mil LltfKAKY HANDBOOK OF THE STEAM-ENGINE CONTAINING ALL THE RCLES REQUIRED FOR THE RIGHT CONSTRUCTION AND MANAGEMENT OF ENGINES OF EVERY CLASS, WITH THE EASY ARITHMETICAL SOLUTION OF THOSE RULES. CONSTIirXIXG A KEY CATECHISM OF THE STEAM-ENGINE.' ILLUSTRATED BY SIXTY-SEVEN WOOD-CUTS, AND NUMEROUS TABLES AND EXAMPLES. BY JOHN BOURNE, C.E., AtmiOK OF 'A TREATISE ON THE STEAM-ENGINE,' 'A TKEATISB ON THl 8CMW-PKOPELLEB, 1 'A CATECHISM OF THE STKAM-K.NGINK, 1 ETC. NEW YORK : D. APPLETON AND COMPANY, 448 & 445 BROADWAY. 1865. TO GEOEGE TUMBULL, ESQ., 'C.E., F.R.A.S., ETC., LATE ENGINEER-IN-CHIEF OF THE EAST INDIAN RAILWAY. MY DEAE MB. TUBNBULL, In dedicating the present Work to you I am moved by two main considerations : First, to testify in the best manner I can my regard and esteem for you personally ; and Second^ to mark my sense of the skill, tact, and abiding integrity which you brought to the onerous duty of constructing the first and greatest of "the Indian railways, and of which, while in India, I had opportunities of forming a just appreciation. The public in this country traditionally so ignorant of India has yet to learn the important fact, that the works carried out under your direction in that country, are greater and more difficult than most of those which are to be found at home ; and that among other achievements, you constructed the largest bridge in the world the great bridge over the St. Lawrence alone excepted. But these technical successes, im- portant as they are, were not more eminent than those which you won over the discouragements and difficulties of the Indian official system ending, too, in gaining the esteem and appro- IV DEDICATION. bation of the Indian Government, as well as of those for whom you zealously labored for so many years in India. Whatever the benefits may be of the Indian railways, their greatest benefit is that they have taken to that country men who have impressed the people with their skill, and who have ac- quired an accurate perception of the physical wants of the country, together with all that practical knowledge of localities which will enable them to carry out with confidence, economy, and success, the numerous improvements still required by that great dependency, and upon which only a comparatively small beginning has yet been made. I remain, my dear Mr. Turnbull, Truly yours, J. BOUENE. PREFAC E. THE present work, designed mainly as a Key to my ' Catechism of the Steam-Engine,' has, during its compo- sition, been somewhat extended in its scope and objects, so as also to supply any points of information in which it appeared to me the Catechism was deficient, or whereby the utility of this Handbook as a companion volume would be increased. The purpose of the Catechism being rather to enun- ciate sound principles than to exemplify the application of those principles to practice, it was always obvious to me that another work which would point out in the plainest possible manner the methods of procedure by which all computations connected with the steam-engine were to be performed illustrated by practical examples of the appli- cation of the several rules was indispensable to satisfy VI PREFACE. the wants of the practical engineer in this department of enquiry. The present work was consequently begun, and part of it was printed, several years ago, but the pressure of other pursuits has heretofore hindered its completion ; and in now sending it forth I do so with the conviction that I have spared no pains to render it as useful as possible to the large class of imperfectly educated engineers to whom it is chiefly addressed. It is with the view of enabling its expositions to be followed by those even of the most slender scientific attainments that I have introduced the first chapter, explaining those several pro- cesses of arithmetic by which engineering computations are worked out. For although there is no want of man- uals imparting this information, there are none of them, that I know of, which have special reference to the wants of the engineer ; and none of them deal with those asso- ciations, by way of illustration, with which the engineer is most familiar. Indeed, engineers, like sailors and other large classes of men, have an order of ideas, and, to some extent, even a species of phraseology of their own ; and the avenues to their apprehension are most readily opened by illustrations based upon their existing knowledge and experience, such as an engineer can best supply. By this familiar method of exposition the idea of difficulty is dis- PREFACE. Vll pelled ; and science loses half its terrors by losing all its mystery. If I might infer the probable reception of the present work from the numerous anxious enquiries addressed to me from all quarters of the world during the last ten years, touching the prospects of its speedy appearance, I should augur for it a wider popularity than any work I have yet written. The questions propounded to me by engineers and others, in consequence of the offer I made in the preface to my ' Catechism of the Steam-Engine,' in 1856, to endeavour by my explanations to remove such difficulties as impeded their progress, have had the effect of showing more clearly than I could otherwise have per- ceived what the prevalent difficulties of learners have been ; and I have consequently been enabled to give such explanations in the present work as appeared best calcu- lated to meet those difficulties for the future. To several of my correspondents I have to acknowledge myself indebted for the correction of typographical errors in my several works, and also for valuable suggestions of various kinds, which I have made use of in every case in which they were available. I may here take occasion to notify that I have lately prepared an Introduction to my ' Catechism of the Steam- Vlll PREFACE. Engine,' which reviews the most important improvements of the last ten years ; and which, for the convenience of persons already possessing the Catechism, may be had separately. These three works taken together form a body of engineering information so elementary as to be intelli- gible by anybody, and yet so full that the attentive student of them will, I trust, be found not to fall far short of the most proficient engineers in all that relates to a knowledge of the steam-engine in its most important applications. J. BOUENE. BERKELEY VILLA, REGENT'S PAKK ROAD, LONDON: 1865. CONTENTS. CHAPTER I. ARITHMETIC OP THE STEAM-ENGINE. PAGE Principles of Numeration ...... 1 Addition ........ 10 Subtraction ........ 13 Multiplication ....... 16 Division ......... 24 Nature and Properties of Fractions .... 30 Addition and Subtraction of Fractions . . . .34 To Reduce Fractions to a Common Denominator . . 35 Multiplication and Division of Fractions . . . .38 Proportion, or Rule of Three ..... 42 Squares and Square Roots of Numbers . * .44 Cubes and Cube Roots of Numbers . . . . 48 On Powers and Roots in General . . . . .49 Roots as represented by Fractional Exponents ... 51 Logarithms ........ 52 Compound Quantities ...... 57 Resolution of Fractions into Infinite Series . . .66 Equations ......... 74 CHAPTER II. MECHANICAL PRINCIPLES OF THE STEAM-ENQINE. Law of the Conservation of Force . . . . .78 Law of Virtual Velocities . . . . . 79 Nature of Mechanical Power . 90 X CONTENTS. PAGE Mechanical Equivalent of Heat. ..... 91 Laws of Falling Bodies . . . . . .93 Motion of Fluids ....... 100 Inertia and Momentum ...... 105 Centrifugal Force ....... 107 Bodies Revolving in a Circle ...... 107 Centres of Gyration and Percussion .... 112 The Pendulum ........ 114 The Governor ....... 116 Friction ......... 118 Strength of Materials ...... 124 Strength of Pillars, Beams, and Shafts . . . .128 CHAPTER III. THEORY OF THE STEAM-ENGINE. Nature and Effects of Heat . . . . . .134 Difference between Temperature and Quantity of Heat . 136 Absolute Zero ........ 136 Fixed Temperatures ...... 137 Thermometers ........ 137 Dilatation ........ 140 Liquefaction ........ 150 Vaporisation ....... 152 Pressure of Steam at Different Temperatures . . . 157 Specific Heat .162 Phenomena of Ebullition . . . . . .168 Communication of Heat ...... 171 Combustion ' . . . . . . . 174 Thermodynamics ....... 180 Expansion of Steam . . > f~' .... 182 Velocity and Friction of Running Water . . ... 199 CHAPTER IV. PROPORTIONS OP STEAM-ENGINES. Nominal Power . . . . . . . 208 General Proportions . .k A; . .212 Steam Ports . >-^ . . , _ . | 216 Steam Pipe . . . ' ~; ! ''" ^ ' -" . . 218 Safety Valves .... >fuuiC . . .219 Feed Pipe .... .-M . . 221 HANDBOOK THE STEAM-ENGINE. CHAPTER I. ARITHMETIC OF THE STEAM-ENGINE. IN this chapter I propose to explain as plainly and simply as I can those principles of arithmetic which it is necessary to know, that we may he ahle to perform all ordinary engineering calcula- tions. In order that my remarks may he generally useful to work- ing mechanics of little education, I shall proceed upon the suppo- sition that the reader is not merely destitute of all arithmetical knowledge, hut that he has no ideas of number or quantity that are not of the most vague and indefinite description. I have known many engineers who were otherwise men of ahility to be in this condition ; and the design of these observations is to enable such, with the aid of their own common sense and their familiar associations, to arrive at tangible ideas respecting the properties of numbers, and to perform with facility all the ordinary engineering calculations which occur in the requirements of engineering prac- tice. These various topics are not beset with any serious diffi- culty. The processes of arithmetic are merely expedients for faci- litating the discovery of results which every mechanic of ordinary ingenuity would find a means of discovering for himself, if really called upon to set about the task ; and it is mainly b- 1 2 ARITHMETIC OF THE STEAM-ENGINE. cause the rationale of these processes has not been much ex- plained in school treatises, hut the results presented as feats of legerdemain performed by the application of a certain rule the reason of which is not made apparent that the idea of difficulty has arisen in connection with such enquiries. The rudest and most savage nations have all some species or other of arithmetic suited to their requirements. The natives of Madagascar, when they wish to count the nuniber of men in their army, cause the men to proceed through a narrow pass, where they deposit a stone for each man that goes through ; and hy subsequently ar- ranging these stones in groups of ten each, and these again in groups of a hundred, and so on, they are enabled to arrive at a precise idea of the number of men the army contains. A la- bourer in counting bricks out of a cart or barge makes a chalk- mark on a board for every ten bricks he hands out ; and these chalk-marks he arranges in groups of five or ten each, so that he may easily reckon up the total number of groups the board contains. These are expedients of numeration which the most moderate intelligence will suggest as conducive to the acquisi- tion of the idea of quantity ; and the rules of arithmetic are merely an extension and combination of such methods as expe- rience has shown to be the most convenient in practice to ac- complish the ends sought. It will be obvious that the number of stones or chalk-marks collected into groups in the preceding examples may either be five, ten, twelve, or any other number ; the only necessary con- dition being that the number in each group shall be the same. The concurrent practice of most nations, however, is to employ groups consisting of ten objects in each group ; no doubt from the circumstances that mankind are furnished with ten fingers, and because the fingers are much used in most primitive systems of numeration. In some cases, however, objects are reckoned by the dozen, or score, or gross ; or, in other words, a dozen, a score, or a gross of objects are collected in each group. But in the ordinary or decimal system of numeration, ten objects or units are supposed to be collected in each group, and ten of these primary groups are supposed to be collected in each higher or DIFFERENT EXPEDIENTS OF NTJMERATION. 3 larger group of the class immediately above, and so on indefi- nitely. The decimal system is so called from the Latin word dccem, signifying ten, and the word unit is derived from the Latin word unus, signifying one. Ten units form a group of ten, and ten of these groups form a group of a hundred, and ten groups of a hundred form a group of a thousand, and so on for ever. The Romans, whose numbers are still commonly used on clock faces, employed a mark or i to signify one ; two marks or n to signify two ; three marks or in to signify three ; and four marks or im to signify four. But as it would have been difficult to count these marks if they became very numerous, they employed the letter v to signify five and the letter x or a cross to signify ten, and v is the same mark as one-half of x, which was no doubt the primary of the two characters. An i appended to the left- hand side of the v or x signified v or x diminished by one, whereas each additional i added to the right-hand side of the v or x, signified one added to v or x. Thus according to the Eoman numeration rv signifies four ; vi signifies six ; ix signifies nine ; xi signifies eleven ; xn signifies twelve ; and so on. A hundred is signified by the letter o, the initial letter of the Latin word centum, signifying a hundred; and a thousand is repre- sented by the letter M, the initial letter of the Latin word mille, signifying a thousand. It is clear that the Eoman numeration, though adequate to the wants of a primitive people, was a very crude and imperfect sys- tem. It has therefore been long superseded for all arithmetical purposes by the system of notation at present in common use, and which has a distinct sign or figure for each number up to 9, and a cipher or 0, which has no individual value, but which af- fects the value of other figures. This system, which came origi- nally from India, was brought into Europe by the Moors ; and in common with most of the oriental languages, it is written from right to left instead of from left to right, like the languages of Europe, so that in performing a sum in arithmetic as in writing a word in Sanscrit or Arabic we have to begin at the right- hand side of the page. In this system the classes or orders of the objects or groups of objects is indicated by the place occu- 4 ARITHMETIC OF THE STEAM-ENGINE. pied by the figures which express their value. Thus in the case of the groups of stones employed in Madagascar, the figure 3 may he employed to designate either three individual stones, or three groups of ten each, or three groups of a hundred each ; hut in using the figure it is quite indispensable that it should appear, by some distinctive mark, which order or class is intended to be designated. "We might use the figure 3 to designate three single stones, and we might use the figure with a circle round it to de- note groups of ten each, and with a square round it to denote groups of a hundred each. But on trial of such a system we should find it to be very cumbrous and perplexing, and the method found to be most convenient is to add a cipher after the three to show that groups of tens are intended to be signified, and two ciphers to show that groups of hundreds are intended to be sig- nified. Three groups of tens, or thirty, are therefore expressed by 30, and three groups of hundreds are expressed by 300. Here the ciphers operate wholly in advancing the 3 into a higher and higher position, which, however, other figures will equally suffice to do if there are any such to be expressed. Three groups of one hundred stones in each, three groups of ten stones in each, and three individual stones, will therefore be represented by the number 333, in which the same figure recurs three times, but which is counted ten times greater at each successive place to which it is advanced, reckoning from the right to the left. Of course, the number three hundred and thirty-three might be represented in an infinite number of other ways, differing more or less from the one here indicated ; and any of the properties belonging to the number would equally hold by whatever expedient of notation it was expressed. But the manner here described is that which the accumulated experience of mankind has shown to be the most convenient; and it is therefore gen- erally adopted, though it is proper to understand that there is no more necessary relation between the number itself and the com- mon mode of expressing it, than there is between the Latin word eqmis, a horse, and that most useful of quadrupeds. In each case the relations are wholly conventional, and might be altered without in any way affecting the object. NATURE OF ARITHMETIC. 5 Arithmetic is the science of numbers. Numbers treat of magnitude or quantity ; and whatever is capable of increase or diminution is a magnitude or quantity. A sum of money, a weight, or a surface, is a quantity, being capable of increase or diminution. But as we cannot measure or determine any quan- tity except by considering some other quantity of the same kind as known, and pointing out their mutual relation, the measure- ment of quantity or magnitude of ah 1 kinds is reduced to this : fix at pleasure upon any one known kind of magnitude of the same spe- cies as that which has to be determined, and consider it as the measure or unit, and determine the proportion of the proposed magnitude to this known measure. This proportion is always expressed by numbers ; so that number is nothing more than the proportion, of one magnitude to that of some other magnitude arbitrarily assumed as the unit. If, for example, we want to determine the magnitude of a sum of money, we must take some piece of known value such as the pound or shilling and show how many such pieces are contained in the given sum. If we wish to express the distance between two cities, we must use some such recognized measure of length as the foot or mile ; and if we wish to ascertain the magnitude of an estate, we must em- ploy some such measure of surface as the square mile or acre. The foot-rule is the measure of length most used for engineering purposes. The foot is divided into twelve inches, and the inch is subdivided into half inches, quarter inches, eighths, and six- teenths. It is clear that two half inches or four quarter inches make an inch, as also do eight eighths and sixteen sixteenths ; and indeed it is obvious that into whatever number of parts the inch is divided, we shall equally have the whole inch if we take the whole of the parts of it. If the inch were to be divided into ten equal parts, then ten of these parts would make an inch. Fractional parts of an inch, or of any other quantity, are ex- pressed as follows : a half, -J-; a quarter, ; an eighth, -J- ; a six- teenth, 7 V ; and a tenth, T V The figure above the line is called the numerator, because it fixes the number of halves, quarters, or eighths, which is intended to be expressed ; and the figure below the line is called the denominator, because it fixes the 6 AKITHMETIC OF THE STEAM-ENGINE. order or denomination of the fraction, whether halves, quarters, eighths, or otherwise. Thus in the fractions f ths and Iths, the figures 3 and 7 are the numerators, and the figures 4 and 8 the denominators; and fths, |ths, or }jjths, are clearly equal to 1. So also |ths, ^ths, and jgths are clearly greater than 1, the first being equal to 1-Jth, the second to Hth, and the third to l T l ff th. The species of fractions here referred to is that which is called vulgar fractions, as being the kind of fractions in common use ; and every engineer who speaks of f ths or fths of an inch, and every housewife who speaks of f of a pound of sugar, or ^ a pound of tea, refers, perhaps unconsciously, to this species of numeration. There is another species of fractions, however, called decimal fractions, not usually employed for domestic pur- poses, but which is specially useful in arithmetical computations, and these fractions being dealt with in precisely the same man- ner as ordinary figures, are very easy in their application. In ordinary figures, the value of each succeeding figure, counting from the right to the left, is ten times greater than the preceding one, in consequence of its position ; and in decimal fractions the value of each succeeding figure, counting from left to right, is ten times less. Thus the figures 1111 signify one thousand one hundred and eleven ; and if after the last unit we place a period or full stop, and write a one after it thus, llll'l, we have one thousand one hundred and eleven and one-tenth. The period, or decimal point, as it is termed, prefixed to any number, im- plies that it is not a whole number but a decimal fraction. Thus '1 means one-tenth, two-tenths, *3 three-tenths, - 4 four- tenths, and so on. So in like manner '11 means one-tenth and one hundredth, or eleven hundredths ; '22 means two-tenths and two hundredths, or twenty-two hundredths; "33, three-tenths and three hundredths, or thirty-three hundredths ; and so on each successive figure of the fraction counting from the left to the right, being from its position ten times less than that which went before it. The number '1111 signifies one thousand one hundred and eleven ten thousandths, the first decimal place being tenths, the next hundredths, the next thousandths, the next ten thousandths, and so on. If we wish to express a hun- NATURE OF DECIMAL FRACTIONS. 7 dredth by this notation, we place a cipher before the unit thus, 01 ; if a thousandth two ciphers, -001 ; and so of all other quan- tities. The multiplication, division, and all the other arithmeti- cal operations required to be performed with decimal fractions, are conducted in precisely the same manner as if they were ordinary numbers the decimal progression being carried down- wards in the one case precisely in the same manner as it is car- ried upwards in the other case ; and it is easy to suppose that the stones used by the natives of Madagascar may not only be collected into groups of tens and hundreds, but that each stone may also be subdivided into tenths, hundredths, or thousandths, so that parts of a stone may be reckoned. Instead of dividing the stone into halves, and quarters, and eighths, and sixteenths, as would be done by the method of vulgar fractions, it is sup- posed by the decimal system of fractions to be at once divided into tenths, whereby the same system of grouping by tens, which is used above unity, is also rendered applicable to the fractional parts below unity to the great simplification of arithmetical processes. In all cases a decimal fraction may be transformed into a vulgar fraction of equal value by retaining the significant figures as the numerator, and by using as the denominator 1, with as many ciphers as there are figures after the decimal point. Thus ! is equal to y 1 ^ ; '11 is equal to ^ ; '01 is equal to T ff ; 001 is equal to y^J 3-1459 is equal to 3 T WffV; and '^ 854 is equal to T V&V In all countries there are certain recognised standards of magnitude for measuring other magnitudes by ; such as the inch, foot, yard, or mile for measuring lengths; the square inch, square yard, or square mile, or square pole, rood, or acre, for measuring surfaces ; the grain, ounce, pound, or ton for measur- ing weights ; and the penny, shilling, and sovereign for measur- ing money. It is, of course, quite inadmissible in conducting any of the operations of arithmetic to confound these different kinds of magnitudes together, and there is as much difference between a linear foot and a square foot as there is between a ton weight and a pound sterling. A square surface measuring an inch long and an inch broad is a square inch. A strip of sur- 8 ARITHMETIC OF THE STEAM-ENGINE. face 1 inch broad and 12 inches or 1 foot long will be equal to 12 square inches ; and 12 such strips laid side by side, and there- fore a foot long and a foot broad, will make 12 times 12 square inches, or 144 square inches. In each square foot, therefore, there are 144 square inches ; and as there are 3 linear feet in a linear yard, there will be in a square yard 9 square feet, as we may suppose the square yard to be composed of three strips of surface, each 3 feet long and 1 foot wide, and therefore contain- ing 3 square feet in each. A cubic inch is a cube or dice measuring 1 inch long, 1 inch broad, and 1 inch deep. A square foot of board 1 inch thick will consequently make 144 cubic inches or dice if cut up. But as it will take twelve such boards placed upon one another to make a foot in depth, or, in other words, to make a cubic foot, it follows that there will be 12 times 144, or, in all, 1,^18 cubic inches in the cubic foot. So, in like manner, as there are 3 lin- ear feet in the linear yard, and 9 square feet in the square yard, there will be 3 times 9 or 27 cubic feet in the cubic yard the cubic yard being composed of three strata 1 foot thick, contain- ing 9 cubic feet in each. Besides the square inch there is the circular inch by which surfaces are sometimes measured. The circular inch is a circle 1 inch in diameter, and as it is a fundamental rule in geometry that the area of different circles is proportional to the squares of their respective diameters, the area of any piston or safety- valve or other circular orifice will be at once found in circular inches by squaring its diameter, as it is called; or, in other words, by multiplying the diameter of such piston or orifice ex- pressed in inches by itself. Thus as a square foot, or a square of 12 inches each way, contains 144 square inches, so a circular foot, or a circle of 12 inches diameter, contains 144 circular inches. There is a constant ratio subsisting between a circular inch or foot and the square circumscribed around it. The cir- cular inch or foot is less than the square inch or foot by a cer- tain uniform quantity ; and this relation being invariable, it be- comes easy when we know the area of any circle in circular inches to tell what the equivalent area will be in square inches, SQUAEE, CIRCULAK, CUBIC, AND OTHER INCHES. as we have only to multiply by a certain number which will be less than unity in order to give the equivalent area. This number will be a little more than f , or it will be the decimal 7854 ; and if circular inches be multiplied by this number, we shall' have the same area expressed in square inches. Multiplying any quantity by a number less than unity, it may be here re- marked, diminishes the quantity, just as multiplying by a num- ber greater than unity increases it. To multiply by gives the same result as to divide by 2 ; and to multiply by the decimal V854: will have the effect of reducing the number by nearly a fourth, as it is necessary should be done in order to convert cir- cular into square inches ; for, seeing that the square inches are the larger of the two, there must be fewer of them in any given area. Besides the cubic inch there are the spherical, the cylindri- cal, and the conical inch, all having definite relations to one another. The spherical inch is a ball an inch in diameter ; the cylindrical inch is a cylinder an inch in diameter and an inch high ; and the conical inch is a cone whose base is an inch in diameter, and which is an inch hjgh. All these quantities are convertible into one another -just as the pound sterling is con- vertible into shillings or pence, and the ton weight is converti- ble into hundred- weights and pounds. The foundation of all mathematical science must be laid in a complete treatise on the science of numbers, and in an accurate examination of the different methods of calculation which are possible by their means. Now Arithmetic treats of numbers in particular, but the science which treats of numbers in general is called Algebra. In algebra numbers are expressed by letters of the alphabet, and the advantage of the substitution is that we are enabled to pursue our investigations without being em- barrassed by the necessity of performing arithmetical operations at every step. Thus if a given number be represented by the letter a, we know that 2 a, will represent twice that number, and a the half of that number, whatever the value of a may be. In like manner if a be taken from a, there will be nothing left, and this result will equally hold whether a be 6, or 7, or 1* 10 ARITHMETIC OF THE STEAM-ENGINE. 1,000, or any other number whatever. By the aid of algehra, therefore, we are enabled to analyse and determine the abstract properties of numbers without embarrassing ourselves with arithmetical details, and we are also enabled to resolve many questions that by simple arithmetic would either be difficult or impossible. ADDITION. The first process of arithmetic is Addition ; and here the first steps are usually made by counting upon the fingers, as an aid to the perceptions of the total amount of the quantity that has to be expressed. For example, if we hold up 5 fingers of the one hand and 3 of the other, and are asked how much 5 and 3 amount to, we at once see that the number is 8, as we either actually or mentally count the other 3 fingers from 5, designating them as 6, 7, 8 ; when, the whole fingers being counted, we know that tbe total number to be reckoned is 8. Persons even of considerahte arithmetical experience, will often find themselves either counting their fingers or pressing them down successively on the table, in order to assist their memory in performing addition. But the best course is to commit very thoroughly to memory an addition table, just as the multiplica- tion table is now commonly committed to memory by arithmet- ical students as such a table, if thoroughly mastered, will greatly facilitate all subsequent arithmetical processes. A table of this kind is here introduced, and it should be gone over again and again, until its indications are as familiar to the memory as the letters of the alphabet, and until the operation of addition can be performed without the necessity of mental effort. The sign + placed between the figures of the following table is the sign of addition termed plus, and signifies that the numbers are to be added together. The table is so plain as scarcely to re- quire explanation. The figures in the first column are obtained by adding together the figures opposite to them in any of the other columns. Thus 4 and 9 make 13, as also do 5 and 8 or 6 and Y. METHOD OF PERFORMING ADDITION. ADDITION TABLE. 11 2 1 + 1 3 1 + 2 4 1 + 3 2 + 2 5 1+4 2 + 3 6 1 + 5 2 + 4 3 + 3 7 1 + 6 2 + 5 3 + 4 8 1 + 7 2 + 6 3 + 5 4+4 9 1 + 8 2 + 7 3 + 6 4 + 5 10 1 + 9 2 + 8 3 + 7- 4+6 5+5 11 2 + 9. 3 + 8 4 + 7 5 + 6 12 3 + 9 4 + 8 5 + 7 6 + 6 13 4 + 9 6 + 8 4 7 14 5 + 9 6 + 8 >7 + 7 15 6 + 9 7 + 8 16 7 + 9 8 + 8 1 17 8 + 9 18 9 + 9 GENERAL EXPLANATION OF THE METHOD OP PERFORMING ADDITION. Write the numbers to be added under one another in such manner that the units of all the subsequent lines of figures shall stand vertically under the units of the first line the tens under the tens, the hundreds under the hundreds, and so on. Then add together the figures found in the units column. If their sum be expressed by a single figure, -write the figure under the units column, and commence the same process with the tens 12 ARITHMETIC OF THE STEAM-ENGINE. column. But if the sum of the figures in the units column he greater than 9, it must in that case he expressed in more than one figure, and in such event write the last figure only under the units column, and carry to the column of tens as many units as are expressed by the remaining figure or figures. Proceed in the same manner with the column of tens, and so with all the other columns. When the column of the highest order, which is always the first on the left, has been added, including the number carried from the column last added up, then if the sum be expressed by a single figure, place that figure under the col- umn. But if it be expressed in more figures than one, write those figures in their proper order, the last under the column and the others preceding it. Examples. Add togetner 1,904, 9,899, 5,467, and 2,708. The numbers are to be arranged as follows^ 1904 Here, beginning 4* the right-hand column, we say 8 9899 and 7 are 15, and 9 are 4, and 4 are 28. We write the 2jQo 8 unddr the column of units, and carry the 2 tens to the next column of tens. Adding up this column, we have 19,978 the 2 carried from the last column added to 6, which make 8, and 9 are 17. Here we write down the 7 and carry the 1 over to the next column. In the third column we have 1 carried from the last column added to 7, which makes 8, and 4 are 12, and 8 are 20, and 9 are 29. Here we write down the 9 and carry the 2 to the next column. In the fourth col- umn we have the 2 carried from the last column, which added to 2 makes 4, and 5 are 9, and 9 are 18, and 1 are 19, which sum of 19 we write at the foot of the column, the 9 under the other figures and the 1 preceding it. The total sum of these several numbers therefore, when added together, is nineteen thousand nine hundred and seventy-eight. Add together the following numbers : USE OF COMMAS IN NOTATION' SUBTKACTION. 13 2808 1467 2708 5794 1407 5988 5467 9969 9969 2829 9899 1407 5794 9694 1904 2808 19,978 19,978 19,978 19,978 It is usual, for facility of reading the figures, to divide them, when they amount to any considerable number, into groups of three each, by means of a comma interposed. But the comma in no way affects the value of the quantity, but is merely used to save the trouble of counting the figures to make sure whether it is thousands, hundreds of thousands, or what other order of figures is intended to be expressed. Thus with the aid of the comma we see at once that the number 19,000 is nineteen thou- sand, or that the number 190,000 is one hundred and ninety thousand, or that the number 1,900,000 is one million nine hun- dred thousand; whereas, without the aid of the commas, we should have to count the figures to make sure of the real value of the expression. The comma, therefore, has no such signifi- cance as the decimal point, and the number may be written with or without the comma at pleasure ; but if written without it there will be more difficulty in reading the number, just as it would be more difficult to read a book if the stops were left out. SUBTRACTION. Subtraction is the reverse of addition. If we have a bag containing 20 shillings, and if we add thereto 5 shillings, 15 shillings, and 10 shillings, we can easily tell by the operation of addition that we must have 50 shillings in the bag. If, how- ever, we now withdraw the 5 shillings, the 15 shillings, and the 10 shillings, or, in all, if we withdraw 30 shillings, we shall, of course, have the original 20 shillings left ; and the operation of subtraction is intended to tell us, when we withdraw a less number from a greater, how much of the greater number we shall have left. As addition is signified by the sign + or plus, 14 AEITHMETIG OF THE STEAM-ENGINE. so subtraction is signified by the sign or minus ; and two short parallel lines = are employed as a substitute for the words equal to. As the expression, therefore, 5 + 3 means 5 increased by 3, or 8; so the expression 5 3 means 5 diminished by 3, or 2. This in common arithmetical notation would be written 5 + 3 = 8 and 5 3 = 2. "When we have a number of quantities to subtract from a greater quantity, the usual course is to add together first all the quantities to be subtracted, in order that the subtraction may be performed at a single operation. Thus in the case of the bag containing 50 shillings, from which we successively withdraw 5 shillings, 15 shillings, and 10 shillings, we first add together the 5 shillings, the 15 shillings, and the 10 shillings, so as to have in one sum the whole quantity to be subtracted, and then we can suppose the operation to be performed at a single step, as, the subtraction having been performed at different times, will not affect the amount of the sum subtracted or the sum left. Thus 50 30 = 20 ; or if we take the successive stages, we have 50 - 5 = 45, and 45 - 15 = 30, and 30 10 = 20, which is the same result as before. GENERAL EXPLANATION OF THE METHOD OF PERFORMING SUB- TRACTION. "Write the less number under the greater in such manner that the units of the second line of figures shall stand vertically under the units of the first line the tens under the tens, the hundreds under the hundreds, and so on, as in addition. Draw a straight line beneath the lower line of figures, and subtract the units of the lower line of figures from the units of the up- per line, and place the remainder vertically under the units col- umn and beneath the straight line which has been drawn. Sub- tract the tens from the tens in like manner, the hundreds from the hundreds, and so on until the whole is completed; and where there is no figure to be subtracted, the figure of the up- per line will appear in the answer without diminution, as ap pears in following examples : METHOD OP PERFORMING SUBTRACTION. 15 1864 Original number 1864 Original number 64 Number to be subtracted 32 Number to be subtracted 1800 Remainder 1832 Remainder From 7854 From 89764384 From 785068473894 Subtract 6532 Subtract 41341073 Subtract 510054103784 Answer 1322 Answer 48423311 Answer 275014370110 In these examples all the figures of the second line are less than those of the first line, and we at once see what the re- mainder at each step will be by considering what sum we must add to the less number to make it equal to the greater. Thus in subtracting 6532 from 7854, we see that we must add 2 to the 2 of the lower line to make the 4 appearing in the upper ; and we must add 2 to the 3 appearing in the lower line to make the 6 appearing in the upper. In cases , however, in which some of the figures of the lower line are larger than those ex- isting in the upper, we must borrow a unit from the preceding column, which will count as ten in the column into which it is imported, and this unit so borrowed will be added to the sum to be subtracted when that preceding column comes to be dealt with. Thus in the groups of stones used by the natives of Mad- agascar if we have 6 groups of 10 stones in each and 7 stones over, and if we want to withdraw 8 stones from the number, it is clear that, as the 7 stones not arranged in groups will not suffice to supply the 8 stones we have to furnish, we must break up one of the groups of 10 to enable the 8 stones to be surren- dered. We shall then have only 5 groups, but with the 7 stones we had before we can supply the 8 by taking only one stone from one of the groups, leaving 9 stones in it, so that, after tak- ing away the 8 stones, we shall have 5 groups of ten each and 9 stones left. This is expressed arithmetically as follows : 67 Here we say we cannot subtract 8 from 7, so that we 1 must borrow 1 from the previous column, which, when 59 imported into the column of units, will be 10; and we = therefore say 8 taken from 17 leaves 9, which 9 we place 16 ARITHMETIC OF THE STEAM-ENGINE. in the remainder. But as we have taken one of the groups from the preceding column, we have to deduct that from the six groups remaining, and we therefore say 1 from 6 leaves 5. So, in like manner, if we had to take 29 shillings from 42 shil- lings, as we cannot take 9 from 2, we take 9 from 12, borrow- ing as before a unit from the preceding column. But as we have afterwards to return this unit, we do not say 2 from 4, but 3 from 4, which leaves 1 ; or, in other words, 29 taken from 42 leaves 13, as we can easily see must be the case, as 13 added to 29 make 42. To prove the accuracy of an answer in subtrac- tion, it is only necessary to add together the two lower lines, which will produce the top one. Examples. From 1864 From 1864 Subtract 14 Subtract... 97 Remainder 1850 Remainder 1767 From 1864 From 1864 Subtract 975 Subtract. ... 1796 Remainder 889 Remainder 68 It will be seen that, by adding together the last two lines of figures in each of these examples, we obtain the first line. jt MULTIPLICATION. Multiplication is a process of arithmetic for obtaining the sum total of a quantity that is repeated any given number of times, and is virtually an abbreviated species of addition. If, for example, we have 6 heaps of stones, with 1,728 stones in each heap, we might ascertain the total number of stones in the six heaps by writing the 1,728 six times in successive lines, and adding up the sum by the method of procedure already ex- plained under the head of Addition. But it is clear that this would be a very tedious process in cases in which the number MULTIPLICATION A SPECIES OF ADDITION. 17 of heaps was great, and multiplication is an expedient for ascer- taining the total quantity by a much less elaborate method of procedure. All numbers whatever it is clear may be formed by the addi- tion of units. The consecutive numbers 1, 2, 3, 4, 5, &c., may be derived as follows : 1 = 1 1 + 1=2 1+1+1=3 1+1+1+1=4 1+1+1+1+1=5 There are certain numbers which are formed by the contin- ued addition of other numbers than 1 ; and the numbers which are formed by the continued addition of 2 may be shown as fol- lows: 2=2 2 + 2=4 2+2+2=6 2+2+2+2=8 2 + 2 + 2 + 2 + 2=10. In like manner, the numbers shown by the successive addi- tions of 3 and 4 may be thus represented : 3=3 4=4 3+3=6 4+4=8 ^ 3+3+3=9 4+4+4=12 3+3+3+312 4+4+4+4=16 8+3+3+3+3=15 4+4+4+4+4=20 Thus it will be seen that in the series of numbers proceeding upwards from 1, some can only be formed by the continued ad- dition of 1, while others may be formed by the continued addi- tion of 2, 3, or some higher number. The numbers 3, 5, and 7 cannot be produced by the continued addition of any other number than 1, while the intermediate numbers 4 and 6 may be formed, the first by the addition of 2, and the second by the con- tinued addition of 2 or 3. 18 AKITHMETIC OF THE STEAM-ENGINE. Those numbers which, cannot be formed by the continued addition of any other number than 1 are termed p rime numbers, The numbers 3, 5, 7, 11, 13, 17, &c., are prime numbers. All other numbers are termed multiple numbers ; and they are said to be multiples of those lesser numbers by the continued addi- tion of which they may be formed. Thus 6 is a multiple of 2, because it may be formed by the continued addition of 2. But it is also a multiple of 3, because it may be formed by the con- tinued addition of 3. In like manner 12 is a multiple of 2, 3, 4, and 6. In the ascending series of numbers, 1, 2, 3, 4, 5, &c., it will be obvious that each alternate number is a multiple of 2. Such numbers are called even numbers, and the intermediate numbers are called odd numbers. Thus 2, 4, 6, 8, 10, &c., are even num- bers, and 1, 3, 5, 7, 9, &c., are odd numbers. As every even number is a multiple of 2, it is clear that no even number except 2 itself can be a prime number, and every prune number except 2 itself must be an odd number. It by no means follows, however, that every odd number must be prime, and it is clear indeed that 9 is a multiple of 3, 15 of 3 and of 5, and so of other odd numbers, which cannot, therefore, be prime numbers. If we take a strip of paper an inch broad and 12 inches long, like a strip of postage stamps, it is clear that this strip will con- tain 12 square inches ; and if we take three such strips placed side by side, they will manifestly have a collective surface of 36 square inches. Nor will the result be different in whatever way we reckon the squares ; and 12 multiplied by 3 will give just the same number as 3 multiplied by 12. In like manner, 7 multi- plied by 5 is the same as 5 multiplied by 7, and so of all other numbers. In order to be able to perform the operations of multiplication with ease and expedition, it is necessary to commit to memory the product of the multiplications of numbers from 1 to 9 ; and to enable this to be conveniently done, a table of these primary multiplications, called the Multiplication Table, forms part of the course of arithmetical instruction given at schools, where, THE MULTIPLICATION TABLE. 19 however, the tables used commonly carry the multiplications up to 12 times 12. A table containing all the multiplications neces- sary to be remembered is given below ; and it is very material to the subsequent ease of all arithmetical processes, that this table should be thoroughly learned by heart, so as to obviate the hesi- tation and inaccuracy that must otherwise ensue. MULTIPLICATION TABLE. 2 3 4 5 6 I 8 9 9 18 27 36 45 54 63 72 81 8 16 24 32 40 48 56 64 7 14 21 28 35 42 49 6 12 18 24 30 36 5 10 15 20 25 4 8 12 16 3 6 9 2 4 To find the product of two numbers by this table, we must look for the greater number in the first upright column on the left, and for the lesser number in the highest cross row. The product of the two numbers will be found in the same cross row with the greater number, and hi the same upright column with the lesser number. Thus 6 times 3 are 18, 6 times 4 are 24, and 6 times 4 are 20. If we find the number 6 in the first column and pass our finger along the same line until we come vertically under the 3 in the top line, we find the number 18, which is the product required. By the same process we find the numbers 24 and 20. Having committed the multiplication table to memory, we are in a condition for performing any multiplication of common 20 AKITHIUETIC OP THE STEAM-ENGINE. numbers without difficulty. If, for example, we wish to multiply 1,728 by 2, we write the 2 under the 8 and draw a line thus : are 16. We write down the 6 1728 2 and carry the 1, which belongs to the order of tens next 3456 above, to that order. Twice 2 are 4, and the 1 carried - from the 16 of the last multiplication make 5. The num- ber 5 being less than 10, there is no figure to carry in this case. "We therefore say twice 7 are 14, where again we write 4 and carry 1, and twice 1 are 2, and 1 carried over from the last mul- tiplication make 3. It is clear that the number 1,728 is made up of the numbers 1,000, 700, 20, and 8, and the result of the multiplication would not be altered if we were to multiply these quantities separately and add them together. A Saint Andrew's cross or x is the sign of multiplication ; and 1000 x 2=2000 700x2=1400 20x2= 40 8x2= 16 3456 Here, then, we see we have precisely the same result as in the former case. But the first expedient is the simpler, and is there- fore commonly used. We shall also obtain the same result by adding 1,728 to 1,728, thus : j^g In this particular case it is as easy to add the number 1728 to itself as to multiply by 2. But when the multiplica- 3456 tion proceeds to 6, 8, or any greater number of times, it : would be very inconvenient to have to add the number to itself 6 or 8 times, and it is much easier to proceed by the common method of multiplication here explained. The number we multiply with is called the multiplier, and the number we multiply is called the multiplicand, while the number resulting from the multiplication is called the product. In the above ex- ample 2 is the multiplier, 1,728 the multiplicand, and 3,456 the product. MULTIPLIERS CONTAINING CIPHERS. 21 If the multiplier consists of two figures instead of one, the same mode of procedure is pursued, except that the whole of the figures resulting from the multiplication of the higher of the two figures is shifted one place to the left. Thus, if the number 1,Y28 has to be multiplied by 22, the mode of procedure is as follows : Here the arithmetical process of multiplication is o 2 precisely the same with each of the two figures, only that in the case of the second multiplication the result- j ! 6 ing number is set one place more to the left ; and the two lines of partial products are then added together 38,016 f or tb e answer. It is, therefore, a rule in all multipli- cations where the multiplier consists of more figures than one, that the first figure of the product shall be set under that particular figure of the multiplier with which that particular line of multiplication is performed. If instead of 22 the multi- plier had been 222, then the operation would have been as follows : Here, it will be observed, the same partial product -go is repeated in every case, but set one place more to the left ; and the several lines of partial products are then added up for the total product of the multiplication. 34 56 In cases where one of the figures of the multiplier is a cipher, the only effect is to shift the figures over to ' ' ' ' the left one place, and which may be done by adding a cipher to the product if the cipher forms the last figure of the multiplier. Thus, 1,728 multiplied by 20, is 34,560, mul- tiplied by 200 is 345,600, and multiplied by 2,000 is 3,456,000. If the cipher comes in the middle of the multiplier, as in multi- plying by 202, we proceed as follows : Here we pass over the cipher altogether, except that 202 we begin the succeeding line of multiplication one place more to the left than we should have done if the cipher * had not been present ; or, in other words, we begin the line pertaining to the next figure of the multiplier un- ' der that figure, just as would be done if any other figure than a cipher intervened. Indeed we might 22 AEITHMETIC OP THE STEAM-ENGINE. write a line of ciphers as resulting from multiplication by a cipher ; but as this line could not affect the value of the sum total, it is left out altogether. In multiplying numbers termi- nating with ciphers, or in multiplying with numbers terminating with ciphers, the mode of procedure is to perform the multipli- cation as if there were no ciphers, and then to annex as many ciphers to the product as there are ciphers in the multiplier and multiplicand together. Thus 65,000 multiplied by 3,300 is treated as if 65 had to be multiplied by 33, and then five ciphers are added to the product to give the correct answer. GENERAL EXPLANATION OF THE METHOD OF PEEFOBMINa MULTIPLICATION. The foregoing explanations of the method of performing the multiplication of numbers will probably suffice to enable all or- dinary questions in multiplication to be readily performed. But for the sake of clearness, it may be useful to recapitulate the several steps of the process. Place the multiplier under the multiplicand, as in addition. Multiply the multiplicand separately by each significant figure of the multiplier, by which we shall obtain as many partial prod- ucts as there are significant figures in the multiplier. "Write these products under one another, so that the last figure of each shall be under that figure of the multiplier by which it has been produced. Add the partial products thus obtained, and their sum will be the total product. It will often facilitate arithmetical calculations if we have committed to memory the products of numbers larger than those found in the common multiplication tables, and it is very impor- tant that these elementary multiples should be accurately and promptly recollected. In the following table the products of numbers are given as high as 20 times 20 : MULTIPLICATION TABLE EXTENDING TO 20 TIMES 20. 23 CO a -r Cl l-H 00 I 1 = 35 CO o ee co 1 SO CO eq -t co -* -T! 00 .b- o -r co co -M 00 o o co OS X 01 CO O 01 CO ** o co X 00 "M 01 t~ 0-1 -2 1C 01 JO I-l o ^ co 1O 00 01 ~ t- CN IO 1O 01 P ^ 01 JO 01 * 01 CO en 01 - ~ -r. o BC P 1- -^ o o -H O CO o 0) O rH o - OS a 00 T-H fc- C1 ee CO 2 11 co ee 01 t* i X 5 O =-. o Ci rH 00 00 o ce i-H 01 jrj l-H ^ i i CO CO rH X 01 o 01 vH t- - -t CO M CO Cl OS 1 1 01 pH JO 9 x> OS rH OS -H 00 t- ft-. g CO CO CO JO 5 co - -M PH i i 00 c ^H (M O i-l CO C3 -. : -H oo X t- 01 t- 6 ^: o J -H 1-1 00 * 01 * CO co JO O c 1-1 01 o OS 00 1O t- o fe- 1O -S o ee JO IQ "* 5 CO O JO co "* i to fc- 01 t- X (O -* CO te te JO 01 JO S8 4 5 CO '.0 01 '.0 CC 01 * o to , and we may, therefore, say that the fraction ^-=0. Now, we know that if we divide the dividend 1 by the quotient -, which is equal to nothing, we obtain again the divisor oo . Hence we learn that infinity is the quotient arising from the division of 1 by 0. Thus 1 divided by expresses a number infinitely great. But is certainly only the half of f , or the third of ; so that it would appear as if one infinity may be twice or three times greater than another. It will be obvious that as the fractions f , f, f, 1, f , "k , I, 1, &c-, are all equal to one another, each of them being in fact equal to 1, so also the fractions f, I, f , I, Y, , , &c., are all equal to one another, each of them being in fact equal to 2 ; for the numerator of each divided by the denominator gives 2. So likewise the fractions f , I, I, , , &c., are equal to one another, since in fact each of them is equal to 3. Now it is clear that as | is the same as -^-, and as f is the same as J/, both being equal to 3, the value of a fraction will not be changed if we multiply numerator and denominator by the same number. Thus in the case of the fraction , if we multiply numerator and denominator by 4, we shall have f , which is clearly equal to . So also the fractions i, f , f , f, ^0, TV, &, A, M, &c., are equal, each of them being equal to . The fractions t, f I. A, A, A, A. A> A* t*f & c., are also equal, each being equal to 4; and the fractions t, *, I, A> ii, if, M, H, i-f &o., are also equal, each of them being equal to jj. REDUCING FRACTIONS TO LOWEST TERMS. 33 Now of all the equivalent fractions the quantity -| is that of which it is easiest to form a definite idea. It is usual, therefore, when we have any such fraction as f- or Jf, to reduce it to its lowest terms, by dividing numerator and denominator by some number that will divide each without a remainder. This division, it is clear, will not affect the nu- merical value of the fraction ; for if we can multiply both nu- merator and denominator by the same number without affecting the value, so we may divide both without affecting the value ; as by such division we bring back the fraction of which both portions had been multiplied to the original expression. The number by which the numerator and denominator of a fraction may be divided without leaving a remainder is called a common divisor ; and so long as we can find for the numerator and denominator a common divisor, it is certain that the frac- tion may be reduced to a lower form. But if we cannot find such common divisor, the fraction is in its lowest form already. Thus in the fraction -j^, we at once see that both terms are di- visible by 2, and, performing this division, the fraction becomes |4 ; which, if again divided by 2, becomes |-f , and which in like manner, by another division by 2, becomes $. This, it will be obvious, cannot any longer be divided by 2, but it may be by 3, when the expression becomes f ; and as this cannot be divided by any other number than 1, it follows that the fraction is now in its lowest terms. Now 2 x 2 x 2 x 3=24, and instead of the successive divisions by 2, by 2, by 2, and by 3, we may divide at once by the product of these quantities, or 24; and dividing numerator and denominator of /$ by 24, we have as before. The property of fractions retaining the same value, whether we multiply or divide their numerator and denominator by the same number, carries this important consequence that it ena- bles fractions to be easily added or subtracted, after having first brought them to the same denomination. If, for example, we had to add together f , -J, ^, and ^ of an inch, we could not do so easily unless we brought the whole of these quantities to 2* 34 ARITHMETIC OF THE STEAM-ENGINE. thirty-seconds. When so reduced the quantities will be |f , -g^-, ^-, /-%, the sum of which is clearly ff , or, dividing numerator and denominator by 8, the expression becomes f. All whole numbers, it is clear, may be expressed by frac- tions, since any whole number may be divided into any number of parts. For example, 6 is the same as . It is also the sam,e as J /> > "> s *~i an( l an infinite number of other expressions which all have the same value. ADDITION AND SUBTEACTIOK OF FRACTIONS. When fractions have the same denomination there is no more difficulty in adding or subtracting them than there is in adding or subtracting whole numbers. Thus i+-f is manifestly f, and -f- $ is obviously f . So also rkr + TJhr-^fr- i Also i + f=f=l and f -f + J==0. But when fractions have not the same denominators, then, before we can add or subtract them, we must change them for others of equal value which have the same denominators. For example, if we wish to add the fractions -J and -J, we must con- sider that |- is the same as f, and that ^ is equivalent to f . We have, therefore, instead of the fractions first proposed, the equiva- lent fractions f and f , the sum of which is . If the two fractions were united by the sign , we should have -J J or f f = & Again, if the fractions proposed be !+-, then as f is the same as f, the sum will be |- +1= J 8 L== -HK ^ ^ e sum f J- and J were required, then as i=-jV anc ^ i == iV> ^he sum These cases are simple and easily reduced. But we may have a great number of fractions to reduce to a common denom- ination, which require a more elaborate process. For example, we may have &, f , f , |, -|, to reduce to a common denomination, in order that we may add them together. The solution of such a case depends upon finding a number that shall be divisible by TO REDUCE FRACTIONS TO A COMMON DENOMINATION. 35 all the denominators of these fractions. Here we proceed ac- cording to the following rule : TO REDUCE FRACTIONS TO A COMMON DENOMINATION. KFLE. Multiply each numerator into every denominator except its own for a new numerator, and multiply all the denom- inators together for a common denominator. When this operation has been performed, it will be fonnd that the numerator and denominator of each fraction have been multiplied by the same quantity, and consequently that the frac- tions retain the same value, while they are at the same time brought to a common denomination. Example. Eednce ^, f, f, 4, and , to a common denomina- tion. Ix3x4x5x 6=360 and 3606= 60 and 602=30 2x2x4x5x6=480 and 480 6= 80 and 802=40 3x2x3x5x6=540 and 540 6= 90 and 902=45 4x2x3x4x 6=576 and 5766= 96 and 962=48 5x2x3x4x 5=600 and 6006=100 and 1002=50 2x3x4x5x 6=720 and 720-7-6=120 and 1202=60 Here, then, we first multiply 1, which is the numerator of the fraction , by the denominators of all the other fractions in succession. We next multiply the number 2, which is the nu- merator of the fraction f , by the denominators of all the other fractions excepting always its own denominator and we pro- ceed in this manner through all the fractions whatever their number may be. We next multiply all the denominators to- gether for the common denominator. Proceeding in this way we find the first numerator to be 360, the second 480, the third 640, the fourth 576, and the fifth 600 ; while the new denomi- inator we find to be 720. It is clear, however, that these frac- tions are not in their lowest terms, and that the numerator and denominator of each may be divided by some common number without leaving a remainder. We may try 6 as such a divisor, and we shall find that the numerators will then become 60, 80, 90, 96, and 100, and the denominator 120. These numbers, 36 ARITHMETIC OF THE STEAM-ENGINE. however, are still divisible by 2, and performing the division the numerators become 30, 40, 45, 48, and 50, and the denomi- nator becomes 60. The same result would have been attained if we had divided at once by 12. And as we cannot effect any further division upon all of the numbers by one common num- ber, without leaving a remainder in the case of some of them, the fractions, we must conclude, are now in their lowest com- mon terms. To add together these fractions we have only to add together the numerators, and place the common denomina- tor under the sum. Performing this addition we find that in this case we have \\^, and as |$ are equal to 1, it follows that %* are equal to 3 and ||, or 3 f J. It is easy to prove that the fractions i, I, f , *, and | are of precisely the same value as the fractions 50 4$. 4_5 AS. $-0 605 60> 60 605 60 which have been substituted for them. Dividing numerator and denominator of the first term by 30 we obtain ; dividing nu- merator and denominator of the second term by 20 we obtain f ; 15 is the divisor in the case of the third term when we ob- tain f ; 12 is the divisor in the case of the fourth term when we obtain the fraction ; and 10 is the divisor in the last case when we obtain the fraction . Dividing the numerator and denominator of each of the transformed fractions, therefore, by the greatest number that will divide both without a remainder, we get the fractions i, I f, 1 5 and | which, it will be seen, are the fractions with which we set out, and they are now in their lowest terms, but are no longer of one common denomination. The lowest terms with a common denominator are 30 40 45 48 or ,/l 50 Tffl-J TT6) ffin fffl-> m(i -STf as determined above. The subtraction of fractions from one another is accom- plished by reducing them to a common denomination as for ad- ADDITION AND SUBTRACTION OF FRACTIONS. 37 dition, and then by subtracting the less numerator from the greater. Thus if we have to subtract f- from -f-, we must re- duce them to a common denomination by the process already explained, when the first becomes |f, and the second Jf, so that f exceeds f in magnitude by ^ So also if we have to subtract | from f, the first fraction becomes by the process of reduction ff , and the second f -j>, so that f taken from leaves ?\. As whenever the numerator of a fraction is a larger number than the denominator, the value of the fraction is greater than unity, and is equal to unity when numerator and denominator is the same, we have only to divide the numerator by the de- nominator to find the number of integers which the fraction contains. So in subtracting a fraction from a whole number, we must break one or more integers up into fractions of the same denomination as that which has to be subtracted. Thus if we have to take f from 1, we must instead of the 1 wrife , and taken therefrom obviously leaves |g. If we have to add together such sums as 3 and 2f, we see at once that the whole numbers when added will be 5, and the equivalent frac- tions under a common denominator will be f and $ or f , which is 1, so that the total quantity will be 6. The addition and subtraction of decimal fractions are per- formed in precisely the same way as the addition and subtrac- tion of whole numbers the only precaution necessary being to place the decimal point in the proper place. Thus 78963-874+ 83952-2 + 364-003 + 10000-997" are added together as follows: 78963-874 Here, beginning as in the addition of whole 83962-2 numbers with the first column to the right, we find 364-003 that 7 and 3 are 10 and 4 are 14. "We set down 10000*99*7 the 4 beneath the column and carry 1 to the next 173281-074 column. Adding up the next column, we find only two significant figures in it, and we say 1 added to 9 makes 10, which added to 7" makes 17. We set down the 7 and carry the 1 as before to the next column, which when added up we find to be 20. This means 20 tenths, and we set down the and carry the 2 to the next column just as in simple addition. So likewise in subtraction, if we take 2-25 from 4 - 75, the result 38 ARITHMETIC OF THE STEAM-ENGINE. will be 2-50; or if we take T79 from 3, the result is T21. In such a case we write the 3 thus : 3-00 Here we write the 3 with a decimal point after it, 1'W and we add as many ciphers after the decimal point as ~7^j there are decimal figures to be subtracted, or we suppose : those ciphers to be added. This does not alter the value of the 3, as 3 with no fractions added to it is just 3. Perform- ing the subtraction we say 9 from 10 leaves 1, and 8 taken from 10 leaves 2, and 2 from 3 leaves 1, just as in simple subtraction. MtTLTIPLICATION AND DIVISION OF FRACTIONS. If we wish to multiply a fraction any number of times, it is clear that it is only the numerator we must multiply. Thus if we multiply -J of an inch by 3, it is obvious that we shall get f of an inch as the product of the multiplication, or -J- repeated 3 times. "We have already seen that to multiply both terms of a fraction by any number does not alter the value of the frac- tion, and if we were to multiply the numerator and denomina- tor of the fraction by 3 we should get ^, which is just the same as -J-. Thus also 3 times \ makes f or \\. 3 times \ makes f or 1. 3 times \ makes f or . 4 times -A- makes 34 or 1A- or 1. 1 6 1 * 1 * o Instead, however, of multiplying the numerator, we may attain the same end by dividing the denominator, and this is a preferable practice when it can be carried out, as it shortens the arithmetical operation. Thus J multiplied by 2 is f or \. But by dividing the denominator of \ by 2, we obtain the same quantity of -|- at one operation. So also if we have to multiply f by 3 we obtain -^-, or f . But if, instead of multiplying the numerator, we divide the denominator, we obtain the f at one operation. In the same way ^f multiplied by 6 is equal ^, or 3J. Where the integer with which the multiplication is per- formed is exactly equal to the denominator of the fraction, the product will be equal to the numerator. Thus MULTIPLICATION OF FRACTIONS BY FRACTIONS. 39 $ X 2 = 1 f X3=2 f x4 = 3 Having now shown how a fraction may be multiplied by an integer, the next step is to show how a fraction may be divided by an integer ; and just as a fraction may be multiplied by di- viding the denominator, so may a fraction be divided by multi- plying the denominator. It is clear that if we divide half an inch into two parts, each of these parts will be J of an inch, and we divide quarter of an inch into two parts, each of those parts will be of an inch, so that J-i-2 ==J and i-4-2 =, which quantities we obtain by successively multiplying the denomina- tors. "We may accomplish the same object by dividing the nu- merator where it is divisible without a remainder. Thus f di- vided by 2 is clearly , and f divided by 3 is f . Thus also divided by 2 gives ^-, |f divided by 3 gives ^, \\ divided by 4 gives ^. When the numerator is not divisible by the divisor without a remainder, the fraction may be put into some equivalent form, when the division may be effected. Thus if we had to divide f by 2, we might turn it into the equivalent fraction f, which, divided by 2, gives f . But the same number is more conven- iently found by multiplying the denominator instead of by di- viding the numerator. We have next to consider the case where one fraction has to be multiplied by another. Thus if the fraction f has to be mul- tiplied by the fraction f , we have first to remember that the ex- pression f means 2 divided by 3, and we may first multiply by 4, which produces , and then divide by 5, which produces T ^. Hence, in multiplying a fraction by a fraction, we multiply the numerators together for the new numerator, and the denominat- ors together for the new denominator. Thus, J x f gives the product f or', fxi gives A> J x ^ gives | or V> 6 , 40 ARITHMETIC OF THE STEAM-ENGINE. Finally, we have to show how one fraction may be divided by another. If the two fractions have the same number for a denominator, the division takes place only with respect to the numerators. An inch being -^ of a foot, it is clear that -fy is contained in T 9 ^ just as often as 3 inches is contained in 9 inches or 3 times ; and in the same manner, in order to divide -fa by f s , we have only to divide 8 by 9, which gives f . So also / ff is contained 3 times in 4-, and T ^ 9 times in T 4 (i 9 ff . But when the fractions have not the same denominator, then we must reduce them to a common denominator by the method of reduction al- ready explained. This result, expressed in words, will be as follows : Multiply the numerator of the dividend by the denom- inator of the divisor for the new numerator, and the denomi- nator of the dividend by the numerator of the divisor for the new denominator. Thus -f divided by f = y|, and divided by =f or -f, or 1J, and f-f divided by =!}- or -f. This rule is commonly expressed in the following form : Invert the terms of the divisor so that the denominator may be in the place of the numerator. Multiply the fraction which is the dividend by this inverted frac- tion, and the product will be the quotient sought. Thus f divided by \ x f - =1. Also, divided by = x f=H, and | divided by |^|f x|=r||g or f. If we have a line 100 feet long, and if we divide it in half, we shall manifestly have two lines each 50 feet long. So if we di- vide it into lengths of 25 feet, we shall have 4 such lengths ; if we divide it into lengths of 2 feet each, we shall have 50 such lengths ; and if into 1 foot lengths, we shall have 100 of them ; if into lengths of half a foot, we shall have 200 lengths ; and if into lengths of \ of a foot, we shall have 400 such lengths. Hence, 100 divided by 100=1 100 divided by 60=2 100 divided by 25=4 100 divided by 1 100 100 divided by = 200 1 00 divided by J = 400 We see, therefore, that to divide a number by the fraction \ DIVISION OF FRACTIONS BY FRACTIONS. 41 is equivalent to multiplying it by 2 ; to divide by the fraction J is the same as to multiply by 4. So, further, if we divide 1 by the fraction -j-jjVfr, the quotient is 1,000, and 1 divided by ^^-5-5- is 10,000. As, then, the fraction gets smaller and smaller, the quotient gets greater and greater, so that we are enabled to con- ceive that a number divided by will be indefinitely great, since in fact there will be an indefinitely great number of nothings in it. As every number whatever, divided by itself, produces unity, so a fraction, divided by itself, produces unity. Thus -s-f =f x 1 = 1. The multiplication of decimal fractions is performed in pre- cisely the same way as the multiplication of whole numbers, and we must mark off in the product as many decimal places as there are in the multiplier and multiplicand together. Thus 1-0025 multiplied by 2-5 = 2-50625 ; also, '0048 multiplied by -000012 = 0000000576. The division of decimals is performed in the same way as the division of common numbers; and if the number of decimal places in the divisor be the same as in the dividend, the quotient thus obtained will be the quotient required, and will be a whole number. But if the number of decimals in the dividend exceed that in the divisor, mark off in the quotient obtained by this di- vision as many decimal places as make up the difference. But if the number of decimals in the divisor exceed that in the divi- dend, annex as many ciphers to the quotient as make up the dif- ference. Thus -805 divided by 2-3 = '35, and 2-5 divided by 32 = 7-8125. The number 3'045 denotes 3 units, tenths, 4 hundreths, and 5 thousandth, and it might be written 3+ -fs+^-y +Tt&rsi an( l the number 3 -47 might be written 3 +A+r^> or ^ might be written 300+40 + 7 = 347 g ^ m5 = 13 ^ = 13f and 100 100 23-0625 = 23^^ = 23^. Also, 4-35 = 4+ A+T*T, or to * + > or by reducing the fractions to the same denomination o + T ff =jj-. So flHK put in the form of a decimal, will be 5-62, for |^^^+^ +T ^. But H*=l, and there - fore =5 42 ARITHMETIC OF THE STEAM-ENGINE. PROPORTION. The Proportion or Ratio of one quantity to another is the number which expresses what fraction the former is of the lat- ter, and is therefore obtained by dividing the former by the latter. The most distinct idea of proportion is obtained by reference to a triangle such as that here figured, where AB has the same proportion to BO that AD has to DE. It is clear that if the quan- tities AB, AD, and BO are fixed, the quantity DE will also be de- termined, as we have only to draw the line AE through o until Fig. l. it intersects the vertical line DE, which it will thereby cut off to the proper length. Thus also the ratio 108 to 144, or as it is written 108 : 144, is |- = f . A proportion is usually stated as follows : 2 is to 4 as 4 is to 8, or 2 : 4 : : 4 : 8 ; and in all cases of proportion the product of the first and fourth terms are equal to the product of the second and third terms. This is expressed by saying that the product of the extremes is equal to the prod- uct of the means. So 2 x 8 = 4 x 4. Conversely, if the product of any two numbers equal the product of other two, then the four numbers are proportionals. The method by which we find a fourth proportional to three given quantities, by multiplying together the second and third and dividing by the first, is what is termed the KTJLE OF THEEE. If a yard of calico costs 1 shil- ling, it is clear that 20 yards will cost 20 shillings ; and we say, therefore, 1 yard is to 20 yards as 1 shilling is to 20 shillings ; or we say, 3 inches: 12 inches :: 12 inches: 48 inches. Here we obtain the 48 by multiplying together 12 and 12, which makes 144, and which divided by 3 gives 48. Proportion is in fact a mere question of scale. If we make NATURE OF PROPORTION. 43 a model or drawing of a house or a machine, we may make it on the scale of J of an inch to the foot, or an inch to the foot, or 1 inch to the foot, or 1 inches to the foot, or on any scale what- ever. But the object, when constructed of the full size, will be precisely the same on whatever scale the model or drawing has been formed. If the scale be J of an inch to the foot, then it is clear the object when formed of full size will be 48 times larger than the model or drawing that is, it will be 48 times longer, 48 times broader, and 48 times higher. So in like manner if the J- inch scale be employed, the object will be 24 times larger; if the scale be 1 inch, it will be 12 times larger ; and if the scale be 1-J- inches to the foot, it will be 8 times larger. So in like manner 20 bears the same proportion to 1 that 20 shillings bears to 1 shilling. But 20 are 400 shillings, and 1 are 20 shillings. Hence, by transforming the pounds into shillings, we see that 400 shillings bear the same relation to 20 shillings that 20 shillings bear to 1 shilling; or, in other words, 400 : 20 :: 20: 1. If -we take a rectangular figure such as ABCD, say 4 inches long and 1 inch wide, and if we enlarge this figure by making it 4 inches longer and 4 inches broader, we see at a glance that the resulting rectangle AEFG is not of the same shape, and in fact is not the same kind of figure as the original rectangle ABOD. This is because the enlargement was not made proportionally, and the diagonal AF consequently does not lie in the same line as the 44 ARITHMETIC OF THE STEAM-ENGINE. diagonal AO. To make the enlargement proportional, we should only have extended AB 1 inch, when we extended AD 4 inches. Fig. 3. Such an extension is shown by the rectangle AIHG; and the diagonal of that rectangle lies in the same line as that of the original rectangle ABCD. In like manner, if the elliptical figure AB be enlarged by equal quantities in the line AB and in the line CD, each successive ellipse becomes more circular, and to main- tain the original figure the enlargements should be in the pro- portion of the length and breadth. ON THE SQUARES AND SQUARE BOOTS OP NUMBERS. The product of a number multiplied by itself is called a square, and the quotient obtained by dividing this product by the num- ber is the square root of the product. Thus 12 times 12 is 144, which is the square of 12 ; and 144 divided by 12 is 12, which is the square root of 144. In like manner, the square root of 12 is the particular number which, multiplied by itself, produces 12. Such number is neither 3 nor 4, as 3 tunes 3 is 9 and 4 times 4 is 16, of which the one is less than 12 and the other greater. The square root of 12 will be some number between 3 and 4, and what the particular number is it is the object of the process for determining square roots to discover. The origin of the term is traceable to the language of geometry, where a rectangular sur- face is produced by the multiplication of one linear dimension SQUARES OF INTEGERS AND FRACTIONS. 45 with another, or a square is produced by the multiplication of one linear dimension by itself. Thus a piece of board a foot long and a foot broad has a surface of one square foot, or, if we count the dimensions in inches, as the length is 12 inches and the breadth 12 inches, the superficies will be 12 tunes 12, or 144 square inches. The square of 1 is 1, since 1 x 1=1. The square of 2 is 4, since 2 x 2=4. The square of 3 is 9, since 3x3=9. Contrariwise 1, 2, and 3 are the square roots of 1, 4, and 9. If we write the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, and their squares 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 122, 144, 169, it will be seen that if each square number is subtracted from that which immediately follows, we obtain the series of odd numbers 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, &c., in which the numbers go on increasing by 2. The square of a fraction is obtained by multiplying the frac- tion by itself, in the same manner as a whole number. Thus ixi=i; *xi=i; fxf=$; ixJ= T V; and ixl=-fr. So also % is the square root* of J; -J is the square root of , and J is the square root of T ' ff . "When the square of a mixed number, consisting of an integer and a fraction, has to be determined, we may reduce the mixed number to a fraction by multiplying the integer by the denomi- nator, and adding the numerator to form a new numerator with the same denominator for the denominator of the new fraction. Thus 3i=-y-+ Y- and the square of - 3 ^=- 9 A l <> r 15 irV Thtl s also, as the square of -f is f|f, the square root of f f is |, and the square root of 12 or - 4 /-=|r=3. But when the number is not a square, it is impossible to extract its square root precisely, though the root may be approximated to with any required de- gree of nearness. "We have already seen that the square root of 12 must be more than 3 and less than 4. "We have also seen that this root is less than 3 , as the square of 3 is 12. Neither is the root 3 T "V or ff the square of which is VaV- or 12 stj> which 46 ARITHMETIC OF THE STEAM-ENGINE. is still greater than 12. So if we try the number 8/3 or ^j-'V 2 /, we shall find the number to be too small, for 12 reduced to the same -denomination is - 2 yV 2 8 8 '> so that 8/3 is T g- g too small, while 3 T 7 j is too great. The fact is, whatever fraction we annex to 3, the square of that sum will always contain a fraction, and will never be exactly 12; and although we know that 3 T 7 7 is too great, and 3 T fi 3- is too small, we cannot fix upon any intermediate num- ber which multiplied by itself shall produce 12 ; whence it fol- lows that the square root of 12, though a determinate magnitude, cannot be expressed by fractions. There is therefore a kind of numbers which cannot be specified by fractions, but which still are determinate quantities, and of these numbers the square root of 12 is an example. These numbers are called irrational num- bers, and they occur whenever we attempt to find the square root of a number that is not a square. These numbers are also called surds or incommensurables. The square roots of all numbers which are not perfect squares, are indicated by the sign ^/, which is read square root. Hence -^/12 means the square root of 12; ^/2 the square root of 2 ; ^/3 the square root of 3 ; ^/f the square root of , and Ja the square root of a. As, moreover, the square root of a number multiplied by itself will produce the number, y 12 to find how many of such groups there are, or, 71 o g in other words, how many shillings there are in the 1 total number of pence. These shillings we transfer to the shillings column, and as after we have done this there are 6 pence left, we write the 6 beneath the pence column, and then proceed to add up the shillings, beginning with the number of shillings we have brought from the pence column. Having thus ascertained the total number of shillings, we find how many pounds there are in that number of shillings by dividing by 20, there being 20 shillings in the pound sterling; and after having found this number of pounds, we carry it to the pounds column, and the 2 shillings which we find remaining we write under the shillings column. We then proceed to add the pounds column, beginning with the number of pounds in shillings which we have carried from the shillings column. In adding up cwts., quarters, and pounds, the mode of pro- cedure is precisely the same, only as there are 28 Ibs. in 1 quar- ter, 4 quarters in 1 cwt., and 20 cwt. in 1 ton, the divisors we use at each step must vary correspondingly. This will be plain from the following example : Here we find the sum to be 20 cwt. 3 qrs. and 17 Ibs., or 1 ton cwt. 3 qrs. and 17 Ibs. ; for, after adding the first column, and dividing the sum by 28, we have 17 left, and after add- 60 ARITHMETIC OF THE STEAM-ENGINE. cwt. qr. Ibs. ing the second or quarters column with the 3 3 12 addition of the number of quarters in Ibs. that 2 318 we k ave carried over from the Ibs. column, we 6219 divide the number so obtained by four to obtain 2 th e number of cwts. there are in all these quar- 1 ton 3 17 ^ ers< ^ e carry the cwt. so obtained to the :== cwts. column, and write beneath the quarters column the 3 quarters which we find are left. Proceeding in the same way with the cwts. column, we find its sum to be 20 cwts. or 1 ton ; and the total quantity to be 1 ton cwt. 3 qrs. 17 Ibs., as stated above. Subtraction of compound quantities. When we wish to subtract one compound quantity from another, we write the less under the greater, so that the terms of the same kind may be in the same column, as in the case of addition. We then subtract the right-hand term of the lower line from that of the upper, if possible. But if this cannot be done, we must transform a unit of the next higher term into its equivalent number of units of the first term, and then performing the subtraction, we write the difference under the first column, and we increase by 1 the next term to be subtracted to compensate for the unit previously borrowed. In algebra, the usual process of subtraction is to change the signs of the lower line, and then to proceed as in addition. If we had to take 27Z. 8s. 4$d. from 34Z. 17s. 9| 021=1363 < 5Y feet per second, which is the ve- locity with which steam with a pressure of two atmospheres would rash into steam with a pressure of one atmosphere. This consequently may be reckoned as the velocity with which steam of 15 Ibs. pressure above the atmosphere would rush into the atmosphere. Such velocities at different pressures are exhibited in the following table : VELOCITY OF EFFLUX OF HIGn-PEESSTTEE STEAM INTO THE ATMOSPHERE. Pressure of steam above the atmosphere. Velocity of free | efflux in feet per second. Pressure of steam above the atmosphere. Velocity of free efflux in feet per second. Ibs. feet. Ibs. feet. 1 482 50 1791 2 663 60 1838 3 791 70 1877 4 890 80 1919 5 973 90 1936 10 1241 100 1957 20 1604 110 1972 80 1643 120 1990 40 1729 130 2004 This table is computed by taking the difference of the two pressures for the effective pressure, which effective pressure is expressed in pounds per square inch, divided by the weight of a cubic foot of the denser fluid in pounds, and the square root of the quotient is multiplied by 96. The denser the fluids are the less, it is clear, will be the velocity of efflux which a given differ- ence of pressure will create ; for the heights of the columns, and also the difference of their heights, will be small in the propor- tion of the density of the denser fluid. The more dense the fluid is, the larger becomes the mass of matter which a given pressure has to move. With steam of 16 Ibs. pressure flowing into steam or air of 15 Ibs. pressure, the moving pressure is 1 lb., and the velocity of efflux is 482 feet per second. With steam of 101 Ibs. pressure flowing into steam or air of 100 Ibs. pressure, INERTIA AND MOMENTUM. 