily certified besides that the most advan- 
 tageous load for the counterweight of 1*25 Ib. per 
 square inch, is in effect that of 7*65 Ibs. per square 
 inch of the piston, and that the most advantageous 
 combination for the engine consists in giving it at 
 once a counterweight of 1-50 Ib. and a load of 
 
 z 2 
 
340 CHAPTER XII. 
 
 7 -40 Ibs. per square inch ; for, on effecting the ne- 
 cessary calculations, we obtain the three following 
 tables : 
 
 14^ l ' 25 jj4= 7-60 u . E. = 1238710 
 
 7-65 1239370 max. 
 
 7-70 1237690 
 
 ilT = 1>5 7TJ= 7<3 u - E.z=1236005 
 
 144 144 
 
 7-40 1242880 ab. max. 
 
 7-50 1241910 
 
 \ 
 
 =s 1-75 = 7-10 u. E.=1234840 
 
 144 144 
 
 7-20 1237230 max. 
 
 7-30 1235805 
 
 The atmospheric engines had never yet been 
 calculated. The authors who have principally 
 devoted their attention to these matters have re- 
 nounced indicating any formulae with respect to 
 them, nor have they even attempted to apply a 
 method analogous to that of coefficients. The 
 facility with which these engines, as well as all 
 others, are calculated by the theory which we have 
 developed, affords then a final proof of the accuracy 
 of that theory ; and we consequently hope that the 
 formulae which we have deduced from it, will be of 
 some service in the construction as well as in the 
 calculation of every description of steam-engine. 
 
APPENDIX. 
 
 Concise rules designed for persons not familiar with the 
 algebraic signs, and intended to render clear and easy 
 the use of the formula contained in the work. 
 
 AMONG the persons who are engaged either in the con- 
 struction or in the working of steam-engines, and whom 
 this work may consequently interest, there is a great 
 number to whom the algebraic terms are little familiar, 
 and who usually give up the reading of a book as soon 
 as they perceive it step beyond the simple notions of 
 arithmetic. When it is intended to make a work profitable 
 to those persons, the usual practice is to annex to each 
 of the definitive formulae, an explanation, in full words, 
 of the arithmetical operations which it represents. 
 
 With the number of formulae contained in this work, 
 such a proceeding would become almost impracticable, 
 since the explanation of each series of formulae would 
 require a considerable number of pages. We think, 
 however, that the want of such explanation may be very 
 advantageously supplied here, by giving the signification 
 of every sign employed in the formulae ; that is to say, 
 by explaining what are the arithmetical operations re- 
 presented by those signs. With the help of a very few 
 rules on this subject, the persons whom this article may 
 interest, will find that the reading of the formulae is quite 
 
342 APPENDIX. 
 
 as easy in algebraic signs as if they were written in words; 
 since, after all, it is but an abridged way of expressing 
 the same things, and that, moreover, the operations to be 
 performed in order to attain the result, are much more 
 clear and more easy for the mind to seize. Again, a 
 perfect acquaintance with the signification of the signs 
 in common use can require but a few hours of attention, 
 and when once a person shall have made himself master 
 of them, he will be capable of reading not only the for- 
 mulae of this work, but all others that may present them- 
 selves in other works. We deem it then rendering a 
 service to practical men, to add here the few concise rules 
 which follow. 
 
 A, B, , by I, m, n, , 0, &c. The letters 
 
 are an abridged manner of writing the numbers 
 which those letters represent. Thus, when the 
 stroke of the piston has been measured, and found, 
 for instance, to be 17J inches, it would be incon- 
 venient to write in all the formulae, the number 
 17J. But if the length of stroke, whatever it 
 might be, has been represented by a letter, as /, for 
 instance ; then, every time the letter / occurs, there 
 needs only to recollect that it represents the num- 
 ber 17^, and performing with that number, the 
 operations indicated in the formulae, relative to the 
 letter /, the result sought will be attained. 
 