105 the moving pressure is the same, but the velocity of efflux will only be 207 feet per second. INERTIA AND MOMENTUM. "When a body is moved from a state of rest to a state of mo- tion, or from a slow motion to a faster, power is absorbed by the body ; and when a body is brought from a state of motion to rest, or from a fast motion to a slow one, power is liberated by the body. The quality which enables a body to resist the sudden communication of motion is termed its Inertia ; and the quality which enables a body to resist the sudden extinction of motion is termed its Momentum. Whatever power a body ab- sorbs in being put into motion, it afterwards surrenders in being brought to a state of rest ; and the amount of power existing in any moving body is measurable by its weight multiplied by the square of its velocity, or by the height through which it must have fallen by gravity to attain its velocity. A railway carriage of ten tons' weight, therefore, moving at a speed of 20 miles an hour, will have as great a momentum as 4 railway carriages weighing 10 tons each moving at the rate of 10 miles an hour. In like manner the momentum of a cannon ball moving at a velocity of 1,700 feet a second, will be 28,900 times greater than if it moved at a speed of 10 feet per second, since the square of 1,700 is to the square of 10 as 28,900 to 1. Josephus mentions that some of the battering-rams employed by the Romans in Judea were 90 feet long, and weighed 1,500 tal- ents of 114 Ibs. to the talent, or 76-3392 tons. The weight of a cannon ball which has the same amount of mechanical power stored up in it, or which will give the same force of impact when moving at a speed of 1,800 feet per second, as the batter- ing-ram will do when moving at a velocity of 10 feet per second, can easily be determined ; for we have only to multiply 76*3392 tons by the square of 10 and divide by the square of 1,800, which will give '0023561 tons, or 5-12776 Ibs., as the weight of the ball required. 5* 106 MECHANICS OF THE STEAM-EXGINE. TO FIND THE QUANTITY OF MECHANICAL POWER REQUIRED TO COM- MUNICATE DIFFERENT VELOCITIES OF MOTION TO HEAVY BODIES. EULE. Multiply the mass of matter fiy the height due to the velocity it has acquired, supposing that it attained its ve- locity ly falling by gravity. The product is the mechanical power communicated in generating that velocity of motion in the body. Example 1. Suppose a waggon on a railway to weigh 2,500 pounds, what mechanical power must be communicated to it to urge it from rest into motion with a velocity of 3 miles an hour, or 4-4 feet per second ? Now here the height in feet from which a hody must have fallen to acquire any given velocity will be the square of the velocity in feet per second divided by 64J ; or it will be the square of the quotient obtained by dividing the velocity in feet per second by the square root of 64J, or 8'021. Now 4 -4 -f- 8-021 = 5487, the square of which is '301 feet, the height that a body must fall to acquire a velocity of 3 miles an hour. Hence the mechanical power communicated is 2,500 Ibs. x -301 ft. 752*5 Ibs. descending through 1 foot. Example 2. Required the mechanical effect treasured up in a cast-iron fly-wheel, the mean diameter of which is 30 feet with a sectional area of rim of 60 square inches, and making 20 turns in the minute. The diameter of the wheel being 30 feet, the circumference will be 94*248 feet, and, as the wheel makes 20 revolutions in the minute, the velocity of the rim will be 94-248 x 20 = 1884-96 feet per minute, or 31*416 feet per second. Again the cubical content of the rim in cubic feet being 60 x94'248-*-144 = 39*27 cubic feet, and the weight of a cubic foot of cast-iron being 45 3 Ibs., we have 39*27x453^=17794-22 Ibs. as the weight of the rim. Hence the mechanical effect treasured up in the rim of this wheel is 17794-22 x(31*416-*-8*021) 2 =268,650 Ibs. raised one foot high. This it will be observed is about eight actual horse-power. The mechanical energy with which the fly-wheel of an engine is generally endowed, is equal to the power exerted BODIES REVOLVING IN A CHICLE CENTRIFUGAL FORCE. 107 in from four to six half strokes of the engine, or two to three complete revolutions ; so that the fly-wheel above particularized is such as would be suitable for an engine which exerts a power of four actual horses, or four times 33,000 pounds raised one foot high in each revolution, or 80 horses' power. BODIES REVOLVING IN A CIRCLE. When bodies revolve in circles round fixed axes of motion, the different particles can have no motion except in circles de- scribed round such fixed axes ; and the velocities of the particles composing the body must be greater or less, depending upon their distance from the centre round which the body revolves. To apply the laws of falling bodies to this case we must imagine the particles composing such revolving bodies to be divided and collected into several small bodies situated at different distances from the centre, and therefore moving with different velocities ; and then we may determine the power which must be commu- nicated to each of the supposed separate bodies to give it the ve- locity which it actually possesses. The sum of all the powers so determined is the total power which must be communicated to the body, to give to it the velocity of motion with which it actually revolves. Thus a rod moving about one of its extrem- ities may be supposed to be compounded of a number of balls, like a string of beads strung on a wire. The velocity of each of these balls can then be ascertained, which will enable us to compute the mechanical power resident in it, and which will be the same as if it moved in a straight line. The sum of the quan- tities thus ascertained will be the total mechanical power resi- dent in the revolving body. CENTRIFUGAL FORCE. The centrifugal force of a body which revolves in any circle in a given time, is proportional to the diameter of the circle in which it revolves. Thus, in the case of two fly-wheels of the same weight but one of twice the diameter of the other, the 108 MECHANICS OF THE STEAM-ENGINE. larger wheel will have twice the amount of centrifugal force that the small one has. The centrifugal force of a body moving with different veloc- ities in the same circle is proportional to the square of the velocities with which it moves in that circle ; or, what is the same thing, to the square of the number of revolutions per- formed in a given time. Thus, the fly-wheel of any engine will have four times the amount of centrifugal force it possessed before, if driven at twice the speed. In Mr. Watt's engines with sun and planet wheels, in which the fly-wheel made twice the number of revolutions made by the engine, the fly-wheel had four times the centrifugal force that would be possessed by the same fly-wheel if coupled immediately to the crank. The centrifugal force of a body of a given weight, revolving with a certain uniform velocity in a circle of a given diameter, was investigated by the Marquis de 1'Hopital, who gave the rule for ascertaining this force that is now generally followed. It is founded on the consideration of the height from which the body must have fallen by gravity to have acquired the velocity with which its centre of gyration moves in the circle which it de- scribes. Then as the radius of that circle is to double the height due to the velocity, so is the weight of the body to its centrif- ugal force. TO FIND THE CENTRIFUGAL FOEOE OF A BODY OF A GIVEN WEIGHT EEVOLVING IN A CIEOLE OF A GIVEN DIAMETER. EULE. Divide tJie velocity in feet per second, ~by 4'01, and, the square of the quotient is four times the height in feet due to the velocity. Divide this quadrupled height by the diameter of the circle, and the quotient is the centrifugal force when the weight of the tody is 1 / consequently, multiplying it by the weight of the body gives the actual centrifugal force in pounds or tons. Example 1. Suppose that the rim of a fly-wheel 30 feet di- ameter and weighing 15718 Ibs., moves at the rate of 27*49 feet per second, what will be its centrifugal force ? Here we have CENTRIFUGAL FORCE OF FLY-WHEELS. 109 the velocity 27-49-j-4-01=6'85, which, squared, is 46'9225 ; and this, divided by 30, is 1/564 : so that the centrifugal force is 1'564 times the weight of the body, or 10'97 tons. Example 2. Suppose that the rim of a fly-wheel which is 20 feet diameter moves with a velocity of 82 feet per second : then 32-16-5-4-01=8-02, the square of which is 64'32 feet, which is the quadrupled height due to the velocity, and this divided by 20 feet diameter gives 3'216 tunes the weight of the rim as the centrifugal force. ANOTHER ETJLE. Multiply the square of the number of revolu- tions per minute T)y the diameter of the circle of revolution in feet, and divide the product by the constant number 5870; the quotient is the centrifugal force of the body in terms of its weight, which is supposed to be 1. Example 1. Suppose a stone of 2 Ibs. weight is placed in a sling, and whirled round in a circle of 4 feet diameter, at the rate of 120 revolutions per minute : then 120 squared=14400 x 4 feet diameter=57600->5870=9'81 which is the ratio of the cen- trifugal force to the weight ; and, the weight being 2 Ibs., the centrifugal force acting to break the string and escape is 19 -6 Ibs. Example 2. In the case of the first fly-wheel 30 feet diam- eter, referred to above, we multiply the square of the number of revolutions per minute (1T&) by the diameter of the circle in feet (30), and divide the product by 5870 ; which gives the cen- trifugal force in terms of the weight of the body, and 17J 2 x 30 -=-5870=1-564 as before. TO FIND THE BATE AT WHICH A BODY MUST REVOLVE IN ANT OIB- OLE, THAT ITS CEBTrBIffUGAL FOBOE MAY BE EQUAL TO ITS WEIGHT. KULE. Divide the constant number 5870 by the diameter of the circle in feet, and the square root of the quotient is the num- ber of revolutions it will make per minute, when the centrif- ugal force is equal to the weight. Example. In a circle of 6'5 feet diameter, a body must re- volve about 30 times a minute that its centrifugal force may be 110 MECHANICS OF THE STEAM-ENGINE. equal to its weight; for 5870^-6*5 = 903, the square root of which is 30'05 revolutions per minute. The mechanical power which must be communicated to a solid disc of uniform density, to make it revolve on its axis, is the same as that which must be communicated to one-half of its weight of matter, to give it motion in a straight line with the same velocity with which the circumference of the disc moves in a circle. TO DETERMINE THE BURSTING STRAIN OF A FLY-WHEEL. If we suppose half of a fly-wheel to be securely attached to the axis, while the other half is held only by the rim or by bolts which it tends to break by its centrifugal force, then there will be a velocity at which the centrifugal force of half the rim will overcome the cohesion of the metal of the rim, or of the bolts, and the wheel will be burst by its centrifugal force. In mechanical works it has been usual to reckon the cohesive strength of wr ought-iron within the limits of elasticity at 17,800 Ibs. per square inch of section, and of cast-iron at 15,300 Ibs. per square inch of section ; by which is meant that a bar of wrought-iron one inch square might be stretched by a weight of 17,800 Ibs. without injury, and a bar of cast-iron might be stretched by a weight of 15,300 Ibs. without injury, and though somewhat drawn out by such weights, would, like a spiral spring, again return to the original length on the weight being removed. This estimate for cast-iron is much too high ; and in machinery wrought-iron should not be loaded with more than 4,000 Ibs. per square inch of section, and cast-iron should not be loaded with more than 2,000 Ibs. per square inch of section. The breaking tensile strength of good wrought-iron is about 60,000 Ibs. per square inch of section, and of good cast-iron about 15,000 Ibs. per square inch of section. But both wrought and cast-iron will be broken gradually with much less strain than would be required to break them at once ; and if the limit of elasticity be exceeded, they will undergo a gradual deteri- oration, and will be broken in the course of time. If the velocity POWER IN A REVOLVING DISC. Ill of rotation of a cast-iron fly-wheel be so great that its centrif- ugal force becomes greater than 15,000 Ibs. in each square inch of the section of the rim, it will necessarily burst, as a wrought- iron one would also do if the centrifugal force exceeded 60,000 Ibs. per square inch of section. But to be within the limits of safety, a strain of 4,000 Ibs. per square inch of section should not be exceeded for wrought-iron, and 2,000 Ibs. per square inch of section for cast. TO DETERMINE THE MECHANICAL POWER RESIDENT IN A EE- VOLVING DISC. RULE. Multiply one-half of the weight of the revolving disc by the height due to the velocity with which the circum- ference of the icheel or disc moves ; the product is the me- chanical power communicated. Example 1. Suppose that a grindstone 4'375 feet diameter, weighing 3,500 Ibs., makes 270 revolutions per minute; what power must be communicated to it to give it that motion ? The velocity of the circumference will be 61'83 feet per second, and the height due to this velocity is 59 '4 feet. The mechanical power is 1,750 Ibs. (half the weight) x 59*4 feet = 103'950 Ibs. raised one foot. If the revolving- wheel is not an entire disc or solid circle, but only a ring or annulus, it must first be considered as a disc, and the effect of the part which is wanting must then be calculated and deducted. Example 2. Suppose the rim of a cast-iron fly-wheel to be 22 feet diameter outside, and 20 feet inside, and that the thick- ness of the rim is 6 inches, and that the wheel makes 36 revo- lutions per minute, what power must be communicated to the rim to give it that motion, the weight of the arms being left out of the account? A solid wheel 22 feet diameter and 6 inches thick would contain 190 cubic feet, from which, if we deduct 157 cubic feet, which would be the capacity of a solid wheel 20 feet diameter and 6 inches thick, we have 33 cubic feet as the cubical contents 112 MECHANICS OF THE STEAM-ENGINE. of the annulus. Now in the case of a solid wheel of 22 feet diameter, the velocity of the circumference at 36 revolutions per minute would be 41 '47 feet per second, the height due to which would be 26'8 feet, which multiplied by 95 cubic feet (or half the mass) gives 2,546 cubic feet of cast-iron, raised 1 foot for the power communicated. Then supposing another solid wheel 20 feet diameter, we shall find by a like mode of computation that the power communicated is equivalent to 1,735 cubic-feet of cast-iron raised through 1 foot. This deducted from 2,546 leaves 811 cubic feet raised through 1 foot as the power resident in the annulus ; and if we take the weight of a cubic foot of cast-iron in round numbers as 480 Ibs., we have 389,280 Ibs. raised 1 foot, for the mechanical power which must be commu- nicated to the rim of the fly-wheel in question to give it a ve- locity of 36 revolutions per minute. The mechanical power which must be communicated to solid discs of different diameters, but of the same thickness and den- sity, to make them revolve in the same time, is as the fourth powers of their diameters. CENTRES OF GYRATION AND PERCUSSION. The centre of gyration is a point in bodies which revolve in circles in which the momentum, or energy of the moving mass, may be supposed to be collected. It is in the same point as the centre of percussion of revolving bodies, because a revolving body, if suffered to strike another body that is either at rest or that moves with a different velocity in the same orbit, will neither be deflected to the right nor to the left, but will act just as if the whole mass of matter were collected in that point. In bodies moving forward in a straight line, the centre of percus- sion is in the centre of gravity ; but, in bodies revolving in cir- cles, the part of the body most remote from the centre of the circle moves with a different velocity from the part nearest to the centre of the circle. The centre of percussion, therefore, cannot be in the centre of gravity in such a case, but at some point nearer the circumference of the circle ; and the line traced TO FIND THE CENTRE OP GTKATION. 113 by that point will divide the body into two parts, each having the same amount of mechanical power treasured in them, or each requiring the same amount of mechanical power to put them into revolution at their existing velocity. If the body, therefore, could be divided instantly, and without violence, through the line traced by the centre of gyration, each portion of the body would continue to revolve with its former velocity. The point tracing the line which thus divides the body is the centre of percussion, and also the centre of gyration, and in re- volving bodies these centres are identical. If a given pressure act, through a given space, upon a body at its centre of gyration, in the direction of a tangent to the circle which that centre must describe round the fixed centre of motion, such an amount of power will move the centre of gyration with the same velocity in its circle of revolution, as it would move an equal mass of matter in a right line by acting at the centre of gravity of the mass. If the whole mass of the revolving body could be col- lected into its centre of gyration, the mechanical power resident in the body would be represented by multiplying the total weight of the body by the square of the velocity of the centre of gyration. TO FIND THE DISTANCE OF THE CENTRE OF GYRATION OF ANT REVOLVING BODY FROM THE CENTRE OR AXIS OF MOTION. EULE. Multiply the weight of each particle, or equal small portion of the body, ~by the square of its distance from the axis, and divide the sum of all these products y the square of the main number of revolutions which the governor makes per minute. The quotient is the proper vertical height in inches of the point of suspension of the tails above the plane in which they revolve, when moving with mean velocity. Example. "What is the proper vertical height of the point of suspension above the plane of revolution in the case of a gov- ernor making 30 revolutions per minute? Here 35225-^900 (the square of 30) = 39-139, which is the same height as that of the seconds pendulum. If we have already the vertical height, and wish to know the proper tune of revolution, we must proceed as follows : TO DETEBMINE THE PEOPEE TIME OF BEVOLTTTION OF A GOV- EENOB OF WHICH THE VEBTICAL HEIGHT 18 KNOWN. RULE. Multiply the square root of the height 'by the constant fraction 0-31986, and the product will be the proper time of revolution in seconds. Example. In what time should a governor be made to re- volve upon its axis when the vertical height of the cone in which 118 MECHANICS OF THE STEAM-ENGINE. the arms are required to revolve when in their mean position is 39-1393 inches ? Here 6'256 x 0-31986=2 seconds. FRICTION. When two hodies are rubbed together they generate heat, and consume thereby an amount of power which is the mechan- ical equivalent of the heat produced. Clean and smooth iron drawn over clean and smooth iron without the interposition of a film of oil, or other lubricating material, requires about one- tenth of the force to move it that is employed to force the sur- faces together. In other words, a piece of iron 10 Ibs. in weight would require a weight of 1 Ib. acting on a string passing over a pulley to draw the 10 Ib. weight along an iron table. But if the surfaces are amply lubricated, the friction will only be from ^th to sLth of the weight. The friction of cast-iron surfaces in sandy water is about one-third of the weight. The extent of the rubbing surface does not affect the amount of the friction. The experiments of General Morin on the friction of various bodies without an interposed film of lubricating liquid, but with the surfaces wiped clean by a greasy cloth have been summarised by Mr. Eankine in the following table : GENERAL MORIN's EXPERIMENTS ON FRICTION. No. SURFACES. Angle of repose. Friction In terms of the weight. 1 Wood on wood, dry 14 to 26} 25 to 5 2 11} to 2' 2 to -04 8 Metals on oak, dry 26} to 81 5 to -6 4 " wet 13} to 14} 24 to -26 5 " soapy ii} 2 6 Metals on elm, dry 11} to 14 2 to -25 7 Hemp on oak, dry 28 53 8 " wet 18} 88 9 Leather on oak 15 to 19} 27 to -88 10 Leather on metals, dry 29 }' 56 11 " wet 20' 36 12 " " greasy . . . 18 23 18 oily 8} 15 14 Metals on metals, dry 8} to 11} 15 to -2 15 " " wet 16} 8 16 IT 18 Smooth surfaces, occasionally greased . . . " continually greased... " best results 4 to 4} 8 1} to 2 07 to -03 05 08 to -086 19 Bronze on lignum vita;, constantly wet . 3? 05? LAWS OF FRICTION. 119 The ' Angle of repose,' given in the first column, is the angle which a flat surface will make with the horizon when a weight placed upon it just ceases to move by gravity. The column of ' Friction in terms of the weight ' means the proportion of the weight which must be employed to draw the body by a string in order to overcome its friction ; and the proportional weight is sometimes called the Co-efficient of Friction. In a paper, of which an abstract has appeared in the Comptes Bendus of the French Academy of Sciences for the 26th of April, 1858, M. H. Bochet describes a series of experiments which have led him to the conclusion, that the friction between a pair of surfaces of iron is not, as it has hitherto been believed, absolutely independent of the velocity of sliding, but that it diminishes slowly as that velocity increases, according to a law expressed by the following formula. Let R denote the friction ; Q, the pressure ; t>, the velocity of sliding, in metres per second = velocity in feet per second x 0-3048 ; /, a, 7, constant co-efficients ; then B The following are the values of the co-efficients deduced by M. Bochet from his experiments, for iron surfaces of wheels and skids rubbing longitudinally on iron rails : /, for dry surfaces, 0-3, 0-25, 0'2 ; for damp surfaces, 0.14. a, for wheels sliding on rails, 0'03 ; for skids sliding on rails, 0-07. y, not yet determined, but treated meanwhile as inapprecia- bly small. The friction of a bearing or machine per revolution, is nearly the same at all velocities, the pressure being supposed to be uniform ; but as every revolution absorbs a definite quantity of power, and generates a corresponding quantity of heat, it will be necessary to enlarge the rubbing surfaces at high velocities, both to prevent the wear from being inconveniently rapid, and also to enable the bearing to present a larger cooling surface to 120 MECHANICS OP THE STEAM-ENGINE. the atmosphere, so as to disperse the increments of heat which in the case of high velocities it will rapidly receive. With the same object the lubrication should be ample. The oil should overflow the bearing, in the same manner as the oil in a carcel or moderator lamp overflows the wick to prevent carbonisation ; and, to prevent waste, the oil should be returned by an oil pump so as to maintain a circulation that will both cool and lu- bricate the rubbing parts. It was found by General Morin in his experiments, that the ' Friction of Eest ' was considerably more than the ' Friction of Motion,' or, in other words, that it took a greater force to move a rubbing body from a state of rest than it afterwards took to continue the motion, some of the softer bodies being in fact slightly indented with the weight. But in determining the friction of machinery, it is the friction of motion alone that has to be considered, so that the other need not be here taken into account. In the case of rubbing surfaces which are amply lubricated, the amount of the friction depends more on the nature of the lubricant than upon the material of which the rubbing bodies are composed; and hard lubricants, such as tallow, are more suited for heavy pressures ; and thin lubricants, such as almond oil, are best suited for mechanisms moving with considerable velocity, but on which the strain is small. If too heavy a press- ure be applied to a bearing, the oil will be forced out and the surfaces will heat ; and this will be liable to take place when the pressure on the bearing is much more than 800 Ibs. per square inch on the longitudinal section of the bearing, though in practice the pressure is sometimes half as much again, or about 1,200 Ibs. per square inch in the longitudinal section of the bearing, but such bearings will be liable to heat. Thus in a marine engine with a cylinder of 74^ inches diameter, the crank pin is 9 inches diameter, and the length of the bearing is 10 inches, which makes the area of the longitudinal section of the bearing 95 square inches. The area of the cylinder is 4,359 square inches, and if we take the pressure upon the piston including steam and vacuum at 25 Ibs. per square inch, we PRESSURE FOR A GIVEN VELOCITY. 121 shall have a total pressure on the piston of 108,975 Ibs., and, consequently, this amount of pressure on the crank pin bearing. Now 108,975 Ibs., the total pressure, divided by 95 square inches, the total surface, gives 1,147 Ibs. for each square inch of the parallelogram which forms the longitudinal section of the bearing. In the engines of Messrs. Maudslay, Messrs. Seaward, and most of the London engineers, the pressure per square inch put upon the crank pin is less. Thus in their 120-horse engines, the diameter of the cylinder is 57| inches, giving an area of 2,597 square inches, which multiplied by a pressure of 25 Ibs. per square inch, gives 64,925 Ibs. as the total pressure upon the piston. The crank pin is 8 inches diameter, and the bearing is 8 inches long, giving 68 square inches as the area of the longi- tudinal section ; and 64,925 Ibs., the total pressure, divided by 68 square inches, the total 'area, gives a pressure of 954'771bs. per square inch of section. This is still in excess of the 800 Ibs. per square inch to which it is expedient to limit, the press- ure. But the assumed pressure on the piston is rather large in the case of these engines, and the actual pressures will be found to agree pretty well with the limit of 800 Ibs. on each square inch of 'the longitudinal section of bearings which it is proper to fix as a general rule in the case of engines moving slowly. In the case of fast-moving engines, however, the sur- face should be greater. The proportion in which the surface should vary with the speed is pretty accurately expressed by the following rule : TO FIND THE PRESSURE PER SQUARE INCH THAT MAT BE PUT UPON A BEARING MOVING WITH ANT GIVEN VELOOITT. RULE. To the constant number 50 add the velocity of the 'bear- ing in feet per minute, and reserve the sum for a divisor. Divide the constant number 70,000 ty the divisor found as above. The quotient will le the number of pounds per square inch that may "be put upon the bearing. Example 1. An engine with a cylinder 74 inches diameter, has a crank pin 10 inches diameter. At 220 feet of the piston 6 122 MECHANICS OF THE STEAM-ENGINE. per minute, and with a stroke of 7J feet, the number of revolu- tions per minute will be about 15 ; and as the circumference of the crank pin will be about 30 inches or 2 feet, the surface of the bearing will travel 15 times 2|, or 37 feet per minute. Adding to this the constant number 50, we have 87, and 70,000 divided by 87 = 800, which, at this speed, is the proper press- ure to put on each square inch of the longitudinal section of the bearing. If it is found on trial that this pressure is ex- ceeded, the length or diameter of the pin must be increased or both. Example 2. An engine with a cylinder 42 inches diameter, has a crank pin 8J inches diameter, the circumference of which is 26'7 inches or 2-225 feet. When the engine makes 54-8 revo- lutions per minute, the surface of the crank pin will move with a speed of 54-8 times 2'225 feet per minute, or 121-8 feet per minute. Now 50 + 121-8 = 171 '8, and 70,000 divided by 171'8=407'3, which, at this speed of revolution, is the proper load to place upon each square inch of section in the line of the axis. The pressure of steam and vacuum in this engine was 40 Ibs. per square inch ; and as the area of a piston 42 inches diam- eter is 1385*4 square inches, the pressure urging the piston will be 40 times 1385-4 or 55,416 Ibs. Now 55,416 divided by 407'3 is 136, which must be the number of square inches in the longi- tudinal section of the bearing in order that there may not be more than 407'3 Ibs. on each square inch. To obtain this area, the bearing must be 16 inches long, since 8 inches multiplied by 16 inches is 136 square inches. At 60 revolutions, the speed of the bearing surface per minute is 60 tunes 2-225 feet or 133-5 feet. Now 50 + 133-5=183-5, and 70,000 divided by 183-5=377'4, which is the proper load in Ibs. for each square inch in the lon- gitudinal section of the bearing. At 70 revolutions the speed of the bearings is 70 times 2-225 feet, or 155-75 feet per minute. Now 50 + 155-75=205-75, and 70,000 divided by 205-75=340-2, which is the proper load in pounds to put upon each square inch of the longitudinal section of the bearing at this speed of rotation. PRESSURE FOR A GIVEN VELOCITY. 123 TO FIND THE PROPER TELOCITY FOR THE SURFACE OF A BEAR- ING WHEN' THE PRESSURE PER SQUARE INCH ON ITS LONGITU- DINAL SECTION IS GIVEN. RULE. Divide the constant number 70,000 ly the pressure per square inch on the longitudinal section of the tearing, from the quotient subtract the constant number 50. The remain- der is the proper velocity of the surface of the bearing in feet per second. Example 1. "What is the proper velocity of the surface of a bearing which has the pressure of 800 Ibs. on each square inch of its longitudinal section ? Here 70,000 divided by 800=87'5 ; from which if we take 50 there will remain 37*5, which is the proper velocity of the bearing in feet per second. If we take a hypothetical pressure of 1,400 Ibs. per square inch of section, we get 70,000 divided by 1,400 = 50, and 5050=0; so that with such a pressure there should be no velocity. Even in cases, however, in which there is very little motion, such as in the top eyes of the side rods of marine engines, it is not advisable to have so great a pressure upon the bearing as 1,400 or even 1,200 Ibs. per square inch of section. Example 2. What is the proper velocity of the bearing of an engine which has a pressure upon it of 407'3 Ibs. per square inch of section? Here 70,000 divided by 407*3=1 71 -8, which diminished by 50 is 121*8, which is the proper speed of the sur- face of the bearing with this pressure per square inch of sec- tion. If the diameter of the bearing be 8 inches, its circum- ference will be 2-225 feet, and 121-8 divided by 2-225=54-8 rev- olutions, which will be the speed of the engine with these data. These proportions allow a good margin, which may often be availed of in practice, either in driving the engine faster than is here indicated, or in putting more pressure upon the bearing. But to obviate inconvenient heating and wear, it will be found preferable to adhere, as nearly as practicable, to the proportion of surface which these rules prescribe. 124 MECHANICS OP THE STEAM-EXGINE. STRENGTH OF MATERIALS. The various kinds of strain to which materials are exposed in machines and structures may be all resolved into strains of extension and strains of compression ; and in investigating the strength of materials there are three fixed points, varying in every material, to which it is necessary to pay special regard the ultimate or breaking strength, the elastic or proof strength, and the safe or working strength. The tensile or breaking strength of wrought-iron, is about 60,000 per square inch of section, whereas the crushing strength of wrought-iron is about 27,000 per square inch of section. In steam-engines where the parts are alternately compressed and extended, it is not proper to load the wrought-iron with more than 4,000 Ibs. per square inch of section ; or the cast-iron with more than 2,000 Ibs. per square inch of section. But in boilers where the strain is con- stantly in one direction, the load of 4,000 Ibs. per square inch of section may be somewhat exceeded. The elastic strength is the strength exhibited by any material without being perma- nently altered in form, or crippled ; for as a piece of iron is finally broken by being bent backward and forward, so by ap- plying undue strains to any material, it will be finally broken with a much less strain than would suffice to break it at once. The elastic tensile strength of wrought-iron is between one-third and one-fourth of its ultimate tensile strength, and to this point the material might be proved without injury. But in proving boilers, and many other objects, it is not usual to make the proving pressure more than twice or three times the working pressure, such proof it is considered involving no risk of strain- ing the material while it is adequate to the detection of acci- dental flaws if such exist. The following table exhibits the te- nacity or tensible strength, and the resistance to compression or crushing strength of various materials : STRENGTH OF MATERIALS. 125 TENSILE AND CRUSHING STRENGTHS OF VARIOUS MATERIALS PER SQUARE INCH OF SECTION. MATERIAL. Tensile strength in Ibs. per square inch of section. Crashing strength in Ibs. per square inch of section. METALS. Wrought-iron bars 60,000 62.000 1 64,000 j 70,000 to f 1 100,000 J 36,500 25,764 ( 100,000 to ) 180,000 18,000 36,000 50,000 19,000 30,000 88,000 60,000 40,997 20,490 4,736 8,137 7,000 1,062 8,000 17,000 12,000 15,000 20,000 18,000 j 10,000 to 1 14,000 20,000 28,000 12,000 16,000 i 8,000 to 1 16,000 f 10,000 to } 19,000 f 9,800 15,000 j 10,666 to | 12,000 27,000 to 37,000 varies as cube of thickness nearly. 100,000 130,000 j. 260,000 10,000 9,000t 9,300 6,400 10,800 10,800 6,875 to ) 6,200 f 7,800 9,900 8,200 10,000 12,000 5,500 to 11,000 f 4,000 to ( 5,000 f 4,000 to) 6,000 f 660 to i 800 j 1,100 1,700 Wrought-iron plates Wrought-iron hoops (best best) Wrought-iron wire* Cast-Iron (average) Cast-iron (toughened) Steel Cast brass Brass wire Cast copper Silver (cast) Gold Tin (cast) Bismuth (cast) Zinc Antimony Lead (sheet) WOODS. Ash Beech Birch Box Elm Fir (red pine) Hornbeam Lance-wood Lignum Yltse Locust Mahogany Oak Pear ... Teak STONES. Granite Limestone Slate Sandstone Brick (weak) Brick (strong) Brick (fire) Glass 9,600 60 Mortar * Mr. Pole found the German steel wire used for pianoforte* to bear a> much u 268,800 Iba, per iqnare Inch, t Those value* are for dry wood. In wet wood the crushing strength U only half a* great. 126 MECHANICS OF THE STEAM-ENGINE. It will be remarked that there are very large variations in the amount of the strength recorded in this table ; and there are so many varieties in the quality of the materials experimented upon that it is hopeless to expect any absolute agreement in the results of different experiments. It will be useful, under these circumstances, to set down the main results arrived at by a few of the principal experimentalists, leaving the reader to select such value as he may consider most nearly agrees with the cir- cumstances with which he has to deal, The following are the strengths of various metals ascertained by Mr. George Eennie, in 1817: TENSILE STRENGTHS OF METALS BY RENNIE. KIND OF METAL. Tearing weight in Ibs. of a bar one inch square. Length of bar one inch square in feet that would break by its own weight. Cast steel 134,256 39,455 Swedish malleable iron 72,064 19,740 English malleable iron 55,872 16 938 Cast-iron 19,096 6 110 Cast copper 19,072 5,003 Yellow brass 17,958 5,180 Case tin 4,736 1,496 Cast lead 1,824 348 Professor Leslie, in his Natural Philosophy, thus explains the law of the extension of iron by weights : 4 A bar of soft iron will stretch uniformly by continuing to append to it equal weights till it can be loaded with half as much as it can bear ; beyond that limit, however, its extension will become doubled by each addition of the eighth part of the disruptive force. Suppose the bar to be an inch square and 1,000 inches in length; 36,000 Ibs. will draw it out 1 inch, but 45,000 will stretch it 2 inches; 54,000 Ibs. 4 inches; 63,000 8 inches; and 72,000 16 inches, where it would finally break.' This popular explanation of the law agrees pretty nearly TENSILE AND CRUSHING STRENGTHS. 127 with the subsequent deductions of Hodgkinson and other enquirers. The cohesive strength of woods varies still more than that of metals in different specimens, and varies even in different parts of the same tree. Thus in Barlow's experiments he found the cohesive strength of fir to vary from 11,000 to 13,448 Ibs. per square inch of section ; of ash from 15,784 to 17,850 ; oak from 8,889 to 12,008; pear from 8,834 to 11,537, and other woods in the same proportions. The following fair average values may he adopted : TENSILE STRENGTHS OP WOODS BY BARLOW. KIND OF WOOD. Tearing weight in Ibs. of a rod one inch square. Length in feet of a rod one inch square that would break by its own weight. Teak 12,915 36,049 Oak 11,880 32,900 Sycamore 9,630 35,800 Beech 12,225 38,940 Ash 14,130 42,080 Elm 9.720 39,050 9,540 40,500 Christiana Deal 12,846 55,500 Larch 12,240 42,160 The crushing strength of wood, as of most other materials, is very different from its tensile strength, and is greatly affected by its dryness. The following table exhibits the results of the experiments made by Mr. Hodgkinson, to ascertain the crush- ing strengths of different woods per square inch of section. The specimens crushed were short cylinders, 1 inch diameter and 2 inches long, flat at the ends. The results given in the first column are those obtained when the wood was moderately dry. Those in the second column were obtained from similar specimens which had been kept two months longer in a warm place : 128 MECHANICS OF THE STEAM-ENGINE. STEEXGTHS OF WOODS BY IIODGKINSON. KIND OF WOOD. Crushing strength per square inch of section. Alder 6831 to 6960 Ash 8683 " 9363 Bay . . 7518 " 7518 Beech 7733 " 7363 English Birch 3297 " 6402 5674 " 5863 Red Deal 5748 " 6586 White Deal 6781 " 7293 Elder 7451 " 9973 Elm " 10331 Fir (Spruce) 6499 " 6819 Mahogany 8198 " 8198 Oak (Quebec) 4231 ' 5982 Oak (English) 6484 ' 10058 Pine (Pitch) 6790 ' 6790 Pine (Red) 5395 ' 7518 Poplar 3107 ' 5124 Plum (Dry) 8241 ' 10493 Teak ' 12101 Walnut 6063 " 7227 Willow 2898 " 6128 The crushing strength of cast-iron is 98,922 Ihs., or, say 100,000 per square inch of section. The strength of wooden columns of different lengths and diameters to sustain weights has not been conclusively deter- mined, and the longer a column is the weaker it is. But, how- ever short it may be, the load upon it should not be above one- third of the crushing load, as given above. LAW OF THE STRENGTH OF PILLARS. The theory of the strength of pillars propounded by Euler is that the strength varies as the fourth power of the diameter divided by the square of the length ; and the recent investiga- tions of Hodgkinson and others show that this doctrine is nearly correct. Thus, in the case of hollow cylindrical columns of cast- iron, it is found experimentally that the 3 - 55th power of the in- ternal diameter subtracted from the 3'55th power of the external LAW OF STRENGTH OF PILLARS. 129 diameter, and divided by the l'7th power of the length, will give the strength very nearly. In the case of hollow cylindrical columns of malleable iron, it is found that the 3 '5 9th power of the internal diameter, subtracted from the 3'59th power of the external diameter, and divided by the square of the length, will represent the strength ; but this rule only holds when the load does not exceed 8 or 9 tons per square inch of section. The power of plates to resist compression varies as the cube or more nearly as the 2'878th power of their thickness. But this law only holds so long as the pressure applied does not exceed 9 to 12 tons per square inch of section. If the load is made greater than this, the metal is crushed and gives way. It has been found experimentally that in malleable iron tubes of the respective thicknesses of '525, '272 and '124 inches, the resistances to com- pression per square inch of section are 19'17, 14-47, and 7'47tons respectively. Moreover, in wrought-iron tubes 1 inches diam- eter and th of an inch thick, the crushing strength is only 6 '55 tons per square inch of section, while in tubes of nearly the same length and thickness, but about 6 inches diameter, the crushing strength is 16 tons per square inch of section. The strength of a pillar fixed at both ends is twice as great as if it were rounded at both ends. The crushing strength of a single square cell or tube of wrought-iron of large size, with angle-irons at the cor- ners, of the construction adopted in tubular bridges, is when the thickness of the plate is not less than one-thirtieth of the di- ameter of the cell, about 27,000 Ibs. per square inch of section ; and where a number of such cells are grouped together so as to prevent deflection, the crushing strength rises to nearly 36,000 Ibs. per square inch of section, which is also the crushing strength of short wrought-iron struts. The length of independent pillars should not be more than 25 tunes the diameter. The weight in Ibs. which a square post of oak of any length will with safety sustain may be determined as follows : TO DETERMINE THE PEOPEE LOAD FOE OAK POSTS. RULE. To 4 times the square of the breadth in inches add half the square of the length in feet, and reserve the sum for 6* 130 MECHANICS OF THE STEAM-ENGINE. a divisor. Multiply the cube of the 'breadth in inches ~by 3,960 times the length in feet, and divide the product by the divisor found as above. The quotient is the weight in Ibs. which the oak post or pillar will with safety sustain. Example. "What weight will a column of oak 6 inches square and 12 feet long sustain with safety? Here the breadth of the post is 6 inches, the square of which is 36 ; and 4 times 36 is 144. The length heing 12 feet, the square of the length is 144, half of which is 72 ; and 72 added to 144 gives 216 for the divisor. The breadth being 6 inches, the cube of the breadth is 216, and the length being 12 feet, we get 12 times 3,960 which is 47,520. Then 216 times 47,520 is 10,- 264,320, which divided by 216 gives 47,520, which is the weight in Ibs. that the post will with safety sustain. The following table is computed from the rule given above : SCANTLINGS OF SQUARE POSTS OF OAK. With the weights they will support and the extent of surface of flooring they will safely sustain, allowing 1 cwt., If cwt., or 2 cwts. to the superficial foot of flooring, and calculated for a height of 10 feet. NOTE. These Scantlings may "be safely used -up to ~i2feet in height; 'but above that a little extra thickness should be allowed. Scantlings. Weight Extent of surface of flooring supported. 1 cwt. per foot. l cwt. per foot. 2 cwt per foot. Inches. Tons. Cwts. Square feet. Square feet. Square feet. 3x3 5 10 110 82^ 55 4x4 9 18 198 148 99 5x5 14 14 294 220 147 6x6 19 12 892 294 196 7x7 24 12 492 369 246 8x8 29 10 590 442 295 9x9 34 8 688 516 344 10x10 39 4 784 588 392 11x11 44 880 660 440 12x12 48 16 976 732 488 13x13 53 10 1070 802^ 535 14x14 58 4 1164 873 582 15x15 62 16 1256 942 628 LAW OP STRENGTH OF PILLARS. 131 Similar calculations of the dimensions and loads proper for rectangular columns of other woods may be determined hy a reference to their relative crushing strengths given in page 128. The formula given by Mr. Hodgkinson for determining the breaking -weight of square oak posts where the length exceeds 30 times the thickness is W=2452 . 2" where W is the breaking weight in Ibs. ; d the side of the square base in inches ; and I the length of the post in feet. TO DETEBMDTE THE PEOPEB LOAD TO BE PLACED UPON SOLID PILLABS OF CAST-IKON. The load which may be safely placed upon round posts, or solid pillars of cast-iron, may be ascertained by the following rule : EULE. To 4 times the square of the diameter of the solid pillar in inches, add 0*18 times the square of the length, of the pillar in feet, and reserve the sum for a divisor. Multiply the fourth power of diameter of the pillar in inches Try the constant number 9562 and divide the product oy the divisor found as above. The quotient is the weight in Ibs. which the solid cylinder or post of cast-iron will with safety sustain. Mr. Hodgkinson's formula for the breaking strength in tons of solid pillars of cast-iron in the case of pillars with rounded ends is /Z 3 * Strength in tons=14'9 ; and in pillars with flat ends Strength in tons=44'16^ where d is the diameter in inches, and I the length in feet. The loads in cwts. which may be put upon solid cylinders or columns of cast-iron of different diameters and lengths are ex- hibited in the following table : 132 MECHANICS OF THE STEAM-ENGINE. WEIGHT IN CWTS. SUSTAINABLE WITH SAFETY BY SOLID CYL- INDERS OR COLUMNS OF CAST-IRON OF DIFFERENT DIAM- ETERS AND LENGTHS. Diameter LENGTH OP COLTTMN IK FEET. of column in inches. 6 8 10 12 14 16 cwts. cwts. cwts. cwts. cwts. cwts. 2 61 50 40 32 26 22 2i 105 91 77 65 55 47 3 163 145 128 111 97 84 si 232 214 191 172 156 135 4 310 288 266 242 220 198 *i 400 379 354 327 301 275 5 601 479 452 427 394 365 6 592 573 550 525 497 469 7 1013 989 959 924 887 848 8 1315 1289 1259 1224 1185 1142 In hollow pillars nearly the same laws obtain as in solid. Thus in the case of hollow pillars, with rounded ends or movable ends, like the cast-iron connecting-rod of a steam-engine, the formula is- n^-tf* Strength in tons=13 _ and in the case of hollow pillars, with flat ends T)S'6 x73-6 Strength in tons=44'3 . ; where D is the external and d the internal diameter. The strength of a pillar with a cross section of the form of a cross was found to be only about half as great as that of a cylindrical hollow pillar. It was also found that in pillars of the same dimensions, but of different materials, taking the strength of cast-iron at 1,000, that of wrought-iron was 1,745, cast steel 2,518, Dantzic oak 108-8, and red deal 78-5. Mr. Hodgkinson's rule for the breaking weight of cast-iron beams is as follows : STRENGTH OF CAST-IKON BEAMS AND SHAFTS. 133 STEENGTH OF CAST-IEON BEAMS. KULE. Multiply the sectional area of the bottom flange in square inches by the depth of the learn in inches, and divide the product by the length between the supports also in inches. Then 514 times the quotient will be the breaking weight in cwt. 8TBEXGTH OF SHAFTS. 44 Ibs. acting at a foot radius will twist off the neck of a shaft of lead 1 inch diameter, and the relative strengths of other materials, lead being 1, is as follows: Tin, 1-4; copper 4'3 ; yellow brass, 4'6; gun metal, 5 ; cast-iron, 9 ; Swedish iron, 9 '5 ; English iron, lO'l ; blistered steel, 16'6; shear steel, 17; and cast steel, 19'5. The strength of a shaft increases as the cube of its diameter. CHAPTER HI. THEORY OP THE STEAM-ENGINE. THE Steam-Engine is a machine for extracting mechanical power from heat through the agency of water. Heat is one form of mechanical power, or more properly, a given quantity of heat is the equivalent of a determinate amount of mechanical power ; and as heat is capable of producing power, so contrariwise power is capable of producing heat. The nature of the medium upon which the heat acts in the production of the power whether it be water, air, metal, or any other sub- stance is immaterial, except in so far as one substance may be more convenient and manageable in practice than another. But with any given extremes of temperature, and any given expen- diture of heat, the amount of power generated by any given quantity of heat will be the same, whatever be the nature of the substance on which the heat is made to act in the generation of the power. And just in the proportion in which power is gen- erated so will the heat disappear. "We cannot have both the heat and the power; but as the one is transformed into the other, so it will follow that the acquisition of the one entails a proportionate loss of the other, and this loss cannot possibly be prevented. It has been already explained that, as in all cases in which power is produced in a steam-engine, there must be a dif- ference of pressure on the two sides of the piston, or between the boiler and the condenser ; so in all cases in which power is pro- NATURE AND EFFECTS OF HEAT. 135 duced in any species of caloric engine, there must be a difference of temperature between the source of heat and the atmosphere or refrigerator. The amount of this difference will determine the amount of power, up to a certain limit, which a unit of heat will generate in any given engine. But as it has been al- ready explained that the mechanical equivalent of the heat con- sumed in heating 1 Ib. of water 1 Fahrenheit would, if utilised without loss, raise a weight of 772 Ibs. 1 foot high, it will fol- low that in no engine whatever can a greater performance be obtained than this, whatever difference of temperature we may assume between the extremes of heat and cold. A weight of 772 Ibs. raised 1 foot for 1 Fahrenheit is equivalent to a weight of 1389*6 Ibs. raised 1 foot for 1 Centigrade; and for conven- ience the term foot-pound is now very generally employed to de- note the dynamical unit, or measure of power, expressed by a weight of 1 Ib. raised through 1 foot. A horse-power, or as it is now commonly termed an actual or indicator horse-power to distinguish it from a nominal horse-power, which is a mere measure of capacity is a dynamical unit expressed by 33,000 Ibs. raised 1 foot high in a minute ; or it is 650 foot-pounds per sec- ond; 33,000 foot-pounds per minute; or 1,980,000 foot-pounds per hour. This unit takes into account the rate ofworTc of the machine. Heat, like light, is believed to be a species of motion, and there are three forms of heat of which a work of this nature re- quires to take cognisance Sensible Heat, Latent Heat, and Speciffc Heat. Sensible Heat is heat that is sensible to the touch, or measur- able by the thermometer. Latent neat is the heat which a body absorbs in changing its state from solid to liquid, and from liquid to aeriform, without any rise of temperature, or it is the heat ab- sorbed in expansion. And Specific Heat is an expression for the relative quantity of heat in a body as compared with that in some other standard body of the same temperature. There is a constant tendency in hot bodies to cool, or to transfer part of their heat to surrounding colder bodies ; and contiguous bodies are said to be of equal temperatures when there ceases to be any 136 THEORY OF THE STEAM-ENGINE. transfer of heat from one to the other. The most prominent phenomena of heat are Dilatation, Liquefaction, and Vaporisa- tion. Difference between temperature and quantity of heat. It is quite clear that two pounds of boiling water have just twice the quantity of heat in them that is contained in one pound of boil- ing water. But it does not by any means follow, nor is it the case, that two pounds of boiling water at 212 contain twice the quantity of heat that is contained in two pounds of water at 106. Experiment indeed shows, that when equal quantities of water at different temperatures are mixed together, the resulting temperature is the mean of the two, so that if a pound of water at 200 be mixed with a pound of water at 100, we have a re- sulting two pounds of water of 150. But before we could sup- pose that a pound of water at 200 has twice the quantity of heat in it that is contained in a pound of water at 100, it would be necessary to conclude that water at or zero, has no heat in at whatever. This, however, is by no means the case ; and tem- peratures much below zero have been experimentally arrived at, and even naturally occur in northern latitudes. A pound of ice at a temperature below zero, rises in temperature by each suc- cessive addition of heat, until it attains the temperature of 32, when it begins to melt ; and, notwithstanding successive addi- tions being made to its heat, its temperature refuses to rise above 32 until liquefaction has been completed. So soon as all the ice has been melted, the temperature of the resulting water will continue to rise with each successive increment of heat, until the temperature of 212 has been attained, when the water will boil, and all subsequent additions to the heat will be expended in evap- orating the water or in converting it into steam. Although, therefore, a pound of water in the form of steam has only the same temperature as a pound of boiling water, it has a great deal more heat in it, as is shown by the fact that it will heat to a given temperature a great many more pounds of cold water than a pound of boiling water would do. Absolute zero. The foregoing considerations lead naturally to the inquiry whether, although bodies at the zero of Fahren- DIFFEKENT THERMOMETERS COMPARED. 137 heit's scale are still possessed of some heat, there may not, never- theless, be a point at which there would be no heat whatever, and which point therefore constitutes the true and absolute zero. Such a point has never been practically arrived at. But the law of the elasticity of gases and their expansion by heat, leads to the conclusion that there is such a point, and that it is situated 461 '2 Fahrenheit below the zero of Fahrenheit's scale, or in other words that it is 461 '2 Fahrenheit, 274 Centigrade, or 219'2 Reaumur. Mr. Eankine has shown, that by reckon- ing temperatures from this theoretical zero, at which there is sup- posed to be no heat and no elasticity, the phenomena dependent upon temperature are more readily grouped and more simply ex- pressed than would otherwise be possible. Fixed Temperatures. The circumstance of the temperatures of liquefaction and ebullition being fixed and constant, enables us to obtain certain standard or uniform temperatures, to which all others may easily he referred. One of these standard tem- peratures is the melting-point of ice, and another is the boiling- point of pure water under the average amospheric pressure of 14*7 Ibs. on the square inch, 2116'8 Ibs. on the square foot ; or un- der the pressure of a vertical column of mercury 29*922 inches high, the mercury being at the density proper to the tempera- ture of melting ice. Thermometers. Thermometers measure temperatures by the dilatation which a certain selected body undergoes from the appli- cation of heat. Sometimes the selected body is a solid, such as a rod of brass or platinum ; at other times it is a liquid, such as mercury or spirits of wine ; and at other tunes, again, it is a gas, such as air or hydrogen. In a perfect gas the elasticity is proportionate to the compression, whereas in an imperfect gas, such as carbonic acid, which may be condensed into a liquid, the rate of elasticity diminishes as the point of condensation is ap- proached. Every gas approaches more nearly to the condition of a perfect gas the more it is heated and rarefied, but an abso- lutely perfect gas does not exist in nature. Common air, how- ever, approaches sufficiently to the condition of a perfect gas, to be a just measure of temperatures by its expansion. 138 THEORY OF THE STEAM-ENGINE. Air and all other gases expand equally with equal increments of temperature ; and it is found experimentally that a cubic foot of air at the temperature of melting ice, or 32, will form 1*365 cubic feet of the same pressure at the temperature of boiling water, or 212. Thermometers, however, are not generally con- structed with air as the expanding fluid, except for the measure- ment of very high temperatures. The most usual species of ther- mometer consists of a small glass bulb filled with mercury, and in connection with a capillary tube. The bulb is immersed in the substance the temperature of which it is desired to ascertain ; and the amount of the dilatation is measured by the height to which the mercury is forced up the capillary tube. The ther- mometer commonly used in this country is Fahrenheit's ther- mometer, of which the zero or of the scale is fixed at the temperature produced by mixing salt with snow ; and which temperature is 32 below the freezing-point of water. The Cen- tigrade thermometer is that commonly used on the continent of Europe ; and it is graduated by dividing the distance between the point where the mercury stands at the freezing-point of water, and the point where it stands at the boiling-point of water, into 100 equal parts. Of this thermometer the zero is at the freez- ing p.oint of water. Another thermometer, called Beaumur's thermometer, has its zero also at the freezing-point of water ; and the distance between that and the boiling-point of water is divided into eighty equal parts. Hence 80 Eeaumur are equal to 100 Centigrade, and 180 Fahrenheit. The correspond- ing degrees of these thermometers are shown in the following table : DILATATION OF SOLIDS. 139 CEXTIGBADE, EEATTMUR's, AXD FAHEEXHEIT's THEEMOMETERS. Cent. Reau. Fahr. Cent Reau. Fahr. Cent. Reau. Fahr. Cent. Reau. Fahr. 100 so- 212- 64 51-2 147-2 29 23-2 84-2 6 4-8 21-2 99 79-2 210-2 63 50-4 145-4 28 22-4 82-4 7 5-6 19-4 9S 78-4 20S-4 62 49-6 143-6 27 21-6 80-6 8 6-4 17-6 97 77-6 206-6 61 48-8 141-8 26 20-8 7S-8 9 7-2 15-8 96 76-8 204-8 60 48- 140- i 25 20- 77- 10 8- 14- 95 76- 203- 59 47-2 138-2 24 19-2 75-2 11 8-8 12-2 94 75-2 201-2 58 46-4 136-4 23 18-4 73-4 12 9-6 10-4 93 74-4 199-4 57 45-6 134-6 22 17-6 71-6 18 10-4 8-6 92 73-6 197-6 56 44-8 132-8 21 16-8 69-8 14 11-2 6-8 91 72-8 195-8 55 44- 131- 20 16- 68- 15 12- 5- 90 72- 194- 54 43-2 129-2 19 15-2 66-2 16 12-8 3-2 89 71-2 192-2 53 42-4 127-4 18 14-4 64-4 17 13-6 1-4 88 70-4 190-4 52 41-6 125-6 17 13-6 62-6 18 14-4 0-4 87 69-6 188-6 51 40-8 123-8 16 12-8 60-8 19 15-2 2-2 86 68-8 186-8 50 40- 122- 15 12- 59- 20 16- 4- 85 68- 185- 49 39-2 120-2 14 11-2 57-2 21 16-8 5-8 84 67-2 183-2 43 38-4 118-4 13 10-4 55-4 22 17-6 7-6 S3 66-4 181-4 47 37-6 116-6 12 9-6 53-6 23 18-4 9-4 82 65-6 179-6 46 36-8 114-8 11 8-8 51-8 24 19-2 11-2 81 64-8 177-8 45 36- 118- 10 8- 50- 25 20- 13- 80 64- 176- 44 35-2 111-2 9 7-2 48-2 26 20-8 14-8 79 63-2 174-2 43 34-4 109-4 8 6-4 46-4 27 21-6 16-6 78 62-4 172-4 42 33-6 107-6 7 5-6 44-6 28 22-4 18-4 77 61-6 170-6 41 82-8 105-8 6 4-8 42-8 29 23-2 20-2 76 60-8 168-8 40 82- 104- 5 4- 41- 80 24- 22- 75 60- 167' 89 81-4 102-2 4 8-2 89-2 81 24-8 23-8 74 59-2 165-2 88 30-2 100-4 8 2-4 87-4 82 25-6 25-6 73 53-4 163-4 87 29-6 98-6 2 1-6 85-6 83 26-4 27-4 72 57-6 161-6 36 28-8 96-8 1 0-8 83-8 84 27-2 29-2 71 56-8 159-8 85 23- 95- o- 82- 85 28- 81- 70 56- 158- 84 27-2 93-2 _1 0-8 80-2 86 28-8 32-8 69 55-2 156-2 83 26-4 91-4 2 1-6 28-4 87 29-6 34-6 68 54-4 154-4 82 25-6 89-6 3 2-4 26-6 83 80-4 86-4 67 53-6 152-6 81 24-8 87-8 4 8-2 24-8 39 81-2 88-2 66 52-8 150-8 80 24- 86- 5 4- 28- 40 82- 40' 65 25- 149- "Water, in common with molten cast-iron, molten bismuth, and various other fluid substances, the particles of which assume a crystalline arrangement during congelation, suffers an increase of bulk as the point of congelation is approached, and expands in solidifying. But so soon as any of these substances has become solid, it then contracts with every diminution of temperature. Water in freezing bursts by its expansion any vessel in which it may be confined, and ice, being lighter than water, floats upon water. So also for a like reason solid cast-iron floats on molten cast-iron. The point of maximum density of water is 39'1 Fah- renheit, and between that point and 32 the bulk of water in- 140 THEORY OF THE STEAM-ENGINE. creases by cold. A cubic foot of water at 32 weighs 62-425 Ibs., whereas a cubic foot of ice at 32 weighs only 57'5 Ibs. There is consequently a difference of nearly 5 Ibs. in each cubic foot, between the weight of ice and the weight of water. DILATATION. Dilatation of Solids. A solid body of homogeneous texture will dilate uniformly throughout its entire bulk by the applica- tion of heat. Thus, if it be found that a bar of zinc is increased one 340th part of its length by being raised in temperature from 32 to 212, its breadth will also be increased one 340th part, and its thickness will be increased one 340th part. It is found, moreover, that equal increments of heat produce equal augmen- tations of volume in nearly all bodies, at all temperatures, until the melting-point is approached, when irregularities occur. Different solids dilate to different amounts when subjected to the same increase of temperature, and advantage is taken of this property in the arts in the construction of time-keepers and other instruments. Thus, in Harrison's gridiron pendulum, the ball is composed of bars of different metals, some of which ex- pand more than the others at the same temperature ; and as the bars which expand the most are fixed at the lower ends and ex- pand upwards, they compensate for the expansion of the pendu- lum rod in the opposite direction, and maintain the centre of os- cillation in the same place. The following table exhibits the rates of dilatation of various solids, as ascertained by the best authorities : DILATATION PRODUCED BY HEAT. 141 DILATATION OF SOLIDS BY HEAT. Bodies. Dilatation from 32 to 212 C , according t Flint Glass (English) Dilatation in Fractions. Decimal 9 Lavoisier and^ 0-00081166 0-00085655 0-00087199 0-00087572 0-00089694 0-00089760 0-00091750 0-00089089 0-00107880 0-00107915 0-00107960 0-00123956 0-00122045 0-00123504 0-00146606 0-00151361 0-00155155 0-00171220 0-00171733 0-00172240 0-00186670 0-00187821 0-00188970 0-00190868 0-00190974 0-00193765 0-00217298 0-00284836 071. 0-00083333 0-00108333 0-00115000 0-00122500 0-00125833 0-00139167 0-00170000 0-00181667 0-00187500 0-00190833 Vulgar. r ^aplace. T^V? 11 tfT 1 A 6 TuTT ToStr rrW sis TOT Tl7 li-ff -5$? F&T F4T Stt ttZ T5T 6^S 635" -sfe vb ?tr Ti-s T5"0" lir TW TTTT ^ T5ff it* T$-5 "5^5 *4lT TTlbf Platinum (accordin** to Borda) Glass (French) with lead Glass tube without lead .. Ditto Ditto Ditto Glass (St Gobain)- Steel (untempered) Ditto Ditto Steel (yellow temper) annealed at 65 Iron, round wire-drawn Gold Gold (French standard) annealed Gold (ditto) not annealed Ditto Ditto Ditto Silver (French standard) Silver Tin Falmouth According to Smeai Glass, white (barometer tubes) \ Steel Steel (tempered). Iron Copper. Copper 8 parts, tin 1 Brass 16 parts, tin 1 142 THEOKY OF THE STEAM-ENGINE. DILATATION OF SOLIDS BY HEAT Continued. Bodies. Dilatation in Fractions. Decimal. Brass wire 0-00193333 Telescope speculum metal 0-00193333 Solder (copper 2 parts, zinc 1) 0-00205833 Tin (fine) 0-00228333 Tin(grain) 0'00248333 Solder white (tin 1 part, lead 2) 0-00250533 Zinc 8 parts, tin 1, slightly forged 0'00269167 Lead 0-00286667 Zinc 0-00294167 Zinc lenthened -^ by hammering 0-00310833 Palladium ( Wollaston) O'OOl 00000 According to Dulong and Petit. P1 .._ j 32 to 2121 0-00088420 Platmum \ 32 to 572 0-00275482 32 to 212 0-00086133 Glass 32 to 392 0-00184502 32 to 572 0-00303252 T 32 to 212 0-00118210 on 32 to 572 0-00440528 r ( 32 to 212 0-00171820 ' ' ( 32 to 572 0-00564&72 According to Troughton. Platinum 0-00099180 Steel 0-001 18990 Steel wire, drawn 0-00144010 Copper 0-00191880 Silver j 0'00208260 From 32 to 217 according to Roy. Glass (tube) 0-00077550 Glass (solid rod) 0-00080833 Glass cast (prism of) O'OOlllOOO Steel(rod of) 0-00114400 Brass (Hamburg) 0-00185550 Brass (English) rod 0-00189296 Brass (English) angular 0-00189450 DILATATION PRODUCED BY HEAT. 143 Measure of the Force of Dilatation. The force with which solid bodies dilate and contract is equal to that which would compress them through the space they have dilated, or to that which would stretch them through a space equal to the amount of their contraction. Now, as it has been shown to be a phys- ical law that in every substance whatever, the same expenditure of heat, with the same extremes of temperature, will generate the same amount of mechanical power, it will follow that the less a body expands with any given increase of temperature, the more forcible will be the expansion, since the force, multiplied by the space passed through, must, in every case be a constant quantity. Dilatation of Liquids. The rate of expansion of liquids becomes greater as the temperature becomes higher, so that a mercurial thermometer, to be accurately graduated, should have the graduations at the top of the scale somewhat larger than at the bottom. It so happens, however, that there is a similar irregularity in the expansion of the glass bulb, but in an opposite direction ; and one error very nearly corrects the other. Ther- mometers are accordingly graduated by immersing the bulb in melting ice, and marking the point at which the mercury stands. The point at which the mercury stands when the bulb is im- mersed in boiling water is next marked, and the space between the two marks is divided into 180 equal parts, and the graduation is extended above the boiling-point and below the freezing, by continuing the same lengths of division on the scale. The increment of volume which water receives on being raised from 32 to 212 is aVrd. of its bulk at 32. Mercury at 32 expands /,th of its bulk at 32 by being raised to 212 ; and alcohol, by the same increase of temperature, increases in volume ^th of its bulk at 32. Compression and Dilatation of Oases. When a gas or vapour is compressed into half its original bulk, its pressure is doubled ; when compressed into a third of its original bulk, its pressure is trebled ; when compressed into a fourth of its original bulk, its pressure is quadrupled ; and generally the pressure varies inversely as the bulk into which the gas is compressed. So, in 144 THEORY OF THE STEAM-ENGINE. like manner, if the volume be doubled, the pressure is made one- half of what it was before the pressure being in every case reckoned from 0, or from a perfect vacuum. Thus, if we take the average pressure of the atmosphere at 14'T/ Ibs. on the square inch, a cubic foot of air, if suffered to expand into twice its bulk by being placed in a vacuum measuring two cubic feet, will have a pressure of 7*35 Ibs. above a perfect vacuum, and also of 7-35 Ibs. below the atmospheric pressure ; whereas, if the cubic foot be compressed into a space of half a cubic foot, the pressure will become 29'4 Ibs. above a perfect vacuum, and 14'7 Ibs. above the atmospheric pressure. This law, which was first investigated by Mariotte, is called Nanette's law. It has already been stated that a cubic foot of air at 32 becomes 1*365 cubic feet at 212, the pressure remaining constant; or if the volume be kept constant, then the pressure of one atmosphere at 32 be- comes 1-365 atmospheres, or a little over 1J atmospheres at 212. These two laws, which are of the utmost importance in all phys- ical researches, it is necessary fully to understand and remember. The rates of dilatation and compression for each gas are not pre- cisely the same ; but the departure from the law is so small as lo be practically inappreciable. According to M. Eegnault, the dilatation under the same pressure, and the increase of pressure with the same volume of different gases when heated from 32 to 212, is as follows : 00 -EFFICIENTS OF DILATATION OF DIFFERENT GASES. Pressure under constant voume. Dilatation under constant pressure. Hydrogen. . 0-3667 0-3661 Atmospheric air 0-3665 0-3670 Nitrogen 0-3668 u 0-3667 0-3669 Carbonic acid 0-3688 0-3710 Protoxide of nitrogen 0-3676 0.3719 0-3845 0-3903 0-3829 0.3877 The rates of dilatation vary somewhat with the pressure and temperature, and in the case of gases, which are more easily DILATATION PRODUCED BY HEAT. 145 condensable into liquids, the rate of dilatation increases rapidly with the density ; whereas the effect of heat is to remove these irregularities, and to maintain more completely the condition of a perfect gas. If we take the dilatation of atmospheric air when heated 180, or from 32 to 212, at - 367 as determined by M. Eegnault, then the amount of expansion which it will undergo from each increase of one degree in temperature will be 180th of 0-367 = 180th of ^V 18 8 oVo which multiplied by 1000=1366, which will be the volume in cubic inches at 212. Example 2. If the volume of steam at 212 be 1696 tunes the volume of the water which produced it, what will the vol- ume be if the steam be heated to 250-3 degrees Fahrenheit, the pressure remaining constant ? Here by the rule 212+459=671 and 250-3+459=709-3. Moreover, 709-3 divided by 671 and multiplied by 1696=1792-8, which will be the bulk which the 1696 measures of steam will acquire when heated to 250-3 out of contact with water, the pressure remaining the same as at first. If we take the co-efficient of expansion of a perfect gas be- tween 32 and 212 at 0-365 instead of 0'367, the expansion per degree Fahrenheit will be ^.3- of the total bulk=0-0020276 per degree Fahrenheit, instead of ^J^th, as supposed by the rule from which the table is computed. This is equivalent to start- ing from the point of absolute zero, or 461-2 below the zero of Fahrenheit; as 461-2 + 32 =493'2 5 148 THEORY OF THE STEAM-ENGINE. TABLE SHOWING THE MELTING POINTS OF VARIOUS BODIES, IN DE- GEEES OF FAHRENHEIT'S THERMOMETER. Name of Substance. Degrees Fahren. Experimentalist. Platinum 3082 English wrought-iron 2912 French " " 2732 Steel 2552 " another sample 2372 Cast-iron 2192 manganese 2282 brown, fusible 2192 very fusible 2012 white, fusible -. 2012 very fusible 1922 Gold (very pure) 2282 Gold coin 2156 Copper 1922 Brass 1859 Silver (very pure) 1832 Bronze 1652 Antimony 810 700 Zinc 705 680 629 Lead 608 590 518 5?5 480 512 455 Tin 446 442 433 Alloy, 5 parts tin , 1 part lead ' ' Alloy, 4 parts tin oho ' 1 part lead ' ' Alloy, 3 parts tin ,- 1 part lead ' ' Alloy, 2 parts tin . Rrt 1 part lead Alloy, 1 part tin [ 652 3 parts lead f Clarke. Vauquelin. Pouillet. Daniell. PouiUet. Murray. G. Morveau. Pouillet. Person. Pouillet. Irvine. Person. Ermann. Pouillet. Crichton. G. Morveau. Person. PouiUet. Crichton. Ermann. Pouillet. MELTING POINTS OF SOLIDS. 149 TABLE SHOWING THE MELTING POINTS OF VAEIOTTS BODIES, IN DE- GEEES OF FAHRENHEIT'S THERMOMETER continued. Name of Substance. Degrees Fahren. Experimentalist. Alloy, 3 parts tin 392 Pouillet 1 part bismuth Alloy, 2 parts tin 333-9 1 part bismuth * " * Alloy, 1 part tin 286-2 1 part bismuth Alloy, 4 parts tin ) 1 part lead > 246 6 parts bismuth ) 239 Person. 237 225 Dumas. Pouillet. Alloy, 2 parts lead ) 3 parts tin > 212 M 5 parts bismuth ) Alloy, 5 parts lead 3 parts tin 212 M 8 parts bismuth Alloy, 1 part lead 1 part tin 201 (( 4 parts bismuth Soda 194 Ga.y-Lus8ac Potash 162 M 136 111-6 109 Pouillet Person. Pouillet. 100 158 Murray. Pouillet. 164 M Wax unbleached 142 ( Stearine . . . . , 143 120 Person. Pouillet Spermaceti 109 120 M Acetic acid 113 <( Tallow 92 (c Ice 32 Oil of turpentine 14 cc 38.2 {( 150 THEOEY OF THE STEAM-ENGINE. LIQUEFACTION. Solidity is an accident of temperature, as there is every rea- son to believe that there is no substance in nature which may not be melted, and even vaporised, by the application of power- ful heat. There are two incidents attending liquefaction that are wor- thy of special attention : the first is that the liquefaction al- ways takes place at the same temperature in the case of the same substance, so that the melting-point may in fact be used as an index of temperature ; and the second is that during lique- faction the temperature remains fixed, the accession of heat which has been received during the process of liquefaction being consumed or absorbed in accomplishing the liquefaction, or in other words it has become latent. This heat is given out again in the process of solidification. "Water deprived of air and cov- ered with a thin film of oil may be cooled to 20 or 22 below the freezing-point. But on solidification the temperature will rise to the freezing-point. Each different substance has, under ordinary circumstances, its own particular melting-point; but it is found that the electrical condition of a body affects its melt- ing-point, and that electricity will fuse bodies at a low tempera- ture which commonly require for their fusion a very high degree of heat. Tims, platinum may be melted or vaporised by an electrical current, even although the heat generated is small ; and a process for separating metals from their ores by the aid of electricity has been projected by using low temperatures, aided by electricity, instead of high degrees of heat. In Part XV. of Taylor's Scientific Memoirs, page 432, there is a paper ' On the Incandescence and Fusion of Metallic Wires by Electricity,' by Peter Riess, being the substance of a paper read before the Eoyal Society of Berlin ; and in this paper it is shown that electrical fusion and vaporisation may take place at tempera- tures far below those at which metals are red hot. This prop- erty of electricity promises to be of service in the arts both in rendering refractory bodies fusible and in enabling bodies to be melted at low temperatures, which might be injured in their qualities by a subjection to high degrees of heat. Thus wrought- LATENT HEAT OF LIQUEFACTION. 151 iron if heated to a very high temperature, is liable to be burnt, unless carefully preserved from contact with the air ; whereas by sending a current of electricity through it, fusion may be accomplished at a comparatively low temperature, and any in- jury to the metal may thus be prevented. The melting-points of some of the most important substances are given in the pre- ceding table. Latent Heat of Liquefaction. Ice in melting absorbs as much heat as would raise the temperature of the same weight of water 142-65, or as would raise 142-65 times that weight of water 1 degree ; yet, notwithstanding this accession of heat, the ice, during liquefaction, does not rise above 32. If the heat employed to melt ice was applied to heat the same weight of ice-cold water, it would heat it to the temperature of 142-65 + 32 =174-65. The following table shows the amount of heat which becomes latent in the liquefaction of various bodies the unit of latent heat being the amount of heat necessary to raise the same weight of water 1 degree : TABLE SHOWING THE HEAT WHICH BECOMES LATENT IN THE LIQUEFACTION OF VAEIOUS SOLID BODIES, AS ASCERTAINED BY M. PEBSON. Names of Substances. Points of Fusion Fahrenheit Latent Heat for Unity of Weight Chloride of lime 83-3 72-42 Phosphate of soda 97-5 120-24 Phosphorus 111-6 8-48 Bees'-wax (yellow) 143-6 78-32 D'Arcet's alloy 204-8 10-73 239-0 16-61 Tin 456-0 26-74 Bismuth 518-0 22-32 Nitrate of soda 590-9 113-36 Lead 629-6 9-27 Nitrate of potash 642.2 83-12 7934 49-43 By this table we see that the heat which becomes latent in melting a pound of bees' wax would raise the temperature of a 152 THEORY OF THE STEAM-ENGINE. pound of water 78'32 degrees; and the heat which becomes latent in melting a pound of lead would raise the temperature of a pound of water 9-27 degrees. When there is no external source of heat, from which the heat which becomes latent in liquefaction can be derived, and the circumstances are, nevertheless, such as to cause liquefaction to take place, the heat which becomes latent is derived from the substances themselves, and correspondingly lowers their temper- atures. Thus, when snow and salt are mixed together, the snow and salt are dissolved. But, as in melting they absorb heat, and as there is no external source from which the heat is derived, the temperature of the mixture falls very much below that of either of the substances before mixing. So, also, when saltpetre and other salts are dissolved in water, cold is produced, and on this principle the freezing mixtures are compounded which are employed to produce artificial cold in warm climates. A more effectual process, however, is to compress air, which heats it ; and the superfluous heat being got rid of by water, it will follow that when this air is again expanded, it will take back an amount of heat equal to that which it before lost, and which demand for heat may be made to cool surrounding bodies. A very effectual freezing machine is constructed on this princi- ple. But it is material that the air in expanding should be made to generate power, else the friction consequent on its escape will generate heat. VAPORISATION. As the first phenomenon of the application of heat to a solid substance is to dilate it, and the next to melt it, so also the fur- ther application of heat converts it from a liquid into a vapour or gas. The point at which successive increments of heat, in- stead of raising the temperature, are absorbed in the generation of vapour, is called the boiling-point of the liquid. Different liquids have different boiling-points under the same pressure, and the same liquid will boil at a lower temperature in a va- cuum, or under a low pressure, than it will do under a high pressure. As the pressure of the atmosphere varies at different LATENT HEAT OF VAPORISATION. 153 altitudes, liquids will boil at different temperatures at different altitudes, and the height of a mountain may be approximately de- termined by the temperature at which water boils at its summit. Difference between Gases and Vapours. Vapours are sat- urated gases, or gases are vapours surcharged by heat. Ordi- nary steam is the saturated vapour of water, and if any of the heat be withdrawn from it, a portion of the water is necessarily precipitated. This is not so in the case of a gas under ordinary conditions. But if the gas be forced into a very small bulk, so that much of the heat is squeezed out of it, then it will follow that any diminution of the temperature will cause a portion of the gas to condense into a liquid. Surcharged or superheated steam resembles gas in its qualities, and a portion of the heat may be withdrawn from such steam, without producing the pre- cipitation of any part of its constituent water. Liquefaction of the gases. Many of the gases have already been brought into the liquid state, by the conjoint agency of cold and compression, and all of them are probably susceptible of a similar reduction by the use of means sufficiently powerful for the required end. They must, consequently, be regarded as the superheated steams, or vapours, of the liquids into which they are compressed. The pressures exerted by some of these steams or gases are given in the following table : TABLE SHOWING THE TEMPEEATUBE AND PRESSURE AT WHICH THE 8EVEEAL GASES NAMED ABE LIQUEFIED. Name* of Gues condensed. Temperature in degrees Fahrenheit Pressure ID Atmospheres. Temperature in degrees Fahrenheit Pressure In Atmospheres. Sulphurous acid 82 1-5 46'4 2'5 Cyanogen gas 82 2'8 Hvdiiodic acid 32 4'0 Ammoniacal gas 82 4.4. 50 5 Hydrochloric acid Protoxide of azote Carbonic acid 82 32 32 8-0 87-0 32-0 51-8 60 43 45 Latent heat of Evaporation. It has already been stated, that when a liquid begins to boil, the subsequent accessions of heat 7* 154 THEORY OF THE STEAM-ENGINE. which it receives go not to increase the temperature, but to ac- complish the vaporisation. The heat which thus ceases to be discoverable by the thermometer is called the Latent heat of Vaporisation ; and experiments have shown, that if the heat thus consumed had been employed to raise the temperature of the water, instead of boiling it away, the temperature of the water would have been raised about 1,000 degrees Fahrenheit, or it would have raised about 1,000 times the same weight of water that is boiled off 1 degree Fahrenheit. The heat consumed in evaporating the same weight of dif- ferent liquids varies very much, but it does not follow that any of them would, therefore, be better than water as an agent for the generation of power, as the bulk of the resulting vapour in those which require least heat is small, in the proportion of the smaller quantity of heat expended hi accomplishing the evapora- tion. Under the pressure of one atmosphere, or 14*7 Ibs. per square inch, the latent heat of steam from water has been found to be 966'1. Alcohol, which boils at 172-2, has a latent heat of evaporation of 364*3. Ether, which boils at 95, has a latent heat of evaporation of 162 '8, and sulphuret of carbon, which boils at 114-8, has a latent heat of evaporation of 156. The most important of the researches in connection with this subject are those which have reference to the Latent heat of Steam, and this topic has been illustrated by the researches of various experimentalists. At the atmospheric pressure, and starting at the temperature of 212, the following estimates of the latent heat of steam have been formed by the best authorities : Watt 950- Southern 945' Lavoisier 1000- Rumford. . 1008-8 Despretz 955-8 Regnault 966'1 Fabreand ) 9M . g Silbermann ) The experiments which are generally considered to be the most correct in connection with this subject are those of M. Kegnault. The following table, taken from his results, show that there is a difference of about 150 between the total heat of the vapour of water at the pressures corresponding to 32 and 446 respectively . LATENT HEAT OF STEAM. 155 SENSIBLE AND LATENT HEAT OE STEAM. BY M. KEGNAULT. Temperature In degree* Fahrenheit. Latent Het Sum of Sensible and Latent Heats. Temperature in degrees Fahrenheit. Latent Heat Sum of Sensible and Latent Heats. 32 1092-6 1124-6 248 936-6 1187-6 60 1080-0 1130-0 266 927-0 1193-0 68 1067-4 1135-4 284 914.4 1198-4 86 1054-8 1140-8 302 901-8 1203-8 104 1042-2 1146-2 320 889-2 1209-2 122 1029-6 1151-6 838 874-8 1212-8 140 1017-0 1157-0 356 862-2 1218-2 158 1004-4 1162-4 374 849-6 1223-6 176 991-8 1167-8 392 835-2 1227-2 194 979-2 1173-2 410 822-6 1232-6 212 966-6 1178-6 428 808-2 1236-2 230 952-2 1182-2 446 795-6 1241-6 Rules for connecting the temperature and elastic force of saturated steam. Various formula have been at different times propounded for deducing the elastic force of saturated steam from its temperature, and the temperature from the elastic force. The experiments of Mr. Southern, which were made at the in- stance of Boulton and "Watt, led to the adoption of the follow- ing rules, which, though not quite so accurate as some others which have since been arrived at, are sufficiently so for practical purposes, and being intimately identified with engineering prac- tice, it appears desirable to retain them. THE TEMPEBATUEE OF 8ATTJBATED STEAM BEING GIVEN IN DEGEEE8 FAHBENHEIT, TO FIND THE COEBESPONDING ELASTIC FOEOE IN INCHES OF MEBCTJEY BY SOUTHEBN's ETJLE. RULE. To the given temperature add 61 '3 degrees. From the logarithm of the sum subtract the logarithm of 135*767, which is 2-1327940. Multiply the remainder /by 5'13, and to the natural number answering to the sum, add the constant fraction !. The sum will ~be the elastic force in inches of mercury. Example. If the temperature of saturated steam be 250 % 3 Fahrenheit, what will be the corresponding elastic force in inches of mercury? 156 THEORY OF THE STEAM-ENGINE. Here 250-3 x 51-3 = 301-6 Log. 2.4794313 135-767 Log. 2-1327940 subtract. remainder 0-3466373 multiply by 6 -13 Natural number 60-013 Log. 1-7782493 This natural number increased by - 1 gives us 60*113 inches of mercury, as the measure of the elastic force sought. THE ELASTIC FORCE OF SATURATED STEAM BEING GIVEN IN INCHES OF MERCURY, TO FIND THE CORRESPONDING TEMPERATURE IN DEGREES FAHRENHEIT BY SOUTHERN'S RULE. EULE. From the given elastic force subtract the constant frac- tion -1 ; divide the logarithm of the remainder a r cj3 r , in which r=+20, t being the centigrade temperature reckoned fromO . Hence 0=6-2640348; Log. o=l'994049292 ; Log. 0=1-998848862 ; Log. 5=0-1397743, and Log. e=0-6924351. The principal properties of saturated steam as deduced from the experiments of M. Kegnault, exhibiting the pressure, the relative volume, the temperature, the total heat, and the weight of a cubic foot of steam of different densities, are given by Mr. Clark in the following tables : REGNAULT'S EXPERIMENTS ON STEAM. 159 PROPERTIES OF SATURATED STEAM. BY M. BEGNATJXT. Total Pressure per Square Inch. Eelatlve Volume . Temperature. Total Heat. Weight of one Cubic Foot. | Total Pressure per | Square Inch. Relative Volume. Temperature. Total Heat. Weight of One Cubic Foot. Us. Fahr. Fahr. Lbs. Lbs. Fahr. Fahr. Lbt. 15 1669 213-1 1178-9 0373 48 573 278-4 1198-8 1087 16 1572 216-3 1179-9 0397 49 562 279-7 1199-2 1108 17 1487 219-5 1180-9 0419 50 552 281-0 1199-6 1129 18 1410 222-5 1181-8 0442 51 542 282-3 1200-0 1150 19 1342 225-4 1182-7 0465 52 532 283-5 1200-4 1171 20 1280 228-0 1183-5 0487 63 523 284-7 1200-8 1192 21 1224 230-6 1184-3 0510 54 514 285-9 1201-1 1212 22 1172 233-1 1185-0 0532 55 506 287-1 1201-5 1232 23 1125 235-5 1185-7 0554 56 498 288-2 1201-8 1252 24 1082 237-9 1186-5 0576 57 490 289-3 1202-2 1272 25 1042 240-2 1187-2 0598 58 482 290-4 1202-5 1292 26 1005 242-3 1187-9 0620 59 474 391-6 1202-9 1314 27 971 244-4 1188-5 0642 60 467 292-7 1203-2 1335 28 939 246-4 1189-1 0664 61 460 293-8 1203-6 1356 29 909 248-4 1189-7 0686 62 453 294-8 1203-9 1376 30 881 250-4 1190-3 0707 63 447 295-9 1204.2 1396 31 855 252-2 1190-8 0729 64 440 296-9 1204-5 1416 82 830 254-1 1191-4 0751 65 434 298-0 1204-8 1436 33 807 255-9 1192-0 0772 66 428 299-0 1205-1 1456 34 785 257-6 1192-5 0794 67 422 300-0 1205-4 1477 35 765 259-3 1193-0 0815 68 417 300-9 1205-7 1497 36 745 260-9 1193-5 0837 69 411 301-9 1206-0 1516 37 727 262-6 1194-0 0858 70 406 302-9 1206-3 1535 38 709 264-2 1194-5 0879 71 401 303-9 1206-6 1555 39 693 265-8 1195-0 0900 72 396 304-8 1206-9 1574 40 677 267-3 1195-4 0921 73 391 305-7 1207-2 1595 41 661 268-7 1195-9 0942 74 386 306-6 1207-5 1616 42 647 270-2 1196-3 0963 75 381 307-5 1207-8 1636 43 634 271-6 1196-8 0983 76 377 308-4 1208-0 1656 44 621 273-0 1197-2 1004 77 372 309-3 1208-3 1675 45 608 274-4 1197-6 1025 78 368 310-2 1208-6 1696 46 595 275-8 1198-0 1046 79 364 311-1 1208-9 1716 47 584 277-1 1198-4 1067 80 359 312-0 1209-1 1736 160 THEORY OF THE STEAM-ENGINE. PROPERTIES OF SATURATED STEAM Continued. BY M. BEGNAULT. Total Pressure per Square Inch. Kelative Volume. Temperature. Total Heat. Weight of one Cubic Foot. | Total Pressure per Square Inch. Relative Volume. Temperature. Total Heat. Weight of One Cubic Foot. Lb Fahr. Falvr. Lbs. Lbs. Fahr. Fahr. Lbs. 81 355 312-8 1209-4 1756 114 261 337-4 1216-8 2388 82 351 313.6 1209-7 1776 115 259 338-0 1217-0 2406 83 348 314-5 1209-9 1795 116 257 338-6 1217-2 2426 84 344 315-3 1210-1 1814 117 255 339-3 1217-4 2446 85 340 316-1 1210-4 1833 118 253 339-9 1217-6 2465 86 337 316-9 1210-7 1852 119 251 340-5 1217-8 2484 87 333 317-8 1210-9 1871 120 249 341-1 1218-0 2503 88 330 318-6 1211-1 1891 121 247 341-8 1218-2 2524 89 326 319-4 1211-4 1910 122 245 342-4 1218-4 2545 90 323 320-2 1211-6 1929 123 243 343-0 1218-6 2566 91 320 321-0 1211-8 1950 124 241 343-6 1218-7 2587 92 317 321.7 1212-0 1970 125 239 344-2 1218-9 2608 93 313 322-5 1212-3 1990 126 238 344-8 1219-1 2626 94 310 323-3 1212-5 2010 127 236 345-4 1219-3 2644 95 307 324-1 1212-8 2030 128 234 346-0 1219-4 2662 96 305 324-8 1213-0 2050 129 232 346-6 1219-6 2680 97 302 325-6 1213-3 2070 130 231 347-2 1219-8 2698 98 299 326-3 1213-5 2089 132 228 348-3 1220-2 2735 99 296 327-1 1213-7 2108 134 225 349-5 1220-6 2771 100 293 327-8 1213-9 2127 136 222 350-6 1220-9 2807 101 290 328-5 1214-2 2149 138 219 351-8 1221-2 2846 102 288 329-1 1214-4 2167 140 216 352-9 1221-5 2885 103 285 329-9 1214-6 2184 142 213 354-0 1221-9 2922 104 283 330-6 1214-8 2201 144 210 355-0 1222-2 2959 105 281 331-8 1215-0 2218 146 208 356-1 1222-6 2996 106 278 331-9 1215-2 2230 148 205 357-2 1222-9 3033 107 276 332-6 1215-4 2258 150 203 358-3 1223-2 3070 108 273 333-3 1215-6 2278 160 191 363-4 1224-8 3263 109 271 334-0 1215-8 2298 170 181 368-2 1225-1 3443 110 269 334-6 1216-0 2317 180 172 372-9 1227-7 3623 111 267 335-3 1216-2 2334 190 164 377-5 1229-1 3800 112 265 336-0 1216-4 2351 200 157 381-7 1230-3 3970 113 263 336-7 1216-6 2370 KEGNAULT'S EXPERIMENTS ON VAPOURS. 161 M. Regnault extended his researches to the pressure of other vapours, beside that of water. The following are the results he obtained with alcohol, ether, sulphuret of carbon, chloroform, and essence of turpentine : TEMPEBATUBE AND ELASTIC FOBCE OF THE VAPOTJE3 OF DIFFEB- ENT LIQUIDS. BY M. BEGNAULT. [A millimetre is one thousandth part of a metre, or 0-03937 of an inch.] Tensionoftho Va- pour of Alcohol. Tension of the Va- pour of .Ether. Tension of the Va- pour of Sulphuret of Carbon. Tension of Vapour of Chloroform by Tension in Vacua. Tension of the Va- pour of Essence of Turpentine. Temperature in T)e- gNM Ceutrlgrade. 1 & II S P 1 ~ 9 go 2fr 9 SS .2*3 5 s' I* ft *j *i ll t* ll IB H if 1 a 3 3 |! P^ 9 || P || |6 3f? 11 % 3 i! &4 a| i H 1* 33 ;s % "3 i 1 21 20 10 10 20 80 40 50 60 70 80 90 100 110 120 180 140 150 152 8-12 3-34 6-50 12-78 24-08 44-0 78-4 184-1 220-8 860-0 689-2 812-S 1190-4 1685-0 2351-8 8207-8 4331-2 5687-7 7257-8 7617-8 20 10 10 20 80 40 60 60 70 80 90 100 101 69-2 113-2 182-8 286-5 434-8 687-0 918-6 1268-0 1730-8 2309-5 2947-2 8899-0 4920-4 7076-2 16 10 10 20 80 40 60 60 70 90 80 100 110 120 180 136 58-8 79-0 127-8 199-3 298-2 434-6 617-5 862-7 1162-6 1549-0 2080-5 2623-1 8321-3 4186-3 6121-6 6260-6 7029-2 + 10 20 80 36 130-4 190-2 276.1 842.2 10 20 80 40 50 60 70 80 90 100 110 120 180 140 160 160 170 180 190 200 210 220 222 2-1 2-8 4-8 7-0 11-2 17-2 26-9 419 61-2 91-0 184-9 187-8 267-0 847-0 462-8 604-5 777-2 989-0 1225-0 1514-7 1865-6 2251-2 2690-8 2778-5 by the method of ebullition. '86* 40 60 60 70 80 90 100 110 120 130 813-4 864-0 524-8 788-0 976-2 1367-8 1811-6 2354-6 8020-4 8818-0 4721-0 Unit of heat. It is convenient with the view of enabling us to compare the quantities of heat in different bodies to fix upon some thermal unit, by which quantities of heat may be measur- ed ; and the thermal unit employed in this country is the quan- 162 THEORY OF THE STEAM-ENGINE. tity of heat which is required to raise a pound of pure water at its point to maximum density, through one degree Fahrenheit. In France the thermal unit employed is the quantity of heat re- quired to raise a kilogramme of pure water at its point of great- est density through one degree Centigrade. A kilogramme is 2'20462 Ibs. avoirdupois, or a pound avoirdupois is 0-453593 of a kilogramme. A degree Centigrade is 1'8 degrees Fahrenheit ; and a degree Fahrenheit is 0'555 of a degree Centigrade. There are 3'96832 British thermal units in a French thermal unit, and there is 0-251996 of a French thermal unit in a British thermal unit. SPECIFIC HEAT. The specific heat of a substance is an expression for the quan- tity of heat in any given weight of it at a certain temperature, just as its specific gravity is an expression for the quantity of matter in a given bulk. Specific heat is most conveniently ex- pressed by a reference to the number of thermal units consumed in producing a given elevation of temperature in the body under consideration ; or, if the weight of a heated body immersed in water be multiplied by the temperature it loses, and the weight of water be multiplied by the temperature it gains, the quotient obtained by dividing the latter product by the former, will be the specific heat of the body. The specific heats of various substan- ces have been experimentally ascertained and recorded in tables, in which the specific heat of water is reckoned as unity. Thus, the specific heat of air is -2379, or it is 4*207 times less than that of water. An amount of heat, therefore, which would raise a pound of water 1 degree, would raise a pound of air 4'207 degrees. The following tables of specific heats are derived from the experiments of the best authorities, and chiefly from those of M. Eegnault. The specific heat of ice is given on the authority of M. Person. SPECIFIC HEATS OF DIFFEREKT SUBSTANCES. 163 SPECIFIC HEATS OF SOLIDS. The specific heat of water being reckoned as unity. NAME OP SUBSTANCE. Specific Heat. NAJtE OF SUBSTANCE. Specific Heat. Iron 0-11379 Gold 0-03244 Cast-iron (white) 0-12983 Platinum 0-03243 Steel, soft 0-11650 Glass 0-19768 " tempered . . . . 0-11750 Sulphur 0-20259 Copper 0-09515 Silicia 0-19132 Brass 0-09391 Carbon 0-24111 Zinc 0-09555 Coke 0-20200 Lead 0-03140 Diamond 0-14687 Tin 0-05623 Phosphorus 0-18870 Silver 0-05701 ' Ice 0-50400 SPECIFIC HEATS OF GASES AND YAPOTJKS. The specific heat of water leing reckoned, as unity. Specifi : Heat. For equal Weights. For equal Volumes. Oxygen . . 0-2182 0-2412 1-1056 Nitrogen 0-2440 0-2370 0-9713 Hydrogen 3-4046 0-2356 0-0692 Chlorine 0-1214 0-2962 2-4400 Protoxide of nitrogen 0-2238 0-3413 1-5250 Binoxide of nitrogen 0-2315 0-2406 1-0390 Carbonic oxide 0-2479 0-2399 0-9674 Carbonic acid 0-2164 0-3308 1-5290 Sulphuret of carbon 0-1576 0-4146 2-6325 Sulphurous acid G'1553 0-3489 2-2470 Ammonia. 0-5080 0-2994 0-5894 Protocarburet of hydrogen 0-5929 0-32-77 0-6527 Bi-carburet of hydrogen Water vapour, or steam Alcohol vapour 0-3694 0-4750 0-4513 0-3572 0-2950 0-7171 0-9672 0-6210 1-5890 0-4810 1-2296 2-5563 Chloroform vapour 0-1568 0-8310 5-3000 Turpentine vapour 0-5061 2-3776 4-6978 164 THEORY OF THE STEAM-ENGINE. SPECIFIC HEATS OF LIQUIDS. The specific heat of water heing reckoned as unity. NAME OF LIQUID. Specific Heat. NAME OF LIQUID. Specific Heat. Mercury 0-0333 Petroleum 0-4684 Turpentine 0-4672 i Solution Clilo. Lime . . 0-6448 Gin 0-4770 Spirit of Wine at 97.. 0-6588 Olive Oil 0-3096 Acetic Acid 0-6501 It will be observed from the foregoing tables that the specific heat of steam is nearly the same as the specific heat of ice. The specific heat of water, and also of air, occupying the same volume, is found to be the same at all temperatures between boiling and freezing, and the specific heat of air under a constant pressure may be taken at 0-2379. In other words, it requires just the same amount of heat to raise water and air one degree in tem- perature at any one part of the thermometric scale as at any other ; and the heat required to heat a pound of air 1 degree is only -2379, or less than one-fourth of the quantity required to heat a pound of water one degree. If therefore a pound of wa- ter at 60 has transferred to it the heat in a pound of air at 1000, the water will not acquire as much elevation of tempera- ture as the air loses, but only -2379 of that temperature. EATIO OF SPECIFIC HEATS OF GASES TINDER CONSTANT PEESStTEB TO THE SPECIFIC HEATS UNDER CONSTANT YOLUME. When air is compressed it generates heat, as is shown in the syringe in which a piece of tinder is lighted by the heat pro- duced by the sudden compression of air; and, contrariwise, when air or any other gas is expanded it produces cold. When, therefore, a cubic foot of air of the atmospheric pressure is heat- ed until its pressure is doubled, it will have a certain tempera- ture which will fall if the air is suffered to expand into a volume of two cubic feet, and to restore the previous temperature more heat must be added. It will take more heat, therefore, to heat SPECIFIC HEATS AND SPECIFIC GRAVITIES. 165 a cubic foot of air to a given temperature, if it be suffered to ex- pand, than if it be not suffered to expand ; and only that part of the heat is, properly speaking, specific heat, which is shown by the rise of temperature, that which is absorbed in enlarging the volume being, in point of fact, latent heat. Both kinds of' heat, however, are very generally called specific heat, but as the quantities are very different, it follows that there are two kinds of specific heat the one the specific heat under a constant vol- ume, and the other the specific heat under the increased volume to which the body naturally enlarges. It is only in the case of gases that there is a material difference between these specific heats. But in the case of gases the difference is very consider- able, and it is found that the specific heat under a constant press- ure divided by the specific heat under a constant volume, is equal, in the case of air, to 1-408; or, in other words, the speci- fic heat of air under a constant pressure is 1'408 times greater than that of air under a constant volume. The specific heat of air under a constant pressure, may be taken at '2379, which makes the specific heat under a constant volume '169. The fol- lowing tables of the specific heats, and some other properties of solids, liquids, and gases are given by Mr. Bankine : SPECIFIC HEATS AND SPECIFIC GRAVITIES OF METALS. Name of Metal. Weight of a cubic foot In Ib*. Do. Specific Gravity. S.G. Expansion from 32 to 212. E Specific Heat C. Specific Heat in foot-pound*. H Brass 487 to 533 7'8 to 8-5 00216 Bronze 524 8-4 00181 Copper. 537 to 556 8-6 to 8-9 00184 0951 73-8 Gold. 11S6 to 1224 19- to 19-6 0015 0298 28 '0 Iron, cast Iron, wrought.. Lead 444 480 712 7-11 7-69 11-4 0011 0012 0029 1188 0298 87-8 22-6 Platinum Silver 1811 to 1373 655 21 to 22 10-5 0009 002 0814 0557 24-2 43-0 fiteel 490 7-86 0012 Tin 462 7-4 0022 0514 89-7 Zinc 436 7^2 00294 0927 71 '6 Ice 676 0-92 604 889 166 THEORY OF THE STEAM-ENGINE. SPECIFIC HEATS AND SPECIFIC GEAVITIES OF LIQUIDS. Name of Liquid. Do. S. G. E. c. K. Water, pure at 39-1 62-425 1-000 0-04775 1-000 772-0 " sea, ordinary 64-05 1-026 0-05 Alcohol, pure 49-38 0-791 0-1112 " proof spirit ^Ether 57-18 44-70 0-916 0-716 0-517 399-1 Mercury 848-75 13-596 0-018153 0-033 25-5 Naphtha 52-94 0-848 Oil, Linseed 68-68 0-940 0-08 " Olive 57-12 0-915 0-08 " Whale 57-62 0-923 " of Turpentine 54-31 8-870 0-07 Petroleum 54-81 0-878 DENSITIES, VOLUMES, BATES OF EXPANSION, AND SPECIFIC HEATS OF GASES. Name of Gas. Weight cubic foot in Iba. De. Volume in cubic feet of lib. V... Expan- sion from 32" to 21J E. Specific Heat in degrees Fahr. Specific Heat in foot-pounds. Under volume. Cv. Under constant pressure Cp. Under constant volume. Kv. Under constant pressure. Kp. Air 0-080728 0-089256 0-005592 0-05022* 0-2093* 0-2137* 0-12259* 0-12344 0-0795 0-078411 0-563* 12-387 11-204 178-83 19-913* 4-777* 4-679* 8-157* 8-101 12-58 12-753 1-7762* 365 867 366 365* 365* 370 0-169 0-156 2-410 0-365* 0-173 0-238 0-218 3-405 0-475 0-481 0-1575 0-217 0-869 0-244 180-3 120-2 1860-6 281-3* 138-6 188-45 168-3 2628-7 866-7 871-3 121-6 167-0 284-9 188-4 Hydrogen .(Ether vapour.. Bisulph. carbon vapour Carbonic acid (ideal) Ditto (actual).. Olefiant gas. . . . Nitrogen Vapour of Mer- cury An asterisk (*) is affixed to the results computed for the ideal condition of a perfect gas. In these tables the volumes are taken at the temperature of melting ice, or 32 ; except in the case of water, which is taken at the temperature of maximum density, or 39-1. The pressure is taken at the usual atmospheric pressure of 2116-4 Ibs. upon the square foot. D is the density or weight of 1 cubic foot of the substance in SPECIFIC HEATS IN FOOT-POUNDS. 167 Ibs. avoirdupois under the pressure of one atmosphere, or 2116-4 Ibs. on the square foot. V is the volume in cubic feet of 1 pound avoirdupois of the substance at the foregoing temperature and pressure. S.G. is the specific gravity, water being taken as unity. E is the expan- sion of unity of volume for fluids, and unity of length for solids, at the temperature of melting ice, in being raised from the tem- perature of melting ice to the temperature of boiling water un- der the pressure of one atmosphere. C is the specific heat in de- grees Fahrenheit, the specific heat of water being reckoned as unity, and C y is the specific heat under a constant volume, while Op is the specific heat under a constant pressure. K is the speci- fic heat, reckoned not in degrees of temperature, but in the equiv- alent value of pounds raised 1 foot high. It has already been explained that there is as much power in the form of heat ex- pended in raising a pound of water 1 degree in temperature as would raise 772 Ibs. to the height of 1 foot ; and 772 foot-pounds is, consequently, the mechanical equivalent of a pound of water raised 1 degree. Now as the specific heats of all bodies are de- tenninable by the temperature to which a pound of the sub- stance will raise a pound of water, and as the accession of heat which a pound of water receives is transformable into its equiv- alent amount of mechanical power, it follows that the specific heats of all bodies may be represented by the amount of me- chanical power in foot-pounds, which is the equivalent of the heat consumed in raising a pound of any of these bodies through one degree of temperature. Such specific heats, accordingly, are those represented in the tables by the letter K ; the expression K T being the specific heat in foot-pounds of unity of weight under a constant volume, and K p the specific heat of the same weight under a constant pressure. The value of Z p -4- K T , Mr. Kankine states, is in the case of air, 1-408 ; oxygen, 1-4; hydrogen, 1-413 ; nitrogen, 1-409; and steam, considered as a perfect gas, 1-304; or, in other words, the specific heat tinder a constant volume is to the specific heat under a constant pressure as 1 to 1 '4 in the case of oxygen, differing slightly in the case of the other gases. 168 THEORY OF THE STEAM-ENGINE. PHENOMENA OF EBULLITION. Influence of Viscosity or Molecular Attraction. Salts dis- solved in water will raise the temperature of its boiling-point. The attraction of a salt for water being greater than the attrac- tion of the particles of the water for one another, will resist the repellent force of the heat to some extent. Mechanical pressure applied to the water has the same operation. Hence, water boils in a vacuum at a lower temperature than under the pressure of the atmosphere, and it also boils at a lower tem- perature under the pressure of one atmosphere than under a pressure of several atmospheres. Water, which has been well purged of air by boiling, does not pass into the state of steam when heated in clean glass vessels, until it has attained a tem- perature considerably higher than its ordinary boiling point; and when the steam finally forms, it forms rather by a jumping motion, or by a sudden shock, than by a gradual and silent dis- engagement. M. Magnus found that water well cleared of air may be raised to a temperature of 105 or 106 Centigrade before boiling, if the glass vessel in which it was heated were perfectly clean ; but if the vessel were soiled, or if dust or other foreign particles were suffered to enter it, the temperature would fall to the usual boiling point of 100 Centigrade. The sides of metallic vessels, or sawdust, metal filings, or insoluble particles of almost any kind, introduced into a liquid, lower its boiling-point. These particles are not at every point completely moistened by the water, and they have a less attraction for the particles of the fluid than the particles of the fluid have for one another. In the process of ebullition, therefore, the steam chiefly forms around those particles and seems to come out of them, and the boiling-point is lowered by the greater facility they occasion to the disengagement of the steam. M. Donny, by freeing water carefully from air, succeeded in raising it to a temperature of 135 without boiling ; but at this temperature steam was suddenly formed, and a portion of the water was projected forcibly from the tube. M. Donny concludes, from his experiments, that the mutual force of cohesion of the parti- SPHEROIDAL CONDITION OF LIQUIDS. 169 cles of water is equal to a pressure of about three atmospheres, and to this strong cohesive force he attributes the irregular jumping motion observed in ebullition, and also some of those explosions of steam-boilers which heretofore have perplexed en- gineers. It is well known that cases have occurred in which an open pan of boiling water has exploded, producing fatal results, and such explosions cannot be explained on the usual hypothesis. M. Donny says that liquids by boiling lose the greater part of the air which they hold in solution, and therefore the molecular at- traction begins to manifest itself in a sensible manner. The liquid consequently attains a temperature considerably above its normal boiling-point, which determines the appearance of new air-bubbles, when the liquid separates abruptly, a quantity of vapour forms, and the equilibrium is for the moment restored. The phenomenon then recurs again with increased violence, and an explosion may eventually ensue. Spheroidal Condition of Liquids on Hot Surfaces. If a drop of water or other liquid be thrown upon a hot metal plate or other highly heated surface, it does not moisten the surface or diffuse itself over it, but forms a flattened ellipsoidal mass ; and if the drop be sufficiently small, it forms a minute spheroid, which revolves rapidly round a shifting axis, and evaporates very slowly without entering the state of ebullition. From Church's experiments it appears that it is necessary for the liquid to emit vapour before it can assume the spheroidal state. Molten lead dropped upon a very hot platinum plate did not as- sume the spheroidal state, whereas mercury dropped upon this plate assumed the spheroidal state at once. The most remark- able experiments, however, which have been made in illustration of the phenomena of the spheroidal state are those of M. Bou- tigny, and to him engineers are mainly indebted for calling their attention to the subject. One of the most singular results ob- tained by M. Boutigny is the power of making ice in a red hot crocible. A small crucible or capsule of platinum being made white hot, some anhydrous sulphurous acid in the liquid state is poured into it. The boiling-point of this liquid is as low as 14 Fahrenheit ; but as it immediately on being projected into the 8 170 THEORY OF THE STEAM-ENGINE. capsule assumes the spheroidal state, it remains upon the white hot metal without touching it ; and if a few drops of water be now let fall upon the liquid acid, the water will be immediately frozen, and a piece of ice may be turned out of the crucible. M. Boutigny has also shown that if acids and alkalies in solution be poured into a clean red hot platinum crucible they will not unite, but both will assume the spheroidal state and roll about the bottom of the crucible without entering into combination. Not merely the gravitation of the liquid, therefore, but also its chemical affinity, appears to be superseded by the causes which make it assume the spheroidal state. When a liquid assumes the spheroidal state it does not wet the surface, but appears to avoid touching it, like water sprinkled upon grease. Instead of entering into violent ebullition when it reaches the hot surface, its temperature will rise very little, and the drops of liquid will either remain at rest or will acquire a gyratory motion. When the surface is cooled down to 400 to 500, depending on the nature of the surface and also on the nature of the liquid, the liquid will begin to diffuse itself, and will be suddenly scattered in all directions. The requisite tem- perature of a platinum plate to make water at the boiling-point assume the spheroidal state is 120 Centigrade, or 248 Fahren- heit ; but if glass be used instead of platinum, the temperature must be raised to 180 Centigrade, or 324 Fahrenheit. For water at Centigrade, the temperatures required are 400 and 800 respectively. When water assumes the spheroidal state, it is possible by placing the eye on the level of the hot surface to see between the surface and the liquid. The electric circuit, moreover, is in- terrupted, showing that there is no actual contact between the liquid and the plate. The repulsion existing between the liquid and the plate is usually imputed to the existence of an atmos- phere of vapour upon whichj as upon a cushion, the spheroids are supposed to rest. There is no reason to conclude, however, because vapour is raised from a liquid, that therefore its gravity must be suspended, and the cause is rather to be sought for in the motion of the spheroid, or of its internal particles, whereby the motion to which gravity is due is partially counteracted. COMMUNICATION OF HEAT. 171 Spheroidal State of the Water in Sailers. There can be no doubt that the water of boilers is sometimes repelled from the metal in the same manner as would be done if it were in the spheroidal state, and explosions have, no doubt, frequently had their origin in this phenomenon. Land boilers, whether of the cylindrical or waggon form, frequently bend down in the bottom where the strongest heat of the furnace impinges, and lead rivets, inserted in them for purposes of safety, are some- times melted out. The water appears to be repelled from the iron in those parts of the boiler bottom where the heat is great- est, and the iron becomes red hot, and is bagged or bent out by the pressure of the steam. In some boilers the bottom can at any time be made red hot by very heavy firing, and in most fac- tory boilers the bottom will be more or less injured if the stoker urges the fire very much. If gauge cocks be inserted at differ- ent levels, in a small upright cylindrical boiler, so that one cock is near the top, another near the bottom, and the rest in inter- mediate positions, it will follow, that if sufficient water be intro- duced into the boiler to show at the lowest gauge cock, it will continue to show there so long as a moderate heat is maintained. So soon, however, as the fire is made to burn fiercely, so as to impart a strong heat to the bottom, the water will disappear from the bottom cock and show in the top cock, thus proving that the water has been repelled by the heat until it occupies the top part of the boiler instead of the bottom part. COMMUNICATION OF HEAT. Heat may be communicated from a hot body to a cold one in three ways by Eadiation, by Conduction, and by Circulation. The rapidity with which heat radiates varies, other things being equal, as the square of the temperature of the hot body in excess of the temperature of the cold one ; so that a body if made twice as hot will lose a degree of temperature in one-fourth of the time ; if made three times as hot, it will lose a degree of temperature in one-ninth of the time ; and so on, in all other proportions. This explains how it comes that a very small pro- portion of surface in a boiler of which the furnace is maintained 172 THEORY OF THE STEAM-ENGINE. at a high temperature is equivalent to a much larger proportion of surface when the temperature is somewhat lower. Radiant heat may be concentrated into a focus by a reflector, in the same manner as light, and, like light, it may likewise be made to undergo retraction and polarisation. The conduction of heat through different substances varies very nearly in the same proportion as their conducting powers for electricity. Taking the conducting power of silver as 100, the following are the conducting powers of metals according to the best authorities : CONDUCTING POWERS OF METALS. Name of Body. Conductivity for Electricity. Conductivity for Heat. Eiess. Becquerel. Lenz. Wiedemann and Franz. Silver 100-0 66-7 69-0 18-4 10-0 12-0 100-0 91-5 64-9 iV-6 12-35 100-0 73-3 58-5 21-5 22-6 13-0 100-0 73-6 63-2 23-6 14-5 11-9 11-6 8-5 8'4 6-3 1-8 Copper . . Gold Brass Tin Iron . . . . Steel Lead 7-0 10-5 5-9 8-27 7-93 10-7 10-3 ' 1-9 Platinum German Silver Bismuth The conducting power of marble is about the same as the conducting power of bismuth ; and the conducting powers of porcelain and bricks are each about half that of marble. The conducting power of water is very low, and hence heat is trans- mitted downwards through water only very slowly. But up- wards it is transmitted rapidly by virtue of the circulation which then takes place. The efficiency of the heating surface of a boiler will depend very much upon the efficiency of the arrangements which are in force for maintaining or promoting a rapid circulation of the water. In like manner, the rapidity of the circulation which is maintained in the water used for refrigeration in surface con- CONDUCTING POWERS OF DIFFERENT SUBSTANCES. 173 densers will mainly determine the weight of steam condensed in the hour by each square foot of refrigerating surface. Peclet found by a number of experiments that water, when used as the refrigerating fluid, was about ten times more effectual than air ; and he further found that when water was used for refrigera- tion, each square foot of copper surface was able to condense about 21% Ibs. of steam in the hour. Mr. Joule, however, found that a square foot of copper surface might, by maintaining a rapid circulation of the cooling water, be made to condense 100 Ibs. of steam in the hour the cooling water being contained in a pipe concentric with that containing the steam, and flowing in the opposite direction. With this rapidity of refrigeration, the cooling surface of a condenser need only be about one six- teenth of the heating surface of the boiler which supplies the engine with steam. In ordinary land boilers 10 square feet of heating surface will boil off a cubic foot, or 62-J- Ibs. of water in the hour ; and one square foot of heating surface will therefore boil off one-tenth of this, or 6'25 Ibs. of water in the hour. To boil off 100 Ibs. in the hour would at this rate require 16 square feet of heating surface. But the 100 Ibs. of steam thus boiled off will, according to Mr. Joule, be condensed by one square foot of cooling surface ; so that, if this authority be accepted, the surface of a well-constructed condenser need only be about one-sixteenth of the heating surface of the boiler, the steam of which it condenses. The importance of maintaining a rapid circulation in the water of boilers has not yet been sufficiently recognised. It is desirable that solid water and not steam should be in contact with the heating surface, else the metal plating will be liable to become overheated, and any given area of heating surface will be much less effective. The species of boiler invented by Mr. David Napier, called the haystack boiler, and in which the water is contained in vertical tubes, is about the best species of boiler for keeping up a rapid circulation of the water. But it necessary to apply large return pipes or a wide water space all round the exterior of the boiler, with a diaphragm to permit ascending and descending currents, in order that the water car- ried upward by the steam may be immediately returned. 174 THEORY OF THE STEAM-ENGINE. COMBUSTION. Combustion is energetic chemical combination between the oxygen of the air and the constituents of the combustible. The combustibles chiefly used to generate the heat consumed by steam-engines are coal, wood, and sometimes charcoal. Coal consists chiefly of carbon and hydrogen, but the pro- portions in which these elements enter into the composition of different coals is very various. Cannel coal consists of about 60 per cent, of volatile matter, and 40 per cent, of coke and earthy matter, whereas splint coal consists of about 65 per cent, of coke, and 35 per cent, of volatile matter. Air consists of oxy- gen and nitrogen, mixed in the proportions of 8 Ibs. of oxygen to every 28 Ibs. of nitrogen, or 1 Ib. of oxygen to every 3| Ibs. of nitrogen. To accomplish the combustion of 6 Ibs. of carbon, 16 Ibs. of oxygen are necessary, forming 22 Ibs. of carbonic acid, which will have the same volume as the oxygen, and, therefore, a greater density. To accomplish the combustion of 1 Ib. of hydrogen, 8 Ibs. of oxygen are necessary. When, there- fore, we know the proportions of carbon and hydrogen existing in coal, it is easy to tell the quantity of oxygen, and, conse- quently, the quantity of air necessary for its combustion. As a general rule, it may be stated that, for every pound of coal burned in a furnace, about 12 Ibs. of air will be necessary to furnish the oxygen required, even if every particle of it entered into combination. But from careful experiments it has been found, that in ordinary furnaces, where the draught is produced by a chimney, about as much more air will in practice be neces- sary, or about 24 Ibs. per Ib. of coal burned. In the case of furnaces, with a more rapid draught maintained either by a steam jet or a fan blast, a smaller excess of air will suffice, and in those cases about 18 Ibs. of air will be required from the combustion of 1 Ib. of coal. If a cubic foot of air weigh 1-291 oz., then 12 Ibs. or 192 oz. will measure about 150 cubic feet, as 1-291 oz. bears the same proportion to 1 cubic foot, as 192 oz. bears to 150 cubic feet nearly. In ordinary furnaces, with a chimney therefor, which require 24 Ibs. of air per Ib. of coal, TOTAL HEAT PRODUCED BY COMBUSTION. 175 the volume of air necessary for the combustion of 1 Ib. of coal will be about 300 cubic feet, which is equal to the content of a room measuring about 6 feet 8J inches every way. The specific gravity of oxygen is a little more than that of air, being by the latest experiments 1-106, while that of air is 1. Now, as 16 Ibs. of oxygen unite with 6 Ibs. of carbon to form 22 Ibs. of carbonic acid, and, as the volume of the carbonic acid at the same temperature remains only the same as that of the original oxygen, it follows that the density or specific gravity of the carbonic acid must be greater than that of the oxygen, in the same proportion in which 22 is greater than 16. Multiply- ing therefore 1-106, which is the specific gravity of oxygen, by 22, and dividing by 16, we get 1-521, which must be the specific gravity of carbonic acid, if the specific gravity of oxygen is 1-106. Formerly, the specific gravity of oxygen was reckoned at 1-111, but there is reason to believe that 1-106 is the more accurate determination. Total Heat of Combustion. The temperature to which a pound of fuel would raise a pound of water, or the total heat of combustion in thermal units, has been carefully investigated by MM. Favre and Silbennann, whose determinations are reca- pitulated and condensed by M. Rankine as follows : TOTAL HEAT OF COMBUSTION OF 1 Ib. OF EACH OF THE COMBUSTIBLES ENTJMEBATED. Combtntlble, I Ib. of each being burned. Lbe. of Air required. Lb. of Air required. Total Heat in Thermal Unit*. Evaporative Power from 212. Hydrogen cas 8 86 62082 64-2 Carbon, imperfectly burned, ) so as to make carbonic > oxide j n 6 4,400 4-55 Carbon, completely burned, I BO as to make carbonic v acid \ 2! 12 14,500 15-0 Olettant gas 3J 15J 21844 22-1 Various liquid hydrocarbons . . Carbonic oxide, as much as") Is made by the imperfect 1 combustion of 1 Ib. of car- f bon, viz. 24 Ibs J li 6 j from 21,000 ( to 19,000 10,100 (from 22 \ to20 10-45 176 THEORY OF THE STEAM-ENGINE. "With regard to the quantities stated as being the total heat of combustion respectively of carbon completely burned, carbon imperfectly burned, and carbonic oxide, Mr. Eankine says that the following explanation has to be made : The burning of carbon is always complete at first ; that is to say, one pound of carbon combines with 2$ Ibs. of oxygen, and makes 3f Ibs. of carbonic acid ; and although the carbon is solid immediately before the combustion, it passes during the com- bustion into the gaseous state, and the carbonic acid is gaseous. This terminates the process when the layer of carbon is not so thick, and the supply of air not so small, but that oxygen in sufficient quantity can get direct access to all the solid carbon. The quantity of heat produced is 14,500 thermal units per Ib. of carbon, as already stated. . But in other cases part of the solid carbon is not supplied directly with oxygen,- but is first heated, and then dissolved into the gaseous state, by the hot carbonic acid gas from the other parts of the furnace. The. 3f Ibs. of. carbonic, acid gas from 1 Ib. of carbon, are capable of dissolving an additional Ib. of car- bon, making 4f Ibs. of carbonic . oxide -gas ; and the volume of this gas is double of that of the carbonic acid gas which pro- duces it. In this case, the heat produced, instead of being that due to the complete combustion of 1 Ib. of carbon or ... . 1 , . 14,500 falls to the amount due to the imperfect combustion of 2 Ibs. of carbon, or ..." '." . 2x4,400 x 8,800 Showing a loss of heat to the amount of . V'" ""'.' J 5,700 which disappears in volatilising the second pound of carbon. Should the process stop here, as it does in furnaces ill supplied with air, the waste of fuel is very great, as the carbonic oxide which is a species of invisible smoke has a large quantity of carbon in it which is dissipated in the atmosphere without use- ful result. But when the 4$ Ibs. of carbonic oxide gas, contain- ing 2 Ibs. of carbon, is mixed with a sufficient supply of fresh air, it burns with a blue flame, combining with an additional 2f Ibs. of oxygen, making 7 Ibs. of carbonic acid gas, and giving ECONOMIC VALUES OF DIFFERENT COALS. 177 additional heat of double the amount due to the combustion of 1-J- Ib. of carbonic oxide ; that is to say, 10,100 x 2 = 20,200 to which being added the heat produced by the imperfect combustion of 2 Ibs. of carbon, or ... 8,800 there is obtained the heat due to the complete combustion of 2 Ibs. of carbon, or . . . . 2 x 14,500 = 29,000 The evaporative powers of different kinds of coal in practice is given in the following table : TABLE SHOWING THE ECONOMIC VALUES OF DIFFEBENT COALS. BY DE J.A. BECIIE AND PLAYFAIK. I fames of Coal employed In the Experiment*. Economical evaporating her of Ibs., of Water evapo- rated from 21 2 by 1 Ib. of CoaL Weight of 1 cubic foot of the Coal as used for Fuel. Ibs. Space occn- pied by 1 ton of the Coal la cubic feet Rate of eva- poration, or number of Ibs. of Water evaporated per hour. Mean. f Graigola 9-35 60-166 37-28 441-48 Anthracite (Jones & Co.) Oldcastle Fiery Vein Ward's Fiery Vein Binea 9-46 8-94 9-40 9-94 58-25 50-916 57-483 57-08 88-45 43-99 89 89-24 409-37 464-30 629-90 486-95 Llangennech 8-86 56-93 89-34 873-22 Pentrepoth 8'72 57-72 38-80 881-50 Pentrelellin 6-36 66-166 83-85 247-24 1 Duffryn 10-14 58-22 42-09 409-32 Mynydd Newydd 9-52 56-83 89-76 470-69 = Threft-quarter'Rock Vein Cwm Frood Eock Vein. Cwm Nanty-gros 8-84 8-70 8-42 66-388 55-277 56-0 89-72 40-52 40-00 486-86 879-80 404-16 Kesolven 9-53 58-66 88-19 890-25 7-47 55-7 40-216 250-40 Bed was 9-79 60-5 44-32 476-96 Ebbw Vale 10-21 58-3 42-26 460-22 Porthmawr 7-53 58-0 42-02 847-44 LColeshill 8-00 53-0 42-26 406-41 ^ s - 'Dalkeith Jewel Seam... " Coronation ( Seam ) Wallsend Elgin 7-08 7-71 8-46 498 51-66 M-ii 44-98 48-86 41-02 855-18 870-08 435-77 rli Fordel Splint 7-56 55-0 40-72 464-98 Grangemouth 7-40 54-25 40-18 880-49 $ 1 '.mi mill ill 7-80 52'5 42-67 897-78 = - H Lydney (Forest of Dean) Slievardagh (Irish An- I thracite) j 8-52 9-85 54-444 62-8 41-14 85-66 487-19 473-18 "Wylam's Patent Fuel... Warlich'8 " Bell's 8-92 10-86 8-58 65-08 69-05 65-8 84-41 82-44 84-80 418-89 457-84 649-11 8* 178 THEORY OF THE STEAM-ENGINE. Maximum Temperature of the Furnace. When we know the total heat of a combustible in thermal units, the weight of the smoke and ashes or the products of combustion, as they are called, and their specific heat, it is easy to tell what is the high- est temperature that the furnace can attain, supposing that the air is not artificially heated. Thus the chief products of combus- tion of coal being carbonic acid, steam, nitrogen, and ashes, with a certain proportion of residual air, which passes unchanged through the fire ; then, if we reckon the specific heat of carbonic acid at 0-217, of steam at 0*475, of nitrogen at 0-245, of air at 0*238, and of ashes at 0*200, and take into account the quantities of each which are present, the mean specific heat of the prod- ucts of combustion may be taken, without much error, as about equal to the specific heat of air. Now, as 12 Ibs. of air are re- quired for the combustion of a pound of carbon, even if every particle of the oxygen be supposed to enter into combination, the weight of the products of combustion will on that supposi- tion be 12 Ibs. + 1 lb., or 13 Ibs. If we take the total heat of combustion of carbon or charcoal at 14,500, and the mean speci- fic heat of the products of combustion at 0-238, then the specific heat multiplied by the weight will be 3*094 ; and 14,500 divided- by 3*094 = 4689, which will be the temperature to which the furnace would be raised in degrees Fahrenheit, supposing every atom of oxygen that entered the furnace entered into com- bination. If, however, as will be the case in ordinary furnaces, twice that quantity of air necessarily enters, then the weight of the products of combustion of 1 lb. of coal will be 25 lb., which, multiplied by the specific heat = 5*95, and 14,500 divided by 5-95 = 2,437, which is the temperature in degrees Fahrenheit that, on this supposition, the furnace would have. If 18 Ibs. of air be supplied per lb. of coal, as suffices in the case of furnaces with artificial draught, then the weight of the products of com- bustion will be 19 Ibs., which, multiplied by the specific heat, gives 4-522, and 14,500 divided by 4-522, gives 3,207 as the tem- perature of the furnace in degrees Fahr. This in point of fact may be taken as a near approach to the temperature of hot fur- naces, such as that of a locomotive boiler. RATE OP COMBUSTION. 179 The increased volume which any given quantity of air at 32 will acquire, by raising its temperature through any given num- ber of degrees, can easily be determined by the rule already given for that purpose. Mi*. Rankine has computed the volume in cubic feet, which 121bs. of air, 18 Ibs., and 24 Ibs., will respec- tively acquire, when heated to different temperatures, by com- bining with 1 Ib. of carbon in a furnace ; the volume of 12 Ibs. at 32, and at the atmospheric pressure, being taken at 150 cubic feet, of 18 Ibs. at 225 cubic feet, and of 24 Ibs. at 300 cubic feet. The results are as follows : TEMPERATURES OF COMBUSTION AND VOLUMES OP PRODUCTS. Supply of Air In pounds per Ib. of fuel. Temperatures. 12 Ibs. 18 Ibs. 24 Ibs. Volume of Air or Gases In cubic feet at each Temperature. 32 150 225 300 68 161 241 322 104 172 258 344 212 205 307 409 892 259 389 519 572 314 471 628 752 369 553 738 1112 479 718 957 1472 588 882 1176 1832 697 1046 1395 2500 906 1359 1812 3275 1136 1704 4640 1551 Rate of Combustion. The rate of combustion, or the quan- tity of fuel burned in the hour upon each square foot of fire- grate, varies very much in different classes of boilers. In Cor- nish boilers it is 3 Ibs. per square foot ; in the older class of land boilers, 10 Ibs. ; in more recent land boilers, 13 to 14 Ibs. ; in modern marine boilers, 16 to 24 Ibs., and in locomotive boilers* 80 to 120 Ibs. on each square foot of fire-grate in the hour. 180 THEORY OF THE STEAM-ENGINE. THERMO-DYNAMICS. It has been already stated that heat and power are mutually convertible, and that the power in the shape of heat which is necessary to raise a pound of water through one degree Fahren- heit, would, if utilised without waste in a thermo-dynamic en- gine, raise 7T2 Ibs. through the height of 1 foot. A pound of water raised through a degree centigrade is equivalent to 1390 Ibs. raised through the height of 1 foot. In every heat engine, the greater the extremes of temperature, or the hotter the boiler or source of heat and the colder the condenser or refrigerator, the larger will be the proportion of the heat utilised as power. In a perfect steam engine, if a be the temperature of the boiler, reckoning from the point of absolute zero, and & be the temperature of the condenser, reckoning also from the point of absolute zero, the fraction of the entire heat communicated to the boiler which will be converted into mechanical effect, will be . Now it is clear if a = 5, or if the temperature of the a a j boiler and condenser are the same, the value of becomes a equal to 0, or there is none of the heat utilised as power, whereas, on the other hand, if a be taken larger and larger, the value of the fraction becomes continually greater, indicating that by in- creasing the difference of the temperatures of the boiler and condenser, a great quantity of the heat expended is converted into mechanical effect, and by taking a= o>, or infinity, the limit to which the fraction approaches is found to be unity, showing that in such a case, if it were possible of realisation, the whole of the heat would be converted into power. The formula given by Professor Thomson for determining the power generated by a perfect thermo-dynamic engine, is as follows : If S be the temperature of the source of heat, and T be the temperature of the refrigerator, both expressed in centigrade degrees ; and if H denote the total heat in thermal units centi- grade, entering the engine in a given time ; and J be Joule's POWER PRODUCIBLE IN A PERFECT ENGINE. 181 equivalent of 1390 Ibs. raised one foot high by a centigrade de- gree ; then the power produced, or "W the work performed, is S T W=JH- -' S + 274 This formula may be expressed in words, as follows : TO FIND THE POWEE GENERATED BY A PERFECT ENGINE IMPELLED BY THE MOTIVE POWER OF HEAT. KFLE. From the temperature of the source or ~boiler, subtract the temperature of the condenser ; divide the remainder ty the sum of the temperature of the source and 274, and multi- ply the quotient ty the total heat communicated to the en- gine per minute, expressed in the number of degrees through which it would raise one pound of water. Finally, multiply this product by 1390. The result is the number of pounds that the engine will raise a foot high in the minute. The temperatures are all taken in degrees centigrade. Example. In a steam-engine working with a pressure of 14 atmospheres, the temperature of the steam in the boiler will be 215 centigrade, and the temperature of the condenser may be taken at 44-44 centigrade. If a gram of coal be burned per minute, the heat imparted every minute to a pound of water win be -905 centigrade. Now 215 44-44 = 170-56 and 215 + 274 = 489, and 170-56 divided by 489 = 0-35, which mul- tiplied by -905 and by 1390 = 440 Ibs. raised 1 foot high every minute, which as a grain of coal is burned every minute, is very nearly the same result as that before indicated. Cheapest Source of Motive Power. The cheapest source of a mechanical power that will be available in all situations, is, so far as we yet know, the combustion of coal. Electricity and galvanism have been proposed as motive powers, and may be used as such, but they are much more expensive than coal. Mr. Joule has ascertained by his experiments that a gram of zinc, consumed in a galvanic battery, will generate sufficient power to raise a weight of 145-6 Ibs. through the height of one foot; 182 THEORY OF THE STEAM-ENGINE. whereas a grain of coal, consumed by combustion, will generate sufficient power to raise 1261-45 Ibs. to the height of 1 foot. Moreover, it appears certain that Mr. Joule's estimate of the heating power of coal is too small. A pound of coal will, under favourable circumstances, evaporate 12 Ibs. of water, which is equivalent to a pound of water being heated 2 degrees Fahren- heit by a grain of coal, or it is equivalent to 1544 Ibs. raised through 1 foot. This is more than ten times the power gener- ated by a pound of zinc. But as thermo-electric engines, it is estimated, expend their energy about four times more bene- ficially than heat engines, the dynamic efficacy of a pound of zinc may be taken as about 4-10ths of that of a pound of coal. A ton of zinc, however, costs fifty or sixty times as much as a ton of coals, while it is not half so effective. There does not ap- pear, therefore, to be the least chance of heat engines being superseded by electro-dynamic engines, of which zinc or some other metal supplies the motive force. EXPANSION OF STEAM. When air is compressed into a smaller volume, a certain amount of power is expended in accomplishing the compression, which power, as in the case of a bent spring, is given back again when the pressure is withdrawn. If, however, the air when compressed is suddenly dismissed into the atmosphere, the power expended in compression will be lost ; and there is a loss of power, therefore, in dispensing with that power, which is re- coverable by the expansion of the air to ita original volume. Now the steam of an engine is in the condition of air already compressed; and unless the steam be worked in the cylinder expansively which is done by stopping the supply from the boiler before the stroke is closed there will be a loss of a cer- tain proportion of the power which the steam would otherwise produce. If the flow of steam to an engine be stopped when the piston has performed one-half of the stroke, leaving the rest of the stroke to be completed by the expanding steam, then the efficacy of the steam will be increased 1-7 times beyond what it MODE OF COMPUTING BENEFIT OF EXPANSION. 183 would have been had the steam at half-stroke been dismissed without extracting more power from it ; if the steam be stopped at one-third of the stroke, the efficacy will be increased 2-1 times; at one-fourth, 2'4 times; at one-lifth, 2'6 times; at one- sixth, 2 -8 times; at one- seventh, 3 times; and at one-eighth, 3*2 times. TO FIND THE INCREASE OF EFFICIENCY ARISING FROM WORKING STEAM EXPANSIVELY. RULE. Divide the total length of the stroke ~by the distance (which call 1) through which the piston moves before the steam is cut off. The Neperian logarithm of the whole stroke ex- pressed in terms of the part of the strolce performed with the full pressure of steam, represents the increase of efficiency due to expansion. Example 1. Suppose that the steam be cut off at ^ ths of the stroke : what is the increase of efficiency due to expansion ? Here it is plain that -j-^ths of the whole stroke is the same as T ? T of the whole stroke. The hyperbolic logarithm of 7'5 is 2*015, which increased by 1, the value of the portion performed with full pressure, gives 3'015 as the relative efficacy of the steam when expanded to this extent, instead of 1, which would have been the efficacy if there had been no expansion. If the steam be cut of at |, f , f , |, f , , or $th of the stroke, the respective ratios of expansion will be 8, 4, 2-66, 2, 1-6, 1'33, and 1-14, of which the respective hyperbolic logarithms are 2-079, 1-386, 0-978, 0-693, 0*470, 0-285, and 0-131 ; and if the steam be cut off at T V, -?*, T 3 , T\> tV TIT> T 7 i A, or rVhs of the stroke, the respective ratios of expansion will be 10, 5, 3-33, 2-5, 1-66, 1'42, 1-25, and 1-11, of which numbers the respective hyperbolic logarithms are 2-303, 1-609, 1-203, 0-916, 0-507, 0-351, 0'223, and 0'104. With these data it will be easy to compute the mean pressure of steam of any given initial pressure when cut off at any eighth part or any tenth part of the stroke, as we have only to divide the initial pressure of the steam in Ibs. per square inch by the ratio of expansion, and to multiply the quo- 184 THEORY OF THE STEAM-ENGINE. tient by the hyperbolic logarithm, increased by 1, of the number representing the ratio, which gives the mean pressure through- out the stroke in Ibs. per square inch. Thus, if steam of 100 Ibs. be cut off at half stroke, the ratio of expansion is 2, and 100 divided by 2 and multiplied by T693 = 84'65, which is the mean pressure throughout the stroke in Ibs. per square inch. The terminal pressure is found by dividing the initial pressure by the ratio of expansion; thus, the terminal pressure of steam of 100 Ibs. cut off at half stroke will be 100 divided by 2 = 50 Ibs. per square inch. Example 2. "What is the mean pressure throughout the stroke of steam of 200 Ibs. per square inch cut off at ~th of the stroke ? Here 200 divided by 10 = 20, which, multiplied by 3-303 (the hyperbolic logarithm of 10 increased by 1) gives 66'04, which is the mean pressure exerted on the piston throughout the stroks in Ibs. per square inch. If the steam were cut off at ]th of the stroke instead of ^Vth, then we should have 200 divided by 8 = 25, which, multiplied by 3'079 (the hyperbolic logarithm of 8 increased by 1), gives TS'O'TS Ibs., which would be the mean pressure on the piston throughout the stroke in such a case. If the initial pressure of the steam were 3 Ibs. per square inch, and the expansion took place throughout ths of the stroke, or the steam were cut off at |th, then 3 -j- 8 = '375, which x by 3-079 = 1'154625 Ibs. per square inch of mean pressure. There are various expedients for stopping off the supply of steam to the engine at any desired point of the stroke, which are described in my ' Catechism of the Steam Engine,' and which, consequently, it would be superfluous to recapitulate here. One mode is by the use of an expansion valve, and another mode is by extending the length of the face of the or- dinary slide valve by which the steam is let into and out of the cylinder, which extension of the face is called lap or cover. For the purposes of this work it will be sufficient to recapitu- late the mean pressure of the steam on the piston of an engine throughout the whole stroke, supposing the steam to be cut off PRESSURES AT DIFFERENT RATES OF EXPANSION. 185 at different successive points of the stroke, counting first by eighths, and next by tenths, and to explain what amount of lap answers to a given expansion, and what expansion follows the use of a given proportion of lap. The mean pressure of the steam throughout the stroke, with different initial pressures of steam and different rates of expansion, or, in other words, the equivalent constant pressure that would be exerted throughout the stroke if such a pressure were substituted for the varying pressures to which the piston is in reality subjected, are exhib- ited in the following tables, in one of which the pressures are those which would ensue if the expansion took place during so many eighths of the stroke, and in the other during so many tenths of the stroke : MEAN PRESSURE OF STEA.M AT DIFFERENT DENSITIES AND BATES OF EXPANSION. The column headed 0, contains the Initial Pressure in Ibs., and the remaining columns contain the Mean Pressure in Ibs., with different amounts of Expansion. Proportion of the Stroke through which Expansion takes place. i s 1 I V 8 2-96 2-89 2-75 2-53 2-22 1-789 1-154 4 8-95 8-85 8-67 8-38 2-96 2-386 1-589 5 4-948 4-S18 4-598 4-232 8-708 2-982 1-921 6 6-987 5-782 5-512 5-079 4-450 8-579 2-309 7 6-927 6-746 6-431 5-925 5-241 4175 2-694 8 7-917 7-710 7-350 6-772 6-984 4-772 8-079 9 8-906 8-678 8-268 7-618 6-675 6-868 8-463 10 9-896 9-637 9-187 8-465 7-417 5-965 3-848 11 10-885 10-601 10-106 9-311 8-159 6-561 4-288 12 11-875 11-565 10-925 10-158 8-901 7-158 4-618 18 12-865 12-528 11-948 11-004 9-642 7-754 6-008 14 18-854 13-492 12-862 11-851 10-884 8-581 r>-:!ss 15 14-844 14-456 13-781 12-697 11-126 8-947 5-778 16 15-834 15-420 14-700 13-544 11-868 9-544 6-158 17 16-828 16-883 15-618 14-890 12-609 10-140 6-542 18 17-818 17-847 16-587 16-287 18-851 10-787 6-927 19 18-702 18-811 17-448 16803 14098 11-338 7-812 20 19-792 19-275 18-875 17-970 14-885 11-930 7-697 25 24-740 24-093 22-968 21-162 18-548 14-912 9-621 80 29-688 28-912 27'52 25-895 22-252 17-895 11-546 85 84-686 88-781 88-156 29-627 25-961 20-877 18-470 40 39-585 8S-550 86-750 88-860 29-670 23-860 15-895 45 44-588 48-368 41-348 88-092 88-878 26-842 17-819 50 49-481 48-187 45-987 42-825 87-067 29-825 19-248 186 THEOKY OF THE STEAM-ENGINE. MEAN PEESSURE OF STEAM AT DIFFERENT DENSITIES AND EATE3 OF EXPANSION. The column Jieaded contains the Initial Pressure in Ibs., and the remaining columns contain the Mean Pressure in Ibs., with different amounts of Expansion. Proportion of the Stroke through which Expansion takes place. Q 4 5 g 1 O 1 U 1 1 10 10 1 To 10 'A 2-980 2-930 2-830 2-710 2-539 2-299 1-981 1-668 0-990 4 3-974 3-913 3-780 3-614 3-3S6 3-065 2-642 2-087 1-320 5 4-968 4-892 4-725 4-518 4-232 8-832 8-303 2-609 1-651 6 5-961 5-870 5-670 5-421 5-079 4-598 3-963 8-130 1-981 7 6-955 6-848 6-615 6-325 5-925 5-364 4-624 8-652 2-811 8 7-948 7-827 7-560 7-228 6-772 6-131 5-284 4-174 6-641 9 8-942 8-805 8-505 8-132 7-618 6-897 5-945 4-696 2-971 10 9-936 9-784 9-450 9-036 8-465 7-664 6-606 5-218 8-302 11 10-929 10-762 10-395 9-939 9-311 8-430 7-266 5-739 3-632 12 11-923 11-740 11-340 10-843 10-158 9-196 7-927 6-261 8-962 13 12-856 12-719 12-285 11-746 10-994 9-963 8-587 6-783 4-292 14 IS 310 13-967 13-230 12-650 11-851 10-729 9-248 7-305 4-622 15 14-904 14-676 14-175 13-554 12-697 11-496 9-909 7-827 4-953 16 15-897 15-654 15-120 14-457 13-544 12-262 10-569 8-348 5-283 17 16-891 16-632 16-065 15-361 14-051 13-028 11-230 8-870 5-613 18 17-884 17-611 17-010 16-264 15-237 13-795 11-890 9-392 5-944 19 18-878 18-589 17-955 17-168 16-083 14-561 12-551 9-914 6-273 20 19-872 19-568 18-900 18-072 16-930 15-328 18-212 10-436 6-600 25 24-840 24-460 23-625 22-590 21-162 19-100 16-515 13-040 8-255 30 29-808 29-352 28-350 27-108 25-895 22-992 19-818 15-654 9-006 85 84-776 34-244 88-075 81-626 29-627 26-824 23-121 18-263 11-557 40 39-744 39-136 37-800 36-144 33-860 30-656 26-224 20-872 13-208 45 44-912 44-028 42-525 40-662 88-0921 34-888 29-727 23-481 14-859 50 49-630 48-920 47-250 45-180 42-325: 38-320 83-030 26-090 16-510 Example. If steam be admitted to the cylinder at a pressure of 3 Ibs. per square inch, and be suffered to expand during -Jth of the stroke, the mean pressure during the whole stroke will be 2*96 Ibs. per square inch. In like manner, if steam at the press- ure of 3 Ibs. per square inch were cut off after the piston had gone through the ^th of the stroke, leaving the steam to expand through the remaining ths, the mean pressure during the whole stroke would be 1*154 Ibs. per square inch. EELATIONS BETWEEN THE LAP OF THE VALVE AND THE AMOUNT OF EXPANSION. The rules for determining the relations between the lap of the valve and the amount of the expansion are as follows : EFFECTS OF LAP ON THE VALVE. 187 TO FIND HOW MUCH LAP MUST BE GIVEN ON THE STEAM SIDE, IN ORDER TO CUT THE STEAM OFF AT ANY GIVEN PAET OF THE 6TBOKE. RULE. From the length of the strode of the piston subtract the length of that part of the stroke that is to be made before the steam is cut off. Divide the remainder by the length of the stroke of the piston, and extract the square root of the quotient. Multiply the square root thiis found by half the length of the stroke of the valve, and from the product take half the lead, and the remainder will be the amount of lap required. TO FIND AT WHAT PAET OF THE STEOKE ANY GIVEN AMOUNT OF LAP ON THE STEAM SIDE WILL CUT OFF THE STEAM. RULE. Add the lap on the steam side to the lead : divide the sum by half the length of stroke of the valve. In a table of natural sines find the arc whose sine is equal to the quo- tient thus obtained. To this arc add 90, and from the sum of these two arcs subtract the arc whose cosine is equal to the lap on the steam side divided by half the stroke of the valve. Find the cosine of the remaining arc, add 1 to it, and mul- tiply the turn by half the stroke of the piston, and the prod- uct is the length of that part of the stroke that will be made by the piston before the steam is cut off. TO FIND HOW MUCH BEFOEE THE END OF THE STEOKE THE EX- HAUSTION OF THE STEAM IN FEONT OF THE PISTON WILL BE OUT OFF. RULE. To the lap on the steam side add the lead, and divide the sum by half the length of the stroke of the valve. Find the arc whose sine is equal to the quotient, and add 90 to it. Divide the lap on the exhausting side by half the stroke of the valve, and find the arc whose cosine is equal to the quo- tient. Subtract this arc from the one last obtained, and find the cosine of the remainder. Subtract this cosine from 188 THEORY OF THE STEAM-ENGINE. 2, and multiply the remainder l>y lialf the stroke of the pis- ton. The product is the distance of the piston from the end of the stroke when the exhaustion is cut off. TO FIND HOW FAB THE PISTON IS FROM THE END OF ITS STEOKE, WHEN THE STEAM THAT IS PROPELLING IT BY EXPANSION 13 AL- LOWED TO ESCAPE TO THE CONDENSER. RULE. To the lap on the steam side add the lead ; divide the sum by half the stroke of the valve, and find the arc whose sine is equal to the quotient. Find the arc whose cosine is equal to the lap on the exhausting side, divided by half the stroke of the valve. Add these two arcs together, and sub- tract 90. Find the cosine of the residue, subtract it from 1, and multiply the remainder by half the stroke of the piston. The product is the distance of the piston from the end of its stroke, when the steam that is propelling it is allowed to es- cape to the condenser. NOTE. In using these rules all the dimensions are to be taken in inches, and the answers will he found in inches also. It will readily be perceived from a consideration of these rules that supposing there is no lead the point of the stroke at which the steam is cut off is determined by the proportion which the lap on the steam side bears to the stroke of the valve. Whatever the absolute dimensions of the lap may be, therefore, it will follow that, in every case in which it bears the same ratio to the stroke of the valve, the steam will be cut off at the same point of the stroke. As some of the foregoing rules are difficult to be worked out by persons unacquainted with trigonometry, it will be conven- ient to collect the principal results into tables, which may be applied without difficulty to the solution of any particular ex- ample. This accordingly has been done in the three following tables, the mode of using which it will now be proper to ex- plain. RELATIONS OF LAP AND EXPANSION. 189 I. PROPORTION OF LAP REQUIRED TO ACCOMPLISH VARIOUS DEGREES OP EXPANSION. Distance of the piston "I from the termina- 21 j* ?* i' 5* tion of its stroke, when the steam is V or 5*t or or or or A cut off, in parts of the length of its | i i i 8 A stroke J Lap on the steam side ~) of the valve, In de- I clmal parts of the | 289 270 250 228 204 ITT 144 102 length of its stroke. J Example. In the first line of the first table will be found eight different parts of the stroke of the piston designated ; and directly below each, in the second line, is given the quantity of lap requisite to cause the steam to be cut off at that particular part of the stroke. The different amounts of the lap are given in the second line in decimal parts of the length of the stroke of the valve ; so that, to get the quantity of lap corresponding to any of the given degrees of expansion, it is only necessary to take the decimal in the second line, which stands under the frac- tion in the first, that marks that degree of expansion, and mul- tiply that decimal by the length we intend to make the stroke of the valve. Thus suppose we have an engine in which we wish to have the steam cut off when the piston is a quarter of the length of its stroke from the end of it, we look in the first line of the table, and we shall find in the third column from the left, \. Directly under that, in the second line, we have the decimal, '250. Suppose that we consider that 18 inches will be a convenient length for the stroke of the valve, we multiply the decimal -250 by 18, which gives 4J. Hence we learn, that with an 18-inch stroke for the valve, 4 inches of lap on the steam side will cause the steam to be cut off when the piston has still a quarter of its stroke to perform. Half the stroke of the valve should always be at least equal to the lap on the steam side added to the breadth * of the port ; consequently, as the lap in this case must be 4 inches, and as * By the ' breadth ' of the port, is meant its dimensions in the direction of the. valro's motion ; in short, its perpendicular depth when the cylinder is upright 190 THEORY OF THE STEAM-ENGINE. half the stroke of the valve is 9 inches, the efficient breadth of the port cannot he more than 9 4 = 44- inches, since half of it is covered over hy the lap. If this breadth of port is not suffi- cient to give the required area to let the steam in and out, we must increase the stroke of the valve ; by which means we shall get both the lap and the breadth of the port proportionally in- creased. Thus, if we make the length of valve-stroke 20 inches, we shall have for the lap '250 x 20=5 inches, and for the breadth of the port 10 5 = 5 inches. This table, as we have already intimated, is computed on the supposition that the valve is to have no lead ; but, if it is to have lead, all that is necessary is to subtract half the proposed lead from the lap found from the table, and the remainder will be the proper quantity of lap to give to the valve. Suppose that, in the last example, the valve was to have J inch of lead, wo should subtract inch from the 5 inches, found for the lap by the table. This would leave 4$ inches for the quantity of lap that the valve ought to have. n. LAP IN INCHES REQUIRED ON THE STEAM SIDE OF THE VALVE TO CUT THE STEAM OFF AT ANY OF THE UNDER-NOTED PARTS OF THE STROKE. Length of stroke of the valve in inches. Proportion of the stroke at which the steam is cut off. 3 JL t ft i i A A 24 6-94 6-48 6-00 5-47 4-90 4-25 3-47 2-45 23* 6-79 6-34 5-88 5-36 4-79 4-16 3-39 2-39 23 6-65 6-21 5-75 5-24 4-69 4-07 3-32 2-34 22* 6-50 6-07 5-62 5-13 4-59 3-98 8-25 2-29 22 6-36 5-94 5-50 5-02 4-49 3-89 3-13 2-24 21* 6-21 5-80 5-38 4-90 4-39 3-80 3-10 2-19 21 6-07 5-67 5-25 4-79 4-28 3-72 3-03 2-14 20* 5-92 5-53 5-1.2 4-67 4-18 3-63 2-96 2-09 20 5-78 5-40 5-00 4-56 4-08 3-54 2-89 2-04 19* 5-64 5-26 4-87 4-45 3-98 3-45 2-82 1-99 19 5-49 5-13 4-75 4-33 3-88 3-36 2-74 1-94 18* 5-34 4-99 4-62 4-22 3-77 3-27 2-67 1-88 18 5-20 4-86 4-50 4-10 3-67 3-19 2-60 1-83 17* 5-06 4-72 4-37 3-99 3-57 3-10 2-53 1-78 17 4-91 4-59 4-25 3-88 3-47 3-01 2-45 1-73 16* 4-77 4-45 4-12 3-76 3-36 2-92 2-38 1-68 16 4-62 4-32 4-00 3-65 3-26 2-83 2-31 1-G3 15* 4-48 4-18 3-87 3-53 3-16 2-74 2-24 1-58 PROPORTIONS OF LAP FOR EXPANSION. TABLE Continued. 191 Length of stroke of the valve in inches. Proportion of the stroke at which the steam is cut off. a -h 4 A * i A & 15 4-33 4-05 3-75 3-42 3-06 2-65 2-16 1-53 14* 4-19 3-91 3-62 3-31 2-96 2-57 2-09 1.-48 14 4-05 3-78 3-50 3-19 2-86 2-48 2-02 1-43 13^ 3-90 3-64 3-37 3-03 2-75 2-39 1-95 1-37 13 3-76 3-51 3-25 2-96 2-65 2-30 1-88 1-32 12* 3-61 3-37 3-12 2-85 2-55 2-21 1-80 1-27 12 3-47 3-24 3-00 2-74 245 2-12 1-73 1-22 11* 3-32 310 2-87 2-62 2-35 2 ; 03 1-66 1-17 11 3-18 2-97 2-75 2-51 2-24 1-95 1-58 1-12 10* 3-03 2-83 2-62 2-39 2-14 1-86 1-51 1-07 10 2-89 2-70 2-50 2-28 2-04 1-77 1-44 1-02 9* 2-65 2-56 2-37 2-17 1-93 1-68 1-32 96 9 2-60 2-43 2-25 2-05 1-84 1-59 1-30 92 8* 2-46 2-29 2-12 1-94 1-73 1-50 1-23 86 8 2-31 2-16 2-00 1-82 1-63 1-42 1-15 81 H 216 2-02 1-87 1-71 1-53 1-33 1-08 76 7 2-02 1-89 1-75 1-60 1-43 1-24 1-01 71 H 1-88 1-75 1-62 1-48 1-32 1-15 94 66 6 1-73 1-62 1-50 1-37 1-22 1-06 86 61 H 1-58 1-48 1-37 1.25 1-12 97 79 56 5 1-44 1-35 1-25 1-14 1-02 88 72 51 ** 1-30 1-21 1-12 1-03 92 80 65 46 4 1-16 1-08 1-00 91 82 71 58 41 *t 1-01 94 87 80 71 62 50 35 3 86 81 75 68 61 53 44 30 The above table is an extension of tho first, for the purpose of obviating, in most cases, the necessity of even the very small degree of trouble required in multiplying the stroke of the valve by one of the decimals in the first table. The first line of the second table consists, as in the first table, of eight fractions, in- dicating tho various parts of the stroke at. which the steam may be cut off. The first column on the left hand consists of various numbers that represent the different lengths that may be given to the stroke of the valve, diminishing by half inches from 24 inches to 3 inches. Suppose that we wish the steam to be cut 192 THEORY OF THE STEAM-ENGINE. off at any of the eight parts of the stroke indicated in the first line of the table (say at % from the end of the stroke), we find % at the top of the 6th column from the left. "We next look for the proposed length of stroke of the valve (say 17 inches) in the first column on the left. From 17, in that column, we run along the line towards the right, and in the sixth column, and directly under the % at the top, we find 3 '47, which is the amount of lap required in inches to cause the steam to be cut off at from the end of the stroke, if the valve has no lead. If we wish to give it lead (say J inch), we subtract the half of that, or ='125 inch, from 3-47, and we have 3'47125=3'345 inches, the quantity of lap that the valve should have. To find the greatest efficient breadth that we can give to the port in this case, we have, as before, half the length of stroke, 8 3'345=5'155 inches, which is the greatest efficient breadth we can give to the port with this length of stroke. It is scarcely necessary to observe that it is not at all essential that the port should be so broad as this; indeed, where great length of stroke in the valve is not inconvenient, it is always an advan- tage to make it travel further than is just necessary to make the port open fully ; because, when it travels further, both the ex- hausting and steam ports are more quickly opened, so as to al- low greater freedom of motion to the steam. The manner of using this table is so simple, that we need not trouble ourselves with more examples, and may pass on, therefore, to explain the use of the third table. Suppose that the piston of a steam-engine is making its downward stroke, that the steam is entering the upper part of the cylinder by the upper steam port, and escaping from below the piston by the lower exhausting port ; if, as is generally the case, the slide-valve has some lap on the steam side, the upper port will be closed before the piston gets to the bottom of the stroke, and the steam above then acts expansively, while the communication between the bottom of the cylinder and the con- denser still continues open, to allow any vapour from the con- densed water in the cylinder, or any leakage past the piston, to escape into the condenser ; but, before the piston gets to the EFFECTS OF LAP ON EDUCTION. 193 bottom of the cylinder, this passage to the condenser will also be cut off by the valve closing the lower port. Soon after the lower port is thus closed, the upper port will be opened towards the condenser, so as to allow the steam that has been acting expan- sively to escape. Thus, before the piston has completed its stroke, the propelling power is removed from behind it, and a resisting power is opposed before it, arising from the vapour in the cylinder, which has no longer any passage open to the con- denser. It is evident, that if there is no lap on the exhausting side of the valve, the exhausting port before the piston will be closed, and the one behind it opened, at the same time ; but, if there is any lap on the exhausting side, the port before the pis- ton will be closed before that behind it is opened ; and the in- terval between the closing of the one and the opening of the other, will depend on the quantity of lap on the exhausting side of the valve. Again, the position of the piston in the cylinder, when these ports are closed and opened respectively, will depend on the quantity of lap that the valve has on the steam side. If the lap is large enough to cut the steam off when the piston is yet a considerable distance from the end of its stroke, these ports will be closed and opened at a proportionably early part of the stroke ; and in the case of engines moving at a high speed, it has been found that great benefit is obtained from allowing the steam to escape before the end of the stroke. The third table is intended to show the parts of the stroke where, under any given arrangement of slide valve, the eduction ports close and open respectively, so that thereby the engineer may be able to estimate how much, if any, of the efficiency he loses, while he is trying to add to the power of the steam by in- creasing the expansion. In this table there are eight columns marked A, standing over eight columns marked B, and at the heads of these columns are eight fractions as before, representing so many different parts of the stroke at which the steam may be supposed to be cut off. 194 THEOBY OF THE STEAM-ENGINE. The columns marked A express the distance of the piston in parts of its stroke from the end of the stroke when the educ- tion port before it is shut, and the columns marked B, and which stand immediately under the columns marked A, express the distance of the piston from the end of its stroke when the ex- hausting port behind it is opened also measured in parts of the stroke.* III. PBOPOETION OF THE STROKE AT WHICH THE EDUCTION POBT IS SHUT AND OPENED. Lap on the ductLn aide of the valve, in parts of the length of its stroke. Proportion of the stroke at which the steam is cut off. i A i A i ? A A l-8th 1-1 6th l-32nd A 178 130 113 092 A 161 118 101 082 A 143 100 085 067 A 126 085 069 055 A 109 071 053 043 A 093 058 043 088 A 074 043 033 022 A 053 027 024 Oil l-8th l-16th l-32nd B 033 060 073 092 B 026 052 066 082 B 019 040 051 067 B 012 030 042 055 B 008 022 033 044 B 004 015 023 033 B 001 008 013 022 B 001 002 004 on Suppose we have an engine in which the slide valve is made to cut the stem off when the piston is l-3rd from the end of its * In locomotive and other fast-moving engines it is very important to open the eduction passage before the end of the stroke, so as to give more time for the steam to escape, and in locomotive valves the lap of the valve is usually made a little over Jth of the travel, and the lead is usually made ^th of the travel. In engines moving slowly the same necessity for an early eduction does not exist, and in such engines there will be a loss from opening the eduction much before the end of the stroke, as the moving pressure urging the piston is thus removed before the stroke tenninates. "When the valve is closed before the piston previously to the end of the stroke, the attenuated vapour in the cylinder will be compressed, and sometimes the compression will be carried so far that the pressure resisting the piston at the end of the stroke will exceed the pressure of the steam in the boiler. The indicator diagram will in such cases appear with a loop at its upper corner, which shows that the pressure before the end of the stroke exceeds the pressure of the steam, and that the first effect of opening the communication between the cylinder and the boiler is to enable the cylinder to discharge its highly compressed vapour backward into the boiler. The act of compressing the steam is what is called cushioning ; and in all ordinary diagrams this action may bo more or less perceived. EFFECTS OF LAP ON EDUCATION. 