 This sign signifies equal to; it expresses that a 
 
 quantity sought is equal to the number resulting 
 from certain operations performed on other quan- 
 tities known. Thus, for instance, if we find the 
 
 expression 
 
 Vzi60 v, 
 
 this will signify that the quantity V is equal to 60 
 times the quantity v. Consequently, if we know 
 besides that the letter v represents the number 
 100, it will follow that the unknown quantity V 
 will have for its value 60 times 100, or 6000. 
 
APPENDIX. 343 
 
 This sign signifies plus (more). Placed between 
 two letters or two numbers, it indicates that they 
 are to be added together. If, for instance, there 
 be in a formula an expression of the form 
 
 it means that to the number 1 must be added the 
 number & If, then, we know besides, that the 
 letter $ represents the number *14, it follows that 
 the expression 1 + will have for its value 
 
 l-t-2=l+-14=M4. 
 
 ............ This sign indicates minus (less). Thus, when an 
 
 expression occurs of the form 
 
 P-/-2118, 
 
 the expression amounts to saying that, from the 
 number P the numbers f and 2118 are to be suc- 
 cessively subtracted. If, then, we know that the 
 letter P represents the number 9360, and that the 
 letter / represents the number 144, the expression 
 will have for its value 
 
 P/--2118=9360 144 2118=7098. 
 
 X ............ This sign expresses multiplied by. Thus the ex- 
 
 pression 
 
 aXv 
 
 indicates that the two numbers represented by the 
 letters a and v, are to be multiplied one by the 
 other ; and the product of that multiplication will 
 be the quantity expressed here by a X v. This mul- 
 tiplication to be performed, is equally expressed 
 by a point between the two letters, or by writing 
 the two letters simply together without any sign 
 interposed ; so that the expressions 
 
 amount to the same, all three expressing the result 
 of the multiplication of the numbers represented 
 by a and v. If, for instance an expression occur 
 like the following, 
 
 a r y, 
 
344 APPENDIX. 
 
 and it be known that the letter a expresses the 
 number 1-57, the letter r the number 2640-96, and 
 the letter v the number 300, the expression a r v 
 will have the value 
 
 a r vl-57 X 2640-96 X 3001243800. 
 .This sign denotes divided by. Thus the expression 
 
 
 
 a 
 
 expresses S divided by a, or the quotient resulting 
 from the division of the number expressed by S, 
 by the number expressed by a. 
 
 For instance, if we have Sir -6 7 and a~ 1-57, it 
 
 is plain that the term - will have for its value 
 a 
 
 ? =-^.=-4268. 
 a 1-57 
 
 A fraction may have its numerator or its de- 
 nominator composed of several numbers, on which 
 divers operations are indicated. In that case, those 
 operations must first be performed ; so as to reduce 
 the numerator and the denominator to single num- 
 bers, before performing the division of the one by 
 the other, as has just been said. 
 
 If, for example, we have the fraction 
 
 10000 
 
 1-492 + -002415 P' 
 
 and know besides that the letter P represents the 
 number 9360 ; we shall first perform the multipli- 
 cation of the number 9360 by the number '002415, 
 and then add to the product the number 1-492. 
 The result will be the number 24-0964, which will 
 therefore represent the denominator of the frac- 
 tion. The fraction may then be written under the 
 
 form 
 
 10000 
 24-0964' 
 
 and consequently it is reduced to the simple in- 
 dication of the quotient of two numbers, as in the 
 preceding case. 
 
APPENDIX. 345 
 
 If two fractions occur, separated by the sign of 
 addition, or that of subtraction, or that of multi- 
 plication, the meaning is that, after having sought 
 separately the quotient indicated by each of those 
 fractions, they are either to be added together, or 
 one deducted from the other, or one multiplied by 
 the other. Thus, the expression 
 
 S 10000 
 
 ~li' 1-492 + -002415 P 
 
 signifies that, after having sought the quotient indi- 
 cated by each of the two fractions, the first of these 
 quotients is to be multiplied by the second. Sup- 
 posing the letters to be of the same numerical value 
 as in the preceding cases, the product of the two 
 fractions would here be the definitive number 176. 
 It would be the same if we were to find one 
 fraction divided by another. Each of them should 
 be first reduced to a single number by finding the 
 quotient they represent, and then the one of these 
 quotients divided by the other. 
 
 ( ) or [ ] or { } "-Parentheses indicate that the different quan- 
 tities contained between them, are to be reduced 
 to a single number before performing the other 
 operations indicated in the formula. 
 
 Thus, for instance, if we find in a formula the 
 expression 
 
 this means that it is the expression (1 -J- 8) entire, 
 which is to be multiplied by v. The sum then of 
 1 +Hs first to be formed, and afterwards multiplied 
 by the number v ; whereas had we only 
 
 i-M* 
 
 this would mean that the product v is first to be 
 formed, and afterwards the number 1 added to it. 
 
 There may occur several parentheses comprised 
 one within the other, but their signification is always 
 the same. The expression 
 
 002415 [(1+ $) 
 
346 
 
 APPENDIX. 
 
 denotes that the sum of 1 +5 is to be formed first, 
 this to be multiplied by r, and the product added 
 to the quantity f, which gives the number repre- 
 sented by the outer parenthesis ; and finally that 
 this number is to be multiplied by '002415. 
 Lastly, when there occurs in the formulae a letter with a small 
 figure or exponent above it, it is the same thing as 
 writing that letter as many times successively as 
 there are units in the figure or exponent. 
 For instance, the expression 
 
 v 2 
 
 is equivalent to the expression v X v, or v written 
 twice ; that is to say, it is the product of v by 
 itself. If then v were known to be equal to 300, 
 the quantity represented by v 2 would be 
 
 = 90000. 
 
 These short explanations are all that is necessary, in 
 order to read and perfectly understand all tbe practical 
 formulae contained in this work. Replacing each of the 
 signs that are met with in a formula, by the periphrasis 
 which the sign represents, you read the formula such as 
 it ought to be expressed, and effecting the arithmetical 
 operations indicated by those signs, you attain the result 
 sought. A formula is then nothing more than an abridged 
 manner of writing the series of operations to be performed, 
 in order to arrive at the result which we want to obtain. 
 
 We will subjoin to this explanation some examples, 
 taken from the practical formulae of high-pressure engines 
 (pages 151 and 153). 
 
 I. Suppose we have the formula 
 
 _ S 10000 
 
 ~ 
 
 a 6-6075 + '00241 5 
 which is intended to determine the unknown value of v ; 
 and let it be supposed that we know, besides, that the 
 other letters comprised in this formula have the following 
 value (page 153): 
 
APPENDIX. 347 
 
 S = -67 
 a = 1-57 
 S = -14 
 r = 2641 
 =1 144*. 
 
 First form the sum (1+5), indicated in the inner parenthesis, 
 which will be 
 
 1+5=1-14. 
 
 Then multiply this number by r or 2641, and the result will be 
 (l+J)r=M4x2641=S010. 
 
 Add to this / or 144, and the sum will consequently be the 
 quantity indicated by the outer parenthesis, viz. 
 
 [(l+5)r+/]=3154. 
 
 Now multiply this sum by the number -002415, and the product 
 will evidently be 
 
 002415 [(1 +5) r+/]=-002415X3154=7-6170. 
 Add to this last result the number 6 f 6075, and you obtain 
 6-6075 + -002415 [(1+5) r+/]zz6'6075 + 7-6170=14-2245. 
 
 This is then the denominator of the fraction which forms the 
 second member of the formula. Performing the division of 
 the number 10000 by the number just obtained, the quotient 
 will be 
 
 _ 1QOQQ _ _ 10000 = 703.04 
 6-6075 + -002415 [(1 + 5) r+f] ~~ 14-2245 " 
 
 On the other hand, dividing S by a, or the number -67 by the 
 
 C 
 
 number 1-57, you have the value of the fraction , viz. 
 