195 stroke, and that the lap on the eduction or exhausting side of the valve is l-8th of the whole length of its stroke. Let the stroke of the piston be 6 feet, or 72 inches. "We wish to know when the exhausting port before the piston will be closed, and when the one behind it will be opened. At the top of the left- hand column marked A, the given degree of expansion (l-3rd) is given, and in the extreme left column we have at the top the given amount of lap (l-8th). Opposite the l-8th in the first column, marked A, we have '178, and in the first column, marked B, '033, which decimals, multiplied respectively by 72, the length of the stroke, will give the required positions of the piston: thus 72 x '178 = 12'8 inches = distance of the piston from the end of the stroke when the exhaustion-port before the piston is shut : and 72 x '033 = 2'38 inches = distance of the piston from the end of its stroke when the exhausting-port behind it is opened. To take another example. Let the stroke of the valve be 16 inches, the lap on the exhausting side inch, the lap on the steam side 3J inches, and the length of the stroke of the piston 60 inches. It is required to ascertain all the particulars of the working of this valve. The lap on the exhausting side is evi- dently 5 V of the length of the valve stroke. Then, looking at 16 in the left-hand column of the table in page 190, we find in the same horizontal line, 3'26, or very nearly 3J, under \ at the head of the column, thus showing that the steam will be cut off at one-sixth from the end of the stroke. Again, under % at the head of the sixth column from the left in the table in page 194, and in a line with ^ m * ne left-hand column, we have '053 un- der A, and -033 under B. Hence, -053 x 60 = 3-18 inches = dis- tance of the piston from the end of its stroke when the exhaust- ing-port before it is shut, and '033 x 60 = 1'98 inches = distance of the piston from the end of its stroke when the exhausting- port behind it is opened. If in this valve the lap on the ex- hausting side were increased say to 2 inches or | of the stroke, the effect would be to cause the port before the valve to be shut sooner in the proportion of -109 to '053, and the port behind it later in the proportion of -008 to -003. Whereas, if the lap on 196 THEORY OF THE STEAM-ENGINE. the exhausting side were removed entirely, the port before the piston would be shut and that behind it opened at the same time. The distance of the piston from the end of its stroke at that time would be '043 x 60 = 2'58 inches. An inspection of the third table shows us the effect of in- creasing the expansion by the slide valve in augmenting the loss of power occasioned by the imperfect action of the eduction pas- sages. Eeferring to the bottom line of the table, we see that the eduction passage before the piston is closed, and that behind i' opened, thus destroying the whole moving power of the engine, when the piston is '092 from the end of its stroke, the steam being cut off at -^ from the end. "Whereas if the steam is only cut off at ^Y from the end of the stroke, the moving power is not withdrawn till only -Oil of the stroke remains uncompleted. It will also be observed that increasing the lap on the exhausting side has the effect of retaining the action of the steam longer behind the piston, but it at the same time causes the eduction port before it to be closed sooner. A very cursory examination of the action of the slide valve is sufficient to show that the lap on the steam side should always be greater than on the eduction side. If they were equal, the steam would be admitted on one side of the piston at the same time that it was allowed to escape from the other ; but universal experience has shown that when this is the case a very con- siderable part of the power of the engine is destroyed by the re- sistance opposed to the piston, by the escaping steam not getting away to the condenser with sufficient rapidity. Hence we see the necessity of the lap on the eduction side being always less than the lap on the steam side ; and the difference should be the greater the higher the velocity of the piston is intended to be, because the quicker the piston moves, the passage for the waste steam requires to be the larger, so as to admit of its getting away to the condenser with as great rapidity as possible. In locomo- tive or other engines, where it is not wished to expand the steam in the cylinder at all, the slide valve is sometimes made with very little lap on the steam side ; and in these circumstances, in order to get a sufficient difference between the lap on the steam EFFECTS OF LAP ON EDUCTION. 197 and the eduction sides of the valve, it may be necessary not only to take away all the lap on the eduction side, but to take off still more, so as to cause both eduction passages to be, in some de- gree, open, when the valve is at the middle of its stroke. This, accordingly, is sometimes done in such circumstances as we have described ; but, when there is a considerable amount of lap on the steam side, this plan of taking more than all the lap off the eduction side ought never to be resorted to, as it can serve no good purpose, and will materially increase an evil we have al- ready explained : viz., the opening of the eduction port behind the piston before the stroke is nearly completed. In the case of locomotive or other engines moving rapidly, it is very con- ducive to efficiency to begin the eduction before the end of the stroke, as the piston moves slowly at that time ; and a very small amount of travel in the piston at that point corresponds to a considerable additional time given for the accomplishment of the eduction. The tables apply equally to the common short-slide three-ported valves, and to the long D valves. The extent to which expansion can be carried conveniently by means of lap upon the valve is about one-third of the stroke ; that is, the valve may be made with so much lap that the steam will be cut off when one-third of the stroke has been performed, leaving the residue to be accomplished by the agency of the ex- panding steam ; but if much more lap be put on than answers to this amount of expansion a distorted action of the valve will be produced, which will impair the efficiency of the engine. By the use of the link motion, however, much of this distorted action can be compensated. If a farther amount of expansion than this is wanted, where the link motion is not used, it may be attained by wire-drawing the steam, or by so contracting the steam pas- sage that the pressure within the cylinder must decline when the speed of the piston is accelerated, as it is about the middle of the stroke. Thus, for example, if the valve be so made as to shut off the steam by the time two-thirds of the stroke have been performed, and the steam be at the same time throttled in the Bteam pipe, the full pressure of the steam within the cylinder cannot be maintained except near the beginning of the stroke, 198 THEORY OF THE STEAM-ENGINE. where the piston travels slowly ; for as the speed of the piston increases, the pressure necessarily subsides, until the piston ap- proaches the other end of the cylinder, where the pressure would rise again but that the operation of the lap on the valve by this time has had the effect of closing the communication between the cylinder and steam pipe, so as to prevent more steam from entering. By throttling the steam, therefore, in the manner here indicated, the amount of expansion due to the lap may be doubled, so that an engine with lap enough upon the valve to cut off the steam at two-thirds of the stroke, may, by the aid of wire-drawing, be virtually rendered capable of cutting off the steam at one-third of the stroke. The Link Motion. The rules and proportions here given, are equally applicable, whether the valve is moved by a single eccentric, or by the arrangement called the link motion, and which has now been very generally introduced into steam en- gines. In the link motion there are two eccentrics, one of which is set so as to drive the engine in one direction, and the other is set so as to drive the engine in the opposite direction, and when the stud in communication with the valve is shifted to one end of the link, that stud partakes of the motion of the for- ward eccentric, whereas, when it is placed at the other end of the link, it partakes of the motion of the backing eccentric. A common length of the link is three times the stroke of the valve. Generally the stud is placed either at one end of the link or the other, not by moving the stud but by moving up or down the link ; and it is better that this movement should be vertical, and be made by means of a screw, than that the movement should be produced by a lever travelling through an arc. The point of suspension should be near the middle of the link where its motion is the least. The link connects together the ends of the two eccentric rods, and is sometimes made straight, but gen- erally curved, the curvature being an arc of such radius that the link may be raised up or down without sensibly altering the position of the stud with which the valve is connected. But the link should be convex or concave towards the valve, according as the eccentric rods are crossed or uncrossed when the throw VELOCITY OF RUNNING WATER IN CONDUITS. 199 of the eccentrics are turned towards the link. In the case of new arrangements of engine, it is advisable to make a skeleton model in paper of the link and its connexions, so as to obtain full assurance that it works in the best way. VELOCITY OF WATER IN RIVERS, CANALS, AND PIPES, ANSWERABLE TO ANY GIVEN DECLIVITY. When a river runs in its bed with a uniform velocity, the gravitation of the water down the inclined plane of the bed, is just balanced by the friction. In the case of canals, culverts, and pipes, precisely the same action takes place. The head of water, therefore, which urges the flow through a pipe, may be divided into two parts, of which one part is expended in giving to the water its velocity, and the other part is expended in over- coming the friction. If water be let down an inclined shoot, its motion at the top will be slow, but will go on accelerating until the friction generated by the high velocity will just balance the gravitation down the plane, and after this point has been attained, the shoot may be made longer and longer without any increase in the velocity of the water taking place. In the case of a ball falling in the air or in water, the velocity of the descent will go on increasing until the resistance becomes so great as to balance the weight ; and, in the case of a steam vessel propelled through the water, the speed will go on increasing until the resistance just balances the tractive force exerted by the engines, when the speed of the vessel will become uniform. In all these cases the resistance increases with the speed ; and as the speed increases, the resistance increases also, until it becomes equal to the ac- celerating force. The resistance which is occasioned by the friction of water increases more rapidly than the increase of the velocity. In other words, there will be more than twice the friction with twice the velocity. It is found by experiment that the friction of water increases nearly as the square of its velocity, so that there will be about four times the resistance with twice the speed. This law, however, is only approximately correct. The 200 THEORY OF THE STEAM-ENGINE. friction does not increase quite so rapidly at high velocities as the square of the speed. It is easy to determine the friction in Ihs. per square foot of any given pipe or conduit, with any given velocity of the stream, when the slope or declivity of the surface of the water is known. For as the gravitation down the inclined plane of the conduit just balances the friction, the friction in the whole length of the conduit will be equal to the whole weight of the water in it, re- duced in the same proportion as any other body descending an inclined plane. Thus, if the conduit be 2,000 feet long, and have 1 foot of fall in that length, the total friction will be equal to the total weight of the water divided by 2,000, and the friction per square foot will be equal to this 2000th part of the weight of the water divided by the number of square feet exposed to the water in the conduit. The friction will in all cases vary as the rubbing surface, or, what is the same thing, as the wetted perimeter As a cylindrical pipe has a less perimeter than any other form, it will occasion less resistance than any other form to water passing through it. In like manner, a canal or a ship with a semi- circular cross section will have the minimum amount of friction. The propelling power of flowing water being gravity, the amount of such power will vary with the magnitude of the stream ; but the resisting power being friction, which varies with the amount of surface, or in any given length with the wetted perimeter, it will follow that the larger the area is relatively with the wetted perimeter, the less will be the resistance rela- tively with the propelling power, and the greater will be the velocity of the water with any given declivity. Now, as the circumference or perimeter of a pipe increases as the diameter, and the area as the square of the diameter, it is clear, that with any given head, water will run more swiftly through large pipes than through small ; and in like manner with any given propor- tion of power to sectional area, large vessels will pass more swiftly than small vessels through the water. The sectional area of a pipe or canal divided by the wetted perimeter, is what is termed the hydraulic mean depth, and this depth is what would result if we suppose the perimeter to be bent out to a VELOCITY OP RUNNING WATER IN CONDUITS. 201 straight line, and the sectional area to be spread evenly over it, so that each foot of the perimeter had its proper share of sec- tional area above it. The greater the hydraulic mean depth, the greater with any given declivity will be the velocity of the stream. With any given fall, therefore, deep and large rivers will run more swiftly than small and shallow ones. The hy- draulic mean depth of a steam vessel will be the indicated power divided by the wetted perimeter of the cross section. TO DETERMINE THE MEAN VELOCITY WITH WHICH WATER WILL FLOW THROUGH CANALS, ARTERIAL DRAINS, OR PIPES, RUN- NING PARTLY OR WHOLLY FILLED. KULE. Multiply the hydraulic mean depth- in feet ty twice the fall in feet per mile ; take the square root of the product and, multiply it by 55. The result is the mean velocity of the stream in feet per minute. This again multiplied by the sec- tional area in square feet gives the discharge in cubic feet per minute. Example. What is the mean velocity of a river falling a foot in the mile, and of which the mean hydraulic depth is 8 feet ? Here 8 x 2 = 16, the square root of which is 4, and this multiplied by 55 = 220, which will be the mean velocity of the stream in feet per minute. In cylindrical pipes running full, the hydraulic mean depth is one-fourth of the diameter. For the hydraulic mean depth being the area divided by the wetted perimeter,it is - = * , 4' * The surface, bottom, and mean velocities of rivers have fixed relations to one another. Thus, If the surface velocity in inches per second be denoted by V, the mean velocity will be (V + 0-5) yT and the bottom velocity by (V + 1) 2 V V. With surface velocities therefore of 4,16, 82, 64, and 100 inches per second, the cor- responding mean velocities will be 2-5, 12-5, 26-8, 56-5, and 90'5 inches per second, and the corresponding bottom velocities will be 1, 9, 21-6, 49, and 81 inches per second. The common rule for finding the number of cubic feet of water delivered each minute by a pipe of any given diameter is as follows : Divide 4-72 times the square root of the fifth power of the diameter of the pipe in inches by the square root of the quotient obtained by dividing the length of the pipe in feet by the head of 'water in feet. Hawksley's rule for ascertaining the delivery in gallons per hour is as folio ws -.Multiply 15 times theffth pcncer of the diameter of the pipe 9* 202 THEORY OF THE STEAM-ENGINE. M. Prony has shown by a comparison of a large number of experiments that if H be the head in feet per mile required to balance the friction, V the velocity of the water through the pipe in feet per second, and D the diameter of the pipe in feet, then _ 2-25V' 2 T~- This equation is identical with that which has been used by Boulton and "Watt in their practice for the last half century, and which is as follows : If I be the length of the main in miles, V the velocity of the water in the main in feet per second, D the diameter of the pipe in feet, and 2 -25 a constant, 2-25ZV 2 then pj = feet of head due to friction. This equation put into words gives us the following Kule: TO DETERMINE THE HEAD OF WATER THAT WILL BALANCE THE FRICTION OF WATER RUNNING WITH ANT GIVEN VELOCITY THROUGH A PIPE OF A GIVEN LENGTH AND DIAMETER. KULE. Multiply 2'25 times the length of the pipe in miles Tjy the square of the velocity of the water in the pipe in feet per second, and divide the product hy the diameter of the pipe in feet. The quotient is the head of water in feet that will "balance the friction. The law indicated by this Rule is expressed numerically in the Tables on pp. 204, 205. in inches 'by the head of water in feet, and divide the product by the length of the pipe in yards. Finally, extract the square root of the quotient, which givet the delivery in gallonsper hour. The annual rain-fall in England varies from 20 to 70 inches, the mean being 42 inches, and it is reckoned that about T * 5 ths of the rain-fall on any given area may be collected for storage. A cubic foot of water is about 6J gallons, and it is found in supplying towns with water that about on the average 16 gallons per head per day are required in ordinary towns, and 20 gallons per head per day in manufac- turing towns, but the pipes should be large enough to convey twice this quantity. In the rainy districts of England collecting reservoirs should contain 120 days' sup- ply, and in dry districts 200 days' supply. Service reservoirs are usually made to contain 3 days' supply. The mean daily evaporation in England is '08 of an inch, and the loss from the overflow of storm water is reckoned to be about 10 per cent. FRICTIOX AND DISCHARGE OF WATER. 203 Explanation of the Tables. The top horizontal row of figures represents either the diameter of a cylindrical pipe, or four times the area of any other shaped pipe divided by the cir- cumference, or four times the area of the cross section of a canal, divided by the sum of all its sides, or bottom and sides, all being in inches. The first vertical column indicates the slope of the pipe or canal, that is, the whole length of the pipe or canal, divided by the perpendicular fall. Any number in any other column indicates the velocity, in inches per second, with which water would run through a pipe of such a diameter as the number at the head of such column expresses, having such a slope as that number in the first column expresses which is horizontally against such velocity. Example 1. With what velocity will water run through a pipe of 16 inches diameter, its length being 8,000 feet, and fall 16 feet? Here the slope manifestly is 8,000-5-16=500. Against 600 in the first column, and under 16, the diameter in the top row of figures, the number 29'8 is found, which is the velocity in inches per second. Example 2. With what velocity will water pass through a pipe of 21 inches diameter, having a slope of 900 ? 21 is not found in the head of the Table, in which case such a number must be found in the top row as will bear such proportion to 21 as some other two numbers in the top row bear to each other, and these latter numbers should be as near to 21 as they can be found. In this case it will be seen that 18 is to 21 as 6 is to 7, or (for compliance with the indication just mentioned) rather as 12 to 14, or still better as 24 to 28. Then say as the velocity (against 900, the slope) under 24 is to 28 (28'7), so is the velocity under 18 (22'7) to that of 21 (viz. 24-7) the velocity in inches per second. By the same process may the velocity for slopes be found or assigned, which are not to be found in the first column of the Table, proceeding with proportions found in the vertical col- umn instead of the horizontal rows; the first vertical column 204 THEORY OF THE STEAM-ENGINE. VELOCITY IN INCHES PEK SECOND OF WATER FLOWING THROUGH PIPES WITH VARIOUS SLOPES AND DIAMETERS. BY BOCTLTON, WATT & CO. Slope or Length divided by FalL INTERNAL DIAMETERS OF THE PIPES IN INCHES. 1 2 3 4 5 6 7 8 9 10 12 14 10 68-3 96-0 121- 142- 161- 180- 193- 208- 221- 234- 255- 280- 20 41-8 63-4 80- 93-6 104- 118- 127- 187- 146- 154- 170- 185- 30 82-8 49-5 62-6 73-4 83-3 931 99-7 107- 114- 121- 133- 145- 40 27-5 41-5 52-5 61-5 69-8 78-0 83-7 901 95'8 103- 112- 121- 50 23'9 36-3 45'7 53'7 60-9 68*0 73'0 787 83'5 88-4 97'5 106' 60 21-6 82-7 41-2 48-3 54-8 61-2 65-7 70 -8 75-8 79-8 87-8 95-4 70 19-6 29-7 87-5 44-0 49-8 55-7 59-7 64-4 68-5 72-5 80-0 86-6 80 181 27-4 84-7 40-7 461 51-5 65-8 59-5 63-8 671 74-0 80-2 90 16-9 25-6 32-4 37-9 43-0 48-0 51-5 55-5 59-0 62-5 69-0 74-8 100 15'8 24-0 30'3 35'5 40-2 45'1 48'4 52'2 55'5 58'7 64-8 70-3 200 10-5 16-0 20-2 23-7 26-8 80-0 82-2 34-7 86-9 39-0 431 46-7 300 8-43 12-7 16-1 18-6 21-4 23-8 25-6 27-7 29-4 80-6 34-3 87-7 400 7-11 10-8 136 15-9 181 20'2 21-6 23-3 24-8 26-8 29-0 81-4 500 6-26 9-50 12'0 14'0 15'9 17'8 191 20-6 21'9 23*2 25'5 27'7 600 5-64 8-57 10-8 12-7 14-3 161 17-2 IS'6 19-7 20-9 28-0 25-0 700 517 7-85 9-90 11-6 13-2 14-7 15-8 17-0 181 19-2 211 22-9 800 4-81 7-30 9-21 10-8 12-2 13-7 14-7 15-8 16-8 17-8 19-6 21-8 900 4-50 6-83 8-62 101 11-4 12-8 18-7 14-8 15-7 16-6 18-8 19-9 1000 4-25 6'45 8'15 9-54 10-8 12'1 12'9 14'0 14-8 15-7 17-3 18'8 2000 2-88 4-37 6-52 6-48 7-83 8-2 8-77 9-48 101 10-6 11-7 12-7 3000 2-30 8-48 4-40 517 5-86 6-55 7-02 7-57 8-05 8-52 9-40 10-2 4000 1-96 2-97 8-75 4-40 4-98 5-57 5-97 6-44 6-84 7-25 8-00 8-66 5000 173 2'62 3'31 3-88 4-40 4-92 5 '28 5-69 6-04 6'40 7'06 7'66 6000 1-57 2-88 8-00 8-52 8-99 4-45 4-79 515 5-44 5-80 6-40 6-95 7000 1-43 2-17 2-78 8-21 8-68 4-06 4-36 4-70 600 5-29 6-82 6-82 8000 1-32 2-01 2-58 2-97 8-46 8-76 4-03 4-35 4-62 4-90 5-40 5-85 9000 1-24 1-87 2-38 2-79 816 8-53 8-79 4-08 4-84 4-60 5-07 5-50 10000 1'17 1-77 2-24 2-62 2'9 3'32 3'57 3*84 4'08 4'32 476 5-16 being substituted in this case for the top row in the former case. In all cases an addition must be made to the fall equal to that which would generate the existing velocity in a body falling freely by gravity. For instance, in the first case, to the fall of VELOCITY OF WATER IN PIPES. 205 TELOCITY IN INCHES PEK SECOND OP WATER FLOWING/ THROUGH PIPES WITH VARIOUS SLOPES AND DIAMETERS. (Continued.) BY BOUXTON, WATT & CO. Slope or Length divided by Fall. INTERNAL DIAMETERS OF THE PIPES IN INCHES. 16 18 20 24 28 32 36 40 60 80 100 10 301- 320- 338- 872- 403- 432- 459- 484- 597- 692- 775- 20 199- 211- 223- 245- 256- 235- 303- 320- 894- 456- 511- 30 loo- 165- 175- 192- 208- 223- 238' 250- 309- 358- 400- 40 ISO- 138- wo- 161- 174- 187- 199- 210- 259- 300- 836- 50 113' 121' rn 1 140' 152' 163' 173' 183' 226' 261' 293' 60 102- 109- 115- 127- 187- 147- 156- 165- 204- 236- 264- 70 93-2 99-0 104- 115- 124- 134- 142- 150- 185- 214- 240- 80 86-8 91-6 97- 106- 115- 123- 131- 139- 171- 198- 222- 90 80-5 85-5 90-4 99-4 107- 115- 122- 129- 159- 185- 207- 100 75'5 80-2 85'0 93-5 102- 108' 115' 121- 150' 173' 194' 200 BO-3 53-3 56-4 62-0 67-2 72-1 76-7 80-8 99-5 115- 129- 300 89-9 42-5 45-0 49-5 53-6 57-5 61-1 64-4 79-5 92-0 105- 400 88-8 85-9 38-0 41-8 45-2 4S-5 51-5 54-4 67-0 77-7 87-0 500 29'8 31-6 33'4 36-8 39-8 42-8 45-5 47'9 59'0 68-4 767 600 26-8 28-6 80-2 88-2 86-0 88-6 41-0 43-2 63-3 61-7 69-2 700 24-6 28-2 27-7 80-4 83-0 85-8 87-6 39-6 48-8 56-5 68-4 800 22-9 24-3 25-7 28-8 80-6 82-9 84-9 86-S 45-4 52-5 68-9 900 21-4 22-7 24-0 26-4 28-7 30-7 32-7 34-4 42-5 49-1 65-1 1000 20'2 21'5 227 25'0 27'1 29'1 30-9 32'5 40'1 46-4 52-0 2000 18-7 14-5 15-4 16-9 18-8 19-7 20-9 22-0 27-2 81-3 85-8 3000 10-9 11-6 12-8 18-5 14-6 15-7 16-7 17-6 21-7 25-2 28-2 4000 9-32 9-90 10-4 11-5 12-4 13-4 14-2 15-0 18-5 21-4 24-0 5000 8-23 8'73 9'25 10-3 ll'O 11-8 12'5 13'2 16'3 18-9 21'2 6000 7-47 7-98 8-40 9-23 10-0 10-7 11-4 12-0 14-8 17-1 19-2 7000 6-80 7-22 7-65 8-40 9-10 9-76 10-8 10-9 18-5 15-6 17-5 8000 6-80 6-69 7-07 7-78 8-48 9-05 9-62 10-1 12-8 14-5 16-2 9000 5-91 6-28 6-64 7-30 7-92 8-50 9-02 9-62 11-7 13-05 15-2 10000 5'55 5-91 6-25 6'87 7'45 7'98 8-50 8'93 ll'O 12'7 14-3 16 feet we must add the fall which would generate the velocity of 2 9 '8 inches per second, namely, 1'15 inches, which will make the total fall 16 feet 1'15 inches that will be requisite to give such a velocity; but in such cases as this it is evident that tho addition of this small fraction might have been disregarded. 206 THEOKY OF THE STEAM-ENGINE. In some cases Messrs. Boulton and Watt have employed the constant l - 82 instead of 2'25. Mr. Mylne's constant is 1'94; but some careful experiments made by him at the West Middle- sex Waterworks, gave a constant as high as 2 - 62. OTHER TOPICS OF THE THEOKY OF STEAM-ENGINES. It will not be necessary to extend these remarks by an inves- tigation of the theory of the crank as an instrument for convert- ing rectilinear into rotatory motion, since the idea, once widely prevalent, that there was a loss of power consequent upon its use, is now universally exploded. Neither will it be necessary to enter into any explanation of the structure of the numerous rotatory engines which have at different times been projected, since none of those engines are in common or beneficial opera- tion. The proper dimensions of the cold water and feed pumps, the action of the fly-wheel in redressing irregularities of the mo- tion of the engine, and other material points which might prop- erly fall to be discussed under the head of the Theory of the Steam-Engine, and which have not already been treated of, will, for the sake of greater conciseness, be disposed of in the chapter on the Proportions of Steam-Engines, when these vari- ous topics must necessarily be considered. Nor is it deemed ad- visable here to recapitulate the rules for proportioning the vari- ous kinds of parallel motion, since parallel motions have now almost gone out of use, and since also any particular case of a parallel motion which has to be considered, can easily be resolv- ed geometrically by drawing the parts on a convenient scale, the principle of all parallel motions being that the versed sine of an arc, pointing in one direction, shall be compensated by an equal versed sine of an arc pointing in the opposite direction ; and the effect of these opposite motions is to produce a straight line. In the case of the parallel motions sometimes employed in side- lever engines, and in which the attachment is made not to the cross-head but to the side-rod, it is only necessary to provide that the end of the bar connected to the side-rod shall move, not in a straight line, but in an arc, the versed sine of which is equal to the versed sine of the arc described by the point of at- MODE OF DRAWING THE PARALLEL MOTION. 207 tachment on the side-rod. As the bottom of the side-rod is at- tached to the heam and the top to the cross-head, and as the bottom moves in an arc and the top in a straight line, it is clear that every intermediate point of the side-rod mnst describe an arc which -will more and more approach to a straight line, or have a smaller and smaller versed sine, the nearer such point is to the top of the rod. By drawing down the side-rod at the end of the stroke, and also at half stroke, the amount of deviation from the vertical at those positions, can easily be determined for any point in the length of the rod ; and the point of attachment of the parallel bar has only to be such, and the length and travel of the radius crank has also to be such, that the end of the paral- lel bar attached to the side-rod shall describe an arc whose versed sine is equal to the deviation from the perpendicular, or, in other words, to the side-travel of that point of the side-rod at which the attachment is made. Since, then, the side-rod is guided at the bottom by the arc of the beam, and near the top by that less arc described by the end of the parallel bar, which answers to the supposition of the cross-head moving in a vertical line, the result is that the cross-head will be constrained to move in this vertical line ; since only on that supposition can the two arcs already fixed be described. The method of balancing the momentum of the moving parts of marine engines which I introduced in 1852 has now been very generally adopted ; and the practice is found to be very useful in reducing the tremor and uneasy movements to which engines working at a high rate of speed are otherwise subject. Nearly all the engines now employed for driving the screw pro- peller are direct-acting engines, which necessarily work at a high rate of speed to give the requisite velocity of rotation to the screw shaft. The principle on which the balancing is effected is that of applying a weight to the crank or shaft, and when the piston and its connexions move in one direction the weight moves in the opposite with an equal momentum. CHAPTER IV. PROPORTIONS OF STEAM-ENGINES. WE now come to the question how we are to determine the proportions of steam-engines of every class. The nominal power of a low pressure engine is determined by the diameter of the cylinder and length of the stroke, as follows : TO DETERMINE THE NOMINAL POWEE OF A LOW PEESStJKE ENGINE OF WATT'S CONSTRUCTION. RULE. Multiply the square of the diameter of the cylinder in inches "by the cube root of the stroke in feet, and divide the product ~by 47. The quotient is the nominal horse-power of the engine. Example 1. What is the nominal power of a low pressure engine with a cylinder 64 inches diameter and 8-feet stroke ? Here 64 x 64 = 4,096, which multiplied hy 2, the cube root of 8 = 8,192 and -*- 47 = 174-3. The nominal powers of engines of different sizes, both high pressure and low pressure, are given in the following tables : TABLES OF NOMINAL POWERS OF ENGINE. 209 NOMINAL HORSE POWER OF HIGH PRESSURE ENGINES. ^ >- -: 2?J 1=1 Sff LENGTH OF STEOKE IN FKET. 1 1| 2 2i 3 3i 4 5 6 7 8 9 2 25 29 32 35 37 38 40 44 46 49 51 53 2* 39 45 50 54 57 60 63 68 72 76 79 83 8 57 65 72 78 83 87 91 98 1-04 1-10 115 1-20 8* 78 89 98 1-06 1-13 1-19 1-24 1-34 1-42 1-49 1-56 1-G2 4 1-02 1-17 1-29 1-38 1-47 1-56 1-62 1-74 1-86 1-95 2-04 2-10 4* 1-29 1-48 1-63 1-75 1-86 1-96 2-05 2-21 235 2-47 258 2-68 5 1-59 1-83 2-01 2-16 2"28 2-43 2-52 2-73 2-88 3-06 818 3-33 5* 1-93 2-21 2-43 2-62 2-78 2-93 3-12 3-30 8-51 369 3-86 4-01 6 2-28 2-01 2-88 8-12 3-30 3-48 8-66 3-93 4-17 4-41 4-59 4-77 6* 2-ey 3-09 3-39 8-66 8-90 4-08 4-23 4-62 4-89 5-16 5-46 5-61 7 3-12 8-57 3-93 4-23 4-50 4-74 4-95 5-34 5-67 5-97 6"27 6-51 T* 8-60 4-11 4-53 4-86 6-19 5-46 5-70 6-15 6-51 6-87 7-18 7-46 8 4-08 4-68 5-16 5-55 5-88 6-21 (J-4-- 6-99 741 7-80 8-16 8-49 8* 4-62 5-28 5-82 6-27 6-63 6-99 7-32 7-9 8-37 8-82 9-29 9-59 9 5-16 5-91 6-51 7-02 7-47 7-86 8-22 8-85 9-39 9-90 10-35 1077 9* 5-76 6-60 7-26 7-SO 8-37 8-76 9-15 9-84 10-47 11-01 11-52 1-1-98 10 6-89 7-32 8-04 8-67 9-21 9-69 10-14 10-92 11-61 12-21 12-78 13-29 1% 7-05 8-04 8-88 9-54 10-14 10-68 11-16 12-03 12-78 13-47 14-07 14-64 11 7-74 V-:, 9-72 10-47 1181 1173 12-45 13-20 14-04 14-76 15-45 1605 11* 8-43 9-66 10-62 11-46 12-15 12-78 18-80 14-61 15-88 16-14 16-88 17-56 12 9-18 10-53 11-58 12-48 13-26 13-95 14-58 15-72 16-71 17-58 18-89 1911 12* 9-96 11-40 12-57 13-53 14-87 15-15 15-84 17-04 18-12 19-08 1992 20-73 1ST 10-80 12-36 13-59 14-64 15-57 16-38 16-92 18-45 19-59 20-64 21-57 22-44 18* 11-64 18-32 14-64 15-78 16-77 17-67 18-48 19-89 21-15 22-26 2325 2419 1* 400-2 416-4 68 214-7 245-8 270-5 291-4 309-6 825-8 340-8 867-2 8897 410-1 429-8 446-1 60 229-8 263-0 289-5 811-7 881-2 848-9 364-8 898-0 417-6 4896 4596 477-9 70 312-8 857-9 393-9 424-5 451-2 474-9 496-5 5846 6682 598-2 625-5 650-4 210 PROPORTIONS OF STEAM-ENGINES. NOMINAL HORSE POWER OF LOW PRESSURE ENGINES. *'* 00 "-= EO LENGTH OF STROKE IN FEET. 1 H 2 01 *2 3 3* 4 5 6 7 8 9 4 35 39 43 46 49 52 51 58 62 65 G8 70 5 53 61 67 72 76 81 .84 91 96 1-02 1-06 1-14 6 76 87 96 1-04 1-10 1-16 1-22 1-31 1-39 1-47 1-53 1*59 7 1-04 1-19 1-31 1-41 1-50 1-58 1-65 1-78 1-89 1-99 2-09 2-17 8 1-36 1-56 1-72 1-85 1-96 2.07 2-16 2-33 2-47 2-60 2-72 2-83 9 1-72 1-97 2-17 2-34 2-49 2-62 2-74 2-95 3-13 3-30 8-45 3-59 10 2-13 2-44 2-68 2-S9 3-07 3-23 3-38 8-64 3-87 4-07 4-26 4-48 11 2-57 2-95 8-24 3-49 3-77 3-91 4-15 4-40 4-68 4-92 5-15 5-35 12 3-06 3-51 8-86 4-16 4-42 4-65 4-66 5-24 5-57 5-86 6-18 6-37 13 3-60 4-12 4-53 4-68 5-19 5-46 5-64 6-15 6-53 6-88 7-19 7-48 14 4-17 4-77 5-25 5-66 6-01 6-33 6-02 7-13 7-58 7-98 8-34 8-67 15 4-77 5-48 6-03 6-50 6-90 7-27 7-60 8-19 8-70 9-16 9-57 9-96 16 5-45 6-23 6-86 7-39 7-86 8-27 8-65 9-31 9-90 10-42 10-89 11-33 IT 6-15 7-04 7-75 8-35 8-86 9-34 9-76 10-52 11-17 11-76 12-30 12-79 18 6-S9 7-89 8-68 9-36 9-94 10-47 10-94 11-79 12-53 13-19 13-79 14-34 19 7'68 8-79 9-68 10.42 11-17 11-66 12-19 13-13 13-96 14-69 15-36 15-98 20 8-51 9-74 10-72 11-55 12-27 12-92 13-51 14-55 15-46 16-28 17-02 17-70 22 10-30 11-79 12-97 13-98 14-85 15-63 16-62 17-65 18-71 19-70 20-60 21-42 24 12-26 14-03 15-44 16-63 17-67 18-61 19-45 20-95 22-27 23-44 24-51 25-49 26 14-39 16-46 18-12 19-52 20-75 21-84 22-56 24-60 26-14 27-51 28-78 29-92 28 16-68 19-09 21-02 22-64 24-06 25-33 26-48 28-52 80-31 81-90 33-36 84-69 80 19-15 21-92 24-13 25-99 27-62 20-07 80-40 32-74 34-80 36-63 38-30 89-83 82 21-79 24-96 27-51 29-57 31-42 83-08 34-59 87-26 39-59 41-6& 43-57 45-32 84 24-60 28-16 30-99 33-39 35-44 37-34 39-04 42-06 44-69 47-05 49-19 61-16 86 27-57 31-56 34-74 87-42 89-77 41-87 43-77 47-15 50-11 52-75 65-15 57-36 88 30-72 35-17 38-71 41-69 44-66 46-64 48-77 52-54 55-83 58-78 61-45 63-91 40 84-04 88-97 42-89 46-20 49-10 51-69 54-04 58-21 61-86 65-12 68-08 70-81 42 37-53 42-96 47-29 50-94 54-13 56-98 59-58 64-18 68-21 71-78 75-06 78-06 44 41-19 47-15 51-90 55-91 59-38 62-54 66-46 70-44 74-85 78-79 82-38 85-68 46 45-02 51-54 56-72 61-10 64-88 68-19 71-48 76-69 81-81 86-12 90-04 93-64 48 49-02 56-11 61-76 66-53 70-70 74-42 77-82 83-83 89-08 93-78 98-04 102-0 50 53-19 60-89 67-02 72-19 76-71 80-76 84-44 90-96 96-65 101-7 106-4 110-6 52 57-55 65-86 72-48 78-08 83-00 87-35 90-25 98-40 104-5 110-0 115-1 119-6 54 62-04 71-02 78-17 84-20 89-48 94-20 98-49 106-1 112-7 118-7 124-1 129-0 56 66-72 76-88 84-07 90-55 96-23 101-30 105-9 114-1 121-2 127-6 133-4 138-8 58 71-58 81-93 9018 97-14 103.2 108-6 113-6 122-4 129-9 136-7 143-1 148-7 60 76-60 87-68 96.50 108-9 110-4 116-8 121-6 181-0 139-2 146-5 153-2 159-8 62 81-79 93-62 103-04 111-0 117.9 124-18 129-81 139-8 148-6 156-7 163-6 170-8 64 87-15 99-84 110-0 118-8 125-7 182-3 138-8 149-0 158-4 166-7 174-8 181-3 66 92-68 106-1 116-8 125-8 133-6 140-7 147-3 158-5 168-4 177-8 185-4 192-8 68 98-40 112-6 123-9 133-6 141-8 149-4 1562 168-2 178-8 188-2 196-8 204-6 70 104-26 119-3 181-3 141-5 150-4 158-8 165-5 178-2 189-4 199-4 i 208-5 216-8 72 110-30 126-2 139-0 149-7 159-1 167-4 175-1 188-6 200-4 211-0 220-6 229-4 74 116-5 133-4 146-8 158-1 167-9 176-7 185-4 199-2 211-6 223-4 ; 288-0 242-2 76 122-9 140-7 154-8 166-8 178-6 186-6 195-0 210-1 223-3 285-1 245-8 255-6 7S 129-4 148-2 163-1 175-6 186-7 196-5 205-4 221-4 235-2 247-6 258-9 1269-2 80 136-2 155-8 171-6 184-8 196-4 206-7 216-1 282-8 247-4 260-5 i 272-8 283'2 82 143-0 163-8 180-2 194-2 206-2 217-8 226-9 244-6 260-0 278-8 286-1 ;297'6 84 150-1 171-8 189-1 208-8 216-5 227-9 288-3 256-7 272-8 287-1 800-2 812-2 86 157-4 180-1 198-2 218-6 227-0 237-8 247-4 269-1 286-0 '801-0 314-7 327'3 88 164-8 188-6 207-6 228-6 1287-5 1 250-2 261-6 281-7 299-4 815-2 829-5 842-7 90 172-3 197-3 2171 238-9 248-6 261-7 273-6 294-7 813-2 329-7 344-7 858-5 100 212-8 243-5 268-0 288-8 806.8 828-0 887-7 863-8 886-6 407-0 425-5 442'6 RULES FOR FINDING THE HORSES POWER. 211 Example 2. What is the nominal power of a low pressure engine of 40 inches diameter of cylinder and 5-feet stroke. Here 40 x 40 = 1,600, which multiplied by 1'71 which is the cube root of 5 very nearly we get 2,736, which divided by 47 gives 5S - 21 as the nominal horse power. The actual horse power of an engine is determinable by the application of an instrument to determine the amount of power it actually exerts. The mode of determining this will be ex- plained hereafter. Meanwhile it may be repeated that an actual horse power is a dynamical unit capable of raising a load of 33,000 Ibs. one foot high in each minute of time. The nominal poicer of a high pressure engine maybe taken at three times that of a low pressure engine of the same size. The assumed pressure in computing the nominal power of low pressure engines is 7 Ibs. on each square inch of the piston, and the assumed pressure in computing the nominal power of high pressure engines is 21 Ibs. on each square inch of the piston. The assumed speed of the piston varies with the length of stroke from 160 to 256 feet per minute, namely, for a 2 ft. stroke, 160 ft. ; 23- ft., 170 ; 3 ft., 180 ; 4 ft, 200 ; 5 ft, 215 ; 6 ft, 228 ; 7 ft., 245 ; and 8 ft., 256 feet per minute. In point of fact, in all modern low pressure engines the un- balanced pressure of steam upon the piston is much more than 7 Ibs., and in most modern high pressure engines the unbalanced pressure of steam upon the piston is much more than 21 Ibs. The speed of the piston is also frequently much more than 256 feet per minute. In the case of screw engines the Admiralty employs a rule to determine the power, in which the old assumed pressure of 7 Ibs. per square inch is retained, but in which the actual speed of piston is taken into account. This rule is as follows : ADMIBALTY RULE FOB DETERMINING THE NOMINAL POWEE OF AN ENGINE. RULE. Multiply the square of the diameter of the cylinder in inches by the speed of the piston in feet per minute, and divide by 6.000. The quotient is the nominal power. Example. What ia the power of an engine with a cylinder 212 PROPORTIONS OF STEAM-ENGINES. of 42 inches diameter, and 3-J feet stroke, and which makes 85 revolutions per minute ? Here 42 x 42 1,764. The length of a double stroke will be 3-J- x 2 7 feet, and as there are 85 revolutions or double strokes per minute, 85 x 7 = 595 will be the speed of the piston in feet per minute. Now 1,764 x 595 = 1,049,580, which, di- vided by 6,000 = 175 horses power. The area of the piston in circular inches, it will be recollect- ed, is found by multiplying the diameter by itself. Thus a pis- ton 50 inches diameter contains 50 x 50, or 2,500 circular inches. Now as every circular inch is '7854 of a square inch, we must, in order to find the area of the piston in square inches, multiply the diameter by itself and by '7854, which will give the area in square inches. Thus, 2,500 x '7854 = 1,963'5 square inches, which is the area in square inches of a piston 50 inches in diam- eter. The circumference of any circle is obtained by multiply- ing the diameter by 31416. Hence the length of a string or tape that will be required to encircle a piston 50 inches in diam- eter will be 50 x 3'1416 = 157'08 inches. The areas of pumps, pipes, safety-valves, and all other circular objects, is computed in the same way as the areas of circles or pistons. Some valves are annular valves, consisting not of a flat circular plate, but of a ring or annulus of a certain breadth. To compute the area of such a valve we must first compute the area of the outer circle, and then the area of the inner, and subtract the less from the greater, which will give the area of the annulus. So in like manner, in trunk engines, we must subtract the area of the trunk from the area of the piston. GENEKAL CONSIDERATIONS AND INSTRUCTIONS. In proceeding to design an engine for any given purpose, the nominal power may either be fixed or the nominal power may be left indeterminate, and only the work be fixed which the engine has to perform. In the first case we have only to ascertain by the foregoing rules or tables what the dimensions of a cylinder are which correspond to the nominal power, DRAWINGS SUITABLE FOR AXL POWERS. 213 and we have then to make all the other parts of dimensions corresponding thereto, which we shall be enabled to do by the rules here laid down. Of course the engineer settles for himself some particular type of engine which he prefers to adopi as the one that is to govern his practice, and any drawing of an engine of a given size or power is applicable to the construction of a similar engine of any other size or power by merely altering the scale of the drawing. If, therefore, any engineer decides upon the class of land engine, paddle engine, or screw engine which he prefers to construct, and chooses to get a set of drawings of such engine on any given scale lithographed, such drawir^s will be applicable to all sizes and powers of that class of engine by altering the scale in the proportion rendered necessary by the enlarged or diminished diameter of the cylinder answerable to the required power. Thus, if we have a drawing of a marine engine of 32 inches diameter of cylinder and 4-feet stroke, made to the scale of -J-inch to the foot, we may from such drawing construct a similar engine of 64 inches diameter and 8-feet stroke by merely altering the scale to one of ^-inch to the foot, so that every part will in fact measure twice what it measured before. In order to make the same drawing applicable to any size of engine, whether large or small, we have only to divide the diameter of the cylinder into the number of parts that the cylinder is to have of inches, and then we may use the scale so formed for the scale of the drawing. Thus, if we wish the engine to have a cylinder of 30 inches diameter, we must divide the diameter of the cylinder as shown in the drawing into 30 equal parts, each of which will represent an inch, and of course any twelve of them will represent a foot. If we now measure any other part of the engine, such as the diameter of the air pump, diameter of crank shaft, or any other part by this scale, we shall find the proper dimensions of the part in question, If we wish to construct from the drawing an engine of 60 or 100 inches, and of corresponding stroke, we have only to divide the diam- eter of the cylinder into 60 or 100 equal parts, and use each of those parts as an inch of the scale, when the proper dimensions of all the parts will be at once obtained. 214 PROPORTIONS OF STEAM-ENGINES. It will be needless to guard these remarks against the obvious exception that in case of very large and very small engines it will be proper to make such slight modifications in some of the details as will conduce to greater convenience in working or in construction. For instance, as the height and strength of a man are a given quantity, it will obviously not be proper in doubling the size of all the other parts to double the height of the starting handles, or even to double their strength. In the case of oscil- lating engines, again, with a crank in the intermediate shaft, it may be difficult to get a sound crank made in the case of very large engines, and some other expedient may have to be adopted. Again, in the case of very small engines, the flanges and bolts may require to be a little larger than the proportion derived from a drawing of large engines, and the valve chests of the feed pumps and other parts may be too small if made strictly to scale to get the hand into conveniently to clear them out. All such points however are matters of practical convenience, only to be determined by the thoughtfulness and experience of the engineer, and in nowise affect the main conclusion that a draw- ing of an engine of any one size will suffice for the construction of engines of other sizes by merely changing the scale. It will consequently save much trouble in drawing offices to have one certain type of engine of each kind lithographed in all its details, and then engines of all sizes may be made therefrom by adding the proper scale, and by marking upon the drawing the proper dimensions of each part in feet or inches the measurements being taken from a table fixed once for all, either by computation or by careful measurement of the drawing with the different suitable scales. By thus systematising the work of the drawing office, labour may be saved and mistakes prevented. It easy to understand the principle on which the main parts of an engine must be proportioned. "We must in the first place have the requisite quantity of boiler surface to generate the steam, the requisite quantity of water sent into the boiler to keep up the proper supply, and the requisite quantity of cold water to condense the steam after it has given motion to the piston. In common boilers about 10 square feet of heating GENERAL CONSIDERATIONS AND INSTRUCTIONS. 215 surface will boil off a cubic foot of water in the hour, and this in the older class of engines was considered the equivalent of a horse power. At the atmospheric pressure, or with no load on the safety valve, a cubic inch of water makes about a cubic foot of steam ; and at twice the atmospheric pressure, or with 15 Ibs. per square inch on the safety valve, a cubic inch of water will make about half a cubic foot of steam. For every half cubic foot of such steam therefore abstracted from the boiler there must be a cubic inch of water forced into it. So if we take the latent heat of steam in round numbers at 1,000 degrees, and if the condensing water enters at 60, and escapes at 100, the condensing water has obviously received 40 degrees of heat, and it has received this from the steam having 1,000 of heat, and the 112 which the steam if condensed into boiling water would exceed the waste-water in temperature. It follows that in order to reduce the heat of the steam to 100 there must be 1,112 of heat extracted, and if the condensing water was only to be heated 1 degree, there would require to be 1,112 tunes the quantity of condensing water that there is water in the steam. Since, however, the water is to be heated 40, there will only require to be one-fortieth of this, or about ^th the quantity of injection water that there is water in the steam. These rough determinations will enable the principle to be understood on which such proportions are determined. The proportions of the condenser and of the air-pump were determined by Mr. Watt at one- eighth of the capacity of the cylinder. In more modern engines, and especially in marine engines where there are irregu- larities of motion, the air-pump is generally made a little larger than this proportion, and with advantage. The condenser is also generally made larger, and many engineers appear to con- sider that the larger the condenser is the better. Mr. Watt, however, found that when the condenser was made larger than one-eighth of the capacity of the engine the efficiency of the engine was diminished. The fly-wheel employed in land engines to control the irregularities of motion that would otherwise exist, is constructed on the principle that there shall be a revolv- ing mass of such weight, and moving with such a velocity, as to 216 PROPORTIONS OP STEAM-ENGINES. constitute an adequate reservoir of power to redress irregulari- ties. It is found that in those cases where the most equable motion is required, it is proper to have as much power treasured up in the fly-wheel as is generated in 6 half-strokes, though in many cases the proportion is not more than half this. It is quite easy to tell what the weight and velocity of the fly-wheel must be to possess this power. When we know the area of the piston and the unbalanced pressure per sq. inch, we easily find the pressure urging it, and this pressure multiplied by the length of 6 half-strokes represents the amount of power which, in the most equable engines, the fly-wheel must possess. Thus, suppose that the pressure on the piston were a ton, and that the length of the cylinder were 5 feet, then in 6 half-strokes the space described by the piston would be 30 feet. The measure of the power therefore is 1 ton descending through 30 feet, and if there were any circumstance which limited the weight of the fly- wheel to 1 ton, then the velocity of the rim or more correctly of the centre of gyration must be equal to that which any heavy body would have at the end of the descent by falling from a height of 30 feet, and which velocity may easily be determined by the rule already given for ascertaining the velocity of falling bodies. If the weight of the fly-wheel can be 2 tons, then the velocity of the rim need only be equal to that of a body falling through 15 feet, and so in all other proportions, so that the weight and velocity can easily be so adjusted as to represent most conveniently the prescribed store of power. With these preliminary remarks it will now be proper to proceed to recapitulate the rules for proportioning all the parts of steam engines illustrated by examples : STEAM PORTS. The area of steam port commonly given in the best engines working at a moderate speed is about 1 square inch per nominal horse-power, or ^ y th of the area of the cylinder, and the area of the steam pipe leading into the cylinder is less than this, or -66 square inch per nominal horse power. Since however engines PROPER AREAS OP CYLINDER PORTS. 217 are now worked at various rates of speed it will be proper to adopt a rule in which the speed of the piston is made an element of the computation. This is done in the rules which follow both for the steam port and branch steam pipe. TO FIND THE PEOPEB AREA OF THE STEAM OE EDUCTION POET OF THE CYLIXDEE. EULE. Multiply the square of the diameter of the cylinder in inches l>y the speed of the piston in feet per minute and l>y the decimal '032, and divide the product ly 140. The quo- tient is the proper area of the cylinder port in square inches. Example. ~Wh&t is the proper area of each cylinder port in an engine with 64-inch cylinder, and with the piston travelling 220 feet per minute ? Here 64 x 64 = 4,096, which multiplied by 220 = 901,120, and this multiplied by '032 = 28,835-8, which divided by 140, gives 206 inches as the area of each cylinder port in square inches. This is a somewhat larger proportion than is given in some excellent engines in practice. But inasmuch as the application of lap to the valve virtually contracts the area of the cylinder ports, and as the application of such lap is now a common prac- tice, it is desirable that the area of the ports should be on the large side. In the engines of the 'Clyde,' 'Tweed,' 'Tay,' and ' Teviot,' by Messrs. Oaird and Co., the diameter of the cylinder was 74| inches, and the length of the stroke TJ feet, so that the nominal power of each engine was about 234 horses. The cyl- inder ports were 33J- inches long and 6| inches broad, so that the area of each port was 224 '4 square inches, being somewhat less than the proportion of 1 square inch per nominal horse power, but somewhat more than the proportion of ^th of the area of the cylinder. As the areas of circles are in the propor- tion of the square roots of their respective diameters, the area of a circle of one-fifth of the diameter of the piston will have one-twenty-fifth of the area of the piston. One-fifth of 74f ths is 15 nearly, and the area of a circle 15 inches in diameter is 176'7 square inches, which is considerably less than the actual 10 218 PROPORTIONS OF STEAM-ENGINES. area of the port. By the rule we have given the area of the ports of this engine would, at a speed of 220 feet per minute, be about 277" square inches, which is somewhat greater than the actual dimensions. At a speed of the piston of 440 feet per minute the area of the port would be double the foregoing. STEAM PIPE. In the engines already referred to, the internal diameter of each steam pipe leading to the cylinder is 13-f- inches, which gives an area of 145-8 square inches. It is not desirable to make the steam pipe larger than is absolutely necessary, as an increased external surface causes increased loss of heat from radiation. The following rule will give the proper area of the steam pipe for all speeds of piston: TO FIND THE AREA OF THE STEAM PIPE LEADING TO EACH OTLINDEE. KULE. Multiply the square of the diameter of tlie cylinder in inches by the speed of the piston in feet per minute and ly the decimal '02, and divide the product ~by 170. The quotient is the proper area of the steam pipe leading to the cylinder in inches. Example. What is the proper area of the branch steam pipe leading to each cylinder in an engine with a cylinder 74J inches diameter, and with the piston moving at a speed of 220 feet per minute? Here 74-6 x 74-5 = 5,550-25, which multiplied by 220 = 1,221,055. and this multiplied by '02 = 24,421-1, which divided by 170 = 144 square inches nearly. The diameter of a circle of 144 square inches area is a little over 13 inches, so that 13 inches would be the proper internal diameter of each branch steam pipe in such an engine. The main steam pipe em- ployed in steamers usually transmits the steam for both the engines to the end of the engine-house, where it divides into two branches one extending to each cylinder. The main steam pipe will require to have nearly, but not quite, double the area of each of the branch steam pipes. It would require to PROPER AREA OF SAFETY VALVES. 219 have exactly double the area, only that the friction in a large pipe is relatively less than in a small; and as, moreover, the engines work at right angles, so that one piston is at the end of its stroke when the other is at the beginning, and therefore moving slowly, it will follow that when one engine is making the greatest demand for steam the other is making very little, so that the area of the main steam pipe will not require to be as large as if the two engines were making their greatest demand at the same time. SAFETY VALVES. It is easy to determine what the size of an orifice should be in a boiler to allow any volume of steam to escape through it in a given time. For if we take the pressure of the atmosphere at 15 Ibs., and if the pressure of the steam in the boiler be 10 Ibs. more than this, then the velocity with which the steam will flow out will be equal to that which a heavy body would acquire in falling from the top of a column of the denser fluid that is high enough to produce the greater pressure to the top of a column of the same fluid high enough to produce the less pressure, and this velocity can easily be ascertained by a reference to the law of falling bodies. In practice, however, the area of safety valves is made larger than what answers to this theoretical deduction, partly in consequence of the liability of the valves to stick round the rim, and because the rim or circumference becomes relatively less in the case of large valves. One approximate rule for safety valves is to allow one square inch of area for each inch in the di- ameter of the cylinder, so that an engine with a 64-inch cylinder would require a safety valve on the boiler of 64 square inches area, which answers to a diameter of about 9 inches. The rule should also have reference, however, to the velocity of the piston, and this condition is observed in the following rule: TO FIND THE PROPER DIAMETER OF A SAFETY VALVE THAT WILL LET OFF ALL THE STEAM FEOM A LOW PRESSURE BOILER. RULE. Multiply the square of the diameter of the cylinder in ~by the speed of the piston in feet per minute, and 220 PROPORTIONS OF STEAM-ENGINES. divide the product ly 14,000. The quotient is the proper area of the safety valve in square indies. Example. "What is the proper diameter of the safety valve of a boiler that supplies an engine with steam, having a 64-inch cylinder, and with the piston travelling 220 feet per minute? Here 64 x 64 4,096, which multiplied by 220 = 901,120, and this divided by 14,000 = 64'3, which is the proper area of the safety valve in square inches. ANOTHER EULE FOE SAFETY VALVES. Multiply the nominal horse power of the engine ty '375, and to the product add 16'875. The sum is the proper area of the safety valve in square inches, when the boiler is low pressure. Example. What is the proper diameter of the safety valve for a low pressnre engine the nominal power of which is 140 horses ? Here 140 x '375 = 52 - 5, adding to which the constant num- ber 16*875, we get 69*375, which is the proper area of the safety valve in square inches for a low pressure engine. A 60-inch cylinder and 6-feet stroke is equal to 140 nominal horses power, so that this rule gives somewhat more than a square inch of area in the valve for each inch of diameter in the cylinder in that particular size of engine. The opening through the safety valve must be understood to be the effective opening clear of bridges or other obstacles, and the area to be computed is the area of the smallest diameter of the valve. Most safety valves are made with a chamfered edge, which edge constitutes the steam tight surface, and the effective area is what corresponds to the smaller diameter of the valve and not to the larger. All boilers should have an extra or ad- ditional safety valve of the same capacity as the other, which may act in case of accident to the first from getting jammed or otherwise. The dimensions of safety valve here computed is that adequate for letting off all the steam. But in some cases the whole steam is not supplied from one boiler, and a safety valve in such case must be put on each boiler, but of a less area, PEOPEK DIAMETER OF THE FEED PIPE. 221 in proportion to the smaller volume of steam it has to let off. If there are two boilers, the safety valve on each will be half the area of the foregoing ; if three boilers, one-third of the area ; if four boilers, one-fourth of the area ; and so of all other pro- portions. The area of the waste steam pipe should be the same as that of the safety valve. TO FIND THE PROPER DIAMETER OF THE FEED POPE. EULE. Multiply the nominal horse power of the engine as com- puted by the Admiralty rule by '04, to the product add 3 ; extract the square root of the sum. The result is the diam- eter of the feedpipe in inches. Example 1. "What is the proper diameter of the feed pipe in inches of an engine whose nominal horse power is 140 ? 140 = nominal horse power of engine 04 = constant multiplier 5-6 3 = constant to be added 8-6 and v8'6 = 2'93 diameter of feed pipe in inches. Example 2. What is the proper diameter of the feed pipe in inches in the case of an engine whose nominal horse power is 385 ? 385 = nominal horse power of engine 04 = constant multiplier 15-4 3 constant to be added 18-4 and 4/18'4 = 4'29 diameter of feed pipe in inches. 222 PROPORTIONS OF STEAM-ENGINES. TO FIND THE PROPER DIMENSIONS OF THE AIR PUMP AND CONDENSER. In land engines the diameter of the air pump is made half that of the cylinder, and the length of stroke half that of the cylinder, so that the capacity is -Jth that of the cylinder ; and the condenser is made of the same capacity. But in marine en- gines the diameter of the air pump is made '6 of the diameter of the cylinder, and the length of the stroke is made from '57 to 6 times the stroke of the cylinder, and the condenser is made at least as large. In some cases the air pump is now made dou- ble-acting, in which case its capacity need only he half as great as when made single-acting. TO FIND THE PROPER AREA OF THE INJECTION PIPE. RULE. Multiply the nominal horsepower of the engine, as com- puted ty the Admiralty rule, Tjy 0'69, and to the product add 2 4 81. The sum is the proper area of the injection pipe in square inches. Example 1. What is the proper area of the injection pipe in square inches of an engine whose nominal horse power is 140 ? 140 = nominal horse power of engine 069 = constant multiplier 9-66 2-81 = constant to be added Answer 12-47 = area of injection pipe in square inches. ^Example 2. What is the proper area of the injection pipe in square inches of an engine whose nominal horse power is 385 ? 385 = nominal horse power of engine 069 = constant multiplier 26-56 2-81 = constant to be added Answer 29-37 = area of injection pipe in square inches. PROPER AREA OP THE FOOT VALVE PASSAGE. 223 The area of the injection orifice is usually made about l-250th part of the area of the piston, which, in. an engine of 385 horse power, would be about 27*7 inches of area. For warm climates the area should be increased. TO FIND THE PROPER AREA OF THE FOOT VALVE PASSAGE. KUIE. Multiply the nominal horse power of the engine ~by 9, divide the product T>y 5, add 8 to the quotient. The sum is the proper area of foot valve passage in square inches. Example 1. What is the proper area of the foot valve pas- sage in square inches of an engine whose nominal horse power is 140? 140 = nominal horse power of engine 9 = constant multiplier constant divisor 5)1260 252 8 = constant to be added Answer 260 area of foot valve passage in square inches. Example 2. What is the area of foot valve passage in square inches of an engine whose nominal horse power is 385 ? 385 nominal horse power of engine 9 = constant multiplier constant divisor 5)3465 693 8 = constant to be added Answer 701 area of foot valve passage in square inches. The discharge valve passage is made of the same size as the foot valve passage. A common rule for the area of the foot and discharge valve passages is one-fourth of the area of the air pump, and the waste 224 PROPORTIONS OF STEAM-ENGINES. water pipe is made one-fourth of the diameter of the cylinder, which gives a somewhat less area than that through the foot and discharge valve passages. Such rules, however, are only appli- cable to slow-going engines. In rapid-working engines, such as those employed for driving the screw propeller by direct action, and in which the air-pump is usually double acting, the area through the foot and discharge valves should be equal to the area of the air-pump, and the waste water pipe should also have the same area. In all cases, therefore, in which these or other rules dependent on the nominal power are applied to fast-going engines, the nominal power must be computed by the Admiralty rule, in which the speed of the piston is taken into account. TO FIND THE PKOPEE DIAMETER OP THE WASTE WATER PIPE. RULE. Multiply the square root of the nominal horsepower of the engine ty 1/2. The product is the diameter of the waste water pipe in inches. Example 1. What is the diameter of the waste water pipe, in inches, of an engine whose nominal horse power is 140 ? 140 = nominal horse power of engine and |/140=11.83 1'2 = constant multiplier Answer 1419 = diameter of waste water pipe in inches. Example 2. What is the diameter of waste water pipe, in inches, of an engine whose nominal horse power is 385 ? 386 = nominal horse power of engine and y 385 = 19-62 1-2 = constant multiplier Answer 23'64 = diameter of waste water pipe in inches. CAPACITY OF THE FEED PUMP. The relative volumes of steam and water are at 15 Ibs. on the square inch, or the atmospheric pressure, 1,669 to 1 ; at SO Ibs., or PROPER DIMENSIONS OF THE FEED PUMP. 225 15 Ibs. on the square inch above the atmospheric pressure, 881 to 1 ; at 60 Ibs., or 45 Ibs. above the atmospheric pressure, 467 to 1 ; and at 120 Ibs., or 105 Ibs. above the atmospheric pressure, 249 to 1. In every engine, taking into account the risks of leakage and priming in the boiler, the feed pump should be capable of dis- charging twice the quantity of water that is consumed in the generation of steam ; and in marine boilers it is necessary to blow out as much of the supersalted water as the quantity that is raised into steam, in order to keep the boiler free from saline incrustations. But if this water is discharged by leakage or priming, the object of preventing salting is equally fulfilled. Pumps, especially if worked at a high rate of speed, do not fill themselves with water at each stroke, but sometimes only half fill themselves, and sometimes do not even do that. Then in steam vessels, one pump should be able to supply both engines with steam, and the pump is generally only single-acting, while the cylinder is double-acting. If, therefore, we wish to see what size of pump we ought to supply to an engine in which the terminal elasticity of steam in the cylinder is equal to the atmospheric pressure, we know that the quantity of water in the steam is just -j-^g-jth of the volume of the steam ; but. as we require to double the supply to make up for waste, the volume of water supplied will on this ground be y^-y ; and as the pump may only half fill itself every stroke, the capacity of the pump must on this ground be 1 -/ 6 - 5 - of the volume of steam. But then the pump is only single-acting, while the cylinder is double-act- ing, on which account the capacity of the pump must be doubled, in order that it may in a half stroke discharge the water re- quired to produce the steam consumed in a whole stroke. This would make the capacity of the pump T-&-J, or ^fa of the capa- city of the cylinder, and a less proportion than this is inadvisa- ble in the case of marine engines. Even with this proportion, one feed pump would not supply all the boilers, as it ought to be able to do in case of accident happening to the other, unless it should happen that the pump draws itself full of water at each stroke instead of half full, as it will nearly do if the mo- tion of the engine is slow and the passages leading into it large, 226 PROPORTIONS OF STEAM-ENGINES. and if at the same time tlie valves are large and have not much lift. In the case of engines working at a high speed, ^-jro of the capacity of the cylinder for the capacity of the feed pump is scarcely sufficient, especially if there be no air vessel on the suction side of the pump, which in such pumps should always be introduced. In the engines of the 'Clyde,' 'Tweed,' 'Tay,' and ' Teviot,' by Messrs. Caird, the feed pump is ^yth of the capacity of the cylinder. In steam vessels there is no doubt always the resource of the donkey engine to make up for any deficiency in the feed. But it is much better to have the main feed pumps of the engine made of sufficient size to compensate for all the usual accidents befalling the supply of feed water. Of course, the supply of feed water required will vary mate- rially with the amount of expansion with which the steam is worked, and also with the amount of superheating ; and in the old flue boilers with the chimney passing up through the steam chest, there was always a considerable degree of superheating. A rule applicable to all pressures of steam and to moderate rates of expansion is as follows : TO FIND THE PEOPEE CAPACITY OF THE FEED PUMP. RULE. Multiply the capacity of the cylinder in cubic inches by the total pressure of the steam in the toiler on each square inch (or y 4,400. The quotient is the proper capacity of the cold-water pump in cubic feet. Example 1. "What is the proper capacity of the cold-water pump in an engine, having a 60-inch cylinder and a 5J-feet stroke ? Here 60 x 60 3,600, which multiplied by 5 is 19,800, and this divided by 4,400 is 4'5, which is the proper capacity of the cold-water pump in cubic feet. Example 2. "What is the proper capacity of the cold-water pump hi the case of an engine, with a 2-feet cylinder and 3-feet stroke ? Here 24 x 24 = 576, and this multiplied by 3 = 1,728, which divided by 4,400 = '39 cubic feet, or multiplying -39 by 1,728, we get the capacity in cubic inches, which is 673'92. This is a somewhat larger content than is sometimes given in practice. Maudslay's 16-horse land engine has a 24-inch cylinder and 3-feet stroke, and the cold-water pump has a diameter of 6| inches, and a stroke of 18 inches, which gives a capacity of 594 cubic inches, instead of 673, as specified above. The larger di- mension is the one to be preferred. FLY-WHEEL. Boulton and "Watt's rule for finding the sectional area of the fly-wheel rim is as follows : RULE. Multiply 44,000 times the length of the strolce in feet by the square of the diameter of the cylinder in inches, and divide the product by the square of the number of revolu- tions per minute, multiplied by the cube of the diameter of PROPER DIMENSIONS OF THE FLY-WHEEL. 229 the fly-wheel in feet. The resulting number will be the proper sectional area of the fly -wheel rim in square inches. Example. What will be the proper sectional area of the fly-wheel rim in square inches in the case of an engine, with a cylinder 24 inches diameter and 5 feet stroke, the fly-wheel he- ing 20 feet diameter. Here 44,000 multiplied by 5, which is the length of the stroke in feet, is 220,000. The square of the diameter of the cylinder in inches is 576, and 220,000 x 576 =126,720,000. The engine will make about 21 revolutions, the square of which is 441, and the cube of the diameter of the fly-wheel in feet is 8,000, which multiplied by 441 is 3,528,000. Finally 126,720,000 divided by 3,528,000 is 35'8, which is the proper area in square inches of the section of the fly-wheel rim. In an engine constructed by Mr. Oaird, with a 24-inch cylin- der, 5-feet stroke, and 20-foot fly-wheel, the width of the rim was 10 inches, and the thickness 3f inches, giving a sectional area of 37*5 square inches, which is somewhat larger than Boul- ton and "Watt's proportion. Suppose that we take the sectional area in round numbers at 36 square inches, and the circumference of the fly-wheel or length of rim if opened out at 62 feet or 744 inches, then there will be 36 times 744, or 26,784 cubic inches of cast iron in the rim, or dividing by 1,728, we shall have 15-5 cubic feet of cast iron. But a cubic foot of cast iron weighs 444 Ibs. Hence 15J cubic feet will weigh 6,882 Ibs., and this weight revolves with a speed of 21 times 62, or 1,303 feet per minute, or 21-7 feet per second, or 260-4 inches per second. To find the height in inches from which a body must have fallen, to acquire any given velocity in inches per second, we square the velocity in inches, and divide the square by 772*84, which gives the height in inches. Now the square of 260*4 is 67,808, which divided by 772-84 = 87 inches, or 7J feet, so that the energy treasured in the fly-wheel is equal to a weight of 6,882 Ibs. falling through 7J feet, or to a weight of 49,984-5 Ibs. falling through 1 foot. Now the area of the cylinder being in round numbers 452 square inches, the total pressure upon it, if we allow an effec- 230 PROPORTIONS OF STEAM-ENGINES. tive pressure including steam and vacuum of 7 Ibs. per square inch, as was the proportion allowed in Watt's engines, will be 3,164 Ibs., and the length of stroke being 5 feet, we shall have 3,164 Ibs. moved through 5 feet, or 5 times this, which is 15,820 Ibs. moved through 1 foot in each half stroke of the engine. Dividing now 49,984*5 foot-pounds, the total power resident in the fly-wheel at its mean velocity, by 158'20 foot-pounds, which is the power developed in each half stroke of the engine, we get 3'1 as the resulting number, which shows that there is over three times the power resident in the fly-wheel that is devel- oped in each half stroke of the engine. In cases where great equability of motion is required, this power of fly-wheel is not sufficient, and in some engines, the proportion is made six times the power developed in each half stroke, or, in other words, the fly-wheel is twice as heavy as that computed above. GOVERNOR. The altitude of the height of the cone in which the arms re- volve, measuring from the plane of revolution to the centre of suspension, will be the same as that of a pendulum which makes the same number of double beats per minute that the governor makes of revolutions ; or if the number of revolutions per minute be fixed, and we wish to obtain the proper height of cone, we divide the constant number 375'36 by twice the number of revo- lutions, which gives the square root of the height of the cone ; and, consequently, the height itself is equal to the square of this number. These relations are exhibited in the following rules : TO DETERMINE THE 8PEED AT WHICH A GOVERNOR MUST BE DRIVEN, WHEN THE HEIGHT OF THE CONE 18 FIXED IN WHICH THE ARMS REVOLVE. RULE. Divide the constant number 375'36 ~by twice the square root of the height of the cone in inches. The quotient is the proper number of revolutions per minute. Example. A governor with arms 30 inches long, measuring from the centre of suspension to the centre of the ball, revolves PROPER PROPORTIONS OF THE GOVERNOR. 231 in the mean position of the arms at an angle of about 30 degrees, with the vertical spindle forming a cone about 26J inches high. At what number of revolutions per minute should this governor be driven ? Here the height of the cone being 26-5 inches, the square root of which is 5*14, and twice the square root 10'28, we divide 375-36 by 10-28, which gives us 36-5 as the proper number of revolutions at which the governor should be driven. TO DETERMINE THE HEIGHT OF THE CONE IN WHICH THE AEM3 MUST EEVOLVE, WHEN THE VELOCITY OF EOTATION OF THE GOVEENOE IS DETEBMINED. ETJLE. Divide the constant number 375-36 ly twice the number of revolutions which the governor makes per minute, and square the quotient, which will fie the height in inches which the cone will assume. Example. Suppose that a governor be driven with a speed of 36 revolutions per minute, what will be the height of the cone in which the balls will necessarily revolve, measuring from the centre of suspension of the arms to the plane of revolution of the balls? Here 36-5 x 2 = 73, and 375-36 divided by 73 = 5-14, and 5*14 squared is equal to 26-4196, or very nearly 26-5 inches, which will be the height of the cone. When the arms revolve at an angle of 45 degrees with the spindle, or at right angles with one another, the centrifugal force is equal to the weight of the balls ; and when the arms revolve at an angle of 30 degrees with the spindle, they form with the base of the cone an equilateral triangle. STRENGTHS OF LOW-PEESSUEE LAND ENGINES. PISTON BOD. The piston rod is made one-tenth of the diameter of the cylinder, except in locomotives, where it is made one-seventh 232 PROPORTIONS OF STEAM-EXGINES. of the diameter. The piston rod is sometimes made of steel, or of iron converted into steel for a certain depth in. This enables it to acquire and maintain a better polish than if made of iron. MAIN LINKS. The main links are the parts which connect the piston rod with the beam. They are usually made half the length of the stroke, and their sectional area is 113th the area of the piston. AIR-PUMP ROD. The diameter of the air-pump rod is commonly made one- tenth of the diameter of the air-pump. BACK LINKS. The sectional area of the back links is made the same as that of the air-pump rod. END STUDS OF THE BEAM. The end studs of the beam are usually made the same diam- eter as the piston rod. Sometimes they are of cast-iron, but generally now of wrought. The gudgeons of water wheels are generally loaded with about 500 Ibs. for every circular inch of their transverse section, which is nearly the proportion that ob- tains in the end studs of engine beams. But the main centre is usually loaded beyond this proportion. MAIN CENTRE. The strength of this part will be given in the strengths of marine engines. But when of cast-iron it is usually made about one-fifth of the diameter of the cylinder. In a cylinder of 24 inches diameter this will be 4'8 inches, or say 4 inches ; and this proportion of strength will be about nine times the breaking weight, if we suppose the main centre to be overhung as in marine engines. Thus, in a cylinder of 24 inches diameter, and, consequently, of 452 square inches area, the total load on the piston with 20 Ibs. on each square inch is 9,040 Ibs. PEOPER DIMENSIONS OF THE MAIN BEAM. 233 But as the strain at the main centre is doubled from the beam acting as a lever of 2 to 1, it follows that the strain at the main centre will be 18,080 Ibs. The ultimate tensile strength of com- mon cast-iron being 12,000 per square inch of section, and the tensile and shearing strength being about the same, ^th of 12,000, or 1,333 Ibs., will be the proper load to place on each square inch of section ; and 18,080 divided by 1,333 will give the proper sec- tional area in square inches, which will be 13$- square inches nearly. This area corresponds to a diameter of a little over Cl- inches. But the strength is virtually doubled by the circum- stance of the main centre of land engines being supported at both ends. MAIN BEAM. The rules in common use for proportioning the main beams of engines are the same as those which existed prior to Mr. Hodgkinson's researches on the strength of cast-iron girders, which showed that the main element of strength was the bot- tom flange. But as in the case of engine beams the strain is alternately up and down, the top and bottom flanges, or beads of the beam, require to be of equal strength. Cast-iron is a bad material for engine beams, unless the central part be made of open work of cast-iron, and the edge of the beam be encircled by a great elliptical or lozenge-formed hoop, as is done in some of the American engines. But if the beam be made wholly of cast-iron, a much larger proportion of the metal should be col- lected in the top and bottom flanges than is at present the ordi- nary practice. The usual length of the main beam is three times the length of the stroke ; the usual breadth is equal to the diameter of the cylinder, and the usual mean thickness is yj^th of the length. The rule is as follows : TO FIND THE PBOPEK DIMENSIONS OP THE MAIN BEAM OF A LAND ENGINE. RULE. Divide the weight in Ibs. acting at the centre Try 250 and multiply the quotient by the distance between the extreme cen- tres. To find the depth, the breadth being given: Divide the 234 PROPORTIONS OF STEAM-ENGINES. product ~by the breadth in indies, and extract the square root of the quotient, which is the depth. The depth of the beam at the ends is usually made one-third of the depth at the middle. It will be preferable, however, to investigate a rule on the basis of Mr. Hodgkinson's rule for proportioning cast-iron gird- ers, which is as follows : Multiply the sectional area of the bottom flange in inches by the depth of the beam in inches, and divide the product by the distance between the supports also in inches, and 514 times the quotient will le the breaking weight in cwts. If the breaking weight be expressed in tons, the constant number 514 must be divided by 20, which gives the breaking weight as 25- 7, or say 26 tons, whereas experiment has shown that if the flange were to be formed of malleable iron instead of cast, the breaking weight would not be less than 80 tons ; or, in other words, that with the same sectional area of flange, the beam would be more than three times stronger. It is a common practice in the case of girders to make the strength equal to three times the breaking weight when the load is stationary, and to six times the breaking weight when the load is movable. But these proportions are too small, and less than nine or ten times the breaking weight will not give a suf- ficient margin of strength in the case of engines where the mo- tion is so incessant, and where heavy strains may be accidentally encountered from priming or otherwise. In the case of an en- gine, the weight answering to the breaking weight is the load on the piston ; and if we suppose the fly-wheel to be jammed, and the piston to be acting with its full force to lift or sink the main centre, it is clear that the strain on the main centre, and, therefore, on the beam, will be equal to twice the strain upon the piston, since the beam acts under such circumstances as a lever of 2 to 1. The problem we have now to consider is how many times the working weight must be less than the breaking weight to give a sufficient margin of strength in any given beam ; or, in other words, what proportions must the beam have to possess adequate working strength. PROPER DIMENSIONS OF THE MAIN BEAM. 235 To take a practical example from an engine in constant work. The engine with a cylinder of 24 inches diameter has a main beam 15 feet (or 180 inches) long ; 30 inches deep in the middle ; and with a sectional area of flange of 7 square inches.. The breaking weight of such a beam in cwts. will be 7x30x514 divided by 180=600 cwt. nearly, and this multiplied by 112 Ibs. = 67,200 lb., which is the breaking weight in pounds avoirdu- pois. The area of the cylinder in round numbers is 452 square inches ; but as there is a leverage of 2 to 1, this is equivalent to an area of cylinder of 904 square inches set under the middle of the beam and pulling it downwards, the beam being supposed to be supported at both ends. Dividing now 67,200 by 904 we get the pressure per square inch on the piston that would break the beam, which is a little over 74 Ibs. per square inch of the area of the piston, or 58 Ibs. per circular inch. If we suppose the working pressure of steam on the piston to be 6'27 Ibs. per circular inch, or 7'854 Ibs. per square inch, then the working strength of the beam will be about 9 times its breaking strength, which would give an adequate margin for safety. But if we suppose the working pressure to be 12 '54 Ibs. per circular inch, or 15'718 Ibs. per square inch, the working strength would in such case be only about 4 times the breaking strength, and the beam would be too weak. The strength of a cast-iron beam of any given dimensions varies directly as the sectional area of the edge flange ; or, if the sectional area of that flange be constant, the strength of the beam varies directly as the depth, and inversely as the length. If while the sectional area of the flange remains the same the depth of the beam is doubled without altering the length, then the strength is doubled. But if the length be also doubled, the strength remains the same as at first. As the length of an en- gine-beam is doubled when we double the length of the stroke, and as in any symmetrical increase of an engine when we double the length of the stroke we also double the diameter of the cyl- inder, to which the depth of the beam is generally made equal, large beams with the same area of flange, and made in the ordi- nary proportions, would be as strong as small beams, except that 236 PROPORTIONS OF STEAM-ENGINES. the load increases as the square of the diameter of the cylinder, and consequently the area of the edge flange must increase in the same proportion. These considerations enable us to fix the following rule for the strength of main beams : TO FIND THE PKOPER DIMENSIONS OF THE MAIN BEAM OF AN ENGINE. RULE. Make the depth of the beam equal to the diameter of the cylinder, and the length of the beam equal to three times the length of the stroke. Then to find the area of the edge flange : Multiply the area of the cylinder in square inches l>y the total pressure of steam and vacuum on each square inch of the piston, and divide the product l>y 650. The quotient is the proper area of the flange of the beam in square inches. Example 1. "What is the proper sectional area of the flange of the main beam of an engine, with cylinder 24 inches diam- eter and 5-feet stroke, the pressure on the piston being 20 Ibs. per square inch ? Here the area of the cylinder will be 452 inches, which mul- tiplied by 20 gives 9,040, and dividing by 650 we get 13'9 square inches, which is the proper sectional area of the edge bead or flange of the beam. JExample 2. What is the proper sectional area of the flange of the main beam of an engine with a cylinder 60 inches di- ameter, 12 J feet stroke, and with a pressure of steam on the pis- ton of 20 Ibs. per square inch ? The area of a cylinder 60 inches diameter is 2,824 square inches, and 2,824 multiplied by 20=56,480, which divided by 650=87 square inches nearly. Such a flange, therefore, if 14 inches broad, would be 6 inches thick. The beam would be 5 feet deep at the middle, and 37J feet long between the ex- treme centres. PROPER DIMENSIONS OP THE MAIN BEAM. 237 ANOTHER ETTLE FOR FINDING THE SECTIONAL AREA OF EACH EDGE FLANGE OF THE MAIN BEAM. RULE. Multiply the diameter of the cylinder in inches ly one- third of the length of the stroke in inches, and by the total pressure on each square inch of the piston, and divide the product ly 650. The quotient is the proper sectional area in square inches of each flange or head on the edge of the "beam. Example 1. What is the proper sectional area of the flange on the edge of the main beam of an engine with a 24-inch cylin- der, 20 Ibs. total pressure on piston per square inch, and 5 feet stroke ? Here 24 x 20 (which is one-third of the stroke in inches) x 20 (the pressure of the steam and vacuum per square inch) = 9,600, which divided by 650=14'7 sq. in., which is the area required. Example 2. "What is the proper sectional area of the flange on the edge of the main beam of an engine with a 60-inch cylin- der, 12^-feet stroke, and with a pressure on the piston of 20 Ibs. per square inch ? Here 60 x 50 (which is one-third of the stroke in inches) x 20, (the pressure of the steam per square inch) = 6,000, which di- vided by 650 gives 92 as the sectional area of the edge bead in square inches. Such a flange, if 15 inches broad, would be 6 inches thick. These results it will be seen are very nearly the same as those obtained by the preceding rule ; and one in- ference from these rules is that nearly all engine beams are at present made too weak. The purpose of the web of the beam is mainly to connect together the top and bottom flanges, so that there is no advantage in making it thicker than suffices to keep the beam in shape ; with which end, too, stiffening feathers, both vertical and horizontal, should be introduced upon the sides of the beam. The first cast-iron beams were made like a long hollow box to imitate wooden beams, and this form would still be the best, unless an open or skeleton beam, encircled with a great wrought-iron hoop, after the American fashion, be adopted. 238 PROPORTIONS OF STEAM-ENGINES. CONNECTING-ROD. The connecting-rods of land engines are now usually made of wrought-iron, and when so made, the proportions will be the same, or nearly so, as those given under the head of marine en- gines. When made of cast-iron the configuration is such that the transverse section at the middle assumes the form of a cross, this form being adopted to give greater lateral stiffness. The length of the rod is usually made the same as the length of the beam, namely, three times the length of the stroke, and the area of the cross section of the rod at the middle is commonly made ^th of the area of the cylinder, and the sectional area at the ends ^jth of the area of the cylinder. Such a strength is needlessly great, and is quite out of proportion to the strength commonly given to the beam. Thus, in the case of an engine with a 24-inch cylinder, the area of the piston is 452 square inches ; and if we take 20 Ibs. per square inch as the load on the piston, then the total load on the piston will be 9,040 Ibs. If the working load be made ^th of the breaking load, as in the case of the beam, then the breaking load should be 81,360 Ibs., and the strength of the connecting-rod should be such that it would just break with that load on the piston. Now the tensile strength of the weakest cast-iron is about 12,000 Ibs. per square inch of section, while its crushing strength is about five times that amount. Dividing 81,361 Ibs., the total tensile strength of the rod, by 12,000, the tensile strength of one square inch, we get about V square inches as the proper area of the smallest part of the connecting-rod when of cast-iron. But ^th of 452 (which is the area of the cylinder in square inches) is 13 square inches, from all of which it follows that while the main beams of en- gines are commonly made too weak, the cast-iron connecting- rods are commonly made too strong. This, however, is partly done for the purpose of balancing the weight of the piston and its connections. FLY-WHEEL SHAFT. The fly-wheel shaft of land engines is usually made of cast- iron. The following is the rule on which such shafts are usually proportioned : PROPEK DIAMETER OP FLY-WHEEL SHAFT. 239 TO FIND THE DIAMETER OF THE FLY-WHEEL SHAFT AT SMALLEST PAET, WHEN IT IS OF CAST-IRON. RULE. Multiply the square of the diameter of the cylinder in inches by the length of the crank in inches; extract the cube root of the product ; finally multiply the result l>y '3025. The product is the diameter of the fly-wheel shaft at the smallest part in inches. Example 1. "What is the proper diameter of the fly-wheel shaft, when of cast-iron, in the case of an engine with a diameter of cylinder of 64 inches and a stroke of 8 feet ? 64 diameter of the cylinder in inches 64 4096 = square of the diameter 48 = length of crank in inches 196608 68-15 = ^196608 and 58-15 x -3025 = 17-59, which is the proper diameter of the fly-wheel shaft at the smallest part. Example 2. What is the proper diameter, at the smallest part, of the cast-iron fly-wheel shaft of an engine, with a diameter of cylinder of 40 inches, and 5 feet stroke I 40 = diameter of cylinder hi inches 40 1600 = square of diameter of cylinder 80 = length of crank hi niches 48000 86-30 = ^48000 and 36-30 x -3025 = 10-98, which is the proper diam- eter of the shaft in niches. ME. WATT'S BULE FOE THE NECKS OF HIS CEANK SHAFTS. RULE. Multiply the area of the piston in square inches 1y the pressure on each square inch (and which Mr. Watt took at 12 ZJs.), and ~by the length of the crank in feet. Divide the product ~by 31-4, and extract the cube root of the quotient, which is the proper diameter of the shaft in inches. 240 PROPORTIONS OF STEAM-ENGINES. Example 1. What is the proper diameter of the fly-wheel shaft in an engine, with a cylinder 64 inches diameter and 8 feet stroke, the pressure on the piston being taken at 12 Ibs. per square inch ? The area of a cylinder 64 inches diameter is 3,217 square inches, which multiplied by 12 = 38,604, and this multiplied by 4, which is the length of the crank in feet, is 154,416. This divided by 31'4 4,917'7, the cube root of which is 17'01 inches. Example 2. "What is the right diameter, according to Mr. "Watt's rule, of the fly-wheel shaft of an engine, with a 24-inch cylinder, 5 feet stroke, and with a pressure of 12 Ibs. on each square inch of the piston ? The area of the cylinder is 452 square inches, which multi- plied by 12 = 5,424, and this multiplied by 2, which is the length of the crank in feet 13,560, which divided by 31'4 = 431, the cube root of which is 7 inches, which is the proper diameter of the shaft. In Mr. Caird's engine the diameter is 8 inches. TO FIND THE PEOPEB THICKNESS OF THE LAEGE EYE OF THE CBANK FOE FLY-WHEEL SHAFT, WHEN OF OAST-IEON. RULE. Multiply the square of the length of the crank in inches ly 1'561, and then multiply the square of the diameter of the cylinder in inches l>y '1235; multiply the sum of these prod- ucts l>y the square of the diameter of the cylinder in inches ; divide this product 2>y 666*283 / divide this quotient by the length of the crank in inches ; finally extract the cube root of the quotient. The result is the proper thickness of the large eye of crank for fly-wheel shaft in inches, when of cast-iron. Example 1. Required the proper thickness of the large eye of crank for fly-wheel shaft, when of cast-iron, of an engine whose length of stroke is 8 feet, and diameter of cylinder 64 inches. THICKNESS OF LARGE EYE OF CRANK. 241 48 = length of crank in inches 48 2304 = square of length of crank in inches 1-561 = constant multiplier 8596-5 64 = diameter of cylinder in inches 64 4096 -f S( l uare f diameter of cylinder ( in inches 1235 = constant multiplier 505-8 3596-5 4102-3 = sum of products ) 409(5 _ j square of the diameter of the cy- ( Under in niches = 666-283)16803020-8 Length of crank = 48)25219-045 525-397 and ^525-397 = 8-07 which is the proper thickness of the large eye of the crank in inches, when of cast-iron. Example 2. Required the proper thickness of the large eye of the crank for fly-wheel shaft, when of cast-iron, of an en- gine, whose length of stroke is 5 feet, and diameter of cylinder 40 inches. 30 = length of crank in inches 30 900 = square of length of crank in inches 1-561 = constant multiplier 1404-9 11 242 PROPORTIONS OF STEA3I-ENGIKES. 40 = diameter of cylinder in inches 40 1600 1235 = constant multiplier 197-6 1404-9 1602-5 = sum of products 1600 = square of diameter of cylinder I = 666-283)2564000-0 Length of crank 30 inches 3848-2 128-3 and ^/128'3 = 5 - 04 inches is the proper thickness in this engine of the large eye of the crank, when of cast-iron. TO FIND THE PROPER BREADTH OF THE WEB OF THE CRANK AT THE CENTRE OF THE FLY-WHEEL SHAFT, WHEN OF CAST-IRON, SUPPOSING THE BREADTH TO BE CONTINUED TO THE CENTRE ' OF THE SHAFT. RULE. Multiply the square of the length of the crank in inches by 1-561, and then multiply the square of the diameter of the cylinder in inches by "1235 / multiply the square root of the sum of these products by the square of the diameter of the cylinder in inches ; divide the product by 23*04, and finally extract the cube root of the quotient. The final result is the breadth of the crank at the centre of the fiy-wheel shaft, when the crank is of cast-iron. Example 1. What is the proper breadth of the web of the crank at the centre of fly-wheel shaft, when of cast-iron, in the case of an engine, with a diameter of cylinder of 64 inches, and length of stroke 8 feet ? 48 = length of crank in inches 48 2304 = square of length of crank 1*561 = constant multiplier 3596-5 PROPER BREADTH OF WEB OP CRANK. 243 &l =. diameter of cylinder in inches 64 4096 = square of diameter of cylinder 1235 constant multiplier 505-8 3596-5 4102-3 = sum of products v/4102-3= 64-05 nearly 4096 = square of diameter of cylinder in inches. 23-04)262348-80(11395-34 2304 3214 2304 9108 6912 21968 20736 12320 11520 8000 6912 10880 9216 1664 $11395-34 = 22-5 inches, which is the proper breadth of the web of the crank, when of cast-iron, supposing the breadth to be continued to the centre of the fly-wheel shaft. Example 2. "What is the proper breadth of the web of a cast-iron crank at the centre of the fly-wheel shaft (supposing it to be so far extended), in the case of an engine with 40 inches diameter of cylinder and 5 feet stroke ? 244 PROPORTIONS OF STEAM-ENGINES. 80 length of crank in inches 30 900 = square of length of crank in inches 1-561 = constant multiplier 1404-9 40 = diameter of cylinder in inches 40 1600 = square of diameter of cylinder 1235 = constant multiplier 197-6 1602-5 = sum of products v/1602-5 = 40-3 nearly 1600 23-04)64480-0 2798-6 nearly $'2798-6 = 14-09, which is the proper breadth in inches of a cast iron crank in an engine of this size, supposing the breadth to be continued to the fly-wheel shaft. TO FIND THE PEOPEK THICKNESS OP THE WEB OF A OAST-IKON CRANK AT THE CENTEE OF THE FLY-WHEEL SHAFT. RULE. Multiply the square of the length of the crank in inches T>y 1'561, and then multiply the square of the diameter of the cylinder in inches l>y '1235 ; multiply the square root of the sum of these products ly the square of the diameter of the cylinder in inches; divide the product 'by 1'32; finally extract the cube root of the quotient. The result is the proper thicTcness of the web of a cast-iron crank in inches at the cen- tre of the fly-wheel shaft, supposing the thickness to ~be ex- tended to that point. Example 1. Required the proper thickness of the web of a cast-iron crank at the centre of the fly-wheel shaft (supposing it PROPER THICKNESS OF WEB OF CRANK. 245 to be so far extended), in the case of an engine with 64 inches diameter of cylinder, and 8 feet stroke. 48 = length of crank in inches 48 2304 = square of the length of crank 1-561 = constant multiplier 3596-5 64 =: diameter of cylinder in inches 64 4096 =r square of diameter of cylinder 1235 =: constant multiplier 505-8 3596-5 4102-3 = sum of products and ^/4102-3 = 64-05 nearly 4096 = square of diameter of cylinder 1422-33 and v' 1423-33 = 11-25 Example 2. "What is the proper thickness of the web of a cast-iron crank at centre of fly-wheel shaft (supposing it to be so far extended), in the case of an engine with 40 inches diameter of cylinder, and 5 feet stroke? 30 = length of crank in inches 30 _ j - \ square of length of crank in inches 1-561 constant multiplier 1404-9 246 PROPORTIONS OF STEAM-ENGINES. 40 = diameter of cylinder in inches 40 1600 = square of diameter of cylinder 1235 = constant multiplier 197-6 1404-9 1602-5 V 1602-5 = 40-3 nearly 1600 349-8 and ^349-8 = 7'04, which is the proper thickness in inches of the web of a cast-iron crank for this engine, measuring at the centre of the fly- wheel shaft. CRANK PIN. The crank pins of land engines having cast-iron cranks, are generally made of cast-iron, and are in diameter about one-sixth of the diameter of the cylinder. MILL GEARING. Boulton aad Watt, by whom the present system of iron gearing was introduced, proportioned their wheels on the follow- ing consideration : ' That a bar of cast-iron 1 inch square and 12 inches long, bears 600 Ibs. before it breaks ; 1 inch long will bear 7,200 Ibs., and T Vth of this = 480 Ibs., is the load which should be put on the wheel,' for each square inch in section of the tooth. Boulton and Watt's rule for the strength of geared wheels is consequently as follows : If H = the actual horses' power which the wheel has to transmit ; d, the diameter of the wheel in feet, and r, the revolutions of the wheel per minute ; then H x 306 -5 = the strength, and the strength divided by the breadth in inches =p\ or the square of the pitch in inches. PROPER PROPORTIONS OF TOOTHED WHEELS. 247 Hence H =? x * X d X r and p = 4 / H X 3 6 , which equations 306 V 5 x d x r' put into words are as follows : TO FIND THE NUMBER OF ACTUAL HOESES POWER WHICH A GIVEN WHEEL WILL TRANSMIT, ACCORDING TO BOULTON AND WATT'S PRACTICE. RULE. Multiply the square of the pitch in inches ~by the "breadth of the wheel in inches, by its diameter in feet, and by tlie number of revolutions it makes per minute, and divide the product ~by the constant number 306. The quotient is the number of actual horses* power which the wheel will safely transmit, according to Boulton and Waffs practice. TO FIND THE PROPER PITCH OF A WHEEL IN INCHES TO TRANS- MIT A GIVEN POWER, ACCORDING TO BOULTON AND WATT'S PRACTICE. EULE. Multiply the breadth of the teeth in inches by the diam- eter of the wheel in feet, and by the number of revolutions it makes per minute, and reserve the product as a divisor. Next multiply the number of actual horses' power which, the wheel has to transmit by the constant number 306, and divide the product by the divisor found as above. Finally, extract the square root of the quotient, which is the proper pitch of the wheel in inches, according to Boulton and Wattfs practice. Instead, however, of reckoning the strain in horses' power, it is preferable to reckon it as a pressure or weight applied to the acting tooth of the driving wheel. If t = the thickness of the tooth in inches, w = the pressure upon it hi Ibs., and c a con- stant multiplier, which for cast-iron is '025, for brass, '035, and for hard wood, -038, then t = c ^Jw, by which formula we can easily find the proper thickness of the tooth, and twice the thickness of the tooth with the proper allowance for clearance, gives the pitch. This formula put into words is as follows : 248 PROPORTIONS OF STEAM-ENGINES. TO FIND THE PROPER THICKNESS OF TOOTH OF A CAST-IRON WHEEL TO TRANSMIT WITH SAFETY AST GIVEX PRESSURE. EULE. Multiply the square root of the pressure in pounds act- ing at the pitch line T)y the constant number '025. The product is the proper thickness of the tooth in inches. Example 1. What is the proper thickness of the teeth of a cast-iron wheel moved by a pressure of 233*33 Ibs. at the pitch circle ? Here V 233-33 = 15-27, and this multiplied by '025 = -381, which is the proper thickness of the teeth in inches. Example 2. What is the proper thickness of the teeth of a cast-iron wheel which is moved round by a pressure of 46,666'6 Ibs. at the pitch circle ? It will be easiest to solve this question by means of logarithms. As the index of the logarithm is always one less than the number of places above unity filled by the number of which the logarithm has to be found ; and as there are five such places in the number 46,666-6, it follows that the index of the logarithm will be 4, and the rest of the logarithm will be found by looking for the nearest number to 46,666-6 in the tables, and which number will be 4,666, the logarithm answering to which is 668945. The residue 6 - 6, however, has not yet been taken into account, and to include it we must multiply the number found opposite to the logarithm in the column marked D, commonly introduced in logarithmic tables (and which is a column of common differences), by the number we have not yet reckoned, namely, 6-6 ; and cut off a number of figures from the product equal to those in the mul- tiplier, adding the residue to the logarithm, which will thereupon become the correct logarithm of the whole quantity. The com- mon difference in this case is 93, which multiplied by 6-6 gives 613'8, and cutting off the 3*8 we add the 61 to the logarithm already found, which then becomes 4-669006. Dividing this by 2, we get 2-334503, which will be the logarithm of the number that is" the square root of 46,666-6. As the index of the loga- rithm is 2, there will be three places above unity in the number, and looking now in the logarithm tables for the number answer- PROPER PROPORTIONS OF TOOTHED WHEELS 249 ing to the logarithm nearest 334503, we get the number 216, the logarithm of which is 334454. The number 216 is consequently the square root of 46,666'6 very nearly, as to extract the square root by logarithms, we have only to divide the logarithm of the number by 2, and the number answering to the new logarithm thus found will be the square root of the original number. Now 216 multiplied by '025 = 5'400, which consequently is the thick- ness in inches of each of the teeth of this wheel. GENERAL RULES REGARDING GEARING. The pitch should be in all cases as fine as is consistent with the required strength. "When the velocity of the motion exceeds 3 feet per second, the larger of the two wheels should be fitted with wooden teeth, the thickness of which should be a little greater than that of the iron teeth. The breadth of the teeth in the direction of the axis varies very much in practice. But where the velocity does not exceed 5 feet per second, a breadth of tooth in the line of the axis equal to four times the thickness of the tooth will suffice. This is nearly the same thing as a breadth equal to twice the pitch. Where the velocity at the pitch circle is greater than 5 feet per second, the breadth of the teeth should be 5 tunes the thickness of tooth, the surfaces being kept well greased. But if the teeth be constantly wet, the breadth should be 6 times the thickness of tooth at all velocities. The best length of the teeth is fths of the pitch, and the length should not exceed fths of the pitch, and the effective breadth of the teeth should not be reckoned as exceeding twice the length ; any additional breadth being good for wear, but not for strength. In the Soho practice the length of the teeth is made -j^ths of the pitch outside, and ^ihs of the pitch inside of the pitch circle, the whole length being -j^ths or fths of the pitch. The London practice is to divide the pitch into 12 parts, and to adjust the length of the tooth by allowing ^ths without, and within the pitch circle, the entire length of tooth being of the pitch. The projection of the teeth beyond the pitch circle w.ill be |th of the pitch, and the surface in contact between 11* 250 PROPORTIONS OF STEAM-ENGINES. the teeth of the two wheels will be half the pitch. About $th of the pitch should be left unoccupied at the bottom of the teeth for clearance. "With regard to the least number of teeth that is admissible in the smaller of two wheels working together, 12 to 18 teeth will answer well enough in crane work, where a pinion is employed to give motion to a AvheeJ at a low rate of speed. But for quick motions, a pinion driven by a wheel should never have less than from 30 to 40 teeth. The best form of teeth is the epicycloidal, and in general the proper curve is obtained by rolling a circle of wood carrying a pencil on another circle of wood answering to the pitch circle, the point of the tooth being described by the rolling circle trav- ersing the outside of the pitch line, and the root by traversing the inside of the pitch line. The diameter of the rolling circle should be 2*22 times the pitch. Some teeth are not epicycloi- dal, but the roots are radii of the pitch circle, and the points are described with compasses from the pitch centre of the next tooth. In the following table will be found the thickness and pitch of teeth answering to different amounts of load or pressure at the pitch circles. But it may here be remarked that such large pitches as 12 and 13 inches are practically not used. In cases where such large pressures are to be transmitted as answer to pitches over 5 inches or thereabout, it is usual to distribute the load by placing two or more parallel wheels upon the same shaft, working into corresponding pinions ; and it is also usual to set the teeth of each wheel a little in advance of the teeth of the wheel next it, so as to divide the pitch, and thus render the action of the teeth smoother and more continuous. EXAMPLES OP HEAVY GEARING. 251 PROPORTIONS OF THE TEETH OF CAST-IRON WHEELS. Pressure in Ibs. at the pitch circle. Pitch of teeth in inches, allowing one-tenth for clearance. Thickness of teeth in inches. Pressure in Ibs. at the pitch circle. Pitch of teeth in inches, allowing one-tenth for clearance. Thickness of teeth in inches. 283-33 798 88 11666-65 5-6705 2-7002 849-95 981 467 13999-98 6-2118 2-9580 466-66 1-134 540 16883-31 6-7099 8-1952 5S3-32 1-268 604 18666-64 7-1728 3-4156 699-99 1-388 661 20999-97 7-6079 8-6228 816-65 1-5 716 23383-3 8-0194 8-8188 938-32 1-604 763 25666-63 8-4109 4-005? 1049-98 1-7 809 27999-96 8-7848 4-1832 1166-65 1-793 854 30838-29 9-1470 4-3557 1283-31 1-88 895 82666-62 9-4887 4-5184 1399-98 1-964 935 84999-95 9-8218 4-6770 1516-64 2-044 . -973 87883-28 10-1439 4-8304 1683-31 2-121 1-04 89666-61 10-4560 4-9790 1749-9T 2-196 1-045 41999-94 10-7592 5-1284 1866-64 2-268 1-08 44838-27 11-0540 6-2638 1983-3 2-S88 1-118 46666-6 11-8412 5-4006 2099-9T 2-405 1-145 49999-98 11-7381 5-5896 2216-63 2-471 1-177 52383-26 12-0103 5-7192 2333-3 2-588 1-208 54666-59 12-2749 5-8452 2449-96 2-598 1-287 66999-92 12-5341 5-9686 2566-63 2-659 1-266 69833-25 12-7883 6-0897 2683-29 2-720 1-295 60666-58 12-9310 6-1576 2799-96 2-777 1-822 62999-91 13-1773 6-2749 4666-66 8-586 1-7078 65333-24 18-8893 6-8759 6999-99 4-8924 2-0916 67666-57 13-6566 6-5031 9833-32 5-0719 2-4152 69999-99 13-WI1 6-6143 It will be useful to illustrate the application of these rules to the case of heavy gearing by one or two practical examples. In a steamer with engines by Messrs. Penn and Son there are two cylinders of 82 inches diameter and 6 feet stroke, giving motion to a toothed wheel 14 feet diameter consisting of four similar wheels bolted together, the teeth being 12 inches broad and 5'86 inches pitch. The area of a cylinder 82 inches being 5,346 square inches, there will be a total pressure on the piston if we reckon the mean average pressure upon each square inch at 25 Ibs. of 133,650 Ibs. But as there are two pistons, the total pressure on the two pistons will be 267,300 Ibs. Now the diameter of the geared wheel being 14 feet, its circumference will be 44 feet, and as at each movement of the pistons up and down through the length of the stroke, or through a distance of 12 feet, the wheel makes one revolution, or moves through 44 feet, the pressure at the circumference of the wheel will be lesa 252 PROPORTIONS OP STEAM-ENGINES. than that on the pistons in the proportion in which 44 exceeds 12, so that by multiplying 267,300 by 12 and dividing the product by 44 we get the equivalent or balancing pressure at the circumfer- ence of the wheel, and which is 69,073 Ibs. As, however, this load is distributed among four wheels, there will only be one-fourth of 69,673, or 17,418 Ibs. to be borne by each of them. According to the rule we have given, therefore, the square root of 17,418 multiplied by '025 will be the proper thickness of each tooth in inches. Now ./1 7,41 8 132, and 132 x -025 = 3'3, which by our rule is the proper thickness of the tooth in inches, and twice this, or 6'6, with one-tenth or '3 for clearance, will be the pitch = 6'9, whereas the actual pitch is 1 inch less than this. If the multi- plier be made '02, instead of '025, the value obtained will agree more nearly with this example, as 132 x '02 = 2'64, which will be the thickness of tooth, and 2-64 x 2 = 5'28, to which adding ^th of the thickness of the tooth for clearance, or '264, we get 5'544 inches as the pitch. If we take the pressure at 20 Ibs. per square inch on the pistons instead of 25 Ibs., then the total pressure on the two pistons will be 213,840 Ibs., which reduced to the equiv- alent pressure at the periphery of the wheel will be 58,320 Ibs. The fourth of this is 14,580, the logarithm of which is 4'163758, the half of which is 2-081879, the natural number answering to which is 120'7, which multiplied by '025 = 3'1175, which is the proper thickness of the tooth in inches for this amount of strain. It will be seen, therefore, that the strength which our rule gives is somewhat greater than that of this example. Let us now take an example by a different maker, and we select the geared engines of the steamer ' City of Glasgow,' con- structed by Messrs. Tod and Macgregor. There were two cylin- ders in this vessel, each 66 inches diameter and 5 feet stroke, and the motion was communicated from the crank shaft to the screw shaft by means of four parallel wheels, V feet diameter, 8 inches broad, and 4 inches pitch. The area of a cylinder 66 inches diameter is 3,421 square inches, and the area of two such cylin- ders will, consequently, be 6,842 square inches. If we take the pressure urging the pistons at 20 Ibs. per square inch, the total pressure on the pistons will be 136,840, which reduced to the EXAMPLES OF HEAVY GEARING. 253 pressure at the periphery of the wheel which moves 2'2 times faster than the pistons will he 62,200 Ibs. ; and as the pressure is divided among four wheels there will be one-fourth of 62,200, or 15,550 Ibs. on each. The logarithm of this number is 4-191430, the half of which is 2*095715, the natural number answering to which is 124-7, and 124-7 multiplied by -025 = 3-1175, which is half as much again as the actual strength given in these wheels. We may take still another example, and shall select the case of the Tire Queen,' a screw yatch constructed by Messrs. Eobert Nap ler and Sons. In this vessel there are two cylinders, each of 36 inches diameter and 36 inches stroke, and the motion is communicated from the crank shaft to the screw shaft through the medium of three parallel wheels 8 feet diameter placed on the end of the crank shaft. The pitch of the teeth is 3-55 inches, and two of the wheels are 4 inches broad, and one of them 6 inches broad. The two narrow wheels may be reckoned as equiv- alent to one broad one, so we may consider the strain to be divided between two wheels. The area of each cylinder is 1,018 square inches, and if we reckon two cylinders of this area, with a pressure of 20 Ibs. per square inch, urging the piston of each, the total pressure urging the pistons will be 40,720 Ibs. The double stroke of the piston is 6 feet, and the circumference of the wheel is 26 g 7 feet ; and as the wheel revolves once while the pistons are making a double stroke, the relative velocities will be 6 and 26'7, and the relative pressures 26'7 and 6. Multiply- ing, therefore, 40,720 by 6 and dividing by 26-7, we get 9,150 Ibs. as the pressure at the circumference of the wheel ; and as this load is to be divided between two wheels, there will be a load of 4,575 Ibs. upon each. The logarithm of 4,575 is 3-660391, the half of which is 1-830195, the natural number answering to which is 67-64, which multiplied by -025 gives 1-691 as the proper thickness of tooth in this wheel. Twice 1-691 is 3-382, to which if weadd^th of the thickness of the tooth, or -169 for clear- ance, we get 3-55 as the proper pitch of this wheel, and this is the very pitch which is really given. In this case, therefore the rule and the example perfectly correspond. The rule gives 254 PROPORTIONS OP STEAM-ENGINES. sufficient strength to represent the mean thickness of wooden and iron teeth the wooden teeth being a little thicker, and the iron teeth a little thinner than the amount which the rule pre- scribes. MARINE ENGINES. The rules which I have given in my "Catechism of the Steam-Engine " for fixing the proper proportions of the parts of marine engines, take into account the pressure of the steam with which the engine works. But in order that the proportions thus arrived at may be more easily comparable with the proportions subsisting in the engines of different constructors, in which the pressure is assumed as tolerably uniform, it will be more convenient so to frame the rules that a uniform pressure of 25 Ibs. per square inch of the area of the piston shall be supposed to be at all times existing. In cases where it is desired to ascer- tain the dimensions proper for a greater pressure than 25 Ibs., it will be easy to arrive at the right result by taking an imaginary cylinder of as much greater area than the real cylinder as the real pressure exceeds the assumed pressure of 25 Ibs., and then by computing the strengths and other proportions as if for this imaginary cylinder, they will be those proper for the real cylin- der. Thus if it be desired to ascertain the strengths proper for an engine with a cylinder of 30 inches diameter, and with a pressure on the piston of 100 Ibs. on the square inch, the end will be attained if we determine the strengths proper for an engine of 60 inches diameter, and with 25 Ibs. pressure on the square inch ; for the area of the larger cylinder being four times greater than that of the smaller, the same total force will be ex- erted with one-fourth of the pressure. So, in like manner, if it be wished to ascertain the strengths proper for an engine with a cylinder 30 inches diameter, and with a pressure on the piston of 50 Ibs. per square inch, we shall find them by determining the proportions suitable for an engine with an area of piston twice greater than the area of a piston 30 inches diameter, and which area will be that answering to a diameter of 42J inches. DIMENSIONS OP THE CROSSHEAD. 255 By this mode of procedure a table of proportions adapted to the ordinary pressures will be made available for determining the proportions suitable for all pressures, as we have only to fix upon an assumed cylinder which shall have as much more area as the intended pressure has an excess of pressure over 25 Ibs. per square inch, and the proportions proper for this assumed cylin- der will be those proper for the real cylinder with the pressure intended. In this way the strengths fixed for marine engines may also be made applicable to locomotives and to high and low pressure engines of every kind. In the following rules, there- fore, it will be understood the strengths and other proportions are those proper to an assumed pressure on the piston, including steam and vacuum, of 25 Ibs. per square inch, and the computa- tions are for side lever engines, but for the most part are appli- cable to all kinds of engines. CROSSHEAD. TO FIND THE PROPER THICKNESS OF THE WEB OF THE CROSS- HEAD AT THE MIDDLE. KULE. Multiply the diameter of the cylinder in inches ~by '072. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '072 = 2'880 inches, which is the proper thickness of the web of the crosshead at the middle in this en- gine. Example 2. Let 64 inches be the diameter of cylinder. Then 64 inches x '072 = 4' 608 inches, which is the proper thickness of the web of the crosshead at the middle in this en- gine. TO FIND THE PROPER THICKNESS OF THE WEB OF THE CROSS- HEAD AT THE JOURNAL. RULE. Multiply the diameter of the cylinder in inches ly '061. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x -061 = 2-440 inches, which is the proper 256 PROPORTIONS OF STEAM-ENGINES. thickness of the web of the crosshead at the journal in this en- gine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x - 061 = 3'904 inches, which is the proper thickness of the web of the crosshead at the journal in this en- gine. TO FIND THE PROPER DEPTH OF THE WEB OF THE CEOSSHEAD AT THE MIDDLE. RULE. Multiply the diameter of the cylinder in inches ~by '268. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '268 = 10'720 inches, which is the proper depth of the web of the cTosshead at the middle in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x '268 = l'T'152 inches, which is the proper depth of the web of the crosshead at the middle in this engine. TO FIND THE PEOPEK DEPTH OF THE WEB OF THE OEOSSHEAD AT JOURNALS. RULE. Multiply the diameter of the cylinder in inches ~by '101. Example 1. Let 40 inches be the diameter of cylinder. Then 40 inches x '101 = 4'040 inches, which is the proper depth of the web of the crosshead at journals in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x '101 = 6'464 inches, which is the proper depth of the web of the crosshead at journals in this engine. TO FIND THE PROPER DIAMETER OF THE JOURNALS OF THE CROSS- HEAD. RULE. Multiply the diameter of the cylinder in inches by "086. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x -086 = 3*440 inches, which is the proper diameter of the journals of the crosshead in this engine. Example 2. Let 64 inches be the diameter of cylinder. DIMENSIONS OF THE CROSSHEAD. 257 Then 64 inches x '086 = 5'504 inches, which is the proper diameter of the journal of crosshead in this engine. TO FIND THE PEOPER LENGTH OF THE JOURNALS OF THE CROSS- HEAD. The length of the journals of the crossheads should be equal to about li times their diameter, but on the whole it appears to be advisable to make the journals of the crosshead as long as they can be conveniently got. TO FIND THE PEOPER THICKNESS OF THE EYE OF THE CROSSHEAD. RULE. Multiply the diameter of the cylinder in inches by '041. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '041 = 1'640 inches, which is the proper thickness of the eye of the crosshead in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x '041 = 2'624 inches, which is the proper thickness of the eye of the crosshead in this engine. TO FIND THE PEOPER DEPTH OF THE EYE OF THE CEOSSHEAD. RULE. Multiply the diameter of the cylinder in inches ~by '286. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '286 == 11 -440 inches, which is the proper depth of the eye of the crosshead in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x -286 = 18-304 inches, which is the proper depth of the eye of the crosshead in this engine. TO FIND THE PBOPEB DEPTH OF GIBS AND CUTTER PASSING THROUGH THE OEOSSHEAD. RULE. Multiply the diameter of the cylinder in inches ~by '105. 258 PROPORTIONS OF STEAM-ENGINES. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '105 = 4*200 inches, which is the proper depth of the gibs and cutter passing through the crosshead in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x '105 = 6'720 inches, which is the proper depth of the gibs and cutter passing through the crosshead in this engine. TO FIND THE PEOPEB THICKNESS OF THE GIBS AND CtTTTEE PASSING THROUGH THE CEOSSHEAD. BULE. Multiply the diameter of the cylinder in inches ~by *021. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '021 '840 inches, which is the proper thickness of the gibs and cutter passing through the crosshead in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x '021 = 1/344 inches, which is the proper thickness of the gibs and cutter passing through the crosshead in this engine. SIDE RODS. TO FIND THE PEOPEB DIAMETER OF THE CYLINDER SIDE RODS AT THE ENDS. . Multiply the diameter of the cylinder in inches ly '065. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '065 = 2'600 inches, which is the proper diameter of cylinder side rods at ends in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x "065 = 4-160 inches, which is the proper diameter of the cylinder side rods at ends in this engine. The diameter of the side rods at the middle should be about DIMENSIONS OF THE SIDE RODS. 259 one-fourth more than the diameter at the ends. Thus a side rod 5 inches diameter at the ends will be 6J inches diameter at the middle. The area of the horizontal section of iron through the middle of eye of side rod is usually about one-half greater than the sec- tional area of the side rod at ends. TO FIND THE PEOPEE BEEADTH OF THE BUTT OF THE SIDE EOD IN INCHES. EULE. Multiply the diameter of the cylinder in inches ~by '077. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '077 = 3'080 inches, which is the proper breadth of butt of side rod in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x '077 = 4'928 inches, which is the proper breadth of butt in this engine. TO FIND THE PEOPEE THICKNESS OF THE BUTT OF THE SIDE EODS. EULE. Multiply the diameter of the cylinder in inches "by '061. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '061 = 2*440 inches, which is the proper thickness of the butt of the side rod in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x '061 = 3'904 inches, which is the proper thickness of the butt of the side rod in this engine. TO FIND THE PEOPEB MEAN THICKNESS OF THE 8TEAP OF THE SIDE EOD AT THE CUTTEE. EULE. Multiply the diameter of the cylinder in inches by *032. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '032 = 1*280 inches, which is the proper mean thickness of the strap of side rod at the cutter in this engine. 260 PROPORTIONS OF STEAM-ENGINES. Example 2. Let 64 inches be the diameter of the cylinder. " Then 64 inches x '032 = 2'048 inches, which is the proper mean thickness of the strap of side rod at the cutter in this engine. TO FIND THE PROPER MEAN THICKNESS OF THE STEAP OF SIDE ROD BELOW THE CUTTER. RULE. Multiply the diameter of the cylinder in inches ~by '023. Example \. Let 40 inches be the diameter of the cylinder. Then 40 inches x '023 ='92 inches, which is the proper mean thickness of the strap of the side rod below the cutter in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x '023 = 1-472 inches, which is the proper mean thickness of the strap of the side rod below the cutter in this engine. TO FIND THE PROPER DEPTH OF THE GIB3 AND CUTTER OF SIDE ROD. RULE. Multiply the diameter of the cylinder in inches ~by '08. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '08 = 3 - 20 inches, which is the proper depth of gibs and cutter of side rod in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x -08 = 5'12 inches, which is the proper depth of gibs and cutter of side rod in this engine. TO FIND THE PROPER THICKNESS OF GIBS AND CUTTER OF SIDE ROD. RULE. Multiply the diameter of the cylinder in inches ~by *016. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x -016 = -64 inches, which is the proper thickness of gibs and cutter of side rod in this engine. Example 2. Let 64 inches equal the diameter of cylinder. Then 64 inches x '016 = 1'02 inches, which is the proper thickness of gibs and cutter of side rod in this engine. DIMENSIONS OF THE PISTON ROD. 261 PISTON ROD. TO FIND THE PROPER DIAMETER OF THE PISTON ROD. RULE. Divide the diameter of the cylinder in inches ~by 10. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches -s- 10 = 4'0 inches, which is the proper diame- ter of piston rod in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches -r- 10 = 6-4 inches, which is the proper diame- ter of piston rod in this engine. TO FIND THE PROPER LENGTH OF THE PART OF THE PISTON ROD IN THE PISTON. RULE. Divide the diameter of the cylinder in inches ~by 5. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches -j- 5 = 8*0 inches, which is the proper length of the part of the piston rod in the piston in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches -j- 5 = 12'8 inches, which is the proper length of the part of the piston rod in the piston in this engine. TO FIND THE MAJOR DIAMETER OF THE PART OF THE PISTON ROD IN THE PISTON. RULE. Multiply the diameter of the cylinder in inches ~by '14. Example 1. Let 40 inches equal the diameter of cylinder. Then 40 inches x '14 = 5'60 inches, which is the proper major diameter of the part of the piston rod in piston in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x -14 = 8-96 inches, which is the proper major diameter of the part of the piston rod in piston in this engine. TO FIND THE MINOR DIAMETER OF THE PART OF THE PISTON HOD IN THE PISTON. RULE. Multiply the diameter of the cylinder in inches by '115. 262 PROPORTIONS OF STEAM-ENGINES. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x -115 = 4*600 inches, which is the proper minor diameter of the part of the piston rod in piston in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x *115 =7'360 inches, which is the proper minor diameter of the part of the piston rod in piston in this engine. TO FIND THE MAJOR DIAMETER OF THE PART OF THE PISTON ROD IN THE OROSSHEAD. RULE. Multiply the diameter of the cylinder in inches by '095. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '095 = 3'800 inches, which is the proper major diameter of the part of the piston rod in the crosshead in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x *095 = 6'080 inches, which is the proper major diameter of the part of the piston rod in the crosshead in this engine. TO FIND THE MINOR DIAMETER OF THE PART OF THE PISTON ROD IN CROSSHEAD. RULE. Multiply the diameter of the cylinder in inches by '09. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '09 = 3 '60 inches, which is the proper minor diameter of the part of the piston rod in crosshead in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x '09 = 5'76 inches, which is the proper minor diameter of the part of the piston rod in crosshead in this engine. TO FIND THE PROPER DEPTH OF THE CUTTER THROUGH PISTON. RULE. Multiply the diameter of the cylinder in inches T>y -085. Example 1. Let 40 inches be the diameter of the cylinder. DIMENSIONS OP THE CONNECTING-ROD. 263 Then 40 inches x '085 = 3-400 inches, which is the proper depth of the cutter through the piston in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x '085 = 5-440 inches, which is the proper depth of the cutter through the piston in this engine. TO FIND THE PEOPEE THICKNESS OF THE CUTTEE THBOUGH PISTON. EULE. Multiply the diameter of the cylinder in inches ly '035. Example 1. Let 40 inches by the diameter of the cylinder. Then 40 inches x '035 = 1-400 inches, which is the proper thickness of cutter through the piston in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x '035 = 2-240 inches, which is the proper thickness of cutter through piston in this engine. CONNECTING-ROD. TO FIND THE PEOPEE DIAMETEE OF THE CONNECTING-BOD AT THE ENDS. EULE. Multiply the diameter of the cylinder in inches Try -095. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x -095 = 3-800 inches, which is the proper diameter of the connecting-rod at the ends in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x "095 = 6-080 inches, which is the proper diameter of the connecting-rod at the ends in this engine. The diameter of the connecting-rod at the middle will vary with the length, but is usually one-fifth more than the diameter at the ends. Thus a connecting-rod 7*7 inches diameter at the ends will be 9-25 inches diameter at the middle. TO FIND THE MAJOE DIAMETEB OF THE PAET OF CONNECTING- BOD IN THE OEOS8TAIL. EULE. Multiply the diameter of the cylinder in inches ~by -098. 264 PROPORTIONS OF STEAM-ENGINES. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '098 = 3*920 inches, which is the proper major diameter of the part of the connecting-rod in the cross- tail in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x '098 = 6'272 inches, which is the proper major diameter of the part of connecting-rod entering the cross- tail in this engine. TO FIND THE PEOPEE MINOR DIAMETER OF THE PAET OF CON- NECTING-ROD ENTERING THE CEOS8TAIL. KULE. Multiply the diameter of the cylinder in inches ~by '09. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '09 = 3 '60 inches, which is the proper minor diameter of the part of the connecting-rod in the cross- tail in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x '09 = 5'76 inches, which is the proper minor diameter of the part of the connecting rod in the cross- tail in this engine. TO FIND THE PROPER BREADTH OF BUTT OF THE CONNEOTtNG- EOD. KULE. Multiply the diameter of the cylinder in inches ~by -156. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x -156 = 6*240 inches, which is the proper breadth of the butt of connecting-rod in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x '156 = 9'984 inches, which is the proper breadth of the butt of the connecting-rod in this engine. TO FIND THE PROPER THICKNESS OF THE BUTT OF THE OON- NECTING-ROD. RULE. Divide the diameter of the cylinder in inches "by 8. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches -4- 8 = 5-00 inches, which is the proper thick- ness of the butt of the connecting-rod in this engine. DIMENSIONS OF THE CONNECTING-ROD. 265 Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches -j- 8 = 8'00 inches, which is the proper thick- ness of the butt of the connecting-rod in this engine. TO FIND THE PEOPEE MEAN THICKNESS OF THE STEAP OF CON- NECTING-BOD AT THE CUTTEE. RULE. Multiply the diameter of the cylinder in inches ~by '043. .Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '043 = 1'720 inches, which is the proper mean thickness of the connecting-rod strap at the cutter in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x '043 = 2'752 inches, which is the proper mean thickness of the connecting-rod strap at the cutter in this engine. TO FIND THE PEOPEE MEAN THICKNESS OF THE CONNECTING-BOD STEAP ABOVE CUTTER. RULE. Multiply the diameter of the cylinder in inches ly '032. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x -032 = 1-280 inches, which is the proper mean thickness of the connecting-rod strap above the cutter in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x "032 = 2'048 inches, which is the proper mean thickness of the connecting-rod strap above the cutter in this engine. TO FIND THE PBOPEE DISTANCE OF CTJTTEE FEOM END OF STRAP OF CONNECTING-BOD. RULE. Multiply the diameter of the cylinder in inches Tyy '048. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '048 = 1*920 inches, which is the proper distance of the cutter from the end of the strap of the connect- ing-rod in this engine. Example 2. Let 64 inches be the diameter of the cylinder. 12 266 PPOPORTIONS OF STEAM-ENGINES. Then 64 inches x '048 = 3-072 inches, which is the proper distance of the cutter from the end of the strap of the connect- ing-rod in this engine. TO FIND THE PEOPEB DEPTH OF THE GIBS AND CUTTER PASSING THROUGH THE CEOSSTAIL. RULE. Multiply the diameter of the cylinder in inches l>y '105. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '105 = 4'20 inches, which is the proper depth of the gibs and cutter passing through the crosstail in this engine. Example, 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x '105 = 6'720 inches, which is the proper depth of the gibs and cutter passing through the crosstail in this engine. The thickness of the cutters passing through the crosstail will be the same as the thickness of those passing through the cross- head. TO FIND THE PBOPEB DEPTH OF THE GIBS AND OUTTEE THEOUGH THE BUTT OF THE CONNEOTING-EOD. RULE. Multiply the diameter of the cylinder in inches ~by '11. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '11 = 4'40 inches, which is the proper depth of the gibs and cutter passing through the butt of the connecting-rod in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x '11 = 7'04 inches, which is the proper depth of the gibs and cutter passing through the butt of the con- necting-rod in this engine. TO FIND THE THICKNESS OF THE GIB8 AND CUTTER THEOUGH THE BUTT OF THE CONNECTING-BOD. RULE. Multiply the diameter of the cylinder in inches by -029. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x "029 = 1*160 inches, which is the proper DIMENSIONS OF THE SIDE LEVER. 267 thickness of the gibs and cutter passing through the butt of the connecting-rod in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x '029 = 1*856 inches, which is the proper thickness of the gibs and cutter passing through the butt of the connecting-rod in this engine. CROSSTAIL. The crosstail is made in all respects the same as the cross- head, except that the end journals, where the crosstail butts fit on, are made so that the length is only equal to the diameter of the journal, instead of being about ! times, as in the crosshead. But as the crosstail butts do not work on these journals or gudg- eons, but are keyed fast upon them, the shorter length is pre- ferable. The butts of the crosstail have the eyes nearly twice the diameter of the journals, or more accurately 1*8 times, and the butts for the reception of the straps for connecting to the side lever are made of the same dimensions as the butts of the side rods. SIDE LEVER AND STUDS OR CENTRES. The side lever is usually made of cast-iron. But it should be in all cases encircled by a strong wrought-iron hoop, thinned at the edge so that it may be riveted or bolted all along to a flange cast on the beam for this purpose, and forming an extension of the usual edge bead. The proportions given in the rules are those of the common cast-iron side levers as usually constructed. But the strength will be increased three times if wrought-iron be substituted for cast in the top and bottom flanges or edge beads. TO FIND THE PBOPEE DEPTH OF THE SIDE LEVER AOB088 THE CENTRE. ETTLE. Multiply the length of the side lever in feet ly '7423 ; extract the cube root of the product and reserve the root for a multiplier. Then square the diameter of the cylinder in ZOO PROPORTIONS OF STEAM-ENGINES. inches ; extract the cule root of the square. The product of the last result, and the reserved multiplier, is the depth of the side lever in inches across the centre. Example 1. What is the proper depth across the centre of the side lever in the case of an engine with a diameter of cylin- der of 64 inches and length of side lever of 20 feet ? Here 20 = length of side lever in feet 7433 length of multiplier 14-848 and ^ 14.846 = 2'458 nearly Also 64 = diameter of cylinder 64 4096 and ^ 4096 = 16 Hence depth at centre = 16 x 2-458 = 39'30 inches, or be- tween 39 and 39 inches. Example 2. What is the proper depth across the centre of the side lever in the case of an engine with a diameter of cylin- der of 40 inches, and length of side lever of 15 feet. Here 15 length of side lever 7423 11-1345 and ty H'1345 = 2'232 Also 40 = diameter of cylinder 40 1600 and -^ 1600 = 11-69 which x 2'232 = 26'09, or a little over 26 inches. The depth, of the side lever at the ends is determined hy the depth of the eyes round the end studs. The thickness of the side lever is usually made about -g^th of its length, and the breadth of the edge bead is usually made about -^ of the length of the lever between the end centres. TO FIND THE PROPEE DIAMETER OF THE MAIN CENTRE JOURNAL. RULE. Multiply the diameter of the cylinder in inches by -183. DIMENSIONS OF THE SIDE LEVEE. 269 Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '183 = 7'32 inches, which is the proper diameter of the main centre journal in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x -183 = 11-712 inches, which is the proper diameter of the main centre journal in this engine. TO FIND THE LENGTH OF THE MAIN CENTRE JOURNAL. RULE. Multiply ike diameter of the cylinder in inches ~by '275. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x -275 = ll'OO inches, which is the proper length of the main centre journal in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x '275 = 17'60 inches, which is the proper length of the niain centre journal in this engine. TO FIND THE DIAMETEE OF THE END STUD9 OF THE SIDE LETEE. RULE. Multiply the diameter of the cylinder in inches /by '07. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '07 = 2 - 80 inches, which is the proper diameter of the end studs of the side lever in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x '07 = 4'48 inches, which is the proper diameter of the end studs of the side lever in this engine. TO FIND THE PEOPEE LENGTH OF THE END STUDS OF THE SIDE LEVEB. RULE. Multiply the diameter of the cylinder in inches ~by "076. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '076 = 8 '04 inches, which is the proper length of the end of studs of the side lever in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x -076 = 4'86 inches, which is the proper length of the end studs of the side lever in this engine. TO FIND THE PEOPEE DIAMETEE OF THE AIB-PUMP STUDS IN SIDE LEVEE. RULE. Multiply the diameter of the cylinder in inches T>y *045. 270 PROPORTIONS OF STEAM-ENGINES. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x "045 = 1'80 inches, which is the proper diameter of the stud in the side lever for working the air-pump of this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x '045 = 2*88 inches, which is the proper diameter of the air-pump studs in the side levers of this engine. TO FIND THE PROPER LENGTH OF THE AIR-PUMP STUDS SET IN THE SIDE LEVEE. EULE. Multiply the diameter of the cylinder in inches Tiy '049. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '049 = 1'96 inches, which is the proper length of the air-pump studs in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x '049 3'136 inches, which is the proper length of the air-pump studs in this engine. TO FIND THE PROPEE DEPTH OF THE EYE ROUND THE END STUDS OF SIDE LEVEE. KULE. Multiply the diameter of the cylinder in inches l>y '074. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x -OY4 = 2'96 inches, which is the proper depth of the eye round the end studs of the side lever in this en- gine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x '074 = 4'736 inches, which is the proper depth of the eye round the end studs of the side lever in this en- gine. It is clear that the diameter of the end stud added to twice the depth of the metal running round it will be equal to the depth of the side lever at the end Hence 2'1 + twice 2'96 = 8'72 will be the depth in inches of the side lever at the ends in the engine with the 40-inch cylin- der, and 4'48 + twice 4'736 = 13-95 will be the depth in inches of the side lever at the ends in the engine with the 64-inch cyl- inder. DIMENSIONS OF THE CRANK. 271 TO FIND THE THICKNESS OF THE EYE BOUND THE END STUDS OF SIDE LEVEE. KULE. Multiply the diameter of the cylinder in inches ly *052. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '052 = 2'08 inches, which is the proper thickness of eye of side lever round the end studs in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x '052 = 3'328 inches, which is the proper thickness of eye of side lever round the end studs in this engine. THE CRANK. TO FIND THE PEOPEE DIAMETEB OF THE CBANK-PLN JOUENAL8. KULE. Multiply the diameter of the cylinder in inches ly '142. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '142 = 5'680 inches, which is tho proper diameter of the crank-pin journal in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x '142 = 9-080 inches, which is the proper diameter of the crank-pin journal in this engine. TO FIND THE PEOPEE LENGTH OF THE CEANK-PIN JOUBNAI.. KULE. Multiply the diameter of the cylinder in inches ly '16. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '16 = 6*40 inches, which is the proper length of the crank-phi journal in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x *16 = 10'24 inches, which is the proper length of the crank-pin journal in this engine. TO FIND THE PEOPEE THICKNESS OF THE SMALL EYE OF CBANK:. RULE. Multiply the diameter of the cylinder in inches ~by '063. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '063 == 2-52 inches, which is the proper thickness of the small eye of the crank in this engine. 272 TKOPORTIONS OF STEAM-ENGINES. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x '063 = 4'032 inches, which is the proper thickness of the small eye of the crank in this engine. TO FIND THE PROPER BREADTH OF THE SMALL EYE OF THE CRANK. RULE. Multiply the diameter of the cylinder in inches by '187. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x "187 = 7'48 inches, which is the proper breadth of the small eye of the crank in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x *187 = 1T968 inches, which is the proper breadth of the small eye of the crank in this engine. TO FIND THE PEOPER THICKNESS OF THE WEB OF CRANK, SUP- POSING IT TO BE CONTINUED TO CENTRE OF CRANK PIN. EIJLE. Multiply the diameter of the cylinder in inches l>y '11. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '11 = 4'40 inches, which is the proper thickness of the web of crank in this engine, supposing it to be continued so far as centre of pin. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x '11 = 7'04 inches, which is the proper thickness of the web of the crank in this engine, supposing that the thickness were to be continued to the centre of the crank- pin and to be there measured. TO FIND THE PROPER THICKNESS OF THE WEB OF THE CRANK, SUPPOSING THE THICKNESS TO BE CONTINUED TO THE CENTRE OF THE PADDLE-SHAFT. KULE. Multiply the square of the length of the crank in inches ty 1'561, and then multiply the square of the diameter of cylinder in inches l>y *1235. Multiply the square root of the sum of these products ly the square of the diameter of the cyl- inder in inches ; divide this quotient by 360 ; finally extract the cube root of the quotient. The result is the thickness of the web of the crank at paddle shaft centre in inches. Example 1. What is the proper thickness of tho web of crank DIMENSIONS OF THE CRANK. 273 at the centre of the paddle-shaft, supposing the thickness to be continued thither and there measured, in the case of an engine with a diameter of cylinder of 64 inches and stroke of 8 feet. 48 =: length of craiik in inches 48 2304 1-561 constant multiplier 3596-5 505-8 product of 64 2 and -1236 4102-3 64 = diameter of cylinder 64 4096 1235 505-8 and V4102-3 = 64 05 nearly 4096 = square of diameter 360)262348-5 728-75 And ^'728 = 9 nearly, which is the proper thickness in inches of the crank of this engine measured at the centre of the paddle shaft. Example 2. What is the proper thickness of the weh of crank at paddle-shaft centre in the case of an engine with a cylinder 40 inches in diameter and stroke of 6 feet? 30 = length of crank in inches 30 900 = square of length of crank 1-561 = constant multiplier 1404-9 12* 274 PROPORTIONS OF STEAM-ENGINES. 40 =; diameter of cylinder in inches 40 1600 = square of diameter of cylinder 235 = constant multiplier 197-9 1404-9 1602-8 and -j/1602'8 = 40-03 1600 177-9 And ty 177'9 i= 5-62, which is the proper thickness in inches of the web of the crank, supposing the web to be continued to the centre of the paddle shaft. TO FIND THE PROPER BREADTH OF THE WEB OF THE CRANK AT PIN-CENTRE, SUPPOSING IT TO BE CONTINUED TO THE CENTRE OF THE CRANK-PIN. EULE. Multiply the diameter of the cylinder T)y '16. The prod- uct is the proper "breadth of the web of the crank, supposing the web to be continued to the plane of the centre of the crank- pin. Example 1. Let the diameter of the cylinder be 40 inches. Then 40 inches x '16 = 6'4, which is the proper breadth in inches of the web of the crank in the plane of the centre of the crank-pin in this engine. Example 2. Let the diameter of the cylinder be 64 inches. Then 64 inches x *16 = 10'24 inches, which is the proper breadth of the web of the crank at the crank-pin end in this engine. TO FIND THE PROPER BREADTH OF THE CRANK AT PADDLE- CENTRE. KULE. Multiply the square of the length of crank in inches by 1*561, and then multiply the square of the diameter of cyl- inder in inches ly '1235 ; multiply the square root of the sum of these products by the square of the diameter of the cyl- DIMENSIONS OF THE CRANK. 275 inder in inches; divide the product ly 45. Finally, extract the cute root of the quotient. Example 1. What is the proper breadth of the crank at paddle-centre in the case of an engine with a diameter of cylin- der of 64 inches and stroke of 8 feet ? 48 length of crank in inches 48 2304 1-561 constant multiplier 3596-5 505-8 4102-3 64 diameter of cylinder 64 4086 1235 constant multiplier 505-8 and i/4102-S = 64-05 nearly 4096 45)262348-5 5829-97 and ^5829 = 18 nearly, which is the proper breadth in inches of the web of the crank at the shaft-centre in this engine. Example 1 },. What is the proper breadth of crank at paddle- centre in the case of an engine with a diameter of cylinder of 40 inches and stroke of 5 feet ? 30 = length of crank in inches 30 900= square of length of crank 1-561 1404-9 276 PROPORTIONS OF STEAM-ENGINES. 40 = diameter of cylinder 40 1600 = square of diameter of cylinder 1235 197-6 1404-9 1602-5 and ^1602-5 = 40'03 1600 45)64048 1466-7 'and -^1466-7 = 11-24 nearly. The purpose of taking the breadth and thickness of the web of the crank at the shaft and pin-centres is to obtain fixed points for measurement. For, although the web of the crank does not extend either to the centre of the shaft or to the centre of the pin, it can easily be drawn in as if extending to those points, and the breadth and thickness being then laid down at those points the proper amount of taper in the web of the crank will be obtained. TO FIND THE PEOPEE THICKNESS OF THE LAEGE EYE OF THE OEANK. KITLE. Multiply the square of the length of the crank in inches ~by 1-561, then multiply the square of the diameter of the cyl- inder in inches ~by '1235 ; multiply the sum of these products J>y the square of the cylinder in inches ; divide the quotient "by the length of the crank in inches; afterwards divide the product l>y 1828'28. Finally, extract the cube root of the quotient. The result is the proper thicTcness in inches of the large eye of crank. Example 1. What is the proper thickness of large eye of the the crank in the case of an engine with a diameter of cylinder of 64 inches and stroke of 8 feet? DIMENSIONS OF THE CRANK 277 48 = length of crank in inches 48 2304 = square of length of crank 1-561 = constant multiplier 3596-5 505-8 = product of 64* and 1235 4102-3 64 = diameter of cylinder 64 4096 1235 = constant multiplier 505- 4102-3 4096 = square of diameter 48)16803020-8 1828-28)350062-94 191-4Y and JJ/191-47 = 5'7Y nearly, whicn is the proper thickness of the large eye of the crank in inches. Example 2. "What is the proper thickness of the large eye of crank in the case of an engine with a diameter of cylinder of 40 inches and with a stroke of 5 feet ? 80 = length of crank in inches 30 900 = square of length of crank 1-561 constant multiplier 1404-9 278 PROPORTIONS OF STEAM-ENGINES. 40 = diameter of cylinder 40 1600 1235 constant multiplier 197-6 1404-9 add 1602-5 1600 = square of diameter 1828-28)2564000 30)1402-41 46-74 and %/ 46-74 = 3-60, which is the proper thickness in inches of the large eye of the crank in this engine. TO FIND THE PROPER DIAMETER OF THE PADDLE-SHAFT JOURNAL. KULE. Multiply the square of the diameter of the cylinder in inches by the length of crank in inches ; extract the cube root of the quotient. Finally, multiply the result ~by '242. The final product is the diameter of the paddle-shaft journal in inches. Example 1. What is the proper diameter of the paddle- shaft journal in the case of an engine with a diameter of cylin- der of 64 inches and stroke of 8 feet ? 64 = diameter of cylinder in inches 64 4096 square of diameter of cylinder 48 = length of crank in inches 196608 and ^196608 = 58-148, and 58'148 x -242 = 14-07 inches. Example 2. What is the proper diameter of the paddle- shaft journal in the case of an engine with a diameter of cylinder of 40 inches and a stroke of 5 feet? DIMENSIONS OF THE PADDLE-SHAFT. 279 40 = diameter of cylinder 1600 = square of diameter of cylinder 30 = length of crank in inches 48000 and ^48000 = 36'30 and 36-30 x -242 = 8'79 inches. TO FETD THE PEOPEE LENGTH OF THE PADDLE-SHAFT JOUBNAL. EULE. Multiply the square of the diameter of the cylinder in inches by the length of the crank in inches; extract the cube root of quotient; multiply the result by '303. The product is the length of the paddle-shaft journal in inches. Example 1. "What is the proper length of the paddle-shaft journal in the case of an engine with a diameter of cylinder 64 inches and stroke 8 feet? 64 = diameter of cylinder 64 4096 = square of diameter of cylinder 48 = length of crank in inches 196608 and ^196608 = 58'148 Length of journal = 68-148 x -303 = 17'60 inches. Example 2. "What is the proper length of the paddle-shaft journal in the case of an engine with a diameter of cylinder of 40 feet and stroke of 5 feet ? 40 40 1600 30 48000 and V 48000=36-30 x -303=10-99. It will be seen from these examples that the length of the paddle-shaft journals is 1J times the diameter. The paddle- shafts, cranks, and all the other working parts of marine 280 PROPORTIONS OF STEAM-ENGINES. engines are made of wrought-iron, except the side levers, which are of cast-iron, and the air-pump rod, which is of copper or brass. THE AIR-PUMP. TO FIND THE PEOPEE DIAMETEE OF AIE-PUMP. RULE. Multiply the diameter of the cylinder in inches ~by '6. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '6 = 24'0 inches, which is the proper diam eter of the air-pump in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x '6 = 38 - 4 inches, which is the proper diam- eter of the air-pump in this engine. AIR-PUMP ROD. TO FIND THE PEOPEE DIAMETEE IN INCHES OF THE AIE-PUMP EOD WHEN OF COPPER. RULE. Multiply the diameter of the cylinder in inches T)y '067. Example 1. Let the diameter of the cylinder be 40 inches. Then 40 x '067 = 2'68 inches, which is the proper diameter of the air-pump rod when of copper in this engine. Example 2. Let the diameter of the cylinder be 64 inches. Then 64 x '067 = 4'28 inches, which is the proper diameter of the air-pump rod when of copper in this engine. TO FIND THE PEOPEB DEPTH OF GLB8 AND CUTTEE THROUGH THE AIE-PUMP CEOSSHEAD IN INCHES. RULE. Multiply the diameter of the cylinder in inches ly -063. Example 1. Let the diameter of the cylinder be 40 inches. Then 40 x '063 = 2'52 inches, which is the proper depth of gibs and cutter through the air-pump crosshead in this engine. Example 2. Let the diameter of the cylinder be 64 inches. Then 64 x '063 = 4-03 inches, which is the proper depth of gibs and cutter through the air-pump crosshead in this engine. DIMENSIONS OF PARTS OP THE AIR-PUMP. 281 TO FIND THE PEOPEB THICKNESS OF GIBS AND CtJTTEE THEOUGH AIE-PUMP OEOSSHEAD IN INCHES. KULE. Multiply the diameter of the cylinder in inches ly '013. Example 1. Let the diameter of the cylinder be 40 inches. Then 40 x '013 = '52 inches, which is the proper thickness of gibs and cutter through the air-pump crosshead in this engine. Example 2. Let the diameter of the cylinder be 64 inches. Then 64 x '013 = '83 inches, which is the proper thickness of gibs and cutter through the air-pump crosshead in this engine. TO FIND THE PEOPEE DEPTH IN INCHES OF THE CUTTEE THEOUGH THE AIB-PUMP BUCKET. KITLE. Multiply the diameter of the cylinder in inches ~by '051. Example, 1. Let the diameter of the cylinder be 40 inches. Then 40 x '061 = 2'04 inches, which is the proper depth of the cutter through the air-pump bucket in this engine. Example 2. Let the diameter of the cylinder be 64 inches. Then 64 x '051 = 3'26 inches, which is the proper depth of the cutter through the air-pump bucket in this engine. TO FIND THE PEOPEE THICKNESS OF THE CUTTEE THBOUGH THE AIR-PUMP BUCKET IN INCHES. KULE. Multiply the diameter of the cylinder in inches ly '021. Example 1. Let the diameter of the cylinder be 40 inches. Then 40 x "021 = *84 inches, which is the proper thickness of the cutter through the air-pump bucket in this engine. Example 2. Let the diameter of the cylinder be 64 inches. Then 64 x *021 = 1'34 inches, which is the proper thickness of the cutter through the air-pump bucket in this engine. The cutter through the air-pump bucket should be always made of brass or copper, but the gibs and cutter through the air- pump crosshead will be of iron. The air-pump bucket should always be of brass, and it is advisable to insert the rod into the crosshead and also into the bucket with a good deal of taper, so as to facilitate its detachment should the bucket require to be taken out. It is usual to form the part of the rod projecting 282 PROPORTIONS OF STEAM-ENGINES. through the crosshead into a screw, and to screw a nut upon it. This also is a common practice at the top of the piston rod and at the bottom of the connecting-rod. AIR-PUMP CROSSHEAD. TO FIND THE PROPER DEPTH OF THE EYE OF THE AIR-PUMP OROSSHEAD. EULE. Multiply the diameter of the cylinder in inches ~by '171. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '171 = 6'84 inches, which is the proper depth of eye of air-pump crosshead in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x *171 = 10'944 inches which is the proper depth of the eye of air-pump crosshead in this engine. TO FIND THE PROPER DEPTH OF THE AIR-PUMP OROSSHEAD AT THE MIDDLE OF THE WEB. EULE. Multiply the diameter of the cylinder in inches ty *161. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '161 = 6'44 inches, which is the proper depth at the middle of the web of the air-pump crosshead hi this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x "161 = 10'30 inches, which is the proper depth at the middle of the web of the air-pump crosshead in this engine. TO FIND THE PROPER DEPTH OF THE WEB OF THE AIE-PUMP OROSSHEAD AT JOURNALS. KULE. Mutiply the diameter of the cylinder in inches ~by "061. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '061 = 2*44 inches, which is the proper depth of the web of the air-pump crosshead at the journals in this engine. Example 2. Let 64 inches be the diameter of the cylinder. DIMENSIONS OF AIR-PUMP CROSSHEAD. 283 Then 64 inches x '061 = 3'90 inches, which is the proper depth of the weh of the air-pump crosshead at the journals in this engine. TO FIND THE PROPER THICKNESS OF THE EYE OF THE AIR- PUMP OEOSSHEAD. EULE. Multiply the diameter of the cylinder in inches by '025. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '025 = I'OO inches, which is the proper thickness of the eye of the air-pump crosshead in this engine. Example 2. Let 64 inches he the diameter of the cylinder. Then 64 inches x - 025 = 1'600 inches, which is the proper thickness of the eye of the air-pump crosshead in this engine. TO FIND THE PROPER THICKNESS OF THE WEB OF THE AIR-PUMP CEOSSHEAD AT THE MIDDLE. KULE. Multiply the diameter of the cylinder in inches ~by -043. Example 1. Lot 40 inches be the diameter of the cylinder. Then 40 inches x '043 = 1*72 inches, which is the proper thickness of the web of the air-pump crosshead at the middle in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x '043 = 2-75 inches, which is the proper thickness of the web of the air-pump crosshead at the middle in this engine. TO FIND THE PROPER THICKNESS OF THE WEB OF THE AIR-PUMP CROSSHEAD AT THE JOURNALS. EULE. Multiply the diameter of the cylinder in inches "by '037. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '087 = 1'48 inches, which is the proper thickness of the web of the air-pump crosshead at the journals in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x '087 = 2 - 36 inches, which is the proper thickness of the web of the air-pump crosshead at the journals in this engine. 284 PKOPOBTIONS OF STEAM-ENGINES. TO FIND THE PEOPEE DIAMETEE OF THE JOUBNALS OF THE AIE-PUMP CKOSSHEAD. RULE. Multiply tlio diameter of the cylinder in inches ~by '051. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '051 = 2'04 inches, which is the proper diameter of the journals of the air-pump crosshead in this en- gine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x '051 = 3'26 inches, which is the proper diameter of the journals of the air-pump crosshead in this en- gine. TO FIND THE PEOPEE LENGTH OF THE JOUENALS OF THE AIB-PUMP OEOSSHEAD. RULE. Multiply the diameter of the cylinder in inches l>y "058. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '058 = 2'32 inches, which is the proper length of the end journals for the air-pump crosshead in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x '058 = 3'TL inches, which is the proper length of the end journals for the air-pump crosshead in this engine. AIR-PUMP SIDE RODS. TO FIND THE PEOPEE DIAMETEE OF AIE-PUMP SIDE BOD AT THE ENDS. RULE. Multiply the diameter of the cylinder in inches 5y *039. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '039 = 1/56 inches, which is the proper diameter of air-pump side rod at ends in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x '039 = 2*49 inches, which is the proper diameter of air-pump side rod at ends in this engine. TO FIND THE BEEADTH OF BUTT FOE AIE-PUMP SIDE BODS. RULE. Multiply the diameter of the cylinder in inches ~by '046. DIMENSIONS OF AIR-PUMP SIDE RODS. 285 Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '046 = l'S4 inches, which is the proper breadth of butt for air-pump side rod in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x '046 = 2'94 inches, which is the proper breadth of butt of air-pump side rod in this engine. TO FIND THE PBOPEB THICKNESS OF BUTT FOE AIE-PUMP SIDE EOD. RULE. Multiply the diameter of the cylinder in inches ~by '037. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x -037 = 1 '48 inches, which is the proper thickness of butt for air-pump side rod in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x '037 = 2'36 inches, which is the proper thickness of butt for air-pump side rod in this engine. TO FIND THE MEAN THICKNESS OF STRAP AT CUTTEB OF AIB-PUMP BIDE EOD. RULE. Multiply the diameter of the cylinder in inches T>y '019. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '019 = - 76 inches, which is the proper mean thickness of the strap at cutter of air-pump side rod in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x '019 = 1*21 inches, which is the proper mean thickness of the strap at cutter of air-pump side rod in this engine. TO FIND THE PEOPEE MEAN THICKNESS OF THE 8TEAP BELOW OUT- TEE OF AIB-PUMP SIDE EOD. RULE. Multiply the diameter of the cylinder in inches ly '014. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x -014 = -56 inches, which is the proper mean thickness of the strap below cutter in the air-pump side rod of this engine. Example 2. Let 64 inches be the diameter of the cylinder. 286 PROPORTIONS OF STEAM-ENGINES. Then 64 inches x '014 = '89 inches, which is the proper mean thickness of strap below cutter in the air-pump side rod of this engine. TO FIND THE PROPEB DEPTH OP THE GIBS AST) CUTTER FOR AIR- PUMP SIDE ROD. RULE. Multiply the diameter of the cylinder in inches ~by '048. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches x '048 = 1'92 inches, which is the proper depth of gibs and cutter for air-pump side rod in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches x *048 = 3'OV inches, which is the proper depth of gibs and cutter for the air-pump side rod in this engine. TO FIND THE PROPER THICKNESS OF THE GIBS AND CUTTER FOR THE AIR-PUMP SIDE ROD. RULE. Divide the diameter of the cylinder in inches ~by 100. Example 1. Let 40 inches be the diameter of the cylinder. Then 40 inches -5- 100 = '40 inches, which is the proper thickness of the gibs and cutter for the air-pump side rod in this engine. Example 2. Let 64 inches be the diameter of the cylinder. Then 64 inches -5- 100 = '64 inches, which is the proper thickness of the gibs and cutter of the air-pump side rod in this engine. It will be satisfactory to compare the dimensions of the parts of engines with the actual dimensions obtaining in some engines of good proportions which have for some time been in success- ful operation ; and I select for the purpose of this comparison the side-lever engines constructed by Messrs. Caird & Co., for the West India Mail steamers ' Clyde,' ' Tweed,' ' Tay,' and ' Tevoit.' The dimensions of the main parts given by the rules, and the actual dimensions, are exhibited in the following table, touching which it is sufficient to remark that where there is any appreciable divergence between the two, the dimensions given by the rules appear to be the preferable ones : RULES TESTED BY PRACTICAL EXAMPLES. 287 COMPABISON OF DIMENSIONS GIVEN BY THE FOREGOING BtTLES WITH THE ACTUAL DIMENSIONS OF THE MAIN PAETS OF THE SIDE LEVEE ENGINES OF THE STEAMERS ' CLYDE,' ' TWEED,' 'TA.Y,' AND 'TEVIOT,' OF 450 HOESES POWER, CONSTEUCTED BY MESSES. CAIED & CO. Diameter of Paddle-Shaft Journal. Dimensions by Rulea. Actual Dimensions. Diameter of paddle-shaft journal 15-15 15-25 Exterior diameter of large eye of crank 27-84 27-875 10-49 9-5 Exterior diameter of small eye of crank tLength of small eye of crank 19-81T7 13-875 20-625 13-25 Thickness of web of crank at paddle centre at crank pin centre Breadth of crank at crank pin centre 9'8 8-14 12-21 10-5 9-75 15-0 Diameter of piston rod 7-4 7-75 7-03 7-6 9-98 9-25 side rod at ends 4-77 5-0 u " at middle 6-6 6-375 " ey o of crosshead (outside) 13-5 14-5 Depth of eye of crosshead (outside) 21-138 21-25 Diameter of crosshead journal 6-349 6-375 Thickness of web of crosshead at centre 5-8 5-5 tDepth of web of crosshead at centre 19-85 19-5 4-514 4-875 Depth of web of crosshead at journal 7-511 9-75 tDiameter of main centre journal < 0-0367 x P*xD= 13-579 I 11-5 The rules give generally smaller numbers than Messrs. Caird's practice. The difference is greatest in 'Breadth of crank at crank-pin centre,' and in ' Exterior diameter of eye of crosshead,' and 'Depth of web of crosshead of journals.' In five cases above, marked thus t, the rules give greater strength than the example selected of Messrs. Caird's engine, especially in ' Diameter of main centre,' where Messrs. Caird's proportions are quite too small. I have already explained that from any one drawing, all sizes of engines of that particular form may be constructed by merely altering the scale ; and all the dimensions of ships and engines, and, in fact, every quantity whatever which increases or dimin- ishes in a given ratio, or according to a uniform law, may be ex- pressed graphically by a curve, which will have its correspond- ing equation, though sometimes that equation will be too com plicated to be numerically expressible. Mr. "Watt, in his early 288 PROPORTIONS OP STEAM-ENGINES. practice, laid down most of the dimensions of his engines to curves, and, indeed, was in the habit of using that mode of in- vestigation and expression in all his researches. A table of the dimensions of the parts of engines may easily be laid down in the form of a curve ; and the benefit of that practice is, that if we have a certain number of points in the curve, we can easily find all the intermediate ones by merely measuring with a pair of compasses and a scale of equal parts. Thus, for example, we may lay down the table of the diameter of crank-shaft, given in page 294, to a curve as follows : First draw a straight horizon- tal line, which divide into equal parts by any convenient scale, beginning, as in the table, with 20, and ending with 100. If now we erect vertical lines or ordinates at every division of the hori- zontal line, and if, with any given length of stroke, say 2 feet, we know the diameter of shaft proper for some of the diameters of cylinder say for a 20-inch cylinder, 4*08 inches ; for a 24-inch cylinder, 4'66 inches; for a 40-inch cylinder, 6'55 inches; and for an 80-inch cylinder, 10*29 inches we can easily determine the diameters of shaft proper for all the intermediate diameters of cylinders, by marking off with the same scale, or any other, the vertical heights corresponding to all the diameters we know ; and a curve traced through these points will intersect all the other ordinates, and give the diameters proper for the whole series. By thus setting down the known quantities in order to deduce the unknown, we shall at the same time see whether the quantities we set down follow a regular law of increase or not ; for if they do not, instead of all the points marked off falling into a regular curve, some of them will be above the curve and some of them beneath it, thus showing that the quantities given do not form portions of a homogeneous system. If the quantity increases in arithmetical progression, the curve will become a straight angular line. Thus in the case of the diameter of the piston rod, as the increase follows the same law as the increase of the diameter of the cylinder, the law of increase will be ex- pressed by a right-angled triangle, the diameters of the cylinder being represented by the divisions on the base, and the diameters of piston rod by the corresponding vertical ordinates. If to the KULES TESTED BY PKACTICAL EXAMPLES. 