 S - = L = -4268. 
 a 1-57 
 
 Finally, then, multiplying this latter quotient by that obtained 
 immediately above, you have definitively 
 
 S 10000 
 
 a 6-6075 + -002415 
 
 = 4268x703-04 300. 
 
 Thus, it is clear that, by effecting successively the series 
 of calculations indicated by the few signs which we have 
 
348 APPENDIX. 
 
 explained, and proceeding gradually from the most sim- 
 ple terms to the more compounded ones, we arrive without 
 difficulty at the definitive result. 
 
 We will give some other examples of these calculations; 
 but instead of effecting the operations, we will merely 
 express in words the signification of the formula, which 
 amounts to the same. 
 
 II. Suppose we have the formula 
 
 arm 4140750 **- (2736 +/), 
 (14-1)0 l+ 
 
 this signifies that the required value of ar will be obtained 
 by performing the following arithmetical operations : 
 
 Add 1 to the number represented by the letter $, 
 and multiply the sum by the number v. 
 
 Then divide the number S, by the product thus 
 obtained; multiply the quotient of this division by 
 the number 4140750; and write apart this first 
 partial result, which represents the first term of the 
 formula. 
 
 Add again to unity the number $, and by that sum 
 divide the number a. 
 
 Similarly add to the number 2736 the number /, 
 and multiply the sum by the last found quotient ; 
 and set apart this partial result, which represents 
 the second term of the formula. 
 
 Finally, from the first partial result, subtract the 
 second, and the difference will be the quantity a r 
 sought. 
 
 Performing these different operations with the values of 
 S, , 2, r and /, given above, and supposing the case 
 wherein the letter v has the value v=300, you find that 
 the quantity a r will have for its definitive value 
 
 III. If we have the formula 
 
 6-6075+ -002415 
 
APPENDIX. 349 
 
 it will amount to the following arithmetical explanation : 
 
 To the number 1 , add the number 5, and multiply 
 the sum by the number r. 
 
 To this product add the number /, and multiply 
 the resulting sum by the number '002415. 
 
 To the latter product, add the number 6-6075, 
 and keep apart this partial result, which expresses, 
 in one number, what proceeds from all the ope- 
 rations comprised in the great parenthesis. 
 
 Then multiply the number a by the number v, 
 and divide the product by the number 10000, 
 which will give you another partial result, express- 
 ing the portion of the formula situated beyond the 
 parenthesis. 
 
 Finally, multiply the former partial result by the 
 latter, and the definitive product will be the re- 
 quired value of S. 
 
 For the values above attributed to the different letters 
 contained in the formula, the result of the calculation will 
 give S='67. 
 
 IV. If we have the formula 
 
 S 10000 
 
 a 1-492 + -002415 P 
 it will be paraphrased as follows : 
 
 Multiply the number '002415 by the number P, 
 and add to the product the number 1 '492 ; divide 
 the number 10000 by the sum thus obtained, and 
 write the quotient apart. 
 
 Then divide the number S by the number a, 
 which will give a second quotient. 
 
 Finally, multiply the former quotient by the 
 latter, and the resulting product will be the re- 
 quired value of v. 
 
 With the values already indicated for the letters, and 
 moreover, for P^9860, the result of the preceding formula 
 will give T/ 176. 
 
350 APPENDIX. 
 
 V. In fine, as a last example, we will suppose the 
 formula 
 
 ar = 
 
 It plainly will signify as follows : 
 
 From the number P, deduct first the number/, 
 and again from the remainder deduct the number 
 2118. 
 
 Then to the number 1 add the number , and 
 divide the number a by the sum thus obtained. 
 
 Finally, multiply this quotient by the difference 
 before obtained, and the definitive product thus 
 formed will be the required value of a r'. 
 
 The operations thus indicated, would, for the case 
 wherein the letters should have the values already given 
 above, produce for the required value of ar', the quantity 
 9777. 
 
 Thus we see how easy it is to replace all the formulae by 
 their expressions in words, and consequently the sight of 
 algebraic formulae ought in no wise to intimidate persons 
 unfamiliar with algebra. 
 
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