289 curve of diameter of crank shaft for each diameter of cylinder with any given length of stroke, we add below the base another curve pointing downwards, representing the increase of the di- ameter of the shaft due to every increase of the length of the stroke, the diameter of the cylinder remaining the same, the total height of the conjoint ordinates will show the diameter of the shaft for each successive diameter of cylinder and length of stroke. One of the curves will be convex and the other con- cave, and the convexity of the one will be equal to the concavity of the other, so that the ordinates will be the same as those of a triangle. Hence, if we double the diameter of the cylinder, and also double the length of the stroke, we shall double the diam- eter of the shaft ; if we treble the diameter of the cylinder, and also treble the length of the stroke, we shall treble the diameter of the shaft, and so on in all other proportions. By referring to the table in page 294, we shall see that these relations are there preserved. Thus a 20-inch cylinder and a 2-feet stroke has a shaft of 4'08 inches in diameter ; a 40-inch cylinder and a 4-feet stroke, a shaft of 8'16 inches diameter; a 60-inch cylinder and a 6-feet stroke, a shaft 12'25 inches diameter, and so on. If this were not so, an engine drawn on any one scale would not be ap- plicable to any other of a different size ; whereas we know that any one drawing will do for all sizes of engines by merely chang- ing the scale. It is very convenient in making drawings of engines to adopt some uniform size for the drawing-boards and drawings, and to adhere to them on all occasions. The best arranged drawing- office I have met with is that of Boulton and Watt, which was originally settled in its present form by Mr. Watt himself, who brought the same good sense and habits of methodical arrange- ment to this problem that he did to every other. The basis of Boulton and Watt's sizes of drawings is the dimensions of a sheet of double elephant drawing paper; and all their drawings are either of that size, of half that size, or of a quarter that size, leaving a proper width for margin. The drawing-boards are all made with a frame fitting around them, so that it is not neces- sary to glue the paper round the edges; but the damped sheet 13 290 PROPORTIONS OF STEAM-ENGINES. TABLE OF THE DIMENSIONS OF THE PBINCIPAL MARINE ENGINES OF NAMES OF PABTS. NOMINAL PH' rf o PM rf' o (H w o e* PL,' H S Diameter of Cylinder in. 20 2 12 If n 2i 1* 4 5 8* 2 li 2* 9 2 24 12 6 6 4 5 11 4* 2* 8} 4 It 29} 83 21 66 i} 2 13 14 5 1 in. 24 21 15 If 1} 2} ? 6 4} 2 P 5* 11 2* 80 15 7} 7* 4} 6* in. 27 1? 2 3 lf 2 ? 5 2} I 6} 11 2* 80 15 71 8 6 6} 1J 5J 8* 4i H 5J- 6} 8T} 42i 25} 76 10 2 2i 15* 19 6} fl in. 29} 8 17} I 8J ? P 1 I 2} 83 16} 8 9 5} 2 6} 4 5 If 5} 6f 89} 45 26 80 11 2} 2} 17 21 jf Piston rod Air-pump Air-pump rod. Steam-pipe Waste-water pipe Beam gudgeon Crank-pin Main shaft Paddle-wheels, in feet Weight-shaft bearings Stroke of Piston Air-pump bucket Feed-pump plunger Cylinder cross/lead Depth of boss Depth of middle Air-pump crosshead Depth of boss 5 ? 1| 4* 5J 84} 39 23 72 81 U 2 14 18 * Thickness Columns Diameter at top Diameter at bottom Centre to centre of fort valve passage Depth Width Seam Breadth at middle Breadth at ends Thickness EXAMPLES OF APPROVED DIMENSIONS. 291 PABTS OF MESSES. MATTDSLAY, SONS, AND FIELD 8 DIFFERENT POWEES. POTTEB OP ENGLSE. ft Pi ft 3 W 2 Pi ft S Pi H PL,' ft a Pi w o TH 1-1 Pi S in. in. in. in. in. in. in. in. in. in. 82 86} 40 43 46 48 50 52} 55} 57 3} 3* 4 4} 4* 41 4J 5 5} 5* 21 23 24 26 28 80 81} 84 2} 2* 21 25 3 3} 8* 81 4 4} 2 2} 2* 21 3 3} 8} 8J 8* 8* 4 4} 4* 5 4 6 6* 7 7* 2} 2* 2* 21 8 8} 8} 3* 8* 4 6* 7 71 8* 9} 10 10* 11 11* 12 8 9 9* 10 11* 12} 18 13* 14 5* 6 6* 7 7* 8 8* 9 9} 9} 3} 8* 4 4} 4* ft 5 5} 6} 5* 2 2} 21 2* 3 4 8} . 4 4* 5 5* 6* 6* 7 7f 7} 8 7 8* 9} 10 10} 10* 11* 12 12} 18 18 15 17 17 19 19 21 21 23 2f 2} 21 21 3 8} 3} 8* 81 8} 86 36 42 48 52 56 60 63 66 72 18 13 21 24 26 28 80 31* 88 86 9 9 10* 12 13 14 15 16 16* 18 9* 10* 12 13 14 14* 15 16 17 17} 6 61 7* 8 81 9 9* 10 11 12 7} 8* 9* 10} 11* Hi 12* 18 13* 14 2} 2* 21 8 3} 8* 3* 3} 4f 61 8 9 10 10* 10} 11 H* 12 12* % 8 7* 5} 61 8 6 8} 6* 8* 9* S 7* 9* 1* H 2 2 s 2} 2} 2* 21 21 6 7 8 8} 91 9} 9* 10 10} 10* 61 ft 9 10* lot 11 11* "I 12 42} 47* 53 551 60} 63 67 68} 70 72 48 54 60 63 69 69 72 78 80 88 27 80 84 84 40 40 42 44 45 46 84 83 96 100 108 108 112 126 128 180 11* 18 15 18* 18* 19 19 20 20 21 21 3 8 4 4 4} 4* 41 4} 3} 81 4 4* 5 5} 5* 6 6* 7 18 20 24 26 28 28 29 81 81 32 28 25 28 29 83 34 85 86 88 89 8 81 10 10* 12 12} 12} 12 15 15* 1* H H 2 2} 2f 2* 2* 2* 21 292 PROPORTIONS OF STEAM-ENGINES. TABLE OF THE DIMENSIONS OF THE PRINCIPAL ENGINES OF DIF- .5 .4 3 i s | i| id 9 w ^s 1 a 3 . 'S, * ^ a g I *& $ S-o 13 1 It II ! (0 ^- ' M I* i H 1-i a fl 3% F "3 I 0, 5 s H if *& I* 1 & i p* s 5 Q 5 fig. a* 5 ft. in. ft. in. in. in. in. in. in. 10 20 12 2 9 4 7 2 li 1 4 15 24 15 2 6 11 4} 8* 2* 1* li 5 20 27 17 2 6 11 6f 10 3 If 1* 5} 30 31} IS* 3 13 6f 10} 3} If If 6* 40 3G* 20 3 13 7* 11* 3* 2} 11 7 50 39} 22 3 6 15 8} 12} 3} 2* 2 7* 60 43 24 4 17 9 13 4 2} 21 8* 70 46 25* 4 3 17 8| 13} 41 3 2* 9} 80 48 27 4 6 19 10 14* 4* 81 2} 10 90 50 23 4 9 19 10* 15} 4} 8* 8 10* 100 52* 5 21 11 16 5 8f Si 11 110 55 5 21 11* 16} 5} 4 81 11* 120 57* 5 6 23 12* 17* 5* 41 8* 12 is laid upon the board, which it somewhat overlaps, and the frame then comes down and turns over the edges of the paper upon the sides of the board, and the frame being then fixed so that its face is flush with the paper, the paper by being thus bound all round the edges is properly stretched when dry. In Mr. Watt's time the drawings were made with copying ink, and an impression was taken from them by passing them through a roller press, so as to retain the original in the office, while a duplicate of it was sent out with the work ; and the copying press was invented by Mr. Watt for this purpose. The whole of the drawings pertaining to each particular engine are placed in a small paper portfolio by themselves ; and these small port- folios are numbered and arranged in drawers, with a catalogue to tell the particular engine delineated in the drawings of each portfolio. In this way I have found that the drawings illustra- tive of any engine, though it may have been made in the last EXAMPLES OF APPROVED DIMENSIONS. 293 PARTS OF MESSES. SEAWARD AND CQ.'s MARIXE FEREXX POWERS. I i , O to M j II 1 M || . If c s J if 1 1 s J l-s |4 o 5, to-2 S a 11 ii 1^ "S s i i bo JS3, *&= II J 3 J a i i *f |. S J in. in. ft. in. in. in. in. in. in. in. In. 26 5 1* 6 2 8 1 2* 2 2* 2f 28 6 H 7 21 4 1* 2* 21 3 31 30 7 2 8 2* 5 It 3 2* 8* 3J 85 8 8i 8 8 2f 5* 2 3* 2* 4 41 38 9 2* 10 3 6 21 3* 2} 4* 41 40 9* 2t 10 6 81 e* 2* 3i 2f 5 M 44 10 8 11 6 8* 7 3f 4 2J 5* 6 60 10 31 12 6 8} 7* 3 41 8 6 6* 66 11 3* 13 4 8 31 4* 3* 6* 7 68 12* 3i 13 6 41 8* 3* 4t 81 7 7* 62 13 4 16 4* 9 3| 5 3* 7| 7* 66 18* 41 16 4t 9* 8t 51 3t 7t 8i 70 14 4* 17 6 5 9J 35 5* 3t 8 8* century, could be produced to me in a few minutes ; and the system is altogether more perfect and more convenient than any other with which I am acquainted. The portfolios are not large, which would make them inconvenient, but are of such size that a double elephant sheet has to be folded to go into one of them ; but most of the drawings are on small sheets of paper, which is a much more convenient practice than that of drawing the de- tails upon large sheets. It will be interesting to compare with the results given in the foregoing rules the actual sizes of some side lever engines of approved construction. Accordingly I have recapitulated, in the tables introduced above, the principal dimensions of the marine engines of Messrs. Maudslay and Messrs. Seaward. These tables ore so clear, that they do not require further explanation, and the same remark is applicable to the tables which follow. 294 PROPORTIONS OF STEAM-EXGINES. DIAMETER OF WEOTJGHT-IEON CKAXK-SHAFT JOUKNAL. ~ Ji 3-r 3 jo LENGTH OF STROKE IN FEET. 2 2.L 3 3^ 4 ^ 5 ^ 6 7 8 9 20 4-03 ! 4-39 4-67 4-91 5-14 5-34 5-53 5-72 5-83 6-19 6-43 6-73 21 4-23 4-54 4-82 5-05 5-30 5-50 5-71 5-89 6-07 6-39 6-63 6-95 22 4-37 4-63 4-9G 5-20 5-46 5-66 5-83 6-07 6-25 6-53 6-83 7-1G 23 4-52 4-81 5-11 5-35 5-G2 5-83 6-05 6-25 6-43 6-77 7-08 7-37 24 4-66 4-95 5-25 5-51 5-73 5-99 6-22 6'43 6-62 6-96 7-23 7-58 25 4-31 5-09 5-40 5-60 5-94 6-16 6-40 6-60 6-80 7-15 7-49 7-78 26 4-95 5-22 5-54 5-81 6-10 6-33 6-57 6-78 6-98 7-35 7-69 7-99 27 510 5-36 5-69 5-9G 6-26 6-49 6-74 6-96 7-16 7-54 7-89 8-20 28 5-24 5-49 5-83 6-11 6-42 6-66 6-91 7-14 7-35 7-73 8-09 8-41 29 5-33 5-53 5-98 6-26 6-53 6-32 7-08 7-31 7-53 7-92 8-29 8-62 80 5-42 5-67 6-12 6-42 6-74 6-99 7-26 7-49 7-72 8-12 8-49 8-83 31 5-47 5-88 6-25 6-56 6-34 7-14 7-41 7-65 7-88 8-29 8-67 9-02 32 5-59 6-00 6-88 6-69 6-93 7-29 7-56 7-81 8-04 8-46 8-85 9-21 33 5-71 6-12 6-51 6-83 7-12 7-43 7-71 7-97 8-20 8-63 9-03 9-39 34 5-83 6-24 6-64 6-96 7-26 7-53 7-37 8-13 S-37 8-30 9-21 9-58 35 5-95 6-37 6-77 7-10 7-41 7-73 8-02 8-28 8-53 3-97 9-39 9-76 36 6-07 6-49 6-90 7-23 7-55 7-83 8-17 8-44 8-69 9-14 9-57 9-95 87 6-19 6-61 7-03 7-37 7-69 8-03 8-32 8-60 8-85 9-31 9-75 10-14 38 6-31 6-73 7-16 7-50 7-33 8-17 8-47 8'76 9-02 9-48 9-93 10-33 39 6-43 6-85 7-29 7-64 7-97 8-32 8-63 8-91 9-18 9-65 10-11 10-51 40 6-55 6-93 7-42 7-78 8-16 8-47 8-79 9-07 9-35 9-38 10-29 10-70 41 6-57 7-19 7-54 7-90 8-29 8-60 8-93 9-21 9-50 9-99 10-45 10-87 42 6-67 7-20 7-66 8-03 8-42 8-74 9-07 9-36 9-65 10-15 10-62 11-04 43 6-77 7-31 7-78 8-15 8-55 8-87 9-21 9-50 9-80 10-31 10-78 11-21 44 6'87 7-42 7-90 8-28 8-63 9-01 9-35 9'65 9-95 10-47 10-95 11-38 45 6-97 7-54 8-02 8-40 8-81 9-14 9-50 9-79 10-10 10-63 11-11 11-55 46 7-06 7-65 8-15 8-53 8-94 9-28 9-64 9-94 10-25 10-78 11-28 11-72 48 7-26 7-88 8-40 8-78 9-20 9-55 9-92 10-23 10-54 11-09 11-61 12-06 50 7-48 8-10 8-61 9-02 9-46 9-82 10-20 10-51 10-84 11-42 11-94 12-41 52 7-70 8-31 8-83 9-26 9-71 10-08 10-47 10-79 11-12 11-71 12-25 12-78 54 7-90 8-52 9-05 9-50 9-96 10-34 10-74 11-06 11-40 12-00 12-56 13-05 56 8-09 8-73 9-27 9-73 10-21 10-60 11-01 11-33 11-69 12-29 12-87 13-37 58 8-29 8-94 9-50 9-97 10-45 10-86 11-28 11-61 11-97 12-58 18-18 18-69 60 8-49 9-15 9-72 10-20 10-70 11-12 11-55 11-89 12-25 12-89 18-48 14-02 62 8-67 9-35 9-93 10-40 1089 11-34 11-79 12-14 12-51 18-17 13-77 14-82 64 8-86 9-55 10-14 10-60 11-08 11-56 12-03 12-39 12-78 13-45 1406 14-62 66 9-04 9-74 10-35 10-79 11-27 11-78 12-28 12-64 13-04 18-73 14-85 14-98 68 9-22 9-94 1056 10-99 11-45 11-99 12-52 12-89 18-30 14-01 14-64 15-28 70 9-41 10-14 10-77 11-19 11-64 12-20 12-77 18-16 13-57 14-29 14-94 15-54 72 9-59 LO-33 10-97 11-41 11-90 12-44 13-01 13-40 13-82 14-55 15-22 15-83 74 9-76 10-52 11-17 11-62 12-16 12-67 13-25 18-64 14-07 14-82 15-50 1612 76 9-93 10-71 11-36 11-83 12-43 12-91 13-49 13-89 14-32 15-09 15-78 16-41 78 10-11 10-89 11-56 12-05 12-69 13-15 13-78 14-13 14-57 15-85 16-06 16-70 80 10-29 11-08 11-76 12-27 12-96 13-38 13-96 14-38 14-84 15-62 16-33 16-98 82 10-46 11-26 11-96 12-49 13-17 13-61 14-19 14-62 15-08 15-88 16-59 17-26 84 10-63 11-44 12-15 12-71 18-38 13-84 14-42 14-85 15-32 16-18 16-86 17-54 86 10-80 11-61 12-35 12-92 13-59 14-07 14-65 15-09 15.56 16-38 17-13 17-82 83 10-97 11-79 12-54 13-14 13-80 14-80 14-88 15-82 115-80 16-63 17-40 18-10 90 11-13 11-99 12-74 ;13-34 14-00 14-54 15-10 15-56 16-05 16-89 17-66 18-37 92 11-29 12-17 12-92 13-47 114-21 14-75 15-32 15-79 16-28 17-14 17-92 18-64 94 96 11-45 11-61 12-34 12-51 13-10 13-60 13-29 13-73 14-42 14-63 14-96 15-18 15-54 16-02 15-76 ! 16-25 16-51 16-74 17-38 17-63 18-18 18-44 18-91 19-18 98 11-77 12-68 13-47 13-86 14-84 15-39 15-98 16-48 16-97 17-87 18-70 19-45 100 11-93 12-85 13-66 14-01 15-04 15-61 16-20 16-71 17-22 18-12 18-95 19-71 TABLES OF NOMINAL POWERS OF ENGINE. 295 LENGTH OF WEOUGHT-IEOX CBANK-SHAFT JOUTCNAL. SSli i'2- a S=M B LENGTH OP STROKE IN FEET. 2 2i 3 3* 4 4* 5 5* 6 7 8 9 20 5-10 5-49 5-84 j 6-15 6-43 6-69 6-93 7-16 7-;:'; 7-75 8-11 S-43 21 5-37 5-69 6-03 6-31 6-62 6-87 7-14 7-36 7-59 7-99 8-86 8-69 22 5-63 5-85 6-21 6-50; 6-82 7-03 7-35 7-59 7-82 8-23 8-61 8-96 23 5-68 6-02 6-39 6-69 7-02 7-29 7-57 7-S1 8-05 8-47 8-86 9-23 24 5-84 6-19 6-57 6-88 7-22 7-50 7-78 8-04 8-28 8-71 9-11 9-50 26 5-99 6-36 6-75 7-07; 7-42 7-70 8-00 8-26 851 8-95 9-36 9-76 26 6-15 6-53 6-931 7-26 7-62 7-91 821 8-48 8-74 9-19 9-61 10-02 27 6-30 6-TO 7-11 7-45 7-82 8-11 8-43 8-70 8-97 9-43 9-86 10-28 23 6-46 6-S7 7-29 7-64 8-02 8-32 8-64 8-92 9-20 9-67 10-11 10-54 29 6-61 7-04 7-47 7-83 8-22 8-53 8-S6 9-14 9-43 9-91 10-36 10-SO 80 6-77 7-21 7-65 8-02 8-42 8-74 9-07 9-36 965 10-15 10-61 11-04 81 6-92 7-37 7-82 8-19 8-60 8-93 9-27 9-56 9-86 10-37 10-84 11-28 82 7-06 7-52 7-98 8.36 8-73 9-11 9-47 9-76 10-06 10-58 11-07 11-52 83 7-20 7-67 8-14 S-53 8-96 9-30 9-56 9-96 10-27 10-79 11-30 11-76 84 7-84 7-82 6-30 8-70 9-14 9 -48 9-75 10-15 10-47 11-00 11-53 11-99 35 7-48 7-97 8-46 8-87 9-31 9-67 9-94 10-35 10-68 11-22 11-76 12-22 86 7-62 8-12 8-63 9-04 9-49 9-85 1013 10-55 10-SS 11-43 11-98 12-45 87 7-76 8-27 8-79 9-21 9-67 10-04 1082 10-74 11-09 11-64 12-20 12-68 83 7-90 842 895 9-38 9-85 10-22 10-51 10-94 11-29 11-86 12-42 12-91 89 8-05 8-57 9-11 9-55 10-03 10-41 10-SO 11-14 11-49 12-07 12-64 13-14 40 8-19 8-72 9-27 9-72 10-20 10-53 10-99, 11-34 11-69 12-29 12-86 13-37 41 8-32 8-86 9-42 9-88 10-37 10-75 11-17 11-53 11-88 12-49 13-07 13-59 42 8-44 9-00 9-57 10-03 10-54 10-92 11-34 11-71 1206 12-69 13-28 13-81 48 8-56 9-18 9-72 10-18 10-70 11-09 11-52 11-89 1225 12'S8 13-49 14-03 44 8-68 9-27 9-87 10-37 10-86 11-26 11-69 12-07 12-43 13-03 13-70 14-25 45 8-80 9-40 10-02 10-49 11-02 H-43 11-87 12-25 12-62 18-28 13-91 14-46 46 8-92 9-54 10-17 10-64, 11-18 11-59 12-04 12-43 12-80 13-47 14-12 14-67 43 9-16 9-81 10-47 10-94 11-50 11-93 12-39 12-79 13-17 13-87 14-52 15-09 50 9-40 10-08 10-76 11-26 11-82 12-27 12-75 13-15 13-55 14-27 1492 15-51 5-2 9-65 10-34 1104 11-56 12-13 12-59 18-09 13-50 13-90 14-64 15-32 15-92 54 9-89 10-60 11-82 11-86 12-44 12-91 13-43 13-85 14-25 15-02 15-71 16-32 56 10-18 10-86 11-60 12-16 12-75 13-23 13-77 14-19 14-60 15-39 16-09 16-72 53 10-87 11-13 11-88 12-46 13-06 13-55 14-11 14-53 14-95 15-76 16-47 17-12 60 10-61 11-37 12-15 12-75 13-87 18-87 14-44 14-86 15-81 16-11 16-85 17-52 62 10-84 11-62 12-41 13-00 13-61 14-15 14-76 15-18 15-64 16-47 17-23 17-90 64 11-07 11-86 12-67 18-25 13-85, 14-42 15-06 15-50 15-97 16-83 17-59 1828 66 11-80 12-11 12-98 18-50 14-09 14-70 15-86 15-82 16-80 17-18 1795 18-66 63 11-58 12-35 18-19 18-75 14-83 14-97 15-66 16-14 16-63 17-52 18-31 19-04 70 11-76 12-60i 18-46 14-01 14-55 15-25 15-96 16-46 16-96 17-86 18-67 19-42 72 11-93 12-84 13-71 14-29 14-88 15-56i 16-26 16-77 17-28 18-20 19-03 19-78 74 12-20 18-07 13-96 14-58 1521 15-871 16-56 17-08 17-60 18-54 1939 20-14 76 12-42 18-80 14-20 14-87 15-54 16-18 16-86 17-89 17-92 18-87 19-73 20-50 78 12-64 18-581 14-44 15-15 15-87 1649 17-16 17-70 18-24 19-19 20-07 20-86 80 12-86 13-76 14-70! 15-44 16-20 16-81 ! 17-45 18-00 18-55 19-52 20-41 21-22 82 18-07 13-99 14-95 15-70 16-46 17-08' 17-73 18-29 18-86 19-84 20-75 21-58 84 18-28 14-22 15-19 16-95 16-72 17-85 18-03 18-58 1916 20-16 21-09 21-94 86 18-49 14-45 15-48 16-20 16-98 17-62 1881 13-87 19-46 20-48 21-43 22-28 88 18-70 14-68, 15-67 16-45 17-24 17-89 18-59 19.16 19-75 20-80 21-75 22-62 90 38-91 14-91 15-92 16-76 17-50 18-15 18-87 19-47 20-06 21-11 22-07 22-96 92 14-11' 16-14 16-15 16-95 17-76 18-42 19-14 19-75 20-36 2142 22-40 28-80 94 14-81 15-86 16-88 17-19 18-02 18-69) 19-40 20-01 20-65 21-73 22-78 23-64 96 14-51 15-59 16-61 1743 18-28 18-96 19-67 20-38 20-94 22-03 2805 28-98 98 14-71 15-82 16-84 17-67 18-54 19-28 19'95 20-57 21-28 2288 28-37 24-32 100 14-91 16-09 17-07 17-92 18-80 19-52 20-25 20-83 21-52 22-65 28-69 24-64 i 296 PROPORTIONS OF STEAM-ENGINES. BREADTH OF WEB OF CRANK, SUPPOSING IT TO BE CONTINUED TO PADDLE-SHAFT CENTRE. "S.S iJ-Sj LENGTH OF STROKE IN FEET. ~6" 2 2i 3 3^ 4 4^ 5 B 6 7 8 9 20 5-28 5-45 5-62 6-04 6-46 6-72 6-98 7-19 7-40 7-7S 8-12 8-44 21 5-51 5-68 5-S5 6-27 6-68 6-94 7-21 7-42 7-63 8-04 8-39 8-71 22 5-74 5-91 6-03 6-49 6-90 7-16 7-44 7-65 7-86 8-30 8-65 8-98 23 5-96 6-14 6-31 6-72 7-12 7-38 7-67 7-88 8-09 8-55 8-90 9-25 24 6-18 6-36 6-54 6-94 7-33 7-60 7-89 8-11 8-32 8-SO 9-16 9-52 25 6'40 6-58 6-77 7-16 7-54 7-S2 8-11 8-34 8-55 9-04 9-41 9-79 20 6'62 6-80 7-00 7-38 7-75 8-04 8-33 8-57 8-78 9-28 9-67 10-06 27 6-84 7-03 7-23 7-60 7-96 8-26 8-55 8-80 9-01 9-51 9-92 10-33 28 7'06 7-25 7-46 7-82 8-17 8-48 8-77 9-03 9-24 9-74 10-18 10-59 29 7-28 7-48 7-09 8-04 8-38 8-70 8-99 9-26 9-47 9-97 10-43 10-85 30 7'50 7-71 7-92 8-26 8-00 8-92 9-24 9-49 9-74 10-22 10-68 11-10 31 7-65 7-88 8-11 8-46 8-80 9-13 9-44 9-72 9-96 10-44 10-92 11-34 82 7-80 8-U5 8-30 8-66 9-00 9-34 9-64 9-94 10-28 10-67 11-16 11-58 88 7'95 8-22 8-49 8-85 9-20 9-54 9-S4 10-15 10-50 10-89 11-39 11-82 84 8-10 8-39 8-68 9-04 9-40 9-74 1004 10-35 10-71 11-12 11-62 12-06 35 8'25 8-56 8-87 9-23 9-59 9-94 10-24 10-55 10-92 11-34 11-85 12-29 36 8-40 8-78 9-06 9-42 9-79 10-14 10-44 10-75 11-13 11-56 12-OS 12-53 3T 8-55 8-90 9-25 9-61 9-99 10-34 10-64 10-95 11-34 11-78 12-31 12-76 83 8-70 9-07 9-44 9-80 10-18 10-54 10-84 11-15 11-55 12-00 12-54 12-99 39 8-85 9-24 9-62 9-89 10-37 10-73 11-04 11-35 11-76 12-23 12-77 13-23 40 9-00 9-40 9-80 10-18 10-56 10-90 11-24 11-56 11-88 12-46 12-98 13-43 41 9-19 9-59 9-99 10-37 10-75 11-08 11-41 11-73 12-05 12-67 13-22 13-71 42 9-38 9-78 10-17 10-56 10-94 11-25 11-57 11-89 12-22 12-87 13-46 13-94 43 9-57 9-97 10-36 10-75 11-18 11-42 11-72 12-05 12-39 13-08 18-70 14-16 44 9-76 10-15 10-55 10-94 11-32 11-59 11-87 12-21 12-56 13-28 18-98 14-38 45 9-94 10-33 10-74 11-13 11-51 11-76 12-02 12-37 12-78 13-49 14-17 14-60 46 10-12 10-51 10-93 11-82 11-70 11-98 12-17 12-53 12-90 13-69 14-40 14-82 48 10-48 10-S7 11-30 11-70 12-08 12-27 12-47 12-85 13-24 14-10 14-87 15-26 50 10-S6 11-27 11-68 12-07 12-46 12-61 12-76 13-17 13-58 14-54 15-34 15-70 52 11-20 11-63 12-06 12-44 12-84 13-03 13-22 13-63 14-04 14-94 15-71 16-14 54 11-54 11-98 12-42 12-81 13-22 13-45 13-68 14-09 14-49 15-34 16-08 16-56 56 11-88 12-32 12-78 13-18 18-60 13-87 14-15 14-55 14-95 15-74 16-45 16-98 58 12-20 12-66 18-15 18-56 13-98 14-29 14-62 15-01 15-41 16-14 16-82 17-41 60 12-52 18-01 13-51 13-94 14-87 14-73 15-10 15-47 15-84 16-54 17-19 17-82 62 12-96 13-41 18-89 14-82 14-73 15-10 15-48 15-85 1 0-2-2 16-92 17-59 18-23 64 13-38 18-81 14-26 14-69 15-09 15-57 15-86 16-23 16-60 17-31 17-99 18-61 66 13-80 14-21 14-63 15-05 15-45 15-94 162-3 16-61 16-98 17-70 18-38 19-05 68 14-22 14-61 15-00 15-41 15-81 16-31 16-60 16-98 17-36 18-09 18-78 19-45 70 14-64 15-01 15-88 15-77 16-17 16-57 16-97 17-35 17-74 18-48 19-18 19-85 72 15-02 15-89 15-75 16-08 16-56 16-03 17-30 17-71 18-12 18-86 19-58 20-26 74 15-40 15-77 16-12 16-46 16-95 17-29 17-63 18-06 18-49 19-24 19-97 20-67 76 15-78 16-15 16-49 16-84 17-34 17-65 17-97 18-41 18-86 19-62 20-35 121-07 78 16-16 16-58 16-87 17-22 17-78 18-01 18-30 18-76 J19-23 20-00 20-73 21-48 80 16-54 16-89 17-24 17-68 18-12 18-37 18-61 19-11 19-61 20-38 12111 21-88 82 16-94 17-20 17-60 18-04 18-47 18-75 19-03 19-51 19-99 20-76 21-51 22-27 84 17-32 17-65 17-98 18-40 18-82 19-18 19-45 19-90 20-36 21-14 21-89 22-65 86 17-71 18-03 18-36 18-76 19-17 19-51 19-85 20-29 '20.78 21-52 22-28 23-03 88 18-09 18-41 18-73 19-12 19-52 19-89 20-26 20-68 ,21-10 21-89 22-67 23-41 90 18-47 18-79 19-11 19-49 19-87 20-27 20-67 21-07 21-47 22-26 23-03 23-79 92 18-87 19-18 19-49 19-87 20-25 20-64 21-04 21-44 121-84 22-64 23-41 24-17 94 19-26 19-56 19-87 20-25 20-62 21-01 21-41 21 -SI 22-21 123-01 23-79 24-56 96 19-64 19-94 20-25 20-63 20-99 21-38 21-78 22-18 22-48 23'88 24-16 24-93 98 20-02 20-82 20-62 21-00 21-36 21-75 22-15 22-55 22-85 23-75 24-53 25-81 100 20-40 20-70 21-00 21-36 J21-73 22-12 22-52 22-92 28-32 J24-12 124.90 25-68 DIMENSIONS OF THE CKANK. 297 THICKNESS OF \VEB OF CRANK, SUPPOSING IT TO BE CONTINUED TO PADDLE-SHAFT CENTRE. fe"s j LENGTH OF STROKE IN FEET. HI S&" 3 2 2* 3 3 4 4i 5 5 6 7 8 9 20 2-64 2-72 2-81 3-02 3-2.3 8-36 8-49 3-59 8-70 8-89 4-06 4-22 21 2-75 2-84 2-93 3-14 3-34 8-47 8-61 8-71 3-82 4-02 4-19 4-36 22 2-86 2-96 ] 3-05 3-25 3'45 3-58 3-73 8-S8 8-94 4-14 4-32 4-50 23 2-97 8-08 i 8-17 8-36 8-56 8-69 3-85 8-95 4-06 4-26 445 4-64 24 8-08 8-19 8"29 8-47 3-67 8-80 8-97 4-07 4-18 4-38 4-58 4-77 25 8-19 3-80 8-41 3-58 8-78 8-91 4-09 4-19 4-30 4-50 4-71 4-90 26 3-30 8-41 8-52 8-69 8-S9 4-02 4-20 4-30 4-42 4-62 4-84 5-03 27 3-41 8-52 8-63 8-80 4-00 4'18 4-31 4'41 4-53 4-74 4-97 5-16 28 8-52 8-63 8-74 8-91 4-10 4-24 4-42 4-52 4-64 4-86 5-10 5-29 29 8-63 3-74 3-85 4-02 4-20 4-35 4-53 4-63 4-75 4-98 5-22 5-42 SO 8-75 3-85 8-96 ! 4-18 4-30 4-46 4-62 4-74 4-87 5-11 5-34 5-55 81 8-38 8-94 4-06 1 4-28 4-89 4-56 4-72 4-85 4-98 5-23 5-46 5-67 32 8-91 4-03 4-16 4-33 4-49 4-66 4-82 4-96 5-09 5-84 5-57 5-79 83 8-99 4-12 4-26 4-48 4-59 4-76 5-92 5-07 6-20 5-45 5-69 5-91 34 4-07 4-21 4-36 4-53 4-69 4-86 5-02 618 5-31 657 5-80 6-03 35 4-15 4-30 4-45 4-62 4-79 4-96 5-12 5-28 5-42 5-68 5-92 6-15 86 4-22 4-88 4-54 4-71 4'89 5-06 P-22 6'38 5-53 5-79 6-03 6-27 87 4-29 4-46 4-63 4-80 4-98 5-16 5-32 5-48 5-64 5-90 6-15 6-39 83 4-86 4-54 4-72 4-89 5-08 5-26 6-42 6-58 5-74 6-01 6-26 6-51 39 4-43 4-62 4-81 4-.S 6-18 5-36 5-52 5-68 6-84 6-12 6-88 6-62 40 4-50 4-70 4-90 5-09 5-28 5-45 5-62 6'78 5-94 6-23 6-49 6-74 41 4-60 4-80 6-00 6-19 5'88 5-54 5-70 5-86 6-08 6-34 6-61 6-86 42 4-70 4-90 5-10 5-29 5'48 5-63 5-78 5-94 6-11 6-45 6-78 6-97 43 4-80 6-00 5-20 5-89 5'58 5-72 6-86 6-02 6-20 6-56 6-85 7-08 44 4-89 5-09 5-30 5-49 6-68 5-81 5-94 610 6-28 6-67 6-97 7-19 45 4-98 5.18 5-39 6-58 5'78 5-90 6-02 6-18 6-37 6-77 7-09 7-30 46 6-07 6-27 5-48 5-67 5-87 5-98 6-10 6'26 6-45 6-87 7-21 7-41 46 6-25 5-45 6-66 5-85 6-05 6-14 6-26 6'42 6-62 7-07 7-45 7-63 50 5-43 6-68 6-84 6-08 6-23 6-30 6-38 6-58 6-79 7-27 7-67 7-85 52 6-61 5-81 6-03 6-28 6-48 6'52 6-62 6-81 7-08 7-47 7-87 8-07 54 6-78 5-98 6"21 6-42 6-68 6-74 6-86 7-04 7-27 7-67 8-05 8-29 56 5-94 6-14 6-39 6-60 6'82 6-96 7-10 7-27 7-50 7-87 8-23 8-51 58 6-10 6-30 6-57 6-78 7-00 7-16 7-88 7'50 7-72 8-07 8-41 8-71 60 6-26 6-50 6-75 6-96 7-18 7-86 7-55 7*78 7-92 8-27 8-59 8-91 62 6-43 6-71 6-95 7-16 7-86 7-56 7-75 7-98 8-11 8-47 8-79 9-11 64 6-70 6-91 7-15 7-84 7-54 7-74 7-94 8-18 8-30 8-65 8-97 9-31 66 6-92 7-10 7-33 7-52 7-72 7-92 8-12 8-31 8-49 8-84 9-17 9-52 68 7-12 7-80 751 7-70 7-90 8-10 8-80 8-49 8-68 9-04 9-87 9-72 70 7-32 7-50 7-69 7-88 8-08 8-28 8-48 8-67 8-87 9-24 9-59 9-92 72 7-51 7-69 7-87 8-07 8-28 8-46 8-66 8-85 9-07 9-48 9-88 10-12 74 7-70 7-83 8-06 8-26 8-47 8'64 8-82 9-08 9-26 9-71 10-06 10-82 76 7-89 8-07 8-25 8-46 8-67 8-82 8-98 8-21 9-44 9-98 10-28 10-53 78 8-08 8-26 8-44 8-65 8-87 9-00 9-14 9-89 9-62 10-16 10-51 10-78 80 8-27 8-44 8-62 8-84 9-06 8-18 9-30 9-55 9-80 10-87 10-75 10-94 82 8-46 Ml 8-81 9-02 9-24 9-87 9-41 9-75 9-99 10-58 10-91 11-13 84 8-65 8-82 9-00 9-20 9-42 8-56 9-72 9-94 10-18 ; 10-69 11-07 11-31 86 8-85 9-01 8-18 8-88 9-60 8-74 9-92 10-18 10.86 10'84 11.28 11-50 88 9-04 9-20 9-36 8-56 9-75 8-88 10-18 10-38 10-54 10-98 11-37 11-69 90 9-28 '.>"?.> 9-55 9-74 9-98 10-18 110-88 i 10-58 10-78 11-18 11-51 11-89 92 8-48 '.>:>-, 9-74 8-98 10-11 10-82 10-51 10-71 10-92 11-32 : ll-69 12-OS 94 8-68 9-78 9-93 10-12 10-80 10-50 J10-70 10-90 11-11 11-50 11-89 12-27 96 8-82 9-97 10-12 10-80 10-49 10-69 10-89 11-09 11-29 ; 11-68 12-08 12-46 98 10-01 S 10-16 10-81 10-49 10-68 10-88 11-08 11-27 11-48 111-87 12-27 12-64 100 10-20 10-35 10-50 10-68 10-86 11-06 11-26 11-46 11-66 12-06 12-45 12-84 13* 298 PBOPOETIONS OF STEAM-ENGINES. THICKNESS OF LARGE EYE OF CRANK. s-a * f "i-s e-.g 5 LENGTH OF STROKE IN FEET. 2 H 3 i 4 4* 5 Bi 6 7 8 9 20 1-71 1-80 1-89 1-97 2-05 2-12 219 2-25 2-32 2-44 2-55 2-64 21 1-77 1-87 1-95 2-04 2-13 2-20 2-27 2-32 2-40 2-52 2-64 2-75 22 1-83 1-93 2-01 2-11 2-20 2-28 2-85 2-39 2-48 2-60 2-72 2-86 23 1-89 1-99 2-07 2-18 2-28 2-36 2-43 2-46 2-56 2-68 2-80 2-97 24 1-95 2-06 2-14 2-25 2-35 2-44 2-51 2-54 2-64 2-76 2-88 3-08 25 2-01 2-12 2-21 2-32 2-43 2-52 2-59 2-62 272 2-84 2-96 3-18 26 2-07 2-19 2-28 2-39 2-50 2-59 266 2-70 2-80 2-92 3-04 3-28 27 2-18 2-25 2-35 2-46 2-58 2-66 2-73 2-78 2-87 2-99 812 3-38 28 2-19 2-31 2-42 2-53 2-65 2-78 2-80 2-86 2-94 8-06 8-20 3-43 29 2-25 2-37 2-49 2-60 2-73 2-80 2-87 2-94 8-01 813 8-28 3-58 80 2-30 2-48 2-56 2-68 2-80 2-87 2-94 8-01 8-08 818 3-36 3-68 31 2-36 2-50 2-63 2-74 2-87 2-94 8-00 3-07 8-15 8-26 3-43 3-73 82 2-42 2-56 2-69 2-80 2-94 8-01 3-06 813 8-22 8-84 8-50 3-78 83 2-49 2-62 2-75 2-86 3-00 8-08 312 8-20 8-29 8-41 8-57 8-83 84 2-55 2-69 2-81 2-92 3-06 815 313 8-27 3-36 8-48 3-64 3-88 85 2'61 2-75 2-87 2-98 3-12 8-22 8-25 8-34 8-48 8-55 8-71 4-93 86 2-67 2-81 2-93 8-04 8-18 8-29 832 3-41 3-50 8-62 3'78 4-98 87 2-74 2-87 2-99 8-10 8-24 8-85 889 8-48 8-57 8-69 8-85 4-03 88 2-81 2-94 3-05 3-16 8-30 8-41 846 3-55 8-64 3-77 8-91 4-07 39 2-87 8-00 8-11 3-22 8'36 8-47 8-53 3-62 3-71 3-84 8-97 411 40 2-93 8-05 8-17 3-29 3-42 3-51 8-60 3'69 8-78 8-90 4-03 415 41 8-08 8-13 8-24 8-37 8-49 8-58 8-67 3-75 3-85 3-98 411 4-24 42 3-13 8-22 3-31 3-45 8-56 3-65 8-74 8'81 3-92 4-06 419 4-82 43 8-23 8-80 8-39 3-53 8-63 3-72 3-81 8'87 8-99 414 4-27 4-40 44 8-83 8-39 8-47 8-60 8-69 3-79 8-88 8-94 4-06 4-22 4-85 4-48 45 8-43 347 8-55 367 8-75 8-86 8-95 4-01 413 4-30 4-43 4-56 46 8-53 8-56 8-68 8-74 8-81 3'93 4-02 4-08 4-19 4-88 4-51 4-64 48 8-73 3-73 8-79 8-88 3-93 4'06 415 4-22 4-32 4-52 4'66 4-80 50 3-93 3-92 3-95 4-02 4-09 4-18 4-27 4-36 4-46 4-64 4-81 4-96 52 4-10 4-05 4-09 4-18 4-24 4-31 4-41 4-48 4-60 4-78 4-95 5-10 54 4-28 4-19 4-24 4-33 4-38 4-45 4-55 4-61 4-74 4-92 5-09 5-24 56 4-45 4-83 4-40 4-47 4-52 4-59 4-69 4-75 4-88 5-06 5-23 5-38 58 4-62 4-46 4-56 4-61 4-66 4-73 4-83 4-89 5-00 5-19 5-37 552 60 4-79 4-60 4-72 4-76 4-80 4-87 4-95 5-03 512 5-21 5-49 5-66 62 4-98 4-80 4-88 4-90 4-96 5-02 5-09 517 5-26 5-45 5-63 5-80 64 5-16 5-00 5-04 5-03 510 516 5-23 5-31 5-40 5-59 5-77 5-94 66 5-84 5-20 5-20 6-15 5-24 5-30 5-37 5-45 5-54 5-73 5-90 6-08 68 5-52 5-40 5-85 5-27 588 5-44 5-51 5-59 5-68 5-87 6-04 6-22 70 5-70 5-60 5'49 5-40 5-52 5-58 5-64 5-72 5-80 5-98 616 6-35 72 5-90 5-78 5-67 5-56 5-68 5-74 5-80 5-86 5-94 612 6-80 6-49 74 6-09 6-96 5-84 5-72 584 589 5-95 6-00 6-08 6-26 6'44 6-63 76 6-27 6-14 6-00 5-88 6-98 6-03 6-09 614 6-22 6-40 6-58 6-75 78 6-46 6-32 6-16 6-02 6-12 617 6-23 6-28 636 6-53 6-71 6-87 80 6-66 6-49 6-82 6-16 6-26 6-31 6-87 6-43 6-60 6-66 6-84 7-02 82 6-86 6-69 6-50 6-35 6-42 6-46 6-51 6-57 6-64 6-80 6-98 716 84 7-06 6-88 6-68 6-54 6-58 6-61 6-65 6-71 6-78 6-94 712 7-30 86 7-27 7-06 6-84 6-78 6-74 6-76 6-79 6-85 6'90 7-08 7'2fi 7-44 88 7-47 7'24 7'00 6-92 6-90 6-91 6-93 6.99 7'02 7-21 7-39 7-57 90 7-67 7-42 718 7-11 7-05 7-06 7-08 711 714 7-34 7-51 7-69 92 7-88 7-62 7-37 7-29 7-21 7-22 7-28 7-27 7-80 7-48 7-65 7-83 94 8-09 7-81 7-55 7-47 7-87 7-88 7-88 7-43 7-45 7-62 7'79 7-97 96 8-31 7-99 7-78 7-65 7-58 7-54 7-58 7-59 7-60 7-76 7'93 811 98 8-52 8-17 7-91 7-81 7-69 7-70 7'68 7-78 7-76 790 8-06 8-24 100 8-72 8-41) 8-09 7-97 7-86 7-84 7-83 787 7-91 8-04 819 8-30 DIMENSIONS OF AIR-PUMP AND PISTON ROD. 299 1 1 DIMENSIONS OP THE SEVERAL PARTS OF PISTON ROD IN INCHES. !!< ->. o I i 5 PH . 3 1 43 S -M , 8 * . "SfU u || II id 3 1 ;jfl 11 a '$ ~ o 8 a **Js , ^3 3 ji a H a S&-J S'Z 51 5 3 31 5-1 1 igd laS" H : : a S 6 ' Jf- S- a !|| go g J3"g -5? s I J - J-S a 2 5 3 5 >3 S 5 S a * 0*" E" 1 3 20 12-0 2-0 4-0 1-90 1-80 2-80 2-30 2-11 42 1-70 70 1-34 21 12-6 2-1 4-2 1-99 1-89 2-94 2-41 2-21 44 1-78 73 1-40 22 13-2 2-2 4-4 2-09 1-98 3-08 2-53 2-32 46 1-87 77 1-47 23 8-8 2-3 4-6 2-18 2-07 3-22 2-64 2-42 48 1-95 80 1-53 24 14-4 2-4 4-8 2-28 2-16 8-36 2-76 2-53 50 2-04 84 1-60 25 15-0 2-5 5-0 2-37 2-25 8-50 2-87 2-63 52 212 87 1-67 26 15-6 2-6 5-2 2-47 2-34 8-64 299 2-74 54 2-21 90 1-73 27 16-2 2-7 5-4 2-56 2-43 8-78 3-11 2-84 57 2-29 94 1-80 28 16-8 2-8 5-6 2-66 2-52 8-92 8-22 2-95 59 2-38 97 1-87 29 17-4 2-9 5-8 2-75 2-61 4-06 M ;U 3-05 61 2-46 1-00 1-94 80 18-0 8-0 6-0 2-85 2-70 4-20 3-45 8-16 63 2-55 1-04 2-00 81 18-6 8-1 6-2 2-94 2-79 4-84 8-57 8-26 65 2-03 1-07 2-07 82 19-2 8-2 6-4 8-04 2-S8 443 8-68 8-87 67 2-72 1-10 2-14 83 19-8 8-3 6-6 8-18 2-97 4-62 8-80 8-47 69 2-80 1-14 2-21 84 20-4 8-4 6-8 8-23 8-06 4-76 8-91 8-57 71 2-89 1-19 2-27 85 21-0 8-5 7-0 8-32 8-15 4-90 4-02 8-67 73 2-97 1-22 2-33 86 21-6 8-6 7-2 8-42 8-24 6-04 414 8-78 75 8-06 1-26 2-40 87 22-2 8-7 7-4 8-51 8-33 518 425 8-88 78 3-14 1-29 2-47 88 22-8 8-8 7-6 8-61 8-42 5-32 4-36 8-99 80 3-23 1-33 2-54 89 23-4 8-9 7-8 8-70 8'51 5-46 4-48 4-09 82 8-31 1-86 2-60 40 24-0 4-0 8-0 8-80 8-60 5-60 4-59 4-20 84 8-40 1-40 2-67 41 24-6 4-1 8-2 8-89 8-69 5-74 4-70 4-80 86 8-48 1-43 2-74 42 25-2 4-2 8-4 8-99 8-78 6-88 4-82 441 89 8-57 1-47 2-81 48 25-8 4-3 8-6 4-08 8-87 6-02 4-98 4-51 91 8-65 1-50 2-87 44 26-4 4-4 8-8 4-18 8-96 6-16 5-05 4-62 93 8-74 1-54 2-98 45 27-0 4-5 9-0 4-27 4-05 6-80 5-17 4-72 95 8-82 1-57 8-00 46 27-6 4-6 9-2 4-37 414 6-44 5-28 4-83 97 8-91 1-61 8-07 48 28-8 4-8 9-6 4-56 4-82 6-72 5-51 5-04 1-02 4-08 1-68 8-20 50 80-0 5-0 10-0 4-75 4-50 7-00 5-74 6-25 1-07 4-25 1-75 8-33 52 81-2 6-2 104 4-94 4-68 7-28 5-97 6-46 1-11 4-42 1-82 8-47 64 82-4 5-4 10-8 5-13 4-86 7-56 6-21 667 1-15 4-69 1-89 8-60 56 88-6 5-6 11-2 6-32 5-04 7-84 6-44 5-88 1-19 4-77 1-96 8-74 58 84-8 5-8 11-6 5-51 5-22 8-12 6-67 6-09 1-23 4-94 2-03 8-88 60 86-0 6-0 12-0 5-70 5-40 8-40 6-90 6-30 1-27 5-11 2-10 401 62 87-2 6-2 12-4 6-69 5-58 8-68 7-18 6-51 1-81 5-28 2-17 4-14 64 88-4 64 12-8 6-08 576 8-96 7-36 6-72 1-85 6-45 2-24 4-27 66 89-6 6-6 18-2 6-27 594 9-24 7-59 6-98 1-89 5-62 2-81 4-40 68 40-8 6-8 18-6 6-46 612 9-52 7-82 7-14 1-40 5-79 288 4-53 70 42-0 7-0 14-0 6-65 6-80 9-80 8-05 7-85 1-47 5-96 2-44 4-67 72 43-2 7-2 14-4 6-84 6-48 10-08 8-28 7-56 1-51 6-13 2-51 4-80 74 44-4 7-4 14-8 7-03 666 10-86 8-51 7-77 1-55 6-80 2-58 4-93 76 45-6 7-6 15-2 7-22 6-84 |10 64 8-74 798 1-67 6-46 2-66 5-07 78 46-8 7-8 15-6 7-41 7-02 11092 8-97 8-19 1-70 6-63 2-73 5-20 80 48-0 8-0 16-0 7-eo 7-20 11-20 9-20 8-40 1-73 6-80 2-80 6-33 82 49-2 8-2 16-4 7-79 7-88 11-48 9-48 8-61 1-76 6-97 2-87 5-47 84 50-4 S-4 16-8 7-98 7-56 jll-76 9-66 8-82 1-79 7-14 2-94 5-60 6 61-6 8-6 17-2 8-18 7-74 12-04 9-89 9-03 1-82 7-81 8-01 5-73 88 62-8 8-8 17-6 8-87 7-92 12-82 10-12 9-24 1-85 7-48 8'08 5-87 90 54-0 9-0 180 8-66 8-10 12-60 10-84 9-45 1-89 7-66 8-15 6-00 92 55-2 9-2 18-4 8-75 8-28 12-88 10-57 9-66 1-92 7-88 8-22 6-14 94 56-4 9-4 18-8 8-94 846 18-16 10-80 9-87 195 8-00 8-29 6-27 96 57' 9-6 19-2 9-12 8-64 13-44 11-08 10-08 2-00 8-17 8-86 6-41 98 68- 9-8 19-6 9-82 8-82 18-72 11-26 10-29 2-05 8-84 8-43 6-54 100 60- 10-0 20O 9-50 9-00 14-00 11-49 10-50 2.11 8-51 8-50 6-66 300 PROPORTIONS OF STEAM-ENGINES. 1 CBANK PIN. CKANK. PIPES AXD PASSAGES. "3 j * -S 3-3 3-S "a * -2 a --s i gSj ||. a . g-SJ g.3 "3 " i j o S 3 I 1 Is o C-g- I 'f.5 '2 1 'i , ^.ej JL > "i .2 In! *>? J3~-*4 2*o C3 ^ r^ O PH -2-2"- * s> z "3 2 aj a * g.s s s g 1*3 is 2 '&. 8 ? '^ IS . .S i !". j- ST C6 at g 3 g|-S 3 3 .2*9 .20^ OfL{ SMO gw ^2 ? "S 'o P-i * ? pH -3 5T J.s s* ft P S M H H \ s ft < < p m 20 2-84 8-20 1-26 3-75 2-20 ' 3-20 6 2-69 17 8-15 1-79 5-00 21 2-98 3-36 1-33 3-93 2-31 3-36 10 3-32 26 3-49 1-84 5-23 22 3-12 3-52 1-40 4-12 2-42 8-52 15 3-94 85 8-84 1-89 5-45 23 3-24 3-68 1-46 4-31 2-53 3-68 20 4-56 44 4-18 1-94 5-66 24 3-38 8-S4 1-52 4-50 2-64 3-84 25 5-18 53 4-53 1-99 5-86 25 3-52 4-00 1-53 4-68 2-75 4-00 30 5'80 62 4-87 2-04 6-05 26 3-66 4-16 1-65 4-87 2-86 4-16 85 6-42 71 5-22 2-09 6-24 27 8-80 4-32 1-71 5-06 2-97 4-32 40 7-04 80 5-56 2-13 6-42 28 8-94 4-48 1-77 5-25 3-08 4-48 45 7-66 89 5-91 2-18 6-60 29 4-08 4-64 1-88 5-43 3-19 4-64 50 8-28 98 6-25 223 6-80 30 4-26 4-80 1-90 5-62 3-30 4-80 55 8-90 107 6-60 2-28 7-07 81 4-40 4-96 1-98 5-81 3-41 4-96 60 9-50 116 6-94 2-32 7-23 82 4-54 5-12 2-02 5-99 3-52 5-12 65 10-06 125 729 2-36 7-39 33 4-68 5-28 2-08 6-18 3-63 5-28 70 10-56 134 7-63 240 7-55 84 4-83 5-44 2-14 6-37 3-74 5-44 75 10-96 143 7-98 2-44 7-71 85 4-97 5-60 2-21 6-56 8-85 5-60 80 11-31 152 8-32 2-48 7'87 36 5-11 5-76 2-27 6-74 3-96 5-76 85 11-61 161 8-67 2-52 8-03 37 5-26 5-92 2-23 6-93 4-07 5-92 90 11-89 170 9-01 2-56 8-19 88 5-40 6-08 2-39 7-12 4-18 6-08 95 12-09 179 9-36 2-60 S'85 39 5-54 6-24 2-45 7-31 4-29 6-24 100 12-19 188 9-70 2-64 8'51 40 5-69 6-40 2-52 7-50 4-40 6-40 105 12-29 197 10-05 268 8'66 41 5-88 6-52 2-58 7-68 4-51 6-56 110 12-56 206 10-39 2-72 8-80 42 5-97 6-63 2-64 7-87 4-62 6-72 115 12-83 215 10-74 2-76 8'94 43 6-11 6-84 2-71 8-05 4-73 6-S8 120 13-10 224 1108 2-80 9'08 44 6-25 7-00 2-78 8-24 4-84 7-04 125 13-37 283 11-43 2-84 9'22 45 6-39 7-16 2-84 8-42 4-95 7-20 130 13-64 242 11-77 2-88 9-36 46 6-54 7-32 2-91 8-61 5-06 7-36 335 18-91 251 12-12 2-91 9'50 48 6-82 7-63 3-03 8-98 5-28 7-68 140 14-18 260 12-46 2-94 9-63 50 7-11 7-95 3-16 9-35 5-50 8-00 150 14-70 278 13-15 800 9-89 52 7-39 8-27 328 9-72 5-72 8-32 160 15-20 296 13-84 3-07 10-12 54 7-67 8-59 3-41 10-09 5-94 8-64 170 1565 814 14-53 8-14 10-36 56 7-95 8-91 3-58 10-46 616 8-96 180 16-09 332 15-22 3-20 10-60 68 8-24 9-23 8-66 10-84 6-38 9-28 190 16-53 350 15-91 3-26 1084 60 8-52 9-54 8-78 11-20 6-60 9-60 200 16-97 868 16-60 8-82 11-06 62 8-80 9-86 8-91 11-57 6-82 9-92 210 17-39 886 17-29 3-38 11-29 64 9-09 10-18 4-04 11-94 7-04 10-24 220 17-79 404 17-98 8-44 11-51 66 9-37 10-50 4-17 12-31 7-26 10-56 230 18-19 422 18-67 3-50 1173 68 9-65 10-82 4-29 12-68 7-48 1088 240 18-58 440 19-36 3-56 11-95 70 994 11-13 4-42 13-06 7-70 -11-20 250 18-97 456 20-05 3-61 1215 72 10-22 11-45 4-54 13-44 7-92 11-52 260 19-34 476 20-74 3-66 1285 74 10-50 11-77 4-67 18-81 8-14 11-84 270 19-70 494 21-43 3-72 1-2-55 76 10-79 12-08 4-79 14-19 8-36 12-16 280 2006 512 22-12 3-77 12-75 78 11-07 12-40 4-91 14-56 8'58 12-48 290 20-42 580 22-81 3-82 12-95 80 11-36 12-72 5-03 14-94 8-80 '12-80 800 20-78 548 23-50 8-88 13-14 82 11-64 13-04 5-16 15-32 9-02 13-12 810 21-12 566 24-19 8-ya 13-33 84 11-93 18-85 5-29 15-69 9-24 13-44 820 21-46 584 24-83 3-98 18-51 86 12-21 18-67 5-41 16-07 9-46 13-76 830 21-80 602 25-57 4-03 1369 88 12-50 18-99 5-54 16-44 9-68 1 14-03 840 22-14 620 2626 4-07 13-87 90 12-79 14-30 5-67 16-82 9-90 14-40 850 22-46 638 26-95 4-12 14-05 92 13-07 14-62 5-80 17-20 10-12 14-72 360 22-77 656 27-64 4-16 14-^3 94 13-35 14-94 5-92 17-58 10-34 15-04 370 23-09 674 28-88 4-21 14-41 96 J8-64 15-26 6-05 17-95 10-56 15-36 880 23-40 692 29-02 4-26 14v9 93 18-92 15-58 6-17 18-87 10-78 15-68 890 23-70 710 29-72 4-31 14-76 100 14-20 15-90 6-30 13-75 11-00 ;16-00 400 24-00 728 80-41 4-87 14-92 DIMENSIONS PROPER FOR LOCOMOTIVES. 301 I may here repeat that the diameter of cylinder in inches is given in the first vertical column, beginning at 20 inches and ending at 100 inches, while the length of the stroke in feet is given in the first horizontal column, beginning with 2 feet and ending with 9. If, therefore, we wish to find the dimension proper for any given engine, of which we must know the diam- eter of cylinder and length of stroke, we find in the first vertical column the given diameter in inches, and in the first horizontal column the given length of stroke in feet ; and where the vertical column under the given stroke intersects the horizontal column opposite the given diameter, there we shall find the required dimension.* LOCOMOTIVE ENGINES. It would be a mere waste of time and space to recapitulate rules similar to the foregoing as applicable to locomotive en- gines, since the strengths and other proportions proper for loco- motives can easily be deduced by taking an imaginary low pres- sure cylinder of twice the diameter of the intended locomotive cylinder, and therefore of four times the area, when the propor- tions will become at once applicable to the locomotive cylinder with a quadrupled pressure, or 100 Ibs. on the square inch. In locomotive engines the piston rod is generally made ^th of the diameter of the cylinder, whereas by the mode of determining the proportions that is here suggested it would be th. But piston rods are made of their present dimensions, not so much to bear the tension produced by the piston, as to bear the com- pression when they act as a pillar ; and properly speaking the proportionate diameter should diminish with every diminution * For screw or other short-stroke engines working at a high speed, the strengths of shafts given In the foregoing tables should be somewhat increased, and the length of bearing at least doubled. In some recent screw engines an Irregular motion of the engine has been perceived, owing to the elasticity of the shaft For 3 //D\ s ^i * B~ such engines a correspondent suggests the formula A/ ( =- I + -r _ diam- eter of journal in Inches ; where D = diameter of the cylinder In inches and K = radius of crank in inches. 302 PROPORTIONS OF STEAM-ENGINES. in the length of the stroke. In very short cylinders a proportion of ^ of the diameter of the cylinder would suffice in the case of low pressure engines, which answers to Hh of the diameter in locomotives where the stroke is always very short. But in high pressure engines of any considerable dimensions, carrying 100 Ibs. on the inch, the diameter of the piston rod should he 5-th of the diameter, answering to ^ rt th of the diameter in low pressure engines of the common total pressure of 25 Ibs. per square inch. CHAPTER Y. PROPORTIONS OF STEAM-BOILERS. IN proportioning boilers two main requirements have to be kept in view : 1st. The provision of a sufficient quantity of grate- bar area to burn with the intended velocity of the draught the quantity of coals required to generate the necessary quantity of steam ; and 2d. The provision of a sufficient quantity of heat- ing surface in the boiler, to make sure that the heat will be prop- erly absorbed by the water, and that no wasteful amount of heat shall pass up the chimney. Even the quantity of heating surface, however, proper to be supplied for the evaporation of a given quantity of water in the hour will depend to some extent upon the velocity of the draught through the furnace : for upon that velocity will depend the intensity of the heat within the furnace, and upon the intensity of the heat will depend the quantity of water which a given area of surface can evaporate. The first point therefore to be investigated is the best velocity of the draught, and the circumstances which determine that velocity. Here, too, there are two guiding considerations. The first is, that if the velocity of the draught be made too great, the small coals or cinders will be drawn up into the chimney and be precipitated as sparks, causing in many cases serious annoy- ance. The second consideration is, that the temperature of the escaping smoke should be as low as possible, and should in no case exceed 600. While, therefore, it is desirable in land and marine boilers to have a rapid draught through the furnace such as is produced in locomotives by the blast-pipe in order that the heat maybe sufficiently intense to enable a small amount of surface to accomplish the required evaporation, it is at the 304 PROPORTIONS OF STEAM-BOILERS. same time inadmissible to have such a rapid draught in the chimney as will suck up and scatter the small particles of the coal; nor is it desirable that the velocity of the air passing through the grate-bars should be so great as to lift small pieces of coal or cinder and carry them into the flues. No furnace has yet been constructed which reconciles the conditions of a high temperature with a moderate velocity of the entering air : but such a furnace may be approximated to by making the opening through the fire-bridge very small, and by insuring the necessary flow of air through these small openings by the application of a horizontal steam-jet at each opening; as by this arrangement a high temperature may be kept up in the furnace, at the same time that the contraction of the area through or over the bridge will not so much impair the draught as to prevent the requisite quantity of coal from being burnt. The exhaustion which a chimney produces is the effect of the greater rarity of the column of air within the chimney than that of the air outside. If the air be heated until it is expanded to twice its volume, then, its density being half of what it was before, each cubic inch of the hot air will weigh only half as much as a cubic inch of cold air ; and if the hot air be enclosed in a balloon, it will ascend in the cold air with a force of ascent equal to half the weight of the balloon full of cold air. As water is about 773 times heavier than air at the freezing-point, it will require 773 cubic inches of air, heated until they expand to twice their volume, to have ascensional force sufficient to balance a cubic inch of water : or if a syphon-tube be formed with a col- umn of water 1 inch high in one leg, it will require a column of the hot air 1546 inches (or nearly 129 feet) high, in the other leg, to balance the column of water 1 inch high. In other words, a chimney heated until the density of the smoke is only half that of the air entering the furnace, and which will be the case at a temperature under 600, will, if 129 feet high, produce an ex- haustion of 1 inch of water. In land boilers the ordinary ex- haustion or suction of chimneys is such as would support a col- umn of from 1 to 2 inches of water. But in steam-vessels the height of the chimney is limited, and the deficient height has to PBOPER HEIGHTS FOR CHIMNEYS. 305 be made up for by an increased area. In practice, tbe diameter of the chimney of a steam-vessel is usually made somewhat less than the diameter of the cylinder, there being supposed to be one chimney and two cylinders, with the piston travelling at the speed usual in paddle vessels. Boulton and "Watt's rule for proportioning the dimensions of the chimneys of their land engines is as follows : BOULTON AND WATTES BULE FOE FIXING THE PEOPES SECTIONAL AEEA OF A CHIMNEY OF A LAND BOILEE WHEN ITS HEIGHT IS DETERMINED. RULE. Multiply the number of pounds of coal consumed under ' the toiler per hour by 12, and divide the product ~by the square root of the height of the chimney in feet: the quotient is the proper area of the chimney in square inches at the smallest part. Example. "What is the proper sectional area of a factory chimney 80 feet high, and with a consumption of coal in the furnace of 300 Ibs. per hour? Here 300 x 12 = 3,600 ; and divided by 9 (the square root of the height nearly) we get 400, which is the proper sectional area of the chimney in square inches. If therefore the chimney be square, it will measure 20 inches each way within. BOTTLTON AND WATT'S EtTLE FOE FIXING THE PEOPEE HEIGHT OF THE CHIMNEY OF A LAND BOILEE WHEN ITS SECTIONAL AEEA IS DETEEMINED. KULE. Multiply the number of pounds of coal consumed under the boiler per hour by 12, and divide the product by the sec- tional area of the chimney in square inches : square the quo- tient thus obtained, which will give the proper height of the chimney in feet. ^Example. What is the proper height in feet of the chimney of a boiler which burns 300 Ibs. of coal per hour, the sectional area of the chimney being 400 square inches ? Here 300 x 12 = 3,600, which divided by 400 (the sectional 306 PROPORTIONS OF STEAM-BOILERS. area) = 9, the square of which is 81 ; and this is the proper height of the chimney in feet. These rules, though appropriate for land boilers of moderate size, are not applicable to powerful boilers with internal flues, such as those used in steam-vessels, in which the sectional area of the chimney is usually adjusted in the proportion of 6 to 8 square inches per nominal horse-power. This will plainly appear from the following investigation : In a marine boiler suitable for a pair of engines of 110-horse- power, the area of the chimney, allowing 8 square inches per nominal horse-power, would be 880 square inches. Supposing the boiler to consume 10 Ibs. of coal per nominal horse-power per hour, or say 10 cwt. (or 1120 Ibs.) of coal per hour, and that the chimney was 46 feet high, then, by Boulton and "Watt's rule for land engines, the sectional area of the chimney should be 1120 x 12 -,- V 46 = 13,440-?- say 7=1,920 square inches. This, it will be observed, is more than twice the area obtained by allowing a sectional area of 8 square inches per nominal horse- power. Here, therefore, is a discrepancy which it is necessary to get to the bottom of. In Peclet's ' Treatise on Heat ' an investigation is given of the proper dimensions of a chimney, which investigation is recapitulated and ably expanded by Mr. Rankine. But it gives results similar to those deduced from Boulton and Watt's rule for their small land boilers, and the expressions are much more complicated. Thus if w = the weight of fuel burned in a given furnace per second; V =the volume of air at 32 required per Ib. of fuel, and which in the case of common boilers with a chimney draught is estimated at 300 cubic feet ; Ti = the abso- lute temperature of the smoke discharged by the chimney, and which is equal to the temperature shown by the thermometer + 461 '2; T =the absolute temperature of the freezing-point, or 461 '2+ 32; A = the sectional area of the chimney in square feet ; and u = the velocity of the current in the chimney in feet per second : M>Y T. Then u = VELOCITY OF DRAUGHT IN CHIMNEYS. 307 If now I = the length of the chimney and of the flue leading to it in feet ; m = the mean hydraulic depth of the smoke, or the area of the flue divided hy its perimeter, and which for a round flue and chimney is J of the diameter; f= a coefficient of friction, the value of which for a current of gas moving over sooty surfaces Peclet estimates at 0'012 ; G a factor of resistance for the passage of the air through the grate, and which in the case of furnaces burning 20 to 24 Ibs. of coal per hour on each square foot, Peclet found to be 12 ; h = the height of the chim- ney in feet : Then by a formula of Peclet's 2. which formula, with the value that Peclet assigns to the con- stants, becomes and by transposition and reduction where 64 is twice the power of gravity, or 32^. If now the chimney be made 46 feet high and the flue leading to it be 3 feet diameter and 64 feet long, then 64-3 x 46 = 2957-8 ; -012 x 100 = 1'2 ; m = Jth of 3, or f, or '75, and 1-2 H- 75 = 1-6. Hence the equation becomes But ?/ - AT Hence W - ' = 14-23 Now if 1,120 Ibs. of coal be consumed per hour, -31 Ibs. will be consumed per second = w ; and if the temperature of 808 PROPORTIONS OF STEAM-BOILERS. the chimney be G00, then 600 + 461 = 1061 = T,, and 461 + 32 = 493 = T . 31 x 300 x 1001 Hence ~ 493 A ~~ = 14'6 A = 14 square feet, or 2,010 square inches; whereas 1,920 square inches is the area given by Boulton and Watt's rule. Peclet's rule, consequently, gives areas much too great for boilers with internal flues, though it will answer pretty well for small land boilers with external flues : but even here it has the disad- vantage of being too complicated for common use. It is clear that the friction of the smoke passing through internal flues must be much less than the friction of smoke passing through external flues like that which surrounds a wagon-boiler. For as only one side of the external flues is efficient in heating, the flue with the same friction per foot in length will require to be nearly three times as long as in the case of an internal flue of the same area, to give the required amount of heating surface. In steam vessels much heat is wasted, from the height of the chimney being necessarily so limited that but a small portion of the as- censional force due to the temperature of the smoke is obtained. Thus, if a height of chimney of 129 feet will produce an exhaus- tion of an inch of water when the heat is sufficient to expand the air into twice its volume, as will be the case at a tempera- ture considerably under 000, then it is clear that another height of 129 feet, added to the first, would produce an exhaustion equal to a column of two inches of water without any additional expenditure of heat ; and this increase would go on until the velocity of the draught became such that the friction of the ad- ditional height balanced its ascensional force. In steam-vessels, where the chimney is necessarily short, a great part of the ex- hausting or rarefying effect of the heat is lost ; and in steam- vessels, therefore, a chimney-draught is a more wasteful expe- dient for promoting combustion than it is in the case of a land boiler, where a much larger proportion of the ascensional power of the heat may be made available. The proportion of heating surface per nominal horse-power PROPER AREA OF HEATING SURFACE. 309 obtaining in marine boilers varies very much in different exam- ples, being in some boilers 12 square feet, in others 17 square feet, in others 20 square feet, in others 30 square feet, and in some as much as 35 square feet per nominal horse power. In fact, the proportion of heating surface required will depend upon the intended ratio in which the nominal is to exceed the actual power, which is now often as much as 8 or 9 times, and also upon the measure of expansive action which is proposed to be adopted. In marine boilers, as in land boilers, about 9 square feet, or 1 square yard, of heating surface will be required to boil off a cubic foot of water in the hour, and in Boulton and Watt's modern marine tubular-boilers they allow 10 square feet of heat- ing surface to evaporate a cubic foot of water in the hour, 10 square inches of sectional area of tubes, 7 square inches of sec- tional area of chimney, and 14 square inches of area over the furnace bridges. The proportions of modern flue-boilers are not very different, except that there is greater sectional area of flue. But no attempt has yet been made to connect the proportions proper SOT small land boilers, with those proper for large marine boilers, or to construct a rule that would be applicable to every class of flue-boilers. Great confusion has been caused by referring to so indefinite a unit as the nominal power of a boiler, and it is much bet- ter to make the number of cubic feet which the boiler can evaporate the measure of its power. This again depends upon the intensity of the draught. But it may be reckoned that 5 or 6 square feet of surface will evaporate a cubic foot per hour in locomotive boilers, and 9 or 10 square feet in land and marine boilers. The main dimensions and proportions of Boulton and Watt's wagon-boilers of different powers are given in the following table : 310 PROPORTIONS OF STEAM-BOILERS. PROPORTION OP BOITLTON AND WATT'S WAGON BOILERS. Power. Length Boiler. Breadth BoHer. Depth Boiler. Mean Height of Flue. Breadth Flue. Sectional Area of Flue. Sectional Area of Flue per H. P. ft. in. ft in. ft. in. in. in. sq. in. sq. in. 2 4 3 2 4 1 20 9 ISO 90 8 5 3 3 4 4 4 21 9 189 63 4 6 3 6 4 7 22 10 220 55 6 7 3 9 5 1* 27 10 270 45 8 8 4 5 6 81 12 872 44 10 9 4 8 5 91 35 12 400 40 12 10 4 6 6 36 13 468 39 14 10 4 9 6 21 39 13 507 86 16 11 9 5 6 6 40 14 560 85 18 12 8 5 2 6 8 42 14 588 82 20 13 6 5 4 6 11 44 H 616 80 80 16 5 6 7 3 45 15 720 24 45 19 6 8 5 53 16 795 17 These proportions enable us to establish the following rule, which is applicable to flue-boilers of every class : TO DETERMINE THE PROPER SECTIONAL AREA OF THE FLUE IN FLTJE-BOILERS. ECXE. Multiply the square root of the number of pounds of coal consumed, per hour fiy the constant number 300, and di- vide the product ly the square root of the height of the chim- ney in feet : the quotient is the proper sectional area of the flue in square inches. Example 1. "What is the proper sectional area of the flue in a flue-boiler burning 100 Ibs. of coal per hour, the chimney being 49 feet high. Here V 100 = 10, and 10 x 300 = 3000; which divided by 7 (the square root of 49) = 428 square inches, which is the proper area of the flue in this boiler. Example 2. What is the proper sectional area of the flue in a flue-boiler burning 30 Ibs. of coal per hour, the chimney being 81 feet high ? Here V30 = 5 '48, and 5-48 x 300 = 1644; which divided by 9 (the square root of 81) = 183 nearly, which is the proper area of the flue in square inches. BOULTON AND WATT'S PRACTICE. 311 Example 3. "What is the proper area of the flue in a flue- boiler burning 1,000 Ibs. of coal per hour, and with the chimney 49 feet high ? Here VlOOO = 31'78, which x 300 = 9534, and dividing by V (which is the square root of 49), we get 1,362, as the proper area of the flue in square inches. This is equivalent to 13 '62 square inches per horse- power. It is the universal experience with boilers of every class, that large boilers are more economical than small, or, in other words, that a given quantity of coal will boil off more water in boilers of large power than in boilers of small power. Nevertheless, for purposes of classification, it may be convenient to assume the efficiencies as equal. The proper proportions of flue-boilers from 1 to 100 horses power are given in the following Table : PEOPEE PBOPOBTIOXS OF FLTJE-BOILEBS OF DIFFEEESTT POWEE3. Hone Power. Pounds of Coal contained per boar. Sectional Area of Flue in B. & W.'t boilers. Sectional Area of Flue by rule, with chim- ney 49 feet high. Sectional Area of Flue by rule, with chim- ney 81 feet high Heating per"lj! P. Sectional Area of Flue per square ft. of heating surface. Ibs. sq. in. sq. in. Bq. in. sq. ft. sq. in. 1 10 123 106 2 20 'iso 191 149 15 6-0 8 80 189 285 188 13 4-8 4 40 220 270 210 11 5-0 6 50 808 235 6 60 '270 881 258 10-7 4-2 7 70 858 278 8 80 '872 8S8 296 10-2 4-3 9 90 406 816 10 100 "466 428 338 10 4-0 11 110 468 860 12 120 '468 469 865 9-8 8-9 18 180 488 880 14 140 '607 607 894 9-8 8-6 15 150 524 408 16 160 '660 541 421 9-7 8-5 17 170 554 481 18 ISO '588 675 446 9-8 3-2 19 190 590 459 20 200 '616 606 471 10 8-0 80 800 720 724 577 9-8 2-4 45 450 795 909 707 9-6 1-7 60 600 1,049 818 75 750 1,178 912 100 1,000 1,866 1,862 1,059 8 16 312 PEOPORTIONS OF STEAM-BOILERS. Mr. Watt reckoned that in his boilers 8 Ibs. of coal would evaporate a cubic foot of water in the hour, which is equivalent to an actual horse-power in the case of engines working without expansion. Good Welsh coal, however, it has been found, will evaporate 10 Ibs. of water for each pound of coal, which is equivalent to 1*6 cubic feet of water, or 1*6 horse's power in the case of an engine working without expansion ; and if such a measure of expansion be used as will double the efficiency of the steam, then 10 Ibs. of coal burned in the furnace will gene- rate 3 '2 actual horses' power. To attain this measure of effi- ciency, however, the steam would have to be cut off between ^ and of the stroke, and in the best boilers and engines work- ing with the usual rates of expansion it will not be safe to reckon more than 2 (or at most 2) actual horses' power as ob- tainable by the evaporation of a cubic foot of water. When, therefore, engines work up to five times their nominal power, as they now often do, it can only be done by passing through them twice the quantity of steam that answers to their nominal power or, in other words, by making the boilers of twice the propor- tionate size, unless where some expedient for producing an ar- tificial draught is employed. The proper height of chimney where the sectional area of the flue is known can easily be deduced from the foregoing rule. 4/P x 300 , (VP x 300) For if A = - TT then Ji = r yh A which formula put into words is as follows : TO FIND THE PEOPEE HEIGHT OF A CHIMNEY IN FEET WHEN THE NUMBEB OF POUNDS OF COAL CONSUMED PEE HOUB AND ALSO THE SECTIONAL AEEA OF THE FLUE AEE KNOWN. KTJLE. Multiply the square root of the number of pounds of coal consumed per hour by the constant number 300, and di- vide the product by the sectional area of the flue in square inches ; the square of the quotient is the proper height of the chimney in feet. DIMENSIONS OF CHIMNEYS FOR GIVEN POWERS. 313 Example 1. "What is the proper height of the chimney of a hoiler consuming 100 Ibs. of coal per hour, and with a sectional area of flue of 428 square inches. Here 4/100 = 10, and 10 x 300 = 3000, which divided by 428 = 7, the square of which is 49, which is the proper height of the chimney in feet. Example 2. "What is the proper height of the chimney of a flue-boiler consuming 100 Ibs. of coal per hour, and with a sec- tional area of flue of 333 square inches ? Here 4/100 = 10, and 10 x 300 = 3000, which divided by 333 = 9, the square of which is 81, which is the proper height of the chimney in feet. In flue-boilers, the sectional area of the chimney will be the same as that of the flue of a boiler of half the power. Hence in the foregoing Table the proper sectional area of the chimney of a 20-horse boiler the chimney being 49 feet high will be the same as the sectional area of the flue of a 10-horse boiler, name- ly 428 square inches, with a height of chimney of 49 feet ; and the proper sectional area of the chimney of a 30-horse boiler will be the same as that of the flue of a 15-horse boiler, namely, 524 square inches, with a height of chimney of 49 feet. If the chimney be 91 feet high, then the values will become 333 and 408 square inches respectively. As then the area of the chimney should be the same as that of the flue of the boiler of half the power, it is needless to give a separate rule for finding the area of the chimney, as such rule will be in all respects the same as that for finding the proper area of the flue, except that we take half the number of pounds of coal burned per hour instead of the whole. In marine tubular boilers the total capacity or bulk of the boiler, exclusive of the chimney, is about 8 cubic feet for each cubic foot of water evaporated per hour divided in the propor- tion of 6 -5 cubic feet devoted to the water, furnaces, and tubes, and 1'5 cubic foot occupied as a receptacle or repository for the bteam. The common diameter of tube in marine boilers is about 8 inches, and the length is 28 or 30 times the diameter. In lo- comotive loilers the usual diameter of the tubes is 2 inches, and 14 314 PROPORTIONS OF STEAM-BOILERS. the length is about 60 times the diameter. The area of the blast orifice in locomotives is about T Vth of the area of the chimney. The fire-bars are commonly inch thick, and the air-spaces are made 1 inch wide for fast trains. The main dimensions of ma- rine and locomotive boilers required for the evaporation of a cubic foot of water, are given in the following Table : PROPORTIONS OF MODERN BOILERS REQUIRED TO EVAPORATE A CUBIC FOOT OF WATER PER HOUR. Proportion required per Cubic Foot evaporated per hour. Marine Flue. Marine Tubu- lar. locomotive. Square feet of heating surface 8 9 to 10 6 70 70 18 Square inches sectional area of flue or tubes Square inches sectional area of chminey. . . Square feet of heating surface per square foot of fire grate 13 6 16'48 10 7 18-54 81 2-4 48 Pounds of coal or coke consumed on each square foot of fire grate per hour. 16 16 62 The quantity of coal or coke burned on each square foot of fire-grate in the hour to evaporate a cubic foot of water will of course very much depend on the goodness of the coal or coke. In the above Table the average working result of '8 Ibs. of water evaporated by 1 Ib. of coal, or a cubic foot of water evaporated by 7 '8 Ibs. of coal, is taken. The efficiency of a steam vessel is measured by the expendi- ture of fuel necessary to transport a given weight at a given speed through a given space, and one of the most efficient steam vessels of recent construction is the steamer Hansa, built by Messrs. Oaird & Co., to ply between Bremen and America. In this vessel there are two inverted direct-acting engines, with cy- linders 80 inches diameter and 3| feet stroke. There are four tubular boilers, with four furnaces in each, containing a total grate surface of 350 square feet, and a heating surface of 9,200 square feet ; besides which there is a superheater, containing a heating surface of 2,100 square feet. The steam is of 25 Ibs. pressure on the square inch, and it is condensed by being dis- charged into a vessel traversed by 3,584 brass tubes, 1 inch ex- ternal diameter, and 7 feet long. Each tube having 1'75 square SURFACE FOR GENERATING AND CONDENSING. 315 feet of cooling surface, the total cooling surface will be 6,272, or about two-thirds of the amount of heating surface. The cooling water is sent through the tubes by means of two double acting pumps, 21 inches diameter and 2-t inches stroke, worked from the forward end of the crank-shaft. It is much better to send the water through the tubes than to send the steam through them. But standing and hanging bridges of plate-iron should be introduced alternately in the chamber traversed by the tubes, so as to compel the current of steam to follow a zigzag course ; and the steam should be let in at that end of the chamber at which the water is taken off, so that the hottest steam may encounter the hottest water. It would further be advantageous to inject the feed water into a small chamber in the eduction-pipe, so as to raise the feed-water to the boiling-point before being sent into the boiler ; or the feed-pipe might be coiled in the eduction- pipe so as to receive the first part of the heat of the escaping steam. A length of 7 feet appears to be rather great for a pipe an inch diameter, as the water at the end of it will become so hot as to cease to condense any steam, unless the velocity of the flow be so great as to involve considerable resistance from fric- tion. Short pipes, with an abundant supply of cold water, will enable a very moderate amount of refrigerating surface to suffice, as plainly appears from Mr. Joule's experiment, already recited. If we reckon the engines of the Hansa at TOO horses' power, there will be half a square foot of grate-bars per nominal horse- power, and 13*1 square feet of heating surface per nominal horse-power in the boiler, besides 3 square feet in the super- heater, making in all 16'1 square feet of heating surface per nominal horse-power, or 32 '2 square feet of heating surface per square foot of fire-grate. If we take 9 square feet as evapora- ting a cubic foot of water per hour, then the total evaporation of the boilers in cubic feet will be 9,200 -s- 9 = 1,022 cubic feet per hour ; and if we reckon 8 Ibs. of coal as necessary to evapo- rate a cubic foot, then the consumption of coal per hour will bo 8,176 Ibs, or 3'6 tons per hour, supposing the boiler to be work- ing at its greatest power. This is 11*6 Ibs. of coal per nominal horse-power, reckoning the power at 700 ; and at this rate of 316 PROPORTIONS OF STEAM-BOILERS. consumption 23'2 Ibs. of coal will be burned every hour on each square foot of fire-grate, to generate the steam rqeuired for a nominal horse-power, or it will be 16 Ibs. on each square foot every hour to evaporate a cubic foot there being nearly T5 cubic feet of water evaporated for the production of each nomi- nal horse-power. INDICATIONS TO EE FULFILLED IN MAKING BOILERS. In all boilers the expedients for maintaining a proper circu- lation of the water, so that the flame may act upon solid water, and not upon a mixture of water and steam, have been greatly neglected ; and the consequence is that a much larger amount of surface is required than would otherwise be necessary. The metal of the boiler is often bent and buckled by being overheated, and priming takes place to an inconvenient extent. In all tubu- lar boilers the water should be within the tubes, and those tubes should be vertical, so as to enable the current of steam and water to rise upward as rapidly as possible. The best form of steam- boat boiler hitherto introduced is the haystack boiler, for which we are indebted to the fertile ingenuity of Mr. David Napier, and in which boiler the prescribed indications are well fulfilled. In the haystack boiler, which is much used in the smaller class of river-boats on the Clyde but which, like the oscillating en- gine at the earlier period of its history, has not yet been employed in seagoing vessels the tubes are vertical, with the water within them ; and the smoke on its way to the chimney imparts its heat to the water by impinging upon the outsides of the tubes. The late Lord Dundonald (another remarkable mechanical genius) proposed a similar plan of boiler ; and boilers on his principle in which the furnace flue of a common marine flue-boiler is filled with a grove of small vertical tubes on which the smoke impinges on its way to the chimney have been much used on the Continent with good results, and were also introduced in the Collins line of steamers navigating the Atlantic. The Clyde haystack boilers are generally made of the form of an upright cylinder with a hemispherical top, from the centre of which the chimney ascends. The furnace is circular, with a water-space NAPIER'S AND DTJNDONALD'S BOILERS. 317 all around it, and with a circular crown ; so that the furnace forms, in fact, a short cylinder, divided in some cases into four quarters by vertical water-spaces crossing one another. Suitable passages are provided to conduct the smoke from the furnace into a cylindrical chamber situated above it the diameter of this cylinder being the same as that of the shell of the boiler, less the breadth of a water-space which runs round it ; and the height of this cylinder being equal to the length of the tubes. The tubes are set in circles round the chimney ; and the smoke, which is delivered from the furnace near the exterior of the cylindrical chamber, has to make its way among the vertical tubes before it can reach the chimney. The lower tube-plate and the furnace crown are stayed to one another by frequent bolts, and the cylindrical chamber containing the tubes is also bolted at intervals to the shell of the boiler. The water-space intervening between the lower tube-plate and furnace crown is made very wide, so as to hold a large body of water, and also to enable a person to reach in should repairs be required. The only weak part of this boiler is the root of the chimney, which some- times has collapsed from becoming overheated by the flame as- cending the chimney before the steam has been generated ; and the small pressure of the air shut within the boiler when heated has caused the root of the chimney to collapse. This risk is easily prevented by placing several rings of T-iron around the root of the chimney, within the steam-chest, and also by carry- ing down the plating of the chimney for some distance into the tube-chamber, so as to constitute a hanging-bridge that would hinder the hottest part of the smoke from escaping, and retain it in the tube-chamber, until it had given out the principal part of its heat to the water. In all boilers of this construction these precautions should be adopted ; and it would further be useful to place a short piece of pipe in the mouth of every upright tube, so as to continue the tube up to the water-level, whereby the column being elongated its ascensional force would be increased, and the circulation of the water be rendered more rapid. As this species of boiler is likely to come into use both for steam-vessels and for locomotives, it will be proper to indicate 318 PROPORTIONS OF STEAM-BOILERS. the forms which appear to be most suitable for those objects. In steam-vessels it is desirable to combine the introduction of a spe- cies of boiler adapted for working at a higher pressure, with arrangements for burning the smoke, which will be best done by maintaining a high temperature in the furnace ; and a high degree of heat will be best kept up in the furnace by forming it of firebrick instead of surrounding it with water in the usual manner. If, therefore, a square box of iron be taken and lined with firebrick, and if it be divided longitudinally and transversely by these brick walls, and afterwards be arched over, we shall have four furnaces, requiring merely the introduction of the fire- bars to enable them to be put into operation. Suppose that on the top of each of these square boxes a barrel of vertical tubes is placed, the barrel being sufficiently sunk into the brickwork to establish a communication for the smoke between a hole at each of the four top corners of the box and corresponding perforations in the barrel, we shall then have the smoke from each of the four furnaces into which the box is divided escaping from one corner into the chamber containing the tubes, and after travelling among them passing to the chimney. In such a boiler the circu- lation of the water could be maintained by forming the external water-space very thick, and by placing a diaphragm-plate in it ; so that the water and steam could rise upward on the side of the water-space next to the tube chamber, while the solid water de- scended on that side of the water-space next to the boiler-shell. The intervening plate would enable these currents to flow in opposite directions without interfering with one another. In a boiler of this kin d the grate-bars should have a sufficient declivity to enable the coal to advance itself spontaneously upon them ; and if there are two lengths of firebars in the furnace, the front length should be set closer together than the others, so as partially to coke the coal as on a dead-plate, before it enters into combustion. This coking would be affected by the radiant heat of the furnace, to which heat the coal would be exposed. The openings through which the smoke would escape to the tube- chamber might be perforations or lattice openings in the brick- work, BO as to bring every particle of the smoke into intimate IMPORTANCE OF RAPID CIRCULATION. 319 contact with the incandescent material of which the furnace is composed; and these perforations should not have too much area, else the heat would escape to the tubes too rapidly, and the temperature of the furnace would fall. To maintain a suffi- cient draught to bring in the requisite supply of air to the fuel, a jet-pipe of steam could be introduced at the bottom of the chimney; which jet-pipe would open into a short piece of pipe of larger diameter, also pointing up the chimney, and it into another larger piece, and so on. The jet at each of these short pieces of pipe would draw in smoke and form with the previous jet a new jet, which would become of larger and larger volume and less velocity at successive steps, until the dimensions of the jet had enlarged to an area perhaps equal to half the area of the chimney. It will be sufficient if the length of each piece of pipe be a little greater than, its diameter ; and the lower end of each piece, or that end facing the current of smoke, should be opened a little into a funnel shape, the better to catch the smoke and carry it forward, to form with the steam a jet continually en- larging its dimensions. By this mode of construction a powerful draught will be created by the jet with a very small expenditure of steam. The area through the cylindrical hanging-bridge at the root of the chimney should not be large, and the bridge itself should be perforated with holes in some places, so as to establish a sufficient current of the smoke upward among the tubes to pre- vent the heat and flame being swept past direct to the bottom of the chimney without rising among the tubes to impart its heat to them. In the case of locomotive boilers formed with upright tubes, the fire-box would be the same as at present; but that part of the boiler called the barrel, and which is now filled with longi- tudinal tubes, would be formed with flat sides and bottom and a semicircular top, so that it would have the same external form as the external fire-box, and this vessel would be traversed by a square flue, in which the vertical tubes would be set. The sides and bottom of this flue would be affixed to the shell by staybolts in the same manner as the internal and external fire-boxes are stayed to one another ; and the top, being semicircular, would 320 PROPORTIONS OF STEAM-BOILERS. not require staying, while the upper tube-plate forming the top of the square internal flue would be strutted asunder and prevent- ed from collapsing by the tubes themselves, some of which should be screwed into the plates or formed with internal nuts, to make them more efficient in this respect. Such a boiler would have various advantages over ordinary locomotive boilers, and might be made of any power that was desired without any limitation being imposed by the width of the gauge of the railway. Such boilers might also be used for steam-vessels by merely increasing the area of the fire-grate. 8TBENGTH OF BOILEE3. The proportions which a boiler should possess in order to have a safe amount of strength will be determined partly by the pres- sure of the steam within the boiler, and partly by the dimen- sions and configuration of the boiler itself. The best propor- tions of the riveted joints of the plates of which boilers are made are as follows : BEST PBOPOBTION8 OF EIVETED STEAM-TIGHT JOINTS. Thickness of Plate In Inches. Proper Diameter of Rivets in Proper Length in inches of Rivets from Proper distance from Centre to Centre of Proper Quantity of Lap in inches In Single Proper Quantity of Lap in inches in Double Inches. Head. Rivets in Riveted Riveted inches. Joints. Joints. A 1 i H H 2^ i i 1* it l A } H it H H t f If i* 2rV H 1 it 2* 2 ii t i it 2* a* 2J 4& 3. H i 3 ! gj If the strength of the plate iron be taken at 100, then it has been found experimentally that the strength of a single-riveted joint will be represented by the number 56, and a double riveted joint by the number 70. According to the experiments of Messrs. Napier and Sons, the average tensile strength of rolled bars of Yorkshire iron was found to 61,505 Ibs. per square inch of section, STRAINS AND STRENGTHS. 321 and the average strength of bars made by nine different makers (and purchased promiscuously in the market) was found to be 69,276 Ibs. per square inch of section. The tensile strength of 1 cast steel bars intended for rivets was found to be 106,950 Ibs. per square inch of section, of homogeneous iron 90,647 Ibs., of forged bars of puddled steel 71,486 Ibs. and of rolled bars of puddled steel 70,166 Ibs. per square inch of section. The strength of Yorkshire plates Messrs. Napier found to be lengthwise 55,433 Ibs., crosswise 50,462 Ibs., and the mean was 52,947 Ibs. per square inch of section. The tensile strength of ordinary best and lest-lest boiler plates, as manufactured by ten different makers, was found to be lengthwise 50,242 Ibs., crosswise 45,986 Ibs., and the mean was 48,114 Ibs. per square inch of section. Plates of puddled steel varied from 85,000 Ibs. to 101.000 Ibs. per square inch of section, and homogeneous iron was found to have a tensile strength of about 96,000 Ibs. per square inch of section. Experiments have been made to determine the strength of bolts employed to stay the flat surfaces of boilers together ; and it has been found that an iron bolt f ths of an inch diameter, liko the staybolt of a locomotive, screwed into a copper plate f ths of an inch thick, and not riveted, bore a strain of 18,260 Ibs. before it was stripped and drawn out. When the end of the bolt was riveted over it bore 24,140 Ibs. before giving way, when the head of the rivet was torn off, and the bolt was stripped and drawn through the plate. When the bolt was screwed into an iron plate fths of an inch thick, and the head riveted as before, it bore a load of 28,760 Ibs. before giving way, when the stay was torn through the middle. When the staybolt was of copper screwed into copper plate and riveted, it broke with a load of 16,265 Ibs., after having first been elongated by the strain one- sixth of its length. Locomotive fire-boxes are usually stayed with f-inch bolts of iron or copper pitched 4 inches asunder, and tapped into the metal of the outer and inner fire-boxes, and the stays are generally screwed from end to end. These stays give a considerable excess of strength over the shell, but it is necessary to provide for the risk of a bad bolt. 14* 322 PROPORTIONS OF STEAM-BOILERS. "With these data it is easy to tell what the scantlings of a boiler should be to withstand any given pressure. If we take the strength of a single-riveted joint at 34,000 Ibs. per square inch, then in a cylindrical boiler the bursting strength in pounds will be measured by the diameter of the boiler in inches multiplied by twice the thickness of the plate in inches, and by the pressure of the steam per square inch La pounds ; and this product will be 34,000 Ibs. Thus in a cylindrical boiler 3 feet or 36 inches diame- ter and half an inch thick, if we suppose a length of one inch to be cut off the cylinder we shall have a hoop | an inch thick and 1 inch long. If we suppose one-half of the hoop to be held fast while the steam endeavours to burst off the other half, the separation will be resisted by two pieces of plate iron 1 inch long and -J an inch thick ; or, in other words, the resisting area of metal will be one square inch, to tear which asunder requires 34,000 Ibs. The separating force being the diameter of the boiler in inches mul- tiplied by the pressure of the steam on each square inch, and this being equal to 34,000 Ibs., it follows that if we divide the total separating force in pounds by the diameter in inches, we shall obtain the pressure of the steam on each square inch that would just burst the boiler. N"ow 34,000 divided by 36 (which is the diameter of the boiler in inches) gives 944'4 Ibs. as the pressure of the steam on each square inch that would burst the boiler. A certain proportion of the bursting pressure will be the safe work- ing pressure, and Mr. Fairbairn considers that one sixth of the bursting pressure will be a safe working pressure ; but in my opinion the working pressure should not be greater than between one-seventh and one-eighth of the bursting pressure. The rule which I gave in my ' Catechism of the Steam Engine,' for determining the proper thickness of a single-riveted boiler, proceeds on the supposition that the working pressure should be T Jg- of the bursting pressure. That rule is as follows : TO FEND THE PEOPEE THICKNESS OF THE PLATES OF A SINOLE- EIVETED CYLIKDBICAL BOILEE. RULE. Multiply the internal diameter of the toiler in inches ~by the pressure of the steam in Ibs. per square inch above the STRAINS AND STRENGTHS. 323 atmosphere, and, divide the product ~by 8,900: the quotient is the proper thickness of the plate of the boiler in inches. Example 1. "What is the proper thickness of the plating of a single-riveted cylindrical boiler of 3J feet diameter, and intended to work with a pressure of 80 Ibs. on the square inch? Here 42 inches (which is the diameter) multiplied by 80 = 3360, and this divided by 8900 = '377, or a little over f of an inch. The decimal '375 is f of an inch. Example 2. What is the proper thickness of a single-riveted cylindrical boiler 3 feet diameter, intended to carry a pressure of 100 Ibs. on the square inch ? Here 36 inches x 100 = 3600, which divided by 8900 = '4, or, as nearly as possible, ^ and %. As the double-riveted joint is stronger than the single-riveted in the proportion of 70 to 56, it follows that 56 square inches of sectional area in a double-riveted boiler will be as strong as 70 square inches in a single-riveted. This relation is expressed by the following rule : TO FIND THE PEOPEB THICKNESS OF THE PLATES OF A DOTJBLE- EIVETED CYLINDRICAL BOILEB. RULE. Multiply the internal diameter of the boiler in inches by the pressure of the steam in pounds per square inch above the atmosphere, and, divide the product by the constant number 11140 : the quotient will be the proper thickness of the boiler in inches when the seams are double-riveted. Example 1. What is the proper thickness of the plates of a double-riveted cylindrical boiler 42 inches diameter, and intended to work with a pressure of 80 Ibs. per square inch ? Here 42 x 80 = 3360, and this divided by 11140 = '3016, or about ^ of an inch, which is the proper thickness of the plates when the boiler is double-riveted. Example 2. What is the proper thickness of a double-riveted cylindrical boiler 8 feet diameter, intended to carry a pressure of 100 Ibs. on the square inch? 324: PROPORTIONS OF STEAM-BOILERS. Here 36 inches x 100 = 3600, which divided by 11140 = -322, or a little more than -f s of an inch, which will be the proper thickness of the plates of the boiler when the seams are double- riveted. If T = the thickness of the plate in inches, D = the diameter of the cylinder or shell of the boiler in inches,, and P = the pressure of the steam per square inch : Then D P is the formula for the thickness of single-riveted is the formula for double-riveted boilers. boilers, and DP 11140 Moreover, in single-riveted boilers p p 8900 T D So also for double-riveted boilers p p _ 11140 T D These formulas put into words are as follows : TO FIND THE PEOPEB DIAMETEE OF A eiNGLE-EIVETED BOILEE OF KNOWN THICKNESS OF PLATES AND KNOWN PRESSURE OF STEAM. RULE. Multiply the thickness in inches T>y the constant number 8900, and divide ly the pressure of the steam in tts. per square inch. The quotient is the proper diameter of the boiler in inches. JZcample 1. What is the proper diameter of a single-riveted cylindrical boiler composed of plates -377 inches thick, and intended to work with a pressure of 80 Ibs. on the square inch? Here -377 x 8900 = 3355-3, which divided by 80=41-94 inches, or 42 inches nearly, which is the proper diameter in inches. STRAINS AND STRENGTHS. 325 Example 2. What is the proper diameter of a single-riveted boiler composed of plates '4 inches thick, and intended to work with a pressure of 100 Ibs. on the square inch ? Here '4 x 8900 = 3560, which divided by 100 = 35'6 inches, which is the proper diameter of the cylindrical shell of the boiler in this case. TO FIND THE PEESSTJEE TO WHICH A SINGLE-EFVETED CYLINDBIOAL BOILEE MAY BE WOEKED WHEN ITS DIAMETEB AND THE THICK- NESS OF ITS PLATING AEE KNOWN. RULE. Multiply the thickness of the plating in inches 7>y the constant number 8900, and divide the product ty the diameter of the toiler in inches. The quotient is the pressure of steam per square inch at which the boiler may Tie worked. Example 1. What is the highest safe-working pressure in a single-riveted boiler 42 inches diameter, and composed of plates 377 of an inch thick? Here -377 x 8900 = 3355'3, which divided by 42 = 79-8 Ibs. per square inch, which is the highest safe pressure of the steam. Example 2. What is the highest safe- working pressure in the case of a single-riveted boiler 36 inches diameter, and composed of plates -4 of an inch thick ? Here -4 x 8900 = 3560, which divided by 36 = 99 Ibs. per square inch. The rules for double-riveted boilers are in every case the same as those for single-riveted, only that the constant 11140 is used instead of the constant 8900. It will therefore be unneces- sary to repeat the examples for the case of double-riveted boilers. Mr. Fairbairn has given the following table as exhibiting the bursting and safe-working loads of single riveted cylindrical boilers. But I have already stated that I consider Mr. Fairbairn's margin of safety too small. The working pressure, however, which he gives for single-riveted boilers would not be too great for double-riveted boilers, as will appear by comparing those pressures with the pressures which the foregoing rules indicate may bo safely employed. 326 PROPORTIONS OF STEAM-BOILERS. TABLE SHOWING THE BURSTING AND SAFE-WORKING PRESSURE OF CYLINDRICAL BOILERS, ACCORDING TO MR. FAIRBAIRN. Diameter of Boiler. Working pressure for J -inch plates. Bursting pressure for . -inch plates. Working pressure for %-inch plates. Bursting pressure for ,^-lnch plates. ft. in. Ibs. Ibs. Ibs. Ibs. 3 118 708} 157} 944} 3 8 109 653} 145} 871} 8 6 101 607 134} 809} 3 9 94 566} 125} 755} 4 86} 531 118 708} 4 3 as* 500 111 666} 4 6 78} 472 104} 629} 4 9 74} 447} 99} 596} 5 70} 425 94} 566} 5 3 67} 404} 89} 539} 5 6 64} 886} 85} 515 5 9 61} 363} 82 492} 6 59 354 78} 472 6 3 56} 340 75} 453} 6 6 54} 326} 72} 435} 6 9 52} 814} 69} 419} 7 50} 308} 67} 404} 7 8 48} 293 65 396} 7 6 47 283} 62} 377} 7 9 45} 274 60} 365} 8 44 265} 59 354 8 8 42} 257} 57 343} 8 6 41} 260 55} 833} 8 9 40} 242} 54 323} 9 89} 236 52} 314} 9 6 37 223} 49} 298} 10 85} 212} 47 283} It will be useful to compare some of the figures of this tablo with the results given by the rules just recited. For example, according to Mr Fairbairn, a single-riveted boiler, 5 feet diame- ter, and formed of -inch plates, may be habitually worked with safety to a pressure of Q4^ Ibs. on the square inch. Now, by our rule, -5 x 8900 = 4450, which divided by 60, the diameter of the boiler in inches, gives 74 Ibs. as the safe pressure at which the boiler may be worked. If the boiler be double-riveted, then we have '5 x 11140 = 5570, which, divided by 60, gives 93 Ibs. as the pressure per square inch at which the boiler may be safely worked. This differs very little from Mr. Fairbairn's result of 94J Ibs., and his table may therefore be used if the results be re- garded as applicable to double-riveted boilers, but as applied to single-riveted boilers his proportions, I consider, are too weak. The following diameters of boilers with the corresponding thick- COLLAPSING PRESSURE OF FLUES. 327 ness of plates, it will be seen, are all of equal strengths, their bursting pressure being 450 Ibs. per square inch, which answers to 34,000 Ibs. per square inch of section of the iron. Diameter 3 ft., thickness -250 inches ; 3 ft., -291 ; 4 ft,, '333 ; 4i ft., -376 ; 5 ft., -416; 5$ ft., 458 ; 6 ft., -500; 6| ft., -541 ; 7 ft., -583 ; 7i ft., -625 ; and 8 ft., -666. The collapsing pressure of cylindrical flues follows a different law from the bursting pressure, being dependent, not merely upon the diameter and thickness of the tube, but also upon its length ; and Mr. Fairbairn gives the following formula for com- puting the collapsing pressure. If T = the thickness of tlie iron, p = collapsing pressure in Ibs. per square inch, L = length of tube in feet, and D = diameter of tube in inches ; then T 2-19 p = 806,300 f - LD and as to multiply the logarithm of any number is equivalent to raising the natural number to the power which the logarithm rep- resents, we may for T 2 ' 19 write 2*19 log. T. With this trans- formation the equation becomes = 806,300 If now we take the thickness of the plate of the circular flue at 291 inches, and if we make the diameter of the flue 12 inches and its length 10 feet, the equation will become P = 806,800 gl 19 **' 291 . Now '291 being a number less than unity, the index of its loga- rithm will be negative, and for such a number as '291 the index will be 1, the minus being for the sake of convenience written on the top of the figure; whereas for such a number as -0291 the index will be 2 ; for -00291 the index will be 3, and so on. It does not signify, so far as the index is concerned, what the sig- nificant figures are, but only at what decimal place they begin ; and -1 has the same index as -291, and -01 as -0291. Now the logarithm of 291, as found in the logarithmicjtables, is 463893, and the index being "T, the whole logarithm is 1-463893. In multi- 328 PROPORTIONS OF STEAM-BOILERS. plying a logarithm with a negative index, as it is the index alone that is negative, while the rest of the logarithm is positive, we must multiply the quantities separately, and then adding the positive and negative quantities together, as we would add a debt and a possession, we give the appropriate sign to that quantity which preponderates. Now "463893 multiplied by 2-19 = 1-01592567, and 1 multiplied by 2-19 gives 2-19, which is a negative quantity. Adding these products together, we in point of fact subtract the 2'19 from the 1'01592567, which leaves 2'82592567. Now if we turn to the logarithmic tables, we shall find that the number answering to the logarithm 82592567, or the number answering to the nearest logarithm thereto (which is 825945), is 6698; but as the index is negative, this quantity will be a fraction, and the index being 2, the number will begin in the second place of decimals or, in other words, it will be 0-C698. Now 806300 multiplied by '06698 = 54004-974, which, divided by 120, gives 450 Ibs. as the collapsing pressure. If we allow the same excess of strength to resist collapse that we allowed to resist bursting namely, 7'6 times a tube of the dimensions we have supposed will be safe in working at a pres- sure of 60 Ibs. on the square inch. But the strength of tubes to resist collapse may easily be increased by encircling them with rings of T iron riveted to the tube. Cylindrical flues of different dimensions, but of equal strength to resist collapse, are specified in the following table : CYLINDRICAL FLUES OF EQUIVALENT STRENGTH, THE COLLAPSING PEESSUEE BEINa 450 POUNDS PEB SQUARE INCH. Diameter of Flue in inches. Thickness of plates in decimal parts of an inch. For a Fine 10 feet long. For a Flue 20 feet long. For a Flue 80 feet long. 12 18 24 80 86 42 48 291 350 899 442 480 51 548 899 480 548 607 659 707 752 480 578 659 780 794 851 905 AS APPLICABLE TO LOCOMOTIVES. 329 T If p = 806300 -- , then by transformation T 2-l9 = _ t l l i 1 L and 806300 2-19 / P L D T = 806300' If now we put p the collapsing pressure = 460 Ibs., L = 10 feet, and D = 12 inches, the expression becomes T - _ log. -06734 806300 2-19 In like manner the quantities L and D can easily be derived from the formula, and in fact the equations representing them will be 806300 T2-19 and PD 806300 T2-19 It is unnecessary to put these equations into words, as the rule for finding the collapsing pressure of flues is not much re- quired, seeing that in the case of all large internal flues they may be strengthened by hoops of T iron, so as to be as strong as the shell. PRACTICAL EXAMPLE OF A LOOOMOTIYB BOILEB. It will be useful to compare the results given by these com- putations with the actual proportions of a locomotive boiler of good construction, and I shall select as the example one of the outside-cylinder tank engines constructed by Messrs. Sharp and Co. for the North-Western Railway. The diameter of cylinder in this locomotive is 15 inches, and the length of the stroke 20 inches. The pressure of the steam in the boiler is 80 Ibs. per square inch. The barrel of the boiler is 3 feet 6 inches diameter, and 10 feet 8^ inches long, and it is formed of iron plates fths thick. The junction of the plates is effected by a riveted jump-joint, which is equal in strength to a single riveted-joint. The rivets are } 330 PROPORTIONS OF STEAM-BOILERS, inch in diameter. The external fire-box is of iron fths thick, and the internal fire-box is T Vths thick, except the part of the tube-plate where the tubes pass through, which is f inch thick. The internal and external fire-boxes are stayed together by means of copper stay-bolts, inch in diameter, and pitched 4 inches apart. The roof of the fire-box is supported by means of seven wrought-iron ribs If inches thick and 3f inches deep, which rest at the ends on the sides of the fire-box, while the fire-box crown, being bolted to the ribs, is kept up. The ribs are widened out at the bolt-holes, and are also made somewhat deeper there, so that only a surface of about -J inch round each bolt bears on the boiler crown, to which it is fitted steam-tight. To assist in keep- ing up the crown, the cross-ribs are also connected with the roof of the external fire-box. The water space left between the outside and inside fire-box is about 3 inches, and the inside fire- box should always be made pyramidical, to facilitate the disen- gagement of the steam from the surface of the metal. There is a glass tube and three gauge-cocks, for ascertaining the level of the water in the boiler. The lowest gauge-cock is set 3 inches above the roof of the internal fire-box, the next 3 inches above that, and the next 3 inches above that, so that the highest cock is 9 inches above the top of the internal fire-box. There is a lead plug fths of an inch diameter screwed into the top of the fire-box. But the usual course now is to place the lead plug in a cupped brass plug rising a little way above the furnace crown, so that the lead may melt before the plating of the crown gets red-hot, should the supply of water be from any cause intercepted. The boiler is fitted with 159 brass tubes, 10 feet Tf inches long, If inches external diameter, and -j^th of an inch thick, fixed in with ferules only at the fire-box end. Such tubes last from four to five years, and they are now made thickest at the fire-box end, where the wear is greatest. The part of the boiler above the tubes is supported by eight longitudinal stays, running from end to end of the boiler. The back tube-plate is of iron fths of an inch thick. The smoke-box is J inch thick, and the chimney, which is 15 inches diameter at bottom and 12 inches AS APPLICABLE TO LOCOMOTIVES. 331 at top, and rises 13 feet 3 inches above the rails, is jth of an inch thick. The damper for regulating the draught is placed at the front of the ash-pan, and there is another similar damper at tho hack of the ash-pan to he used when the engine is made to travel backward, which tank engines can the better do, as they have no tender. The surface of the fire-grate is lOfths square feet. The steam ports for admitting the steam to the cylinder are 11 inches by If ths, and consequently each has an area of 17'875 square inches. The branch steampipe leading to each cylinder has J less area than this. The blast-pipe is 6f inches diameter, taper- ing to 5 J inches diameter at the top, and within it is a movable piece of taper pipe, which may be raised up when it is desired to contract the blast orifice. The consumption of coke in these engines is 25 Ibs. per mile. The evaporation in locomotive boilers is 7 to 8 Ibs. of water per Ib. of coke, and in locomotive boilers working without expansion the evaporation of a cubic foot of water in the hour will be about equivalent to an actual horse-power. Now if the speed be supposed to be 30 miles an hour, a mile will be performed in two minutes ; and as the con- sumption per two minutes is 25 Ibs., the consumption per one minute will be the half of 25 Ibs., or say 12 Ibs. per minute ; and the consumption in 60 minutes, or one hour, will be conse- quently 720 Ibs. of coke ; and if 8 Ibs. of water are evaporated by 1 Ib. of coke, the water evaporated per hour will be 8 times 720, or 5760 Ibs. Now if we take a cubic foot of water at 62-J- Ibs., and as the evaporation of a cubic foot in the hour is equivalent to a horse-power, 5760 divided by 62-J- = 92, will be the number of actual horse-power exerted by this engine under the circumstances supposed. Practically, however, locomotives of this class are capable of exerting much more than 92 actual horse-power; for all modern locomotives work, to a certain extent, expansively, whereby a given bulk of water raised into steam is enabled to exert more power, and further, the consumption of coal per mile may be increased beyond 25 Ibs., with a corresponding in- crease of the power generated. In all boilers, indeed, whether land, marine, or locomotive, the evaporative power will be 332 PROPORTIONS OF STEAM-BOILERS. greatly increased by every expedient which increases the velocity of the draft, and if arrangements be simultaneously made for in- creasing the temperature of the furnace, by contracting the escaping orifice over the bridge or through the flues, the expen- diture of fuel to accomplish any given evaporation will not be increased. In this way marine boilers have been constructed with only 12 square feet of heating surface per nominal horse power, and in which the consumption was only 2* Ibs. of coal per actual horse power, as will be seen by a reference to page 52 of the Introduction to my ' Catechism of the Steam Engine.' CHAPTER YI. POWER AND PERFORMANCE OF ENGINES. THE manner of determining the nominal power of an engine has heen already explained, and it now remains to show in what manner its actual or indicator horse-power may be determined. Construction of the Indicator. The common form of indica- tor applicable to engines moving at low rates of speed I have al- ready described in my ' Catechism of the Steam-Engine.' But in the case of engines moving at high rates of speed, and, in fact, in the case of all engines to which the steam is quickly admitted, the diagrams formed by this species of indicator are much dis- torted, and the accuracy of the result impaired, by the momen- tum of the piston of the indicator itself, which is shot up sud- denly by the steam to a point considerably higher than what answers to the actual pressure. The recoil of the spring again sends the piston below the point which properly represents the pressure ; and in interpreting the diagram the true curve is sup- posed to run midway between the crests and hollows of the waving line produced by these oscillations. Latterly an im- proved form of indicator, called Richards' indicator, has been introduced, which is represented in fig. 5, of which the main pe- culiarity is that its piston is very light and has a very small amount of motion, so that its momentum is not sufficiently great to disturb the natural line of the diagram. The motion of the piston of the indicator is multiplied sufficiently to give a diagram of the usual height by means of a small lever jointed to the top of the piston rod. To the end of this lever a small link, carry- 334 RICHARDS' INDICATOR. Fig. 5. RICHARDS' INDICATOR. (By Elliot Brothers, Strand.) METHOD OF APPLYING THE INDICATOR. 335 ing the pencil, is attached, and from the lower end of this small link a small steel radius bar proceeds to a fixed centre on a suit- able part of the instrument, so as to form a parallel motion whereby the pencil is constrained to move up or down in a ver- tical direction. The paper is placed upon the drum, shown in the figure with a graduated scale, and the string causing this drum to turn round and back again on its axis is put into con- nection with some part partaking of the motion of the piston in the usual manner. To withdraw the pencil from the paper, the whole parallel motion and the arms carrying it are turned round upon the cylinder, and the pencil is thus made readily accessible. The action of this indicator is precisely the same as that of the common indicator, which, having been described in my ' Cate- chism of the Steam-Engine,' need not be further noticed here. But in this indicator, as the spring is very stiff, and the travel of the piston correspondingly small, there are no inconvenient os- cillations of the pencil such as occur when a long and slender spring is employed. Method of applying the Indicator. The drum being put into communication with some part of the engine possessing the same motion as the piston, but sufficiently reduced in amount to be suitable for the small size of the instrument, the drum will begin to be turned round when the piston begins its forward stroke ; and the string having drawn it round in opposition to the ten- sion of the spring coiled at the bottom of it, it wiU*follow that when the string is relaxed, as it will be on the return stroke of the piston, the drum will turn back again to its original position, and its motion and that of the string will be an exact miniature of the motion of the piston. The pencil, if now suffered to press against the paper, will describe a straight line. But if the cock which connects the cylinder of the indicator with the cylinder of the engine be now opened, the pencil will no longer trace a straight line, but being pressed upward during the forward stroke by the steam, and being sucked downward by the vacuum during the return stroke, if the engine is a condensing one, or being pressed downward by the spring when the pressure of the steam is withdrawn, as it will bo during the return stroke, it is 336 POWER AND PERFORMANCE OF ENGINES. quite clear that the pencil must now describe a figure containing a space or area, and the figure is what is called the indicator di- agram, and the amount of the space is the measure of the amount of the power exerted at each stroke by the engine. This will be more clearly understood by a reference to fig. 6, which is an in- dicator diagram taken from a steam fire-engine constructed by Messrs Shand, Mason and Co., with two high-pressure engines of 6J inch cylinders and 7 inches stroke, with a pressure on the Ion <^ <(%{% Forward stroke. Admission .8-9 r. Steam line. corner. , . ! 130- r *^ 120 110- 100- 90- 90- 70- o o o if> o O \f> 60 o Is o a N N N 5 01 w 50- 40- / 30- v / 20 : \ "~ 1 EDUCATION LIME / 10. Hri3 L_J, ..: L _.._.. L Atmospheric Hne lieturn stroke. Ciompressive corner. DIAGRAM ILLUSTRATIVE OF THE MODE OP COMPUTING THE HORSE-POWER. boiler of 145 Ibs. per square inch, and making 156 revolutions per minute. The total weight of this engine is 24 cwt. 2 qrs., and by a reference to the diagram it will be seen that the mean pressure urging the piston is 117'5 Ibs. per square inch, which mean pressure is ascertained by adding together the pressure at each division or ordinate, and dividing by the number of ordi- nates, which in this case is 10. The mean pressure multiplied by the areas of the cylinders and by the speed of the piston in feet per minute, and divided by 33000 Ibs., gives 18'3 horses as INTERPRETATION OP INDICATOR DIAGRAMS. 337 the power actually exerted by this engine. The weight of the engine is consequently only 1-3 cwt. per actual horse-power. The advantage of taking 10 ordinates instead of 8 or 9 or 11 is, that the division by 10 is accomplished by merely shifting the position of the decimal point; while 10 ordinates are enough to enable the area to be measured accurately enough for all practi- cal purposes. Thus the total amount of the pressures in the di- agram, fig. 6, taken at 10 places, is 1175 Ibs., and the tenth of this, or 11T5 Ibs. per square inch, is the mean pressure on the piston throughout the stroke. It is clear that when we have got the mean pressure on each square inch of the piston, we have only to ascertain the number of square inches in it, and the distance through which it moves in a minute, to determine the power, and the indicator enables us to determine the mean pressure on the piston throughout the stroke in the manner just explained. The indicator is sometimes applied to the air-pump and to the hot well, to determine the varying pressures within them at different parts of the stroke ; and it is virtually the stethoscope of the engine, as it enables us to tell whether all its internal motions and pulsations are properly performed. Mode of reading Indicator Diagrams. In the preceding di- agram the piston moves in the forward stroke in the direction shown by the arrow, and backward on the return stroke in the di- rection shown by the arrow. In all diagrams the top indicates the highest pressure, and the bottom the lowest pressure. But it is quite indifferent whether the diagram is a right-hand or left- hand diagram ; and where two diagrams are shown on the same piece of paper, as is often done, that which represents the per- formance of one end of the cylinder is generally right-hand, and that which represents the performance of the other end of the cylinder is generally left-hand. This arrangement, however, is quite immaterial, that which alone determines the power exert- ed being with any given scale the area shut within the diagram. In fig. 6, the steam being supposed to be let in upon the pis- ton of the engine, presses the piston of the indicator up to the point shown at the 'admission corner,' and as the piston moves forward the steam continues to press upon it with undiminished 15 338 POWER AND PERFORMANCE OF ENGINES. pressure, until close to the end of the stroke, at the ' eduction corner,' the eduction passage is opened ; and as the steam con- sequently escapes into the atmosphere there is no longer the same pressure on the spring of the indicator as hefore, and its piston consequently descends. As, however, the steam cannot instantaneously get away, the pressure does not descend quite so low as the atmospheric line. The eduction passage, it appears by the diagram, begins to be opened when about nine-tenths of the forward stroke has been completed, and it also begins to be shut when about nine-tenths of the return stroke has been corn- Fig. 7. DIAGRAM TAKEN FROM STEAMER ' ISLAND QUEEN.' pleted, as appears by a reference to the ' compression ' corner, which shows that the back pressure begins to rise before the termination of the stroke. The area comprehended between the atmospheric line and the bottom of the diagram shows the amount of back pressure resisting the piston, which in this dia- gram is of the average amount of 5*1 Ibs. ; and this increased back pressure at the ' compression corner ' is* produced by the compression of the steam shut within the cylinder, which is ac- complished by the piston as it approaches the end of its stroke. Various examples of Indicator Diagrams. In the engine of which the diagram is given in fig. 6, the steam works with very little expansion ; but in fig. 7 we have a diagram taken from the steamer 'Island Queen,' which shows a large amount of expan- INTERPRETATION OF INDICATOR DIAGRAMS. 339 sion. This diagram is a left-hand diagram, the former one, shown in fig. 6, being a right-hand diagram. A is the admission corner, and the steam is only admitted until the piston reaches the position answering to that of a vertical line drawn through a, and which is about one-eighth of the stroke. The steam be- ing shut off from the cylinder at a, thereafter expands until the end of the stroke is nearly reached, when the eduction passage is opened, and the pencil then subsides to the point B, at which point the piston begins to return. The straight line drawn across the middle of the diagram is the atmospheric line ; and it is traced by the pencil before the cock of the indicator commu- nicating with the cylinder is opened. The distance of the line B o below the atmospheric line shows the amount of vacuum ob- tained in the cylinder, and the height of A a above the atmos- pheric line shows the pressure of the steam subsisting in the cylinder. This diagram, which is a very good one, is obtained with the aid of a separate expansion valve. The pressure of the steam was 22 Ibs. per square inch, the vacuum 14J Ibs., and the number of revolutions per minute 17. In some high-pressure engines, where the steam is allowed to escape suddenly through large ports, and a large and straight pipe, there is not only no back pressure on the piston, but a partial vacuum is created within the cylinder by the momen- tum of the escaping steam. In ordinary condensing engines the momentum of the steam escaping into the condenser might in some cases be made to force the feed-water into the boiler, in the same manner as is done by a Giffard's injector, which is an instrument that forces water into a boiler by means of a jet of steam escaping from the same boiler. This instrument will not act if the temperature of the feed-water be above 120 Fahr., as in such case the steam will not be condensed with the required rapidity. As the steam is water in a state of great subdivision, and as the particles of this water are moved with the velocity of the issuing steam, which is very great, we have in effect a very small jet of water issuing with a very great velocity, and this small stream would consequently balance a very high head of water, or, what comes to the same thing, a very great pres- 340 POWER AND PERFORMANCE OF ENGINES. sure. Precisely the same action takes place when the steam es- capes to the condenser ; and under suitable arrangements the boiler might be fed by aid of the power resident in the educting steam, and indeed the function of the air-pump might also be performed by the same agency. In fig. 8 we have an example of the diagrams taken from the top and the bottom of the cylinder disposed on the same piece of paper, those on the left-hand side being taken from the top of the cylinder, and those on the right-hand side being taken DIAGRAMS TAKEN AT MOORINGS FROM HOLYHEAD PADDLE-STEAMER ' MUNSTER.' from the bottom of the cylinder. There are three diagrams taken from each end with different degrees of expansion. A is the admission corner of the three diagrams, taken from the top of the cylinder, and a a a are the three several points at which the steam is cut off in these three diagrams. Thereafter the steam continues to expand, and the pressure gradually to fall, until the points 5 I 5 are reached, when the eduction passage is opened to the condenser, and the pressure then falls suddenly to the point B. The line B B' represents the amount of exhaustion DIAGRAMS OF HOLYHEAD STEAMER. 341 attained within the cylinder measured downward from the at- mospheric line M L ; and ccc represent the three points at which compression begins, answering to the three degrees of expansion. The letters A', a', &', B', and c' represent the corresponding points for each of the three diagrams taken from the bottom of the cyl- inder ; and the amount of correspondence in the right-hand and left-hand diagrams shows the amount of accuracy with which the valves are set to get a similar action at each end of the cyl- Fig. 9. DIAGRAMS TAKEN FROM HOLYHEAD PADDLE-STEAMER ' ULSTER* WHBN UNDER WAY. inder. The diagrams given above were taken from the Holy- head steam-packet ' Munster,' the engines of which were con- structed by Messrs. Boulton and Watt. The cylinders are oscil- lating, of 96 inches diameter and T feet stroke. The pressure of steam was 2616 Ibs. per square inch, vacuum 25 Ibs., and the number of strokes per minute 9 the vessel having been at moor- ings at the time. It will be seen by these diagrams that the amount of lead upon the eduction side, or the equivalent distance which the piston is still from the end of the stroke when eduction begins to take place, corresponds in every instance with the amount of 842 POWER AND PERFORMANCE OF ENGINES. the compression, since, in fact, by shifting the eccentric round to let the steam out of the cylinder before the end of the stroke, the valve will be equally shifted to shut the educting orifice be- fore the end of the stroke, and thus to keep within the cylinder any vapour left in it when the valve has been shut, and which is thereafter compressed by the piston until the end of the stroke is reached, or until the valve opens the communication with the boiler. Fig. 9 represents a diagram taken from the top, and another taken from the bottom of one of the cylinders of the Holyhead paddle-steamer 'Ulster,' a vessel of the same power and dimen- sions as the ' Munster,' and the engines also by Messrs. Boulton Fig. 10. DIAGRAMS FROM STEAMER ' ULSTER ' AT 4 1-2 STROKES. (STEAM THROTTLED BY THE LINK.) and "Watt. When these diagrams were taken the pressure of the steam in the boiler was 26 Ibs. per square inch, the vacuum in the condenser 13 Ibs. per square inch, and the engine was mak- ing 23 strokes per minute. The mean pressure on the pistons, obtained by taking a number of ordinates, as in fig. 6, reckoning up the collective pressure at each, and dividing by the number of ordinates, was 28*27 Ibs. It is immaterial what number of ordinates is taken, except that the more there are taken the more accurate will be the result. In fig. 10 we have diagrams taken from top and bottom in the same engine, when slowed to 4J strokes per minute, partly by closing the throttle valve, and partly by shifting the link towards its mid-position. In these diagrams nearly the whole areas are DIAGRAMS OF HOLYHEAD MAIL STEAMER. 343 below the atmospheric line. But on the left-hand corner of one of the figures a loop is formed, which often appears in engines employing the link, and the meaning of which it is necessary to explain. The extreme point of the diagram in every instance answers to the length of the stroke; and if the steam is pent up in the cylinder hy the eduction passage being shut before the end of the stroke, or if it be suffered to enter from the boiler be- fore the stroke is ended, the pencil will be pushed up to its high- est point before the stroke is ended, and as the paper still con- tinues to move onward the upper part of the loop is formed. If the pressure within the cylinder when the piston returns were to be precisely the same as when the piston advances during this part of its course, the loop would be narrowed to a line. But as the advance of the piston when the valve is very little opened somewhat compresses the steam, and as its recession when the valve is very little opened somewhat wire-draws it, the pressures while the piston advances and retires through this small distance, although the cylinder is open to the boiler by means of a small orifice, will not be precisely the same ; and the higher pressure will form the upper part of the loop, and the lower pressure the lower part. In fig. 10, by following the outline of the left-hand diagram, it will be seen that the steam begins to be compressed within the cylinder when about three-fourths of the stroke has been completed ; and the pencil consequently begins to rise somewhat above its lowest point. But as the vapour within the cylinder is very rare, the rise is very little until, when the piston is about one-eighth part of its motion, or about 8 inches from the end of the stroke, the steam-valve is slightly opened, when the piston of the indicator is compelled to ascend to the point answering to the pressure within the cylinder thus produced. As the opening from the boiler continues, and the piston by ad- vancing against the steam, instead of receding from it, compress- es rather than expands the steam admitted into the cylinder, the pressure continues to rise somewhat to the end of the stroke ; when the piston of the engine, having to move in the opposite direction, the steam within the cylinder will be expanded, and any still entering will be wire-drawn in the contracted passage, 344 POWER AND PERFORMANCE OF ENGINES. and the pressure will fall. Under such circumstances a loop will necessarily be formed at the corner of the diagram, such as is shown to exist at the left-hand corner of fig. 10. The reason why there is no corresponding loop at the right-hand corner of the right-hand diagram is simply because the valve is somewhat differently set at one end of the engine from what it is at the other ; and the angles of the eccentric rods will generally cause Fig. 11. DIAGEAM FROM AIR-PUMP OF STEAMER ' ULSTER.' (19 BErOLTTTIONS PEB MINUTE.) some small difference in the action of the valve at the different ends of the engine. Diagrams from the Air-Pump. Fig. 11 is a diagram taken from the air-pump of the ' Ulster,' when the engine was making 19 revolutions per minute. In this diagram the pencil begins to ascend from that point which marks the amount of exhaustion existing in the air-pump, and it rises very slowly until about two-thirds of the stroke of the pump has been performed, when it shoots rapidly upwards, indicating that at this point the water is encountered which has to be expelled. Midway between the DIAGRAMS TAKEN FROM THE AIR-PUMP. 345 atmospheric line and the highest point of ascent, the delivery valve begins to open, and somewhat relieves the pressure ; and there is consequently a wave in the diagram on that point. But the inertia of the water in the hot-well has then to be encoun- tered, and an amount of pressure is required to overcome this inertia, which is measured by the highest point to which the pencil ascends. So soon as the water in the hot-well and waste- water pipe has been put into motion, the motion is continued by its own momentum, without a sustained pressure being re- quired to be exerted by the bucket of the pump ; and the pres- sure in the pump consequently falls, as is shown by the descent of the piston of the indicator towards the end of the stroke. Fig. 12. DIAGRAM FEOM AIR-PUMP OF STEAMER ' ULSTER. (7 STBOKE8 PEB MINUTE.) The effect of partially closing the throttle-valve of an engine so as to diminish the speed, will be to reduce the momentum of the water in the hot-well, and correspondingly to reduce the maximum pressure which the pump has to exert. But the ef- fect will also be to fill the pump with water through a larger proportion of its stroke ; and if the engine were to be slowed very much by shutting off the steam, without correspondingly shutting off the injection, the air-pump at its reduced speed would be unable to deliver all the water, which would conse- quently overflow into the cylinder and probably break down the engine. In fig. 12 we have an air-pump diagram taken from the steamer 'Ulster,' when the speed of the engine was reduced to 346 POWER AND PERFORMANCE OF ENGINES. six strokes per minute ; and it will be observed that we have no longer the same amount of maximum pressure in the pump, nor the same sudden fluctuations. The pump, however, is filled for a greater proportion of its stroke ; and the maximum pres- sure once created,. is constant, and does not rise much above the pressure of the atmosphere, being, in fact, the simple pressure due to the pressure of the atmosphere, and that of the column of water intervening between the level of the air-pump and that of the waste-water pipe. Diagram illustrative of the evils of Small Ports. Fig. 13 is a diagram taken from a pumping-engine in the St. Katherine's Docks, and is introduced mainly to show the detrimental effect Fig. 13. DIAGRAM TAKEN FROM PUMPING-ENGINE, ST. KATHERINE'S DOCKS. of an insufficient area of the eduction passages. The steam is supposed to enter at the left-hand corner, but as the speed of the piston accelerates, as it does towards the middle of the stroke, the pressure falls, from the port being small and the steam wire- drawn. Towards the other end of the stroke the pressure would again rise, but that it is hindered from doing so by the condensa- tion within the cylinder, which is considerable, as the engine works at the low speed of 12 strokes per minute, lifting the wa- ter 9J feet. The eduction corner of the diagram is very much rounded away, from the inadequate size of the ports ; and the eduction will also be impeded by any condensed water within the cylinder, which, unless got rid of by other arrangements, will have to be put into motion by the escaping steam. The mean DIAGRAMS TAKEN FROM THE HOT-WELL. 347 pressure exerted on the piston of this engine is only 12 '45 Ibs. per square inch, although it operates without expansion ; and it may be taken as a fair example of eneligible construction. Diagrams showing the momentum of the Indicator piston. Fig. 14 is a pair of diagrams taken from one of the engines of H. M. S. ' Orontes.' This vessel, which is 300 feet 1 inch long, 44 feet 8 inches broad, and 2,823 tons, has horizontal direct acting engines of 500 horse-power, constructed by Messrs Boul- Fig. 14. DIAGRAM TAKEN FROM H.M. TROOP-STEAMER ' ORONTES.' ton and "Watt. With a midship section of 644 square feet, and a displacement of 3,400 tons, the vessel attained a speed on her offi- cial trial, of 12-622 knots, with a pressure of steam in the boiler of 25 Ibs. per square inch, 61 revolutions per minute, the engines exerting 2,249 horse-power. On one occasion the speed obtained was 13-3 knots. With an area of immersed section of Y81 square feet, and a displacement of 4249 tons, the speed attained was 12*354 knots, with 2,143 horse-power. There are two horizontal 348 POWER AND PERFORMANCE OF ENGINES. engines of 71 inches diameter, and 3 feet stroke. The screw is 18 feet diameter, 25 feet pitch, and 4 feet long, and the slip of the screw was found to vary between 13 and 16 per cent. When the diagrams represented in fig. 14 were taken, the pressure of the steam in the boiler was 2H Ibs. of the vacuum, in the con- denser llf Ibs., and the engine was making 60 revolutions per minute. If ordinates be taken in the case of these diagrams, and the mean pressure be thus determined, it will be found to amount to 25 '22 Ibs. per square inch. In these diagrams the waving line formed by the pencil, owing to the momentum of the piston of the indicator, is very plainly shown ; and although such irregu- larities will not materially impair the accuracy of the result, if a sufficient number of ordinates be taken correctly to measure the irregularity, yet it is greatly preferable to employ an indicator which will be as free as possible from the disturbing influence of the momentum of its own moving parts. In this engine, as in most of Messrs. Boulton and Watt's engines, there is a great similarity in the diagrams taken from each end of the cylinder a result mainly produced by giving a suitable length to the eccentric rods, by moving up or down the links vertically by a screw, instead of by a lever moving in the arc of a circle, and placing the projecting side of the eccentric suitably with the curvature of the link, since, if placed in one position, it will aggra- vate the distortion produced by the angle of the eccentric rods, and if placed in the opposite position it will correct this dis- tortion. Fig. 15 represents a series of diagrams from each end of one of the engines of the ' Orontes,' formed by allowing the pencil to rest on the paper during many revolutions, instead of only dur- ing one. These diagrams show small differences between one another, mainly in the mean pressure of the steam. Fig. 16 represents two diagrams taken from the engines of the iron-clad screw steamer 'Eesearch, fitted with horizontal engines, with 50-inch cylinders, and 2 feet stroke. With a pres- sure of steam in the boiler of 22 Ibs., and with a vacuum in the condenser of 12f Ibs. per square inch, the mean pressure on the piston shown by the diagrams is 24*55 Ibs., the engine making DIAGRAMS FROM DIFFERENT STEAMERS. 349 85 revolutions per minute. This engine is fitted with surface condensers. The serrated deviation at a is caused by the mo- mentum of the piston of the indicator. In fig. 17 we have two diagrams, taken from opposite ends of one of the engines of H.M.S. ' Barossa.' This vessel is 225 feet long, 40 feet 8 inches broad, and 1,702 tons burden. With a mean draught of water 15-J- feet or thereabout, the area of mid- ship section is 466 square feet, and the displacement 1,780 tons. The vessel is propelled by two horizontal engines, with cylinders Fig. 15. DIAGRAMS TAKEN FEOM SCREW STEAMER ' ORONTES.' of 64 inches diameter and 3 feet stroke, the nominal power being 400 horses. On the official trial this vessel realised a speed of 11-92 knots, with a pressure of steam in the boiler of 20 Ibs. per square inch, and with an indicated power of 1798'2 horses, the engine making 66 revolutions per minute. The screw is 16 feet diameter, 24 feet pitch, and 3 feet long, and the slip at the time of trial was 23 - 71 per cent. When the diagrams shown in fig. 17 were taken, the pressure of steam in the boiler was 19 Ibs. per square inch ; vacuum in condenser 12 Ibs. per square inch, the revolutions 66 per minute, and the mean pres- 350 POWER AND PERFORMANCE OF ENGINES. sure on the piston 22*3 Ibs. per square inch. The area of a cylin- der of 64 inches diameter is 3216-2 square inches, the douhle of which (as there are two cylinders) is 6433'8 square inches, and as there 22'3 Ibs. on each square inch, there will be a total pres- sure of 6433-8 times 22-3, or 143,473-74 Ibs. urging the pistons, and as the length of the double stroke is 6 feet, the power ex- erted will be equal to 6 times 143,473-74 Ibs., or 860,840-44 Fig. 16. INDICATOR DIAGRAMS PROM IRON-CLAD STEAMER ' RESEARCH.' foot-pounds per stroke, and as there are 66 strokes per minute, there will be 66 times this, or 56,797,869*04 foot-pounds exerted per minute. As an actual horse-power is 33,000 foot-pounds per minute, we shall, by dividing 56,797,869-04 by 33,000, get the actual power exerted by this engine at the time the above diagrams were taken, and which, by performing the division, we shall find to be 1721-1 horses. Various Diagrams. Fig. 18 is a diagram taken from the air-pump of the ' Barossa,' which is a double-acting pump. The DIAGRAMS FROM DIFFERENT STEAMERS. 351 injection was all on at the time this diagram was taken, and the vacuum was only 11 Ibs. per square inch. In my 'Catechism of the Steam-Engine,' published in 1856, 1 drew attention to the fact of the existence of very imperfect vacuums in engines with Fig. 17. DIAGRAMS TAKEN FROM H. M. STEAMEE ' BAROSSA.' double-acting air-pumps, the buckets of which move at a high rate of speed ; and I also pointed out the cause of this imperfect vacuum, which I showed to be consequent on the lodgment of Fig. 18. tfcr-10 AIR-PUMP DIAGRAM FROM H. 1C. STEAMER ' BAROSSA.' large quantities of water between the foot and delivery-valves at the end of the pump, into which water the pump forced in the air or drew it out without ejecting it from the pump at all. I consequently recommend that in all pumps of this class the bucket 352 POWER AND PERFORMANCE OP ENGINES. and valve-chambers should be so contrived that every particle of water would be forced out of the puinp at every stroke. But up to the present time I do not find that this recommendation has been generally adopted, and in nearly every species of direct- acting screw-engine operating by a jet in the condenser, the vacuum is much worse than it was in the old class of paddle- engines, or even in the land engines made by Watt nearly a cen- tury ago. In fig. 19 we have an example of diagrams taken from the top and bottom of one of the paddle-engines of the steamer ' Great Eastern,' constructed by Messrs. J. Scott Russell and Co. These engines are oscillating engines of 74 inches diameter of cylinder, and 14feet stroke, making 10 revolutions per minute, and there are Fig. 19. DIAGRAMS FROM PADDLE-ENGINES OP ' GREAT EASTERN.' four cylinders, or two to each wheel. The mean pressure on the piston which these diagrams exhibit is 22'2 Ibs. per square inch, from which, with the other particulars, it is easy to compute the power. In fig. 20 we have two different pairs of diagrams. The larger pair is taken from one of the engines of the paddle- steamer ' Ulster,' and the smaller pah 1 represented iu dotted lines is taken from the engines of the paddle-steamer ' Victoria and Albert.' In the case of the ' Ulster ' the pressure of steam in the boiler when the diagram was taken was 26 Ibs. per square inch, and the vacuum in the condenser 13 Ibs. per square inch. The number of strokes per minute was 23, the mean pressure on the piston 28'77 Ibs. per square inch, and indicated horse-power DIAGRAMS FROM DIFFERENT STEAMERS. 353 4,100. The ' Victoria and Albert ' has two oscillating engines, with 88-inch cylinders and 7-feet stroke. The pressure of the steam in the hoilers when the diagrams were taken was 26 Ibs. per square inch ; of the vacuum 12 Ibs. per square inch ; the mean pressure on the piston 22-87 Ibs. per square inch, and the num- ber of strokes per minute 25 '4. The area of an 88-inch cylinder is 6082'! square inches, and the area of two such cylinders is the Fig. 20. COMPARATIVE DIAGRAMS PROM 'ULSTER* AND 'VICTORIA AND ALBERT.' double of this, or 12,164*2 square inches, and as there are 22*87 Ibs. on each square inch, the total pressure urging both pistons will be 12,164-2 times 22*87 or 278,195 Ibs. Now, as the length of Ahe stroke is 7 feet, and as the piston traverses it each way in each revolution, the piston will travel 14 feet for each revolution, and 278,195 multiplied by 14 will give 3,894,730 as the number of foot-pounds exerted in each stroke ; or, as there are 25*4 strokes each minute, there will be 25 '4 times 3,894,- 354 POWER AND PERFORMANCE OF ENGINES. 730, or 98,926,142 foot-pounds exerted each minute. Dividing this by 33,000, we get the power exerted by this engine as equal to 2997'7 actual horse-power. In the diagrams of the ' Victoria and Albert,' it will be re- marked there is a greater disparity in the period of the admission of the steam than in the case of the diagrams of the ' Ulster,' arising from the valves not being so accurately set. Diagram showing wrong setting of Valves. In fig. 21 are given two diagrams, taken from an engine making 200 strokes per minute, applied to work the exhausting apparatus employed by the Pneumatic Despatch Company to shoot letters and par- cels through a tube. These diagrams show that the valve is wrongly set, and that at one end of the cylinder the steam is ad- Fig. 21. DIAGRAMS FEOM ENGINE OF PNEUMATIC DESPATCH COMPANY 1 . mitted too soon, and at the other end too late. By following the right-hand diagram it will be seen that the eduction passage is closed when about half the stroke has been performed, and that the steam is admitted in front of the piston when about one- fourth of the stroke has still to be performed, whereas the left- hand diagram shows that a considerable part of the stroke has been performed before that end of the cylinder begins to get steam. The action in this case would be amended by shifting round the eccentric. The mean pressure on the piston shown by these diagrams is only 10'79 Ibs. per square inch. % Diagram showing the necessity of large Ports for high speeds of Piston. Fig. 22 represents two diagrams taken from the same engine with the unequal action at the different ends of the cylinder corrected. But the diagrams show that the engine has DIAGRAMS FROM FAST EXGINES. 355 not enough lead in the valves, and, moreover, that the passages are too small for the speed with which the engine works. It would be an advantage to increase either the width or the amount of travel of the valve of this engine, or hoth; as also to give more lead, so that the steam would be able to attain and maintain its proper pressure at the beginning of the stroke, and until it is purposely cut off. The mean pressure of steam on the piston shown by the diagrams represented in fig. 22 is 13'36 Ibs. per square inch. Diagrams illustrative of the action of the Link Motion. In fig. 23 we have a diagram taken from a horizontal engine, with 27-inch cylinder and 3-feet stroke, constructed by Messrs. Boulton and Watt, employed to work the Portsmouth Floating Fig. 22. DIAGRAMS FBOM ENGINE OP PNEUMATIC DESPATCH COMPANY. Bridge. The steam is cut off by the link so as to make the ad- mission almost the least possible, so as to test the engine itself before the chains which draw the bridge backward and forward had been applied. With the steam cut off thus early there is necessarily a very large amount of expansion, and also a very large amount of cushioning ; and it will be observed that the steam begins to be compressed at not much less than half-stroke. With this amount of expansion the link is 2 inches from the centre. The pressure of steam in the boiler was 22 Ibs., and that of the vacuum in the condenser 11 Ibs. per square inch, when this diagram was taken ; and the engines ran without the chains at 40 revolutions per minnte. Fig. 24 is another diagram taken from the same engine with 356 POWER AND PERFORMANCE OF ENGINES. the link in the same place. Pressure of steam in boiler, 21 Ibs. per square inch; pressure of vacuum in condenser, lljlbs. per square inch; number of revolutions per minute, 35. In this diagram, and also in the last, we have a small loop formed at the top of the diagram, from causes already explained. In fig. 25 we have another diagram taken from the same en- gine, but in this case the steam is not shut off by the link but by the throttle-valve, and there is consequently very little cushioning, and the loop at the top of the diagram almost dis- Fig. 23. DIAGRAM FROM ENGINE OF PORTSMOUTH FLOATING BRIDGE. (ENGINE THROTTLED BY appears. When the diagram was taken the pressure of steam in the boiler was 22 Ibs., and of the vacuum in the condenser 11J- Ibs. per square inch, and the number of revolutions per minute was 38. Figs. 26, 27, and 28 are diagrams taken by Eichards' indi- cator from Allen's engine, in the United States department of the International Exhibition of 1862. In this engine the diam- eter of the cylinder was 8 inches ; length of stroke, 24 inches ; pressure of steam in boiler, 49 Ibs. per square inch; revolutions per minute, 150. DIAGRAMS FROM FAST ENGINES. 357 Diagrams illustrative of action of Air-pump and Hot-well. -Fig. 29 is a diagram taken from the air-pump of the Duke of Fig. 24. Us. DIAGRAM FROM EXRINK OF PORTSMOUTH FLOATING BRIDGE. (ENGINE THROTTLED BY LINK.) Sutherland's yacht ' Undine,' a vessel fitted with two inverted angular engines, with cylinders 24 inches diameter and 15 inches Fig. 25. DIAGRAM FROM ENGINE OF PORTSMOUTH FLOATING BRIDGE. (ENGINE THEOTTLED BY THROTTLE- VALVE.) 358 POWER AND PERFORMANCE OF ENGINES. stroke. When this diagram was taken the ordinary amount of injection was on, and the engine was working at moorings at 72 strokes per minute. There was also an air-vessel on the hot-well. In fig. 80 we have a diagram taken from the air-pump of the same engine, with an extra amount of injection put on. Fig. 26. DIAGRAM FROM ALLEN S ENGINE. The pump appears to be quite too small for the work it has to do, as is seen hy the different configuration of the diagram from that of the diagrams represented in figs. 11 and 18, which are also diagrams taken from air-pumps. In those diagrams, how- ever, the stroke of the bucket is more than half performed, be- Figs. 27 and 28. DIAGRAMS FROM ALLEN S ENGINE. fore the pressure rises above the atmospheric line ; whereas in fig. 30, the pressure rises above the atmospheric line the moment the bucket begins to ascend, showing that at that time the whole of the pump bai'rel is filled with water. The vacuum must always be inferior where the air-pump is gorged with water. DIAGRAMS TAKEN FROM AIR-PUMPS. 359 Unlike the previous diagrams taken from air-pumps, we see in these figures the pressure or resistance has to be encountered from the beginning, or nearly the beginning of the stroke ; and the vacuum is not good, and the pump overloaded. There is a DIAGRAM FROM AIR-PUMP OF DUKE OF SUTHERLAND'S YACHT. (OBDINABY INJECTION.) worse vacuum with the increased injection than with the ordi- nary injection, showing that it is not the too great heat of the condenser which makes the vacuum bad, but a deficient capacity of pump, or an imperfect emptying of it every stroke. Fig. 30. DIAGRAM FROM AIR-PUMP OF DUKE OF SUTHERLAND S YACHT. (EXTRA INJECTION PUT os.) In fig. 31 we have a diagram illustrative of the diminished load upon the air-pump, caused by putting an air-vessel on the hot-well. A is the atmospheric line, and B is the line represent- 360 POWER AND PERFORMANCE OF ENGINES. ing the ordinary pressure existing in the hot-well when the air- vessel is in operation. By letting out the air the pressure rises to c, showing that the pressure on the pump is less with the air- vessel than without it. If the air-vessel be discarded, an in- creased velocity must be given to the water passing through the waste-water pipe to enable the bucket to ascend, and this im- plies a waste of power. Fig. si. DIAGRAM FROM HOT-WELL OF DUKE OF SUTHERLAND'S YACHT. (AIR- VESSEL ON.) In fig. 32 we have a diagram taken from the hot-well of the Duke of Sutherland's yacht after the air-vessel has been re- moved. In this diagram the pressure begins to rise pretty quickly, as the bucket of the pump ascends ; and the maximum pressure, when reached, is maintained pretty uniform to the end of the stroke. It does not then, however, suddenly fall, but only gradually, owing to the momentum of the water ; and the Fig. 32. DIAGRAM TAKEN FROM HOT-WELL OF DUKE OF SUTHERLAND S TACHT. (AIB- VESSEL OFF.) pencil does not again come down to the atmospheric line until nearly half the downward stroke of the pump has been com- pleted. In fig. 33 we have a diagram taken from the hot- well of the steamer ' Scud,' a vessel fitted with two single-trunk engines, that is, trunk engines with the trunks projecting only at one end, and not at both, as in Messrs. Penn's arrangement. The DIAGRAMS TAKEN FROM WATER-PUMPS. 361 engines are angular, working up to the screw-shaft, and the cylinders are 68 inches diameter, and 4J- feet stroke. The trunks are 41 inches diameter. These engines made 42 strokes per minute, and worked up to 8 times the nominal power. The diagram shows an increase of pressure in the hot- well at Fig. 33. DIAGRAM TAKEN FROM HOT-WELL OF STEAMER ' SCUD. each end of the stroke of the double-acting pump, and the pressure runs up slowly at each end of the stroke, when it slowly falls, forming the loop shown in the diagram. Diagram from Pump of Water-works. Fig. 34 is a diagram taken from the pump of a pumping-engine at the Cork Water- Fig. 34. V7V DIAGRAM TAKEN FROM PUMP OF CORK WATER-WORKS. works. This engine, in common with most pumping-engines of modern construction, is a rotative engine an innovation first effectually introduced by Mr. David Thomson. The engines make 31 revolutions per minute, and work with steam of 40 Ibs. on the square inch. "When the plunger is ascending, the pump 15 362 POWEK AND PERFORMANCE OF ENGINES is sucking ; and when the piston is descending it is forcing, and the diagram shows that both operations are accomplished with much regularity, and without any of those sudden fluctuations which always occasion a loss of power. Having now shown in what manner the indicator may be ap- plied lo ascertain the performance of ordinary engines, I shall proceed to describe the manner of its application in the case of double-cylinder engines. In this class of engines the steam having pressed the first piston to the end of its stroke in the manner of a high-pressure engine, escapes, not into the atmos- phere, but into another engine of larger dimensions, where it expands, and acts as low-pressure steam on the piston of the second engine, being finally condensed in the usual manner. The pressure urging the first, or high-pressure piston, is conse- quently the difference of pressure between the steam in the boiler and that in the second cylinder ; and the pressure urging the second, or low-pressure piston, is the difference of pressure between the steam on the eduction side of the high-pressure cylinder and that of the vapour in the condenser. There will be a small difference between the pressures in the communicating parts of the high and low-pressure engines, just as there is a small difference between the vacuum in the' cylinder and that in the condenser. But in well-constructed high-pressure engines this difference will not sensibly detract from the power. Diagrams from Double-cylinder Engines. In proceeding to determine the power of a double-cylinder engine, we first de- termine by a diagram and a computation, such as I have already given examples of, the power exerted by the high-pressure en- gine ; and then, in like manner, we determine the power exerted by the low-pressure engine. The total power is obviously the sum of the two. An example of the diagrams taken from the high and low- pressure cylinders of a double-cylinder engine, at the Lambeth "Water-works, constructed by Mr. David Thomson, and erected under his direction, will next be given. In a paper read by Mr. Thomson before the Institution of Mechanical Engineers, and a copy of which he has forwarded to me, the main particulars of OF THE DOUBLE CYLINDER KIND. 363 these engines are recited ; and some of the most material points of that paper I shall here recapitulate, as these engines consti- tute a very superior example of the double-cylinder class of engine. These engines are heam-engines, having the double cylinders at one end of the beam, and a crank and connecting-rod at the other end. Four engines of 150 horse-power each are fixed side by side in the same house, arranged in two pairs, each pair working on to one shaft, with cranks at right-angles, and a fly- wheel between them. The strokes of the crank and of the large cylinder are equal ; while the small cylinder, which receives the steam direct from the boiler, has a shorter stroke, and its effec- tive capacity is nearly one-fourth that of the large cylinder. The pumps are connected direct to the beams near the connecting- rod end by means of two side rods, between which the crank works. The pumps are of the combined plunger and bucket construction, and are thus double-acting, although having only two valves. This kind of pump, which is now in general use, was first introduced by Mr. Thomson at the Eichmond and the Bristol Water- works in the year 1848. The following are the principal dimensions of the engines : Diameter of large cylin- der, 46 ins. ; diameter of small cylinder, 28 ins. ; stroke of large cylinder, 8 ft. ; stroke of small cylinder, 5 ft. 6f ins. ; diameter of pump-barrel 23$- ins. ; diameter of pump-plunger, 16$ ins. ; stroke of pump, 6 ft. llf ins. ; length of beam between extreme centres, 26 ft. 6 ins. ; height of beam-centre from floor, 21 ft. 4 ins. The valves are piston- valves, connected by a hollow pipe, through which the escaping steam passes, and are so constructed that one valve effects the distribution of the steam in each pair of cylinders. The cylinder-ports are rectangular, with inclined bars across the faces to prevent the packing-rings of the valve from catching against the edges of the ports ; and the bars are made inclined instead of vertical, in order to avoid any tendency to grooving the valve-packing. The openings of the port extend two-thirds round the circumference of the valve in the ports of the large cylinder ; but they extend only half round in the ports of the 364 POWER AND PERFORMANCE OF ENGINES. small cylinder. The packing of the valve consists of the four cast-iron rings, which are cut at one side exactly as in an ordi- nary piston, the joint being covered by a plate inside. A con- siderably stronger pressure of the rings against the valve-chest is required than was at first expected, because the openings of the steamports extend so far round the valve ; and for this pur- pose springs are placed inside the packing-rings to assist their own elasticity. This construction of valve has the advantage of admitting of great simplicity in the castings of the cylinders ; and also allows of the whole of the valve-work being executed in the lathe, which is generally the cheapest and most correct kind of work in an engineering workshop. These valves are worked by cams. The principal object aimed at in the construction of this piston-valve was a reduction to a minimum of the loss of pres- sure which the steam undergoes in passing from the small cyl- inder to the large one. This is here accomplished by making the passage of moderate dimensions and as direct as possible ; and also by preventing any communication of this passage with the condenser, so that when the steam from the small cylinder enters the passage, the latter is already filled with steam of the density that existed in the large cylinder at the termination of the previous stroke. In constructing the engines some doubt was entertained as to the best size of passage, in order on the one hand to avoid throttling the steam, and on the other to ob- viate as much as possible the loss of steam in filling the passage. The size adopted was a pipe 6 inches in diameter, or l-60th of the area of the large cylinder, for a speed of piston of 230 feet per minute in the large cylinder : and this is believed to be about the best proportion, the entire cubic content of the whole passage in the valve amounting to 3,944 cubic inches. The indi- cator diagrams show that with this construction of valve there is very little or no throttling of the steam, and also that there is but a very moderate drop in the pressure as the steam passes from the small cylinder into the large one. In this respect the valve completely answered the expectations entertained of it, and left little further to be desired on this point. DIAGRAMS FROM DOUBLE CYLINDERS. 365 In figs. 35 and 36 we have diagrams taken simultaneously from the top of the small cylinder and the bottom of the large one, in the double-cylinder engines of the Lambeth Water- works, designed by Mr. Thomson the high-pressure diagram being placed above, and the low-pressure diagram below, with a small space between the two answering to the loss of pressure in the communicating pipe. The dotted line shows the exhaust- line in the small cylinder reversed, so as to tell by direct measure- Figs. 35 and 36. DIAGRAMS FROM DOUBLE-CYLINDER ENGINES, LAMBETH WATER-WORKS. (TAKEN SIMU I.TA M:. .rsi.v FBOM TOP or SMALL v LIN :>! i: AND BOTTOM OF LABOB CYLINDER.) ment between this bottom and the top of the diagram what is the pressure of the steam on the small piston at every part of its stroke. The most material of the results which may be deduced from the indicator diagrams of this engine are as follows : Percent- age of stroke at which steam is cut off in small cylinder, 40 per cent.; total expansion at end of stroke in small cylinder, in terms of bulk before expansion, 2 '41 per cent. ; amount of ex- pansion on passing from small to large cylinder, in terms of bulk 3G6 POWER AND PERFORMANCE OF ENGINES. before escaping from small cylinder, 1'18 per cent.; total expan- sion at end of stroke in large cylinder, in terms of original bulk, 9-66 per cent. ; total amount of efficient expansion, in terms of original bulk, 8'19 per cent. ; total pressure of steam per square inch at point of cutting off, 41 Ibs. ; theoretical total pressure at end of stroke of small piston, 1TO Ibs. ; actual total pressure shown by diagram, 18'0 Ibs. ; excess of actual over theoretical in percentage of actual pressure, 6 per cent. ; theoretical loss of pressure in passage from small to large cylinder, 2'6 Ibs. ; actual loss shown by diagram, 4'5 Ibs. ; theoretical total pressure at end of stroke of large piston, 4- 2 Ibs. ; actual total pressure shown by diagram, 5*5 Ibs. ; excess of actual over theoretical in per- centage of actual pressure, 23 per cent.; mean pressure on crank-pin from both cylinders, 22,400 Ibs. ; maximum ditto, 36,- 058 Ibs. ; ratio of maximum to mean, 1'61 to I'OO ; ratio of max- imum to mean pressure on crank-pin in a single cylinder engine with the same total amount of efficient expansion, the clearances and ports bearing the same proportion to the working capacity of the cylinder, namely, l-40th part (this ratio is calculated from the ordinary logarithmic expansion curve), 2*75 to I'OO ; effi- ciency of steam contained in large cylinder at end of stroke, as shown by diagram, if used without expansion, taken as 1*00 ; actual efficiency of same steam as employed in both cylinders, as shown by diagram, 2 - 90 ; theoretical efficiency of the same steam if expanded to the same degree as the total amount of efficient expansion, 3'10. The engines are fitted with steam- jackets, and these indicator diagrams show that the pressure of the steam at the end of the stroke, instead of falling short of what it ought to be by the theoretical expansion curve, exceeds that amount by about 23 per cent, of the actual final pressure. It might be supposed that the increased pressure at the end of the stroke was due to the heat imparted from the jackets either superheating the steam or converting the watery vapour mixed with it into true steam ; and probably the latter is the cause of a small part of the observed effect ; but Mr. Thomson considers it less likely that sufficient heat could be communicated from the jackets to produce an increase of 23 per cent, in the actual FEATURES OF WATER-WORKS ENGINES. 367 final pressure, especially as on several occasions the condensed water from the jackets has been collected and found not to ex- ceed half-a-gaUon per hour. The experiments made on the quantities of water passed from the boilers give uniformly the result, that a considerably larger quantity of water passes from the boilers than is accounted for by the indicator diagrams, taking the quantity and pressure of the steam just before it escapes to the condenser as the basis of calculation. In some trials made within a few days of these diagrams being taken, the excess of water thus disappearing from the boilers was about 37 per cent. To suppose that the valve was leaking might ac- count for it ;* but besides great care having been taken to avoid this source of error, it can hardly be supposed that the valve was always leaking more than the pistons. To ascertain the amount of friction in these engines Mr. Thomson made many experiments, and found that, when the engines were new, and working at perhaps little more than half their power, the loss in comparing the work done with the indi- cator diagrams amounts to as much as 25 per cent, of the indi- cated power ; but in these cases the pistons have been too tight in the cylinders, and when this error has been corrected, and the engines worked up to their regular work, all the losses were brought down to from 12 to 15 per cent, of the indicated power. This includes the friction of both the engines and the pumps, the working of the air-pumps, feed-pumps, cold-water pumps, and pumps for charging the air-vessels with air. With regard to the economy of fuel attained by these double- cylinder engines, it may be stated that the four pumping-engines at the Lambeth "Water- works are fixed in one house, and are employed in pumping through a main-pipe 30 inches diameter and about nine miles in length ; and when all the engines are working together at their ordinary speed of 14 revolutions per minute, the lift on the pumps, as measured by a mercurial gauge, is equal to a head of about 210 feet of water. Under these cir- cumstances they were tested by Mr. Field soon after being fin- * Some of the disappearance of the heat is no doubt impntable to its transfor- mation into^ower, as explained under the head of thermo-dynamics. 3G8 POWER AND PERFORMANCE OF ENGINES ished, in a trial of 24 hours' duration without stopping. The actual work done by the pumps during this trial was equal to 97,064,894 Ibs., raised one foot high for every 112 Ibs. of coal consumed ; in addition to which this consumption included the friction of the engines and pumps, and the power required to work the air-pumps, feed and charging-pumps, and the pumps raising the water for condensation. The coal used was "Welsh, of good average quality. The economy in consumption of fuel during this trial, and in the subsequent regular working of these engines, together with the satisfactory performance generally of the engines and pump work, induced the Chelsea "Water- works Company, and also the New Eiver Company, each to erect in 1854 a set of four similar engines, which were made almost exactly the same as the Lam- beth Water- works engines already described, with the exception that a jacket of high-pressure steam was in these subsequent en- gines provided under the bottoms of the cylinders, which had not been done with the previous engines. The pumps were also different in size to suit the different lifts. The New Eiver engines were tested soon after being com- pleted, and the result reported was 113 million Ibs. raised one foot high by 112 Ibs. of "Welsh coal. But this duty was obtained from a trial of only seven or eight hours' duration, which is too short to obtain very trustworthy results. The set of engines made for the Chelsea "Water-works was the last finished, and on completion the engines were tested by Mr. Field in the same manner as the Lambeth engines, by a trial of 24 hours' continuous pumping. The coal used was Welsh, as be- fore, and the duty reported was 103'9 million Ibs. raised one foot high by 112 Ibs. of coal. This, as in the previous instance, was the duty got from the pumps in actual work done, no allowance being made for the friction of the engines and pumps, and the power required to work the air-pumps, cold-water pumps, &c. At the time of these engines being tested, the loss by friction and by working the air-pumps, &c., averaged about 20 per cent. of the power, as given by the indicator diagrams ; so that if the duty had been estimated from the indicator diagrams, fts is usual OF THE DOUBLE CYLINDER KIND. 369 in marine engines, it would have been 103'9 x J^, or about 130 million Ibs. raised one foot by 112 Ibs. of coal, which is equiva- lent to a consumption of 1 - T lb. per indicated horse-power per hour. In figs. 37 and 38 we have diagrams taken from a small en- gine called Wenham's double-cylinder engine, working with a pressure of 40 Ibs. per square inch in the boiler, and exhibited at the Great Exhibition in 1862. The average pressure on the Pig. 3T. DIAGRAM FROM HIGH-PRESSURE CYLINDER OF WENHAM's DOUBLE-CYLINDER ENGINE. (CYLINDER THREE INCHES DIAMETER AND TWELVE INCHES STROKE.) piston of the high-pressure engine, which is 3 inches diameter and 12 inches stroke, is 26'61bs. per square inch, and the power it exerts is 3*16 horses. The average pressure exerted on the Fig. 38. DIAGRAM FROM LOW-PRESSURE CYLINDER OF WENHAM's DOUBLE-CYLINDER ENGINE. piston of the low-pressure engine is 8'5 Ibs. per square inch, and the power it exerts is 2*37 horses. The steam in passing from one cylinder to the other is heated anew, as had previously been done by me in the engines of the steamer 'Jumna,' of 400 horse- power. The total power developed in both cylinders of Wen- ham's engine is 6*05 horses. Having now explained how to interpret a diagram, the next thing is to explain how to take one, and here I cannot do better 15* 370 POWER AND PERFORMANCE OF ENGINES. than recite the instructions for this operation issued with Eichards' indicator by the makers, Elliot Brothers, of the Strand. To fix, the Paper. Take the outer cylinder off from the instrument, secure the lower edge of the paper, near the corner, by one spring, then bend the paper round the cylinder, and insert the other corner between the springs. The paper should be long enough to let each end project at least half-an-inch between the springs. Take the two projecting ends with the thumb and finger, and draw the paper down, taking care that it lies quite smooth and tight, and that the corners come fairly together, and replace the cylinder. The spring used on this indicator for holding the paper will be found preferable to the hinged clamp. A little prac- tice, with attention to the above directions, will enable any one to fix the paper very readily. Tlit, marking-point should be fine and smooth, so as to draw a fine line, but not cut the paper. It may be made of a brass wire ; the best material is gun-metal, which keeps sharp for a long time, and the line Fig. 39. made by it is very durable. Lines drawn by German silver points are liable to fade. A large-sized common pin, a little blunted, answers for a marking-point very well indeed ; a small file and a bit of emery cloth used occasionally will keep the point in order. To connect the Cord,. The indicator having been attached, and the correct motion obtained for the drum, and the paper fixed, the next thing is to see that the cord is of the proper length to bring the diagram in its right place on the paper that is, midway between the springs which hold the paper on the drum. In order to connect and disconnect readily, the short cord on the indicator is furnished with a hook, and at the end of the cord coming from the engine a running loop maybe rove in a thin strip of metal, in the manner shown in the preceding cut, by which it can be readily adjusted to the proper length, and taken up from time to time, as it may become stretched by use. On high-speed en- gines, it is as well, instead of using this, to adjust the cord and take up the stretching, as it takes place, by tying knots in the cord. If the cord becomes wet and shrinks, the knots may need to be untied, but this METHOD OP TAKING A DIAGRAM. 371 rarely happens. The length of the diagram drawn at high speeds should not exceed four and a-half inches, to allow changes in the length of the cord to take place to some extent, without causing the drum to revolve to the limit of its motion in either direction. On the other hand, the diagram should never be drawn shorter than is necessary for this purpose. To take the Diagram. Every thing being in readiness, turn the han- dle of the stop-cock to a vertical position, and let the piston of the in- dicator play for a few moments, while the instrument becomes warmed. Then turn the handle horizontally to the position in which the commu- nication is opened between the under side of the piston and the atmos- phere, hook on the cord, and draw the atmospheric line. Then turn the handle back to its vertical position, and take the diagram. When the handle stands vertical, the communication with the cylinder is wide open, and care should be observed that it does stand in that position whenever a diagram is taken, so that this communication shall not be in the least obstructed. To apply the pencil to the paper, take the end of the longer brass arm with the thumb and forefinger of the left hand, and touch the point as gently as possible, holding it during one revolution of the engine, or during several revolutions, if desired. There is no spring to press the point to the paper, except for oscillating cylinders ; the operator, after admitting the steam, waits as long as he pleases before taking the dia- gram, and touches the pencil to the paper as lightly as he chooses. Any one, by taking a little pains, will become enabled to perform this opera- tion with much delicacy. As the hand of the operator cannot follow the motions of an oscillating cylinder, it is necessary that the point be held to the paper by a light spring, and instruments to be used on engines of this class are furnished with one accordingly. Diagrams should not be taken from an engine until some time after starting, so that the water condensed in warming the cylinder, Ac., shall have passed away. Water in the cylinder in excess always distorts the diagram, and sometimes into very singular forms. The drip-cocks should be shut when diagrams are being taken, unless the boiler is priming. If when a new instrument is first applied the line should show a little evidence of friction, let the piston continue in action for a short time, and this will disappear. As soon as the diagram is taken, unhook the cord ; the paper cylin- der should not be kept in motion unnecessarily, as it only wears out the spring, especially at high velocities. Then remove the paper, and minute on the back of it at once as many of the following particulars as you have the means of ascertaining, viz. : The date of taking the diagram, and scale of the indicator. 372 POWER AND PERFORMANCE OF ENGINES. The engine from which the diagram is taken, which end, and which engine, if one of a pair. The length of the stroke, the diameter of the cylinder, and the num- ber of double strokes per minute. The size of the ports, the kind of valve employed, the lap and lead of the valve, and the exhaust lead. The amount which the waste-room, in clearance and thoroughfares, adds to the length of the cylinder. The pressure of steam in the boiler, the diameter and length of the pipe, the size and position of the throttle (if any), and the point of cut- off. On a locomotive, the diameter of the driving-wheels, and the size of the blast orifice, the weight of the train, and the gradient, or curve. On a condensing-engine, the vacuum by the gauge, the kind of con- denser employed, the quantity of water used for one stroke of the en- gine, its temperature, and that of the discharge, the size of the air-pump and length of its stroke, whether single or double acting, and, if driven independently of the engine, the number of its strokes per minute, and the height of the barometer. The description of boiler used, the temperature of the feed-water, the consumption of fuel and of water per hour, and whether the boilers, pipes, and engine are protected from loss of heat by radiation, and if so, to what extent. In addition to these, there are often special circumstances which should be noted. Counter and Dynamometer. There are other instruments besides the indicator for telling the performance of an engine the counter which registers the number of strokes made by an engine being nsed for this purpose, in the case of pumping- engines, working with a uniform load, and the dynamometer being employed in testing the power exerted by small engines. The dynamometer consists of a moving disc well oiled, and en- circled by a stationary hoop, which can be so far tightened as to create sufficient friction to constitute the proper load for the en- gine. The hoop is prevented from revolving with the disc by an arm extending from it, which is connected with a spring, the tension on which, reduced to the diameter of the disc, represents the load which the friction creates ; and the load multiplied by the space passed through per minute by any point on the cir- cumference of the disc will represent the power. Such dyna- NEW FORM OF DUTY METER. 373 mometers, however, cannot be conveniently applied to large en- gines ; and as in steam-vessels, where economy of fuel is most important, the counter will not accurately register the work done, seeing that the resistance is not uniform, and as without some reliable means of determining the power produced in dif- ferent vessels relatively with the fuel consumed, it is impossible to establish such a comparison of efficiency as will lead to emula- tion, and consequent improvement, I have felt it necessary to contrive a species of continuous indicator, or power-metor, for Fig. 40. BOUENK'S DUTT METER. ascertaining and recording the amount of work done by any engine during a given period of time. The outline of one form of this instrument is exhibited in fig. 40 ; but I prefer that the cylinder should be horizontal instead of vertical, and that it should be larger in diameter, and shorter this figure being copied from a photograph of an instrument I had converted from a common M'Naught's indicator, for the sake of readiness of construction. In this instrument one end of the indicator cylin- der communicates with one end of the main cylinder, and the other end of the indicator cylinder with the other end of the 374 POWER AND PERFORMANCE OF ENGINES. main cylinder, so that the atmosphere does not press upon the piston of the indicator at all, but that piston is pressed on either side by steam or vapour of precisely the same tension as that which presses on either side of the piston of the engine. The indicator piston is pressed alternately upward and downward against a spring in the usual manner. A double-ended lever vi- brating on a central pivot, and with a slot carried along it near- ly from end to end, as in the link of a common link-motion, is attached to the side of the cylinder, and from this slot a horizon- tal rod extends to the arm of a ring encircling a ratchet-wheel, there being a number of pawls in this ring of different lengths to engage the ratchets. This link is moved backwards and forwards on its centre, 8 or 10 times every stroke of the engine, by means of the lower horizontal rod which is attached at one end to the lower end of the link, and at the other end to a small pin in the side of a drum, which is drawn out by a string, like the drum for carrying the paper in a common indicator, and is, in like manner, returned by a spring ; but the dimensions of the drum, and the place of attachment of the string, are such that the drum makes a considerable number of turns say 10 for each stroke of the engine, and the link makes the same number of recipro- cations. If there be an equality of pressure on each side of the piston, the end of the rod moving in the slot will be in the mid position ; and as while it is there no amount of vibration of the link will give it any end motion, there will be no motion under such circumstances communicated to the ratchet. If, however, the pressure either upward or downward is considerable, the end of the rod will be moved so much up or down in the link that its reciprocation will give considerable end motion to the rod communicating with the ratchet ; and the amount of motion given to the ratchet every stroke will represent the amount of mean pressure urging the piston. The number of revolutions to be made by the drum every stroke having been once definitively fixed, it is clear that the number of revolutions it will make per minute will depend on the number of strokes made per minute by the engine, and the revolutions of the ratchet-wheel will consequently represent both the mean pressure and the speed of HEATING SURFACE IN MODERN BOILERS. 375 piston or in other words, it will represent the power. The spindle of the ratchet wheel is formed into a screw, which works into the periphery of a wheel that gives motion to other wheels and hands, like the train of a gas-meter ; and on opening the in- strument at the end of any given time, such as at the termina- tion of a voyage of an ocean steamer, the power which the ves- sel has exerted since she started on the voyage will be found to be accurately registered. This being compared with the quan- tity of coals consumed, which can easily be found from the books of the owners, will give the duty of the engine ; and by ascer- taining and publishing the duty of different vessels, a wholesome emulation would be excited among engine-makers and engine tenders, and a vast reduction in the consumption of fuel would no doubt be obtained. For many years past I have urged the introduction of that system of registration in the case of steam- vessels which in the case of the Cornish engines speedily led to such unprecedented economy. But the want of a suitable register- ing apparatus constituted a serious impediment, and I have con- sequently undertaken to contrive the instrument of which a rough outline is given above. Heating Surface in modern Boilers. The quantity of heat- ing surface given in modern boilers per nominal horse-power has been constantly increasing, until, in some of the boilers of recent steam- vessels intended to maintain a high rate of speed, it has become as much as 35 square feet per nominal horse-power ; and such vessels exert a power nine times greater than the nominal power. The nominal power, in fact, has ceased to be any measure of the dimensions of a boiler ; and the best course will be to con- sider only the water evaporated. In modern marine boilers it may be reckoned that a cubic foot of water will be evaporated in the hour by Y Ibs. of coal burned on 70 square inches of fire- bars, and the heat from which is absorbed by 10 square feet of heating surface, so that the consumption of coal per hour, on each square foot of grate, will bo 14'4 Ibs. If the steam be cut off from the cylinder when one-third of the stroke has been per- formed, as is a common practice, the efficiency of the steam will be somewhat more than doubled, or a horse-power will be 376 POWER AND PERFORMANCE OF ENGINES. generated with something less than 3J Ibs of coal. In large boilers and engines, however, the efficiency is greater than in small, and there is further benefit obtained from superheating, and from heating the feed-water very hot. In modern steam- vessels of efficient construction, therefore, the consumption of coal is not more than 2^ Ibs. per actual horse-power. Boulton and "Watt put sufficient lap upon their valves to cut off the steam when two-thirds of the stroke have been performed as a minimum of expansion ; and then, by aid of the link-motion, they can ex- pand still more, if required, so as to cut off when one-third of the stroke has been performed. The area of the back uptake should be 1 5 square inches per cubic foot evaporated ; the area of the front uptake 12 square inches, and the area of the chimney 7 square inches per cubic foot evaporated. These proportions will enable the dimensions of any boiler to be determined when the rate of expansion has been fixed. The proportion in which the actual exceeds the nominal power varies very much in different engines, but about 4 or 4 times appears to be the prevalent proportion in 1865, though, as I have stated, in special cases twice this proportion of power is exerted, and the boilers are proportioned to give the increased supply of steam required. For any temporary purpose the power may be increased by quickening the draught through the furnace by a jet of steam in the chimney; but in such case the consump- tion of fuel per cubic foot of water evaporated will be somewhat in- creased. The first proportion of heating surface, however, which the flame encounters is very much more efficient than the last portion, in consequence of the higher temperature to which it is subjected ; and if the draught be quickened the temperature will be increased, and every square foot of heating surface will thereby acquire a greater absorbing power. The hotter the furnace is, the more heat will be absorbed by the water in the region of the furnace; and the more heat that is absorbed by the furnace the less will be left for the tubes to absorb. It is material, therefore, to maintain high bridges, a rapid draught, and all other aids to a high temperature in the furnace ; as the absorption of heat will thus be more rapid, and the combustion will be more perfect, ADVANTAGES OF HOT FUBNACES. 377 from the high temperature to which the smoke is exposed. It will increase the efficacy of the heating surface, moreover, if the smoke he made to strike against instead of sliding over it ; and this end will he best attained by using vertical tubes, with the water within them, on which the smoke may strike on its way to the chimney. Such tubes, furthermore, are eligible in con- sequence of the facilities they give for the rapid circulation of the water within the boiler ; and this rapid circulation will not merely render the boiler more durable by preventing overheat- ing of the metal, but as the rapidly ascending current, by carry- ing off the steam and presenting a new surface of water to be acted upon, keeps the metal of the tubes cool, they are in a better con- dition for absorbing heat from the smoke than if the metal had become overheated from the entanglement of steam in contact with it, which impeded the access of the water, and prevented the rapid absorption of heat which would otherwise take place. In locomotive boilers, where the temperature of the furnace is very high, as much evaporative efficacy is obtained from 7 Ihs. of coal, with 5 or 6 square feet of heating surface, as is obtained in land and marine boilers with 9 or 10 ; and the reason manifestly is, that as the rapidity of the transmission of heat increases as the square of the temperature, a square foot of heating surface in a fur- nace twice as hot will be four times more effective, so that the tubes are left with comparatively little work to do, from so much of the work having been done in the furnace. Each square foot of tube surface in locomotives will only evaporate as much as each square foot in an ordinary land and marine boiler ; but the mean efficacy of the whole heating surface is, nevertheless, raised very high by the greatly increased efficacy of the fire-box surface, from its high temperature. It is desirable to imitate these con- ditions in marine and land furnaces by making the area fire-grate small, the draught rapid, and the bridges high, to the end that a high temperature in the furnace may be preserved, and a con- sequently rapid generation of steam promoted. It would also be desirable, and not difficult, to feed the furnaces with hot air instead of with cold, which would conduce more to economy than feeding the boiler with hot instead of cold water ; and it would not be dif- 378 POWER AND PERFORMANCE OF ENGINES. ficult to carry out this improvement, by encircling the chimney with air-casing nearly to the top, and conducting the air which would be admitted by openings around the casings at its upper end, past the smoke-box doors, to the end of the furnaces. The only diffi- culty which might be apprehended from this procedure would be the increased heat and diminished durability of the furnace-bars. But this difficulty might no doubt be surmounted by making the bars deep and thin, and by not increasing the temperature of the entering air beyond the point which experience proved it could be raised to with impunity. The area of the casing around the chim- ney would require to be about as great, at the largest part, as the area of the chimney itself. But it could be made conical, or tapering off at the top, and the air might be admitted in vertical slits extending downwards for a certain length, as the heat at the top of the chimney could be abstracted by such a small volume of air as a narrow casing would contain. In this heating of the air entering furnaces there is an expedient of economy available for the engineer which has not yet been brought into force ; and its effect will be both to reduce the consumption of the fuel and to render the existing heating surface more effective. If, for ex- ample, we take the existing temperature of the furnace to be 3,000 Fahrenheit, and if we increase the temperature of the en- tering air by 500, which we might easily do without any new expense, we shall not merely save one-sixth of the fuel, but we shall render the absorbing surface of the furnace more efficacious by raising the temperature from 3,000 to 3,500. Nor will this probably be the limit of benefit obtained ; and as in feeding boilers with boiling water instead of cold, and in surrounding cylinders by steam to keep them hot instead of exposing them to the atmosphere, we obtain a greater benefit than theory would have led us to expect, so in feeding furnaces with hot air instead of cold air we shall in all probability obtain a larger benefit than that which theory indicates. The experience already obtained of the saving effected by using the hot blast in iron smelting furnaces certainly points to the probability of such a realization ; and one manifest effect will be, that the combustion of the coal will be rendered more perfect, and less smoke will be produced. DESIDERATA AT THE PRESENT TIME. 379 The present system of land and marine boilers, however, is altogether faulty, and must be changed completely. When I planned and constructed the first marine tubular boiler in 1838, and which was adapted for working with a high pressure of steam, and which also had the advantage of surface condensation, the innovation was a step in advance, and it has proved successful and serviceable, though up to the present time the system then propounded by me has not been fully wrought out in practice. But we now want something much better than what would have sufficed for our wants in 1838, and I will here briefly recapitulate what we require and must obtain. First, then, we must have a still higher pressure of steam than I contemplated in 1838 ; to obtain which with safety we must have two things ; a very strong boiler, and absolute immunity from salting. The expedient of surface condensation, which I propounded in 1838, as the means of accomplishing the last disideratuni, though effectual for the purpose, and now widely adopted, is less eligible for moderate pressures than the method of preventing salting which I have since suggested, and which consists in the introduction of a small jet in the eduction-pipe, the water of which, though unable wholly to condense the steam, will be itself raised to the boiling point, and be transmitted to the boiler without any means of stopping it off; and the excess of feed- water which, under this arrangement, will always be entering the boiler, will escape through a continuous blow off, and thus prevent the boiler from salting. The column of steam escaping to the condenser will, under suitable arrangements, itself force this water into the boiler ; and in locomotives, in like manner, the water may be forced into the boiler by using a portion of the steam escaping from the blast pipe for that purpose, whereby the boiler will be fed with boiling water by the aid of steam otherwise going to waste. In this way marine boilers may be kept from salting ; for the sulphate of lime which is deposited from sea water at the tem- peratures of high-pressure steam, may be separated by filtration in the feed pipe. On the whole, for high pressures a small sur- face condenser with auxiliary jet seems best. To give a rapid circulation to the water, and render the heating surface efficient 380 POWER AND PERFORMANCE OF ENGINES. in the highest degree, the tubes should he upright with the water within them ; and the furnaces should be fed with coal by self- acting mechanism, which would abridge the labor of firing, and insure the work being better done. To reduce the strain on the engine at the beginning of the stroke, when steam of a high pressure is employed, the stroke should be long, the piston small in diameter, and a considerable velocity of piston should be em- ployed ; or, where there are two engines, the steam may be ex- panded from the cylinder of one engine into the cylinder of the other engine, according to Nicholson's system, whereby twice the expansion will be obtained with only the same apparatus. Relative surface areas of Boilers and Condensers. The evaporative power of land and marine boilers per square foot of heating surface, depends very much upon the structure and con- figuration of the boiler. In some marine engines a performance of six times the nominal power has been obtained with a propor- tion of heating surface in the boiler of only 12 square feet per nominal horse-power ; and as about half of this power was ob- tained by expanding the steam, 1 cubic foot of water was evap- orated by every 4 square feet of heating 'surface, which is a smaller proportion even than that which obtains commonly in locomotives. In such cases the proportion of cooling surface in the condenser has been made equal to the amount of heating surface in the boiler ; and the amount of cooling surface in the condenser relatively to the amount of the heating surface of the boiler should manifestly have reference to the activity of that heating surface. So in like manner it should be influenced by the amount of expansion which the steam undergoes in the cyl- inder ; since the steam, in communicating power, parts with a corresponding quantity of heat. A still more important condi- tion of the action of the condenser is, that the water shall pass through the tubes with rapidity, and that it shall flow in the op- posite direction to the steam, so that the hottest steam shall meet the warmest water ; as warm water will suffice to condense hot steam, which would be quite inoperative in condensing at- tenuated vapour. A common proportion of condenser surface in modern engines is '15 that of the boiler surface. Thus a INTERNAL CORROSION OF BOILERS. 381 boiler with 20 square feet of heating surface will have 15 square feet of heating surface. But the largest part of this surface is required to obtain the last pound or two of exhaustion ; and it is preferable to employ a moderate surface to condense the bulk of the steam, and to condense the residual vapour by a small jet of salt water let in from the sea. It is found advisable to admit a small quantity of salt water on other grounds. For the fresh water in the boiler, as it forms no scale, leaves the boiler subject to the corrosive influence produced by placing a mass of copper tubes on which the sea water acts chemically in connexion with the mass of wet iron which constitutes the boiler ; and, as in Sir Humphrey Davy's arrangement for protecting copper sheathing by iron blocks, the copper tubes are protected at the expense of the boiler, since the communicating pipes and the water within them form an efficient connexion. It would be easy to break the circuit so far as the metal is concerned by in- terposing glass flanges between the flanges of the pipes. But this would not stop the communication by the water itself, and the best course appears to satisfy the corroding conditions by placing blocks of zinc within the condenser, which might be corroded instead of the tubes or the boiler. The present anti- dote to the corrosive action consists in the introduction of a cer- tain proportion of salt water into the boiler, which is intended to shield the evaporating surfaces from corrosive action by de- positing a coating of scale upon those evaporating surfaces. But in this arrangement we have necessarily an excess of water entering the boiler ; for we have not only all the water returned which passes off as steam, but a certain proportion of sea water besides. It will consequently be necessary to provide for the excess being blown out of the boiler ; and the question is, whether, as we must introduce such an arrangement, it would not be ad- visable, with low pressures, to make the proportions such as would enable us to dispense with the surface condenser alto- gether ? If it is retained at all, it should only be retained in such shorn proportions as to condense the grossest part of the steam the water resulting from which should be sent into the boiler quite hot, and the rarer part of the steam should be con- 382 POWER AND PERFORMANCE OF ENGINES. densed by a jet of salt water of about the same dimensions as that already employed. It is very necessary to be careful in the case of surface condensers to prevent any leakage of air, which, if mingled with the steam, would form a wall of air against the refrigeratory surface, which would prevent the contact of the steam and hinder the condensation, precisely as it was found to do in the old engines of Newcomen, where air was purposely admitted to form a stratum between the hot steam and the cold cylinder ; and which diminished the loss from the condensation of the steam within the cylinder to a very material extent. Example of modern marine engine and boiler. As an exam- ple of the proportions of marine engines and boilers and con- densers of approved modern construction, I may here recapitu- late the main particulars of the machinery of the screw steamer ' Rhone,' constructed for the West India Mail Company by the Millwall Iron Company in 1865. These engines are on the inverted cylinder principle of 500 horse-power. There are two cylinders of 72 inches diameter and 4 feet stroke, and the estimated number of revolutions per minute is 52. The cylinders are supported on massive hollow standards resting on a bed plate of the same construction. There are two air-pumps wrought by links and levers from two pins on the ends of the piston rods. The surface condenser is placed between the two air-pumps, and is fitted with brass tubes placed horizontally, and resting in vertical tube plates. The two end plates have screwed stuffing boxes, with cotton washer packing for each tube. The tubes are divided into three groups or sec- tions, through each of which the condensing wftter successively passes ; and the water enters from the lower end of the con- denser and escapes at the upper end, where the steam enters, so that the hottest water meets the hottest steam. The two circu- lating pumps are placed opposite each other, and are wrought by a crank on the end of the crank shaft. The steam is condensed outside the tubes ; and the condensed water flows down to the^ air-pumps, by which it is pumped to the hot well, and from which it is taken to the boilers in the usual way. The crank shaft is of Krupp's cast steel in two pieces, PRINCIPLE OP GIFFARD'S INJECTOR. 383 coupled by flanges. The screw shaft is of iron, covered with brass in the stern tube, and working in lignum vitse bearings in the stern tube and after stern post. The boilers are in four separate parts, and fitted with a superheating apparatus consist- ing of a series of vertical iron tubes 4J- inches bore, on the plan of Mr. Ritchie, the company's superintending engineer. The surface condenser has 3,566 tubes, inches external diameter, and 9 feet 2J inches long between the tube plates. The surface of the tubes is 6,525 square feet, or 13-05 square feet per nominal horse-power. The two circulating pumps are dou- ble acting 25" diameter, with a trunk of 17" diameter on one end of the plungers. The boilers have 20 furnaces 3' 0" wide, with fire bars of 6 feet 8 inches in length. The total area of fire grate is 400 square feet, = 0'8 square feet per nominal horse- power. The number of brass tubes in the boiler is 1,180 of 3 external diameter and 6 feet 8 inches long. The total heating surface in the boilers is 9,800 square feet, or 19*6 square feet per nominal horse-power. In the superheater the surface is 2,160 square feet, or 4*32 square feet per nominal horse-power, making the total heating surface in boiler and superheater 23*92 square feet per nominal horse-power. The area of heating surface in the boiler per square foot of grate is 24'5 square feet, and the area of superheating surface per square foot of grate is 5-4 square feet, making the total heating surface in boiler and superheater 29'9 square feet per square foot of grate. The total area of the condenser surface is '66 of the total heating surface in the boiler, and '54 of the total area of the heating surface of boiler and superheater taken together. These engines are very strong, and manifestly embody the results of the long experience of steam navigation which the West India Mail Company must now pos- sess. The workmanship and materials are of the very first quality; and accurate adjustment and conscientious construc- tion are manifested throughout. GiffarcCs Injector. This instrument, which feeds boilers by a jet of steam discharged into the feed pipe, acts on the principle that the particles of water which obtain a high velocity when they flow oxit as steam retain this velocity when reduced by 384 POWER AND PERFORMANCE OF ENGINES. condensation to the form of water ; and a jet of water of great velocity is capable of balancing a correspondingly high head, or a pressure greater than that which subsists within the boiler. The jet consequently penetrates the boiler, as we can easily un- derstand any jet would do which has a greater velocity than a similar jet escaping from the boiler. These injectors, though very generally employed in locomotives, are not much used for land or marine boilers ; and in their present form they occasion much waste, as the steam by which they are actuated is drawn from the boiler, whereas it ought to be the steam, or a portion of it, which escapes to the condenser or the atmosphere. These injectors, like Bourdon's gauges, and other instruments employed in the steam-engine, are not made by engineers, but are a dis- tinct manufacture; and the manufacturers, on being supplied with the necessary particulars, furnish the proper size of instru- ment in each particular case. The proper diameter of the nar- rowest part of the instrument to deliver into the boiler any given number of gallons per hour, may be found by dividing the num- ber of gallons required to be delivered per hour by the square root of the pressure of the steam in atmospheres, and extracting the square root of the quotient, which, multiplied by the con- stant number '0158, gives the diameter in inches at the smallest part. Contrariwise, if we have the size, and wish to find the delivery, we multiply the constant number 63'4 by the diameter in inches and square the product, which, multiplied by the square root of the pressure of the steam in atmospheres, gives the de- livery in gallons per hour. These rules correspond very closely with the tables of the deliveries of different sizes published by the manufacturers, Messrs. Sharp, Stewart, and Co., of Manchester. POWEE BEQUIBED TO PEBFOEM VAEIOtTS KINDS OP WORK. The power required to obtain any given speed in a given steamer will be so fully discussed in the next chapter that the subject need not be further referred to here ; and in my ' Cate- chism of the Steam-Engine ' I have recapitulated the amount of power, or the size of engine, required to thrash and grind corn, spin cotton, work sugar and saw mills, press cotton, drive piles, EFFICIENCY OF HYDRAULIC MACHINES. 385 dredge earth, and blow furnaces. The subject, however, is so important that I shall here recapitulate other cases for the most part derived from experiments made with the dynamometer in France by General Morin,* whose researches on this subject have been highly interesting, and have been conducted with much care and ability. Comparative efficiency of different machines for rawing water. Of the different pumps experimented upon by General Morin, the result of eight experiments made with pumps draining mines showed that the effect utilised was 66 per cent, of the power expended. But in these cases there was considerable loss from leakage from the pipes. At the salt works of Dreuze the useful effect was 52-3 per cent, of the power expended. In fire-engine pumps employed to deliver the water pumped at a height of from 12 to 20 feet, the proportion of the water delivered to the capacity of the pump was, in the pumps of the following makers Merryweather, Tylor, Perry, Carl-Metz, Letestu, Flaud, and Perrin, respectively, as follows : '920, '887, '910, '974, -910, -920, and '900 ; while the percentage of useful effect relatively with the power expended was 39-7, 39-1, 30-2, 28-7, 27'1, 19'4, and 15 -5, respectively. "With a higher pressure, the efficiency of the whole of the pumps increased ; and when employed in throwing water with a spout-pipe the delivery of water relatively with the effective capacity, or space described by the piston, was, when the names are arranged, as follows : Carl-Metz, Merryweather, Tylor, Letestu, Perry, Flaud, Perrin, and Lamoine, respective- ly, -950, -810, -565, -870, '910, -912, '950, and '900 ; while the proportion of useful effect, or percentage of work done relatively with the power expended, was 80, 57'3, 54'5, 45<2, 37'8, 33-4, 28'8, and 17'5, in the respective cases. In the membrane pump of M. Brule the efficacy was found to be 40 to 45 per cent, of the power expended. In the water- works pumps of Ivry, construct- ed by Cave, the efficiency was found to be 53 per cent, of the power expended ; and in the water- works of St. Ouen, by the same maker, 76 per cent. It is desirable that the buckets of the pumps of water-works should move slowly, otherwise the * Aide Mfmoire, by General Morin, 5th edition, 1864. 17 386 POWER AND PERFORMANCE OF ENGINES. water will go off with considerable velocity, involving a corre- sponding loss of power. The area through the valves should be half the area of the pump, and the area of the suction and forcing pipes ought to be equal to three-fourths of the area of the body of the pump. Waste spaces should be avoided. The loss of water through the valves before they shut is, in good pumps, about 10 per cent. In a chain-pump the efficiency was found to be 38 per cent., but in many chain-pumps the efficiency is much more than this. The efficiency of the Persian wheel was found to increase very much with the height to which the water was raised. For heights of 1 yard it was 48 per cent., for 2 yards 57, for 3 yards 63, for 4 yards 66, and for 6 yards and upwards 70 per cent, of the power consumed. For a wheel of pots the efficiency is 60 per cent. ; Archimedes screw, 65 per cent. ; scoop wheel with flat boards moving in a circular channel, 70 per cent.; improved bucket wheel, 82 per cent., and tympan-wheel, or, as it is some- tunes called, "Wirtz's Zurich machine, 88 per cent. This machine should dip at least a foot into the water to give the best results. In the belt-pump the efficiency was found to be 43 per cent.; in Appold's centrifugal pump, 65 per cent. ; in the centrifugal pump, with inclined vanes, 42 per cent., and with radial vanes, 24 per cent. In Gwynn's pump the efficiency was 30 per cent. In the Archimedes screw the diameter is usually one-twelfth of the length, and the diameter of the newel or central drum should be one-third of the diameter of the screw. It ought to have at least three convolutions, and the line traced by the screw on the enveloping cylinder should have an angle of 67 to 70 with the axis. The axis itself should make an angle of from 30 to 45 with the horizon. There is a sensible advantage ob- tained from working hand-pumps by a crank instead of a lever. Old, French Flour Mill at Senelle. Diameter of millstones, 70 inches ; number of revolutions per minute, 70 ; quantity of corn ground and sifted per hour, 260'7 Ibs. ; power consumed, 3 - 34 horses. The power is in all cases the power actually exert- ed, as ascertained by the dynamometer. English Flour Mill near Metz. Diameter of millstones, 5118 POWER REQUIRED TO DRIVE MILLS. 387 inches; number of revolutions per minute, 110; weight of mill- stones, 1 ton ; corn ground per hour by each pair, 220 Ibs. ; with two pairs of millstones acting, one bolting machine and one win- nowing machine, the power consumed was 8^ horse-power. English Flour Mill near Verdun, Diameter of millstones, 51'18 inches; number of revolutions per minute, 110; quantity of corn ground per hour by each pair, or by each revolving mill- stone, 220 Ibs. ; with two stones revolving the power consumed was 5 "64 horses. The power consumed by one winnowing ma- chine and two bolting machines, with brushes sifting 1,650 Ibs. of flour per hour, was 6 horses. In another mill the number of turns of the millstone was 486 per minute, the quantity of corn ground by each horse-power was 120 Ibs., and the quantity of corn ground per hour was 110 Ibs. of which 72-7 per cent, was flour, 7' 8 per cent, was meal, and 19 -5 per cent, was bran. In a portable flour-mill, with machinery for cleaning and sifting, the total weight was 1,000 Ibs. Barley Mill. Number of revolutions of the millstone per minute, 246; barley ground per hour, 143-68 Ibs. ; motive force ill horses, 3-11 ; barley ground per hour by each horse-power, 48-2 Ibs. The products were, of first and second quality of barley flour, 60-12 per cent., of meal and bran, 30*25 per cent., and of . bran and waste, 9-63 per cent. Eye Mill. Number of revolutions of the millstone per minute, 448 ; rye ground per hour, 92-114 Ibs. ; power expended, 2-86 horses; temperature of flour, 60*8 Fahr. ; products 64-9 per cent, of flour, 9 - l per cent, of meal, and 26 per cent, of bran. In another rye mill the revolutions of the millstones per minute were 232 ; rye ground per hour, 180 Ibs. by 2-19 horse-power, and the rye ground per hour by each horse-power was 82-21 Ibs. The products were 72*5 per cent, of flour; 17'5 per cent, of meal and fine bran, and 10 per cent, of bran and waste. Maize Mill. Number of revolutions of the millstone per minute, 246 ; maize ground per hour, 73-96 Ibs. ; motive force in horses, 2'69 ; maize ground per hour by each horse-power, 27'5 Ibs. Products: first and second quality of flour, 61-1 per cent. ; meal and fine bran, 30-2 per cent. ; bran and waste, 4-7 per cen% 388 POWER AND PERFORMANCE OF ENGINES. Vermicelli Manufactory. External diameter of edge runners, 66'93 inches; internal diameter of edge runners, 62'99 inches; number of revolutions of the arbour of the mill per minute, 4 ; pounds of paste prepared per hour, 77 Ibs. ; power expended, 2'95 horse-power. Sean Mill. Number of revolutions of the millstone per min- ute, 496; power expended per hour, 1'76 horse. Oil Mill. "Weight of edge runners, 6,600 Ibs. ; number of turns of the vertical spindle per minute, 6 ; weight of seed intro- duced every ten minutes, 55 Ibs. ; weight of seed crushed daily, 3,300 Ibs. ; product in oil in 12 hours, 1,320 Jbs. ; power expend- ed, 2-72 horses. S'aw Mill "Weight of the saw frame, 842'6 Ibs. When cut- ting dry oak 8.73 inches thick, with 1 blade in operation, the reciprocations or strokes of the saw were, 88 per minute, the surface cut, -525 square foot, and the power expended 3'3 horses. When cutting the same wood with 4 blades in operation, the number of strokes of the saw per minute was 79 ; the surface cut by each per minute '433 square foot, or 1'73 square foot per minute for the 4; and the power expended was 3*70 horses, which is equivalent to 28 square feet cut per hour by 1 horse-power. When cutting four-year seasoned oak, 12-4 inches thick, with 4 blades, making 90 strokes per minute, the surface cut by each blade was 35 square foot, and the surface cut by the 4 blades, 1/41 square foot. When the saw was run along the middle of a cylindrical log of beech one-year cut, 23'6 inches diameter, the number of strokes of the saw per minute was 88 ; the surface cut per minute, "968 square foot ; and the power expended, 3 horses. In these experiments the breadth of the saw cut was '157 inch, and the experiments show that it does not take more power to drive a frame with one saw than to drive a frame with four, the great- est part of the power indeed being consumed in giving motion to the frame. The common estimate in modern saw mills, when the frame is filled with saws, is, that to cut 45 superficial feet of pine, or 34 of oak per hour, requires 1 indicated horse-power. The crank, which moves the frame up and down, and which is usually placed in a pit under tho machine, .should have balance POWER REQUIRED TO DRIVE SAWS. 389 weights applied to it, the momentum of which weights, when the saw is in action, will be equal to that of the reciprocating frame. In some cases the weight of the saw frame is borne by a vacuum cylinder, and with a 20-inch stroke it makes 120 strokes per minute. Circular Saw. Diameter of saw, 27'5 inches ; thickness of oak cut, 8*73 inches; number of revolutions per minute, 266; surface cut per minute, T93 square foot; power consumed 3*55 horses. When set to cut planks of dry fir, 10'62 inches broad, and one inch thick, the number of revolutions made by the saw per minute was 244; surface cut per minute, 7'67 square feet; and the power expended, 7'35 horses. These results show that in sawing the smaller class of timber one circular saw will do at least as much work as four reciprocating saws, with the same expenditure of power. The surface cut is, in all these cases, under- stood to be the height multiplied by the length, and not the sum of the two faces separated by the saw. The speed of the circular saw here given is not half as great as that now commonly em- ployed. Circular saws now work with a velocity at the peri- phery of 6,000 to 7,000 feet per minute, and band saws with a velocity of 2,500 feet per minute, and it is generally reckoned that 75 superficial feet of pine, or 58 of oak, will be sawn per hour by a circular saw for each indicated horse-power expended. Planing machine cutters move with a velocity at the cutting edge of 4,000 to 6,000 feet per minute, and the planed surface travels forward ^ O th of an inch for each cut. Reciprocating Veneer Saw. Length of stroke of saw, 47'24 inches; thickness of the blade, '01299 inch; breadth of saw cut, 02562 inch; length of teeth for mahogany and other valuable woods, -196 inch; pitch of the teeth, -3939 inch; distance ad- vanced by the wood each stroke, '0196 to '03937 inch ; number of strokes of the saw per minute, 180 ; surface cut per hour counting both faces, 107*64 square feet ; power expended 0'66 horses. Sawing Machine for Stones. Soft sandstone : breadth of saw- cot, inch ; time employed to saw 10 square feet, 5 minutes 25 seconds ; power expended 4'54 horses. Hard sandstone : breadth 390 POWER AND PERFORMANCE OP ENGINES. of saw-cut, J inch; time employed to cut 10 square feet, 1 hour 3V minutes ; power expended 2 horses. Sugar Mill for Canes. A three-cylinder mill, with rollers 5 feet long, 30 inches diameter, and making 2J turns a minute, driven hy an engine of 25 to 30 horse-power, will express the juice out of 130 tons of canes in 12 to 15 hours. An acre of land produces from 10 to 20 tons of canes, according to the age and locality of the canes. The juice stands at 8 to 12 of the saccharometer, according to the locality. The product in sugar varies from 6 to 10 per cent, of the weight of the canes, accord- ing to the locality and mode of manufacture. Well-constructed mills give in juice from 60 to 70 per cent, of the weight of the canes, and one main condition of efficiency is, that the rollers shall travel slowly, as with too great a speed the juice has not time to separate itself from the Avoody refuse of the cane, and much of it is reabsorhed. To defecate 330 gallons of juice 6 boiling-pans, or caldrons, are required, 4 scum presses, and 10 filters ; and to granulate the sugar 2 vacuum pans, 6 feet diam- eter, are required, with 2 condensers, and it is better also to have 2 air-pumps. The steam for boiling the liquor in the vacuum pans is generated in three cylindrical boilers, each 6 feet in diameter. To whiten the sugar there are 10 centrifugal machines, driven by a 12-horse engine, which also drives a pair of crushing-rollers. The sugar in the centrifugal machines is wetted with syrup, which is driven off at the circumference of the revolving cylinders of wire gauze, carrying with it most of the colouring matter of the sugar, which to a great extent adheres to the outside of the crystals, instead of being incorporated in them, and may consequently be washed off. When the sugar is thus cleansed it is again dissolved, and the syrup is passed through deep filters of animal charcoal. Provision must be made to wash the charcoal, both by steam and by water, and two fur- naces, to re-burn the animal charcoal, will be required. The action of animal charcoal in bleaching sugar is not well understood. But it appears to be due to certain metallic bases in the bones, which by burning are brought to or towards the metallic state, from the superior affinity of the carbon present EXAMPLE OF A COTTON MILL. 391 for the oxygen in the base at the high temperature at which the re-burning takes place. When, however, the charcoal is mixed with the syrup, the metallic base endeavours to recover the oxygen it has lost, by decomposing the water, leaving thereby a certain quantity of hydrogen in the nascent state ; and this hy- drogen appears to dissolve the small particles of carbon in the sugar which detract from its whiteness, and to form therewith a colourless compound. When the metallic basis has recovered all its lost oxygen the charcoal ceases to act, and has to be re- burned ; and, after numerous re-burnings, the charcoal appears to be all burned out of the bones, when re-burning ceases to be of service. But their efficacy might be restored by mingling por- tions of wood charcoal. The use of charcoal in sugar refining is not merely a source of expense in itself, but it occasions a loss of sugar, as, when the mass of charcoal becomes effete, it is left saturated with syrup, and the water with which it is washed has to be boiled down, to recover the sugar as far as possible. I consequently proposed several years ago a method of revivifying the charcoal without removing it from the filter. But the method has not yet been practically adopted. The begass, or woody refuse of the cane, is usually employed to generate the steam in the boilers. But it is generally neces- sary to use coal besides. Fans for blowing Air. The indicated power required to work a fan may be ascertained by multiplying the square of the velocity of the tips in feet per second by the collective areas of the escape orifices in square inches, and by the pressure of the blast in pounds per square inch, and finally dividing the product by the constant number 62,500, which gives the indicated power required. The pressure in pounds per square inch may be de- termined by dividing the square of the velocity of the tips in feet per second by the constant number 97,300. Cotton-spinning Mill. Number of spindles, 26,000 ; power consumed, 110 horses; Nos. of yarn spun, 30 to 40; spindles with preparation driven by each horse, 237. It is reckoned that each machine requires 1 horse-power. Another example of a Cotton Mill. Number of spindles, 392 POWER AND PERFORMANCE OF ENGINES. 14,508 ; power required to drive them, 50'5 horses; ISTos. of yarn spun, 30 to 40 ; spindles and preparation driven by each horse- power, 287. Another example of a Cotton Mill. Number of spindles, 10,476; Nos. of yarn spun, 30 to 40; spindles and preparation driven by each horse-power, 235. Details of power required l>y each Machine in Cotton Mills. One beater making 1,100 revolutions per minute, with ventilat- ing fan making half this number of revolutions, cleaning 132 Ibs. of cotton per hour, requires 2'916 horse-power. One beater making 1,200 revolutions per minute, with combing drum 1*23 feet diameter and 2'8 feet long, making 800 revolutions per min- ute, and preparing 132 Ibs. of cotton per hour, requires 800 revo- lutions per minute and 1*767 horse-power. Power required to work the fluted cylinders and endless web of this machine, '812 horse. Twelve double-casing cylinders, with eccentrics, re- quiring 2*697 horses, including the transmission of the motion, or per machine, '225 horse. Transmitting the motion for 26 carding-machines requires 1*82 horse-power. One simple card, consisting of a drum 39'37 inches diameter and 19*68 inches long, making 130 revolutions per minute, and carding 2 Ibs. of cotton per hour, requires '066 horse-power, without reckoning the power consumed in communicating the motion. The same card working empty requires '044 horse-power. One double- carding machine carding 4*18 Ibs. of cotton per hour, requires 207 horse-power. A drawing-frame drawing 119 Ibs. per hour requires 1'835 horse-power. A roving-frame, with 60 spindles, with cards, making 525 revolutions per minute, and producing 42 Ibs. of No. 7 rovings per hour, requires '760 horse-power. One frame with screw-gearing, having 60 spindles, making 550 revolutions per minute, and producing 42 Ibs. of No. 7 per hour, requires '486 horse-power. Two frames with screw-gearing, each containing 96 spindles, making in one case 510 revolutions and the other 500 revolutions per minute, producing 28*6 Ibs. of No. 2*75 to 3 per hour, requires 1*482 horse-power. Two frames with screw-gearing, one containing 78 spindles making 344 revolutions per minute, and the other 60 spindles making 260 POWER REQUIRED TO DRIVE WOOLLEN MILLS. 393 revolutions per minute, and producing 57'2 Ibs. of No. 8 per hour, requiring '797 horse-power. One spinning-frame, with cards, having 240 spindles, making 5,000 revolutions per minute, and producing T65 Ib. of yarn of No. 38 to ]STo. 40 per hour, requires '686 horse-power, and in another experiment '648 horse- power. Three spinning-frames for weft, having each 360 spin- dles, making 4,840 revolutions per minute, and producing 8 Ibs. of No. 30 to No. 40 yarn per hour, require 2*103 horse-power. One retwisting machine, with 120 spindles, making 3,000 revo- lutions per minute, requires 1'19 horse-power. One dressing machine for calico 35 inches wide, with ventilator : speed of the principal arbor, 176 revolutions per minute; speed of the brushes, 45 strokes per minute ; power required, '735 horse. The same machine, with the ventilator not going, requires -206 horse-power. Power-loom Weaving. To drive one power-loom weaving calico 35J inches wide, and 82 to 90 picks per inch, making 105 strokes per minute, requires, taking an average of four experi- ments, '1195 horse-power. Another example of Power-loom Weaving. Number of looms weaving calico driven by water-wheel, 260 ; dressing ma- chines, 15 ; winding machines, 5 ; warping machines, 8 ; small pumps, 6 ; yards of calico produced per month, 283,392 ; power required to drive the mill, 25'6 horses; number of looms, with accessories, moved by 1 horse, 12. Another example of Power-loom Weaving. Total number of looms, 60 ; dressing machines, 5 ; warping machines, 3 ; wind- ing machines, 2; monthly production of cotton cloth called 'Normandy linen,' 47i inches wide, 360 pieces, each 396 yards long ; power consumed, 8 horses ; looms with their accessories moved by each horse-power, 7'8. Wool-spinning Mill. Machines driven : simple cards, 29 ; double cards, 2 ; scribbling beater, 1 ; mules of 240 spindles, 8 ; mules of 200 spindles, 4; lathes, 3; power consumed, 9'76 horses. Also in another experiment with 9 simple and 3 double- carding machines, 2 beaters, and 2 scribbling machines, the power consumed in driving was 3*5 horses. 17* 394 POWER AND PERFORMANCE OF ENGINES. Another example of a Wool-spinning Mill. A wheel exert- ing 10 horse-power drives 6 mules of 240 spindles, 6 of 180, 2 of 192, 2 of 120, and 5 of 100, making in all 3,644 spindles ; also 32 carding and 2 scribbling machines. Another wheel, also ex- erting 10 horse-power, drives 8 mules of 240 spindles, 4 of 120, and 7 of 180, making in all 3,660 spindles; also 31 carding and 2 scribbling machines, and 2 beaters. The spindles, number- ing in all 7,304, make 5,000 revolutions per minute, and the cards 88 to 89, requiring a horse-power for 365 spindles. Prod- uct per day of 12 hours, 1,100 Ibs. of yarn from No. 12 to No. 13. Details of Power consumed in spinning Wool. One winding machine with 16 bobbins, without counting the power expended in the transmission of the motion, requires to drive it '259 horse ; 3 winding machines with 64 bobbins in ah 1 , with power lost by transmission, 1'427 horse; one mule spinning No. 6 warp yarn, with 220 spindles, making 3,650 revolutions per minute, 259 horse. One mule called ' Box-organ,' spinning No. 50 warp yarn with 300 spindles, making 3,200 revolutions per minute, requires 1*273 horse-power. Mill for spinning Wool and weaving Merinos. Nineteen machines to prepare the combed wool, having together 350 rollers; 16 mules with 3,400 spindles ; one winding machine of 60 rollers to prepare the warp ; 2 warping machines ; 2 self- acting feeders; 100 power-looms; 2 lathes for wood and iron, and 1 pump, require in all 30 horse-power. Produce : 13,600 cops of woollen thread, of 45 cops to the lb., each measuring 792 yards. The looms make 115 revolutions per minute, and produce daily 4 pieces of double- width merino of 68 yards each, and 4 pieces of simple merino of 1*2 to 1-4 yard broad, and each 88 yards long. Fulling Mill. In falling the cloths called 'Beauchamps,' each piece being 220 yards long, and -66 yard wide, and weigh- ing from 121 to 127 Ibs., the fuller making 100 to 120 strokes per minute, each piece requires 2 hours to full it, and the expen- diture of 2 horse-power during that time. Flax Manufacture. A machine for retting the flax, having POWER REQUIRED TO DRIVE FLAX MILLS. 395 15 pairs of rollers with triangular grooves, requires 3'376 horse- power, and the heckles '057 horse-power. One fly breaking-card 12 - 59 inches diameter and 47'24 inches long, making 915 revolutions per minute, with a drum of 42'12 inches diameter, and 47'24 inches long, making 76 revolutions per minute ; 4 distributing rollers, having a diameter of 4 inches and a length of 47'24 inches, making 380 revolutions per minute ; 3 travellers, 5 inches diameter and 47'24 inches long, making 10 turns per minute, and one combing cylinder 15 inches diameter and 47'24 inches long, making 6 revolutions per minute, require together 1-939 horse-power, and produce 17 Ibs. of carded flax per hour. One finishing carding cylinder, 40 inches diameter and 47'24 inches long, making 176 revolutions per minute ; 5 distributing rollers, 4 inches diameter, making 23 revolutions per minute ; 4 travellers, 5 inches diameter, making 7'3 revolutions per min- ute ; 1 combing cylinder, 15 inches diameter, making 3 -4 revolu- tions per minute, together require '811 horse-power, and produce 8 Ibs. of carded flax per hour. One spinning-machine, containing 132 spindles, making 2,700 revolutions per minute, spinning yarns from No. 7 to No. 9, re- quires 1*24 horse-power, and produces 3f Ibs. of yarn per hour. One spinning-machine, having 168 spindles, making 2,700 revolutions per minute, and producing 3 Ibs. of No. 18 to 24 yarn per hour, requires 1'96 horse-power. Wet spinning of flax : one drawing-frame drawing a sliver for No. 20 yarn, requires -493 horse; drawing-frame drawing sliver for No. 50 yarn, requires *487 horse ; drawing-frame draw- ing sliver for No. 70 yarn, requires *495 horse. Second drawing-frame, drawing two slivers for yarns Nos. 20 and 30, requires '68 horse ; second drawing-frame, drawing two slivers for yarns Nos. 30 and 40, requires -544 horse ; second drawing-frame, drawing one sliver for No. 60 yarn and one for No. 70, requires '617 horse. Third drawing-frame, drawing two slivers for yarns Nos. 30 to 60, requires '69 iorse. Koving-frame of 8 spindles, preparing the flax for yarn No. 396 POWER AND PERFORMANCE OF ENGINES. 20, requires '608 horse; roving-frame of 8 spindles, preparing the flax for No. 30 yarn, requires '486 horse; frame of 16 spin- dles, preparing the flax for No. 40 yarn, requires '987 horse- power. Paper Manufacture. In some cases the pulp, or stuff of which paper is made, is obtained by beating the rags by stamp- ers ; but more generally it is produced by placing the rags be- tween revolving cylinders stuck full of knives. When produced by stampers, the proportions of the apparatus are as follows : weight of stampers, 220 Ibs. ; distance of the centre of gravity from the axis of rotation, 4 feet ; rise of the centre of gravity each stroke, 3| inches; number of stampers, 16; number of lifts of each stamper per minute, 55 ; weight of rags pounded in 12 hours by each stamper, 33 Ibs. ; weight of stuff produced in 12 hours by each stamper, 122 Ibs. ; power consumed, 2'7 horses. Chopping-cylinders, for preparing the pulp : number of cyl- inders working, 2 ; number of turns of cylinders per minute, 220 ; weight of rags chopped and purified in 12 hours, 528 Ibs. ; power consumed, 4'48 horses. In another instance, 10 cylinders for preparing the pulp, making 200 revolutions per minute, 1 paper-making machine, cutting-machines, pump, and accessories, consumed 50-horse power. The machine made 13 yards of paper per minute, and the produce was 1 ton of printing paper per day of 24 hours. In another instance, 28 pulping-cylinders, and 3 paper-mak- ing machines produced 2 to 3 tons of paper per day of 24 hours, and consumed 113 horse-power. Printing Machinery. Printing large numbers is now per- formed by cylindrical stereotype plates, revolving continuously ; and the ' Tunes ' and other newspapers of large circulation are thus printed. The impressions are taken from the types in papier mache, and in twenty minutes a large stereotype plate is ready to be worked from. The power required to drive this machine varies with the number of impressions required in the hour. For 5,000 impressions per hour, the power required is 3-75 horses ; for 6,000 impressions, 4*77 horses; 7,000 impres- sions, 5-9 horses; 8,000 impressions, 7'03 horses; 9,000 impres- WEAVING BY COMPRESSED AIR. 397 sions, 8'75 horses; and 10,000 impressions, 10-35 horses. The paper should be supplied to such machines in a continuous web, with a cutter to cut off the sheets at the proper intervals, and a steam cylinder to dry and press them. But this has not yet been done. The machine could also be easily made to perforate the paper along the edges of the leaves, and to fold each paper up and put a printed and stamped paper envelope around it, so as to be ready at once to put into the post-office or to distribute by hand. The most expeditious mode of stereotyping would be to use steel types set on a cylinder, against which another cylinder of type-metal is pressed, and the paper would then be printed in the same manner as calico. Glass Works. Mill to grind red lead : to grind 3 tons, the vertical arbor requires to make for the first ton 20 revolutions per minute, for the second 25, and for the third 40, consuming 5'28 horse-power. Vertical millstones, to grind clay and broken crucibles; diameter of the granite stones or runners, 3'V feet; thickness, 1/4 foot; weight, 1 ton; distance of edge runners from central spindle, 4 feet ; number of turns of the arbor per minute, Tg-; power consumed 1'92 horse. In the 12 hours 6 or 8 charges of about 300 Ibs. each of old glass pots are ground, and about 3 tons of dry clay. Wheels for cutting the glass, 1YO ; lathes for preparing the cutting wheels, 5 ; lathes for metal, 2 ; power consumed, 17*9 horses; wheels driven by each horse- power, 9 - 5. Iron- Worlcs. The weekly yield of each smelting furnace in "Wales is from 100 to 120 tons ; pressure of blast, 2 to 3 Ibs. per square inch ; temperature of the blast, 600 Fahr. ; yield weekly of each refining-furnace, 80 to 100 tons; of each puddling-furnace, 18 tons; of each balling-furnace for bars, 30 tons; of each ball- ing-furnace for rails, 80 tons ; iron rolled weekly by puddle rolls, 300 tons ; by rail rolls, 600 tons ; power required to work each train of rail rolls, 250 horses ; to work puddle rolls and squeezer, 80 horses; small bar train, 60 horses; pumping air into each blast-furnace, 60 horses ; into each refining-furnace, 26 horses ; rail saw, 12 horses. Weaving ty compressed air. In common power-looms, the 398 POWER AND PERFORMANCE OF ENGINES. shuttle is driven backward and forward by a lever which imi- tates the action of the arm in the hand-loom. But it has long been obvious to myself and others that it might be shot back- ward and forward like a ball out of a gun, by means of com- pressed air. This innovation has now been practically carried out. But the benefits derivable from the practice have been much exaggerated, and a much more comprehensive improve- ment than this is now required. Indeed, reciprocating looms of all kinds are faulty, as they make much noise, consume much power, do little work, and cannot be driven very fast ; and the proper remedy lies in the adoption of a circular loom in which the cloth will be woven in a pipe, and in which many threads of weft will be fed in at the same time. Circular Loom. The obvious difficulty in a circular loom, is to drive the shuttle round continuously within the walls formed by the warp. One mode of driving proposed by me, is by mag- nets or other suitable form of electro-motive machine, which does not require contact ; and the shuttle should be a circular ring, with many cops placed in it, so that many threads might be woven in at once. The desideratum, however, is to weave a vertical pipe with the bobbins of the weft in the centre of the circle ; and this may be done by depositing the thread between metallic points, like circular heckles, which points will change their positions inward or outward at each time a thread is de- posited. These points would conduct the threads of the warp. CHAPTER VIL STEAM NAVIGATION. STEAM navigation embraces two main topics of enquiry : the first, what the configuration of a vessel shall be to pass through the water at any desired speed with the least resist- ance ; and the second, what shall be the construction of ma- chinery that shall generate and utilise the propelling power with the greatest efficiency. The second topic has, in most of its details, been already discussed in the preceding pages ; and it will now be proper to offer some remarks on the remaining portion of the subject. The resistance of vessels passing through the water is made up of two parts : the one, which is called the bow and stern resistance, being caused partly by the hydrostatic pressure forc- ing back the vessel, arising from the difference of level between the bow and stern, and partly by the power consumed in blunt bows in giving a direct impulse to the water ; while the other part of the resistance, and the most important part, is that due to the friction of the water on the sides and bottom of the ship. The bow and stern resistance may be reduced to any desired extent by making the ends sharper. But the friction of the bot- tom cannot be got rid of, or be materially reduced, by any means yet discovered. When a vessel is propelled through water, the water at the bow has to be moved aside to enable the vessel to pass ; and the velocity with which the water is moved sideways will depend upon the angle of the bow and the speed of the vessel. When 400 STEAM NAVIGATION. these elements are known it is easy to tell with what velocity the water will be moved aside ; and when we know the velocity with which the water is moved, we can easily tell the power consumed in moving it, which power will, in fact, he the weight of the water moved per minute multiplied by the height from which a body must fall by gravity to acquire the same velocity. But as nearly all the power thus consumed in moving aside the water at the bow of a vessel is afterwards recovered at the stern by the closing in of the water upon the run, it is needless to go into this investigation further than to determine what amount of power is wasted by the operation, or in other words, what amount of power is expended that is not afterwards recovered. If the vessel to be propelled is of a proper form, each particle of water will be moved sideways by the bow, in the same man- ner as the ball of a pendulum is moved sideways by gravity, so as to enable the vessel to pass ; and when the broadest part of the vessel has passed through the channel thus created, each particle of water will swing backward again until it comes to rest at the stern. There will be no waste of power in this operation, except that incident to the friction of the moving water ; just as in the swinging of a pendulum there is no expen- diture of power beyond that which is necessary to overcome the friction of the air upon the moving ball. But as the move- ment of the vessel, however well she may be formed, will some- what raise the water at the bow, and somewhat depress the water at the stern, there will be a certain hydrostatic pressure required to be continually overcome as the vessel advances in her course, which opposition constitutes the bow and stern re- sistance ; and this, with the friction of the bottom, make up the whole resistance of the ship. Before, however, proceeding to investigate the amount of this hydrostatic resistance, it will be proper to show how accidental sources of loss may be elim- inated from the problem by the introduction of that particular form of vessel which will make this resistance a minimum ; and I will therefore first proceed to indicate in what way such form of vessel may be obtained. If we take a short log of wood, such as is shown by the FORM OF MINIMUM RESISTANCE. 401 dotted lines A B c r> E F G, in the annexed figure (fig. 41), and if we proceed to enquire in what way we shall mould this log into a model which shall offer the least possible hydrostatic resist- ance in being drawn through the water, we have the following considerations to guide us in arriving at the desired knowledge : We shall, for the sake of simplification, suppose that the cross section of the completed model is to be rectangular, or in other words, that the model is to have vertical sides and a flat bottom ; for although this is not the best form of cross section, as I shall Fig. 41 afterwards show, the supposition of its adoption in this case will simplify the required explanations. "We first draw a centre line x y longitudinally along the top of the model from end to end, and continue the line vertically downward at the ends as at y z, which vertical lines will form the stem and stern post of the model. At right angles to the first line, and at the middle of the length of the model, we draw the line a, which answers to the midship frame ; and midway between a and the ends we draw other two lines 5 5. We may afterwards draw any convenient number of equi-distant cross- 402 STEAM NAVIGATION. lines, or ordinates, as they are termed, that we find to be conven- ient. Now as, by the conditions of the problem, the particles of water have to swing sideways like a pendulum, in order that the resistance may be a minimum, the particle which encounters the stem at x must be moved sideways very slowly at first, like a heavy body moved by gravity, but gradually accelerating until it arrives at 5, midway between x and a, where its velocity will be greatest ; and this point answers to the position of the ball of the pendulum when it has reached the bottom of the arc, and has consequently attained its greatest velocity. Thereafter the motion, which before was continually accelerated, must be now continually retarded, as it is in any pendulum that is ascending the arc in which it beats, or in any ball which is projected up- wards into the air against the force of gravity. When the par- ticle of water has attained the position on the side of the model which is opposite to the midship frame #, it will have come to rest, this being the point answering to the position of the pen- dulum at the top of its arc, and when just about to make the return beat. Thereafter the particle which was before moved outwards, will now move inward with a velocity, slow at first, but continually accelerating, until it attains the position on the side of the model which is opposite to the frame 5, when the velocity again begins to diminish ; and the particle finally comes to rest at the stern. A particle of water that is moved in this way will be moved with the minimum of resistance ; for since it retains none of the motion in it that has been imparted, but surrenders the whole gradually without impact or percussion, by the time it has come finally to rest, there can be no power consumed in moving it except that due to friction only. "Wher- ever the water is not moved in this manner it will either retain some of the motion, which implies a corresponding waste of power, or heat will be generated by impact, which also involves a corresponding waste of power. That the water may be moved in the same manner as a pendulum is moved, is obviously possi- ble, by giving the proper configuration to the sides of the model ; and in fact, if an endless sheet of paper be made to travel vertically behind a pendulum, with a pencil or paint CURVE OF GRAVITY. 403 brush stuck in the ball, the proper form for the side of the model will be marked upon the paper. The curve, however, which is a parabolic one, may be described geometrically as follows : If we compute the height through which a heavy body falls by gravity in any given number of seconds, we shall find that in the first quarter of a second it will have fallen through l T g^ foot, in the second quarter of a second 3^ feet, in the third 9|, in the fourth 16 T L, in the fifth 25^, in the sixth 36 T 3 F , in the seventh 49 T 4 5 9 2, in the eighth 64^, in the ninth 81f J, and in the tenth quarter of a second 100||. The height fallen through, there- fore, or the space described by a falling body in a given time, varies as the square of the time of falling ; and any body which is to be moved in the same manner as a falling body is moved by gravity, must have the motion imparted to it gradually at the same rate of progression. If, then, we draw a line, x y in fig. 42, and which line we may suppose to be the vertical plane of the keel, then if we form the parallelogram A B o D, with the line x y passing through the middle of it, and make this parallelogram one- fourth of the length of the vessel and half the breadth, and divide the line x y into any number of convenient parts or ordinates, say 10, by the vertical co-ordinates numbered from 1 to 10, then if we cause the lengths of these successive and equidistant co- ordinates, measuring from the line x y, to follow the same law of increase that answers to the height through which a body 404: STEAM NAVIGATION. falls by gravity in successive and equal portions of time, a line traced through the ends of these different lines will give the right form for the side of a vessel to have, in order that it may move the water sideways, in the same manner, or according to the same law, by which a heavy body falls vertically by gravity ; and consequently such line is the proper water-line of a ship form- ed under the conditions supposed, in order that it may have a. minimum resistance. The heights of the several vertical ordi- nates which are drawn on a different scale from the lengths, marked on the line x y, are 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100, which, it will be seen, are the squares of the horizontal ordinates 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 ; and the scale by which these vertical ordinates are measured is formed by divid- ing the distance y D, which represents one-fourth of the breadth of the vessel, into 100 equal parts. The ordinate y D is therefore equal to 100 of those parts, the next ordinate to 81 of them, the next to 64 of them, and so on, until the height vanishes at * altogether. We might have divided the line x y into nine equal parts, or into 8, or 7, or any other convenient number. In such case the vertical line y D would have to be divided into 81 equal parts to obtain the vertical scale, or into 64, or into 49, according as 9, 8, or 7 had been the number selected ; but the number of parts into which y D is divided must always be equal to the square of the number of ordinates, or the square of the number of parts into which the horizontal line is divided. As it is difficult to measure the hundredth part of such a small length as y D, we may call the number of parts 10 instead of 100, in which case the length of the next ordinate will be 8*1, of the next 6'4, of the next 4*9, and so on the whole of the squares being divided by 10 ; which proceeding will in no way affect the result, as, in point of fact, the difference is only much the same thing as if we meas- ured in inches instead of in feet. In the figure, x y is five times longer than y D, and x y rep- resents one-fourth of the length of the vessel, and y D one-fourth of the breadth. The curved line x D represents the proper form of the water-line of the front half of the fore body in the case of a vessel of these proportions, and with a rectangular cross-sec- CURVE OF GRAVITY. 405 tion. The water-line of the second half of the fore body is formed by repeating the same curve, but inverted and reversed, this will be made obvious by an inspection of fig. 43, where the first half of the fore body is repeated on a smaller scale ; and the second portion of the fore body is added thereto, thus continuing the water-line to the midship frame a a. Here the rectangle enclosing the water-line of the first half of the vessel is shown in dotted lines, as is also the rectangle enclosing the water-line of the first half of the fore body ; and it is plain that the shaded space a d is the exact duplicate of the shaded space x d; so that if the figure x d has been obtained, we may obtain the figure d a by cutting out of the paper the figure x d, inverting it and re- Fig. 43. versing it, so that the line x d shall coincide with the line d a, and the point x with the point a ; or the figure d a may be con- structed by co-ordinates in exactly the same manner as the figure x d. If the vertical sides of the vessel be formed with the curve shown by the curve line x a, then it will follow that a particle of water encountering the stem at as, will be moved aside slowly at first, and with a rate continually increasing, like a body falling by gravity, until the frame 5 lying midway between the stem and the midship frame is reached, at which point the water will be moving sideways with its greatest velocity. Thereafter the vessel will not move the water, but merely follow up the motion already given to it, and as the water, when no longer impelled sideways by the vessel, will move slower and slower, and gradually come to rest, so the vessel will have less and less following up to do, until at the midship frame a a, the side motion of the water ceases 406 STEAM NAVIGATION. altogether. Thereafter the water begins to move in the opposite direction to fill up the vacuity at the stern left by the progress of the vessel. The water gravitates into the run slowly at first, and the velocity increases until the point midway between the midship frame and the stern is attained, at which point the ve- locity is greatest ; and from thence the velocity of the water, flowing inward, continually diminishes, until it comes to rest at the stern. A rectangular box, such as that shown by the dotted lines A B o D E F G, fig. 41, into which the model exactly fits, is called its circumscribing parallelepiped ; and it will be at once appar- ent, on a reference to fig. 41, that the bulk or capacity of the model is exactly one-half of its circumscribing parallelepiped. The rectangle x d is equal to the rectangle d y, and the shaded space * d being equal to the shaded space d a, the area included between the water-line and the vertical plane of the keel, namely, the area x y a, is clearly equal to the rectangle d d y a. But that rectangle, and the rectangle standing beneath it, are equal to the whole area within the water-line of the fore body, and two similar rectangles are equal to the area within the water-line of the after body. As these four rectangles form just half the area of the circumscribing parallelogram, the total area within the water-line is equal to half the area of the circumscribing parallel- ogram. But the area multiplied by the depth gives the capacity, and as the depth of the model is the same as that of the box, or circumscribing parallelepiped, while the area of the circumscrib- ing parallelogram is twice that of the area of the figure within the water line, it follows that the volume or bulk of the model is just one-half of the circumscribing parallelepiped. This forms a measure of sharpness which in no case it is useful to exceed, if the section be made rectangular, or, in other words, if the vessel be built with a flat bottom and vertical sides. But if the vessel be built with a rising floor the effect is equivalent to a reduction of the breadth, and the circumscribing parallelepiped would, in such case, be that answering to the equivalent breadth. What- ever be the form of the cross-section, however, the sectional area at each successive frame should be equal to that of a vessel with FISHES CONFORM TO THE LAW. 407 a rectangular section having water-lines formed on the principle which has been here explained. There are other curves, no doubt, which equally with that described by a pendulum fulfil the indication of beginning and terminating the motion gradually so as to involve no loss of power, and any of these curves are eligi- ble as the water-line of a ship. But the pendulum curve is the most readily understood, and the most conveniently applicable to practical uses, while it perfectly fulfis the required indica- tions. If in any intended vessel we have a given form of cross- section, and a given ratio of length to breadth, we can easily determine the proper water-lines of such a vessel by taking the case of a hypothetical vessel of rectangular cross-section having Fig. M. the same area of midship-section, and by forming the water-lines for this hypothetical vessel on the principle already explained. The area of cross-section at each successive frame of this hypo- thetical vessel, will be the proper area at each successive frame of the intended vessel. It is obvious that, according to the prin- ciple here unfolded, the form of water-line must vary with every alteration of the cross -section ; and in some cases, although the same rate of displacement as that already indicated is pre- served, the water-lines will cease to be hollow at any part. Thus the cylindrical solid, with pointed ends, shown in fig. 44, is virtu- ally of the same form as that represented in fig. 41, since the area of each successive circular cross-section is the same as those of each rectangular cross-section in fig. 41. This solid is 408 STEAM NAVIGATION. supposed to be wholly immersed. It has, in some cases, been made an objection to the use of hollow water-lines for ships, that in the case of fishes, however fast swimming, no hollow lines are to be found in them. Fig. 44, however, which resembles the form of a fish, shows that fishes form no exception to the appli- cation of the law of progressive parabolic displacement already explained ; and if a fast-swimming fish be cut across at equal distances, and the areas of these sections be computed and laid down with a rectangular outline of uniform depth, it will be found that the skin or covering placed over the ends of these sections or frames will assume the very form which has been de- lineated in the foregoing figures as that proper for a solid in- tended to pass through the water with the least amount of hydro- static resistance. In fig. 44, x y is the axis of the pointed cylindrical solid ; and a is the circle or section which answers to the midship frame, and 5 5 the sections answering to the frames lying midway be- tween the centre frame and the ends. The other lines corre- sponding to those marked on the model shown in fig. 41, and the area of each successive circle is equal to the area of each successive rectangular section of the model delineated in fig. 41. The water consequently will be displaced at the same rate by one solid as by the other. For actual vessels, with rounded bilges and more or less rise of floor, the form of the water-lines will be neither that shown in fig. 41 nor fig. 44, but will be some- thing intermediate between the two ; but such, nevertheless, that the transverse sectional area of that part of the vessel beneath the water-line shall at each successive frame vary in the ratio pointed out. As water is practically incompressible by any force which a ship can bring to bear upon it, the water which a ship displaces must find some outlet to escape ; and it will escape in the line of least resistance, which is to the surface. A particle of water, therefore, on which a ship impinges, will have two kinds of mo- tion one a motion outwards and inwards, such as has been already described as resembling the motion of a pendulum, and the other a motion upwards and downwards, caused by the ne- BEST FORM OF CROSS-SECTION. 409 cessity of the particles beneath the surface rising up towards the surface to allow the vessel to pass, and afterwards of sinking down at the stern to fill the vacuity which the progress of the vessel would otherwise occasion. This last motion also resem- hles that of a pendulum, the particles of water at the stem rising up until they attain their greatest height at the midship frame, and then again subsiding towards the stern. It is not difficult, from these considerations, to deduce the conclusion that the form of vessel with a flat floor is not the best which can be adopted, as will be more clearly understood by a reference to fig. 45, where the rectangle D E F a, represents the Fig. 45. cross-section beneath the water-line of a flat-floored vessel at the point midway between the stem and the midship frame, while the triangle ABO is the cross-section of a sharp-floored vessel at the same point, and with the same sectional area. The draught of water in each case is 10 feet, represented by the figures 1 to 10 ; and the half breadth of the vessel with the rectangular cross-section at this point of the length is 5 feet, which also is one-fourth of the midship breadth. As the water has to be set back from the line of the stem to the line of the side, or in the case of the flat-floored vessel, through a distance of 5 feet, we may represent the power consumed in the operation by 5 feet multiplied by the mean hydrostatic pressure of the water <5n each square foot. The mechanical power required to be ex- 18 410 STEAM NAVIGATION. pended therefore in separating the water in the two sections will be as follows : Rectangular section. Triangular section. 5x1=5 9x1=9 5x2 = 10 8x2 = 16 6x3 = 15 7x3 = 21 5x4 = 20 6x4 = 24 5x5 = 25 5x5 = 25 5x6 = 30 4x6 = 24 5x7 = 35 3x7 = 21 5x8 = 40 2x8 = 16 5x9 = 45 1x9=9 5 xlO = 50 xlO = 275 165 The area of the triangle ABC being equal to that of the rectangle DBF G, the weight of water displaced by a foot in the length of the vessel will be the same whichever form of cross- section is adopted ; and as the areas of the shaded triangles A D x and B r> a;, or of the corresponding triangles B F x and o G a, are also the same, they represent equal amounts of outward motion of the water, and also equal amounts of displacement. In the one case, however, this motion is produced against a much greater hydrostatic pressure than in the other case ; and as by shifting the triangle B F a; into the position o G x whereby we enable the vessel to move outward the same volume of water, but against a less hydrostatic resistance we transform the rectangle H B F G into the triangle H B o, it follows that there is less resistance caused by the movement of the water in the case of triangular cross sections than in the case of rectangular. The rubbing sur- face too is less in the triangular section. By the principles of geometry, applicable to all right-angled triangles,* (BF) S + (F *) 2 = * This is proved by the 47th Proposition of the first book of Euclid, which shows that the area of the square described on the side A o, opposite to the right angle of a right-angled triangle is equal to the sum of the squares described on the other sides A B and B c. BEST FOEM OF CROSS-SECTION. 411 (B a 1 )' 2 . As B F = 5 feet and F x also = 5 feet, then (B F)- = 25, and (F x) 2 = 25, and 25 + 25 = 50, consequently B 2 = ^/50 = 7 nearly. The length of the immersed triangular outline is conse- quently 7 x 4 28 feet, whereas the length of the rectangular outline = 3 x 10 = 30 feet. As the resistance due to the friction of the bottom varies as the quantity of rubbing surface, it follows that, as regards friction, the triangular outline is also the more eligible. Instead, however, of a simple triangle, it is preferable Fig. 46. that the cross-section should be of the order of figure indicated as the best for the horizontal water-lines; and the same con- siderations which led to the conclusion that this form would offer the least resistance in the case of a body moving through stationary water lead also to the conclusion that it will offer the least resistance to water moving upwards past a stationary ob- ject which a ship may be supposed to be relatively to the plane in which she floats. Such a figure is represented in fig. 46, in which the triangular section is shown in dotted lines, and the waving lines pass alternately without and within the dotted lines. The cross-section of the vessel is for the most part of the outline 412 STEAM NAVIGATION. a semi-circle m m m a semicircle being the form which presents the smallest perimeter relatively with the immersed sectional area ; but the triangular portion in n is added both to prevent the vessel from rolling inconveniently, and to bring the outline into the waving curve which other considerations point out as the most eligible. One of these considerations, as already men- tioned, is that it best fulfils the condition of beginning the up- ward displacement slowly, and another is that it eifects the least possible alteration' in the shape of the displaced water. In Fig. 47. altering the form of a liquid, as in altering the form of a solid, there is a certain expenditure of force ; and although this ex- penditure in the case of a liquid is relatively very small, it is large enough to be worthy of attention in a case where large amounts are consumed in giving motion to water. It hence be- comes better, since the displaced fluids must assume the form of a wave, to effect the displacement so that this form shall be at once acquired, instead of some other form being first given to it which is subsequently changed by the action of other forces. This reasoning will be better understood by a reference to fig. BEST FORM OF CROSS-SECTION. 413 47, where w L is the water-level, m n the cross-section of half the vessel, and A A the wave which would be raised if there were no outward motion of the water, but only an upward motion. The outward motion reduces the altitude to some such small elevation as a a. Nevertheless it is advisable that the outline of the wave a a should be the same order of figure as the outline of the wave A A, only laterally extended. Such indeed is the shape it will necessarily assume ; and there will be less change of shape and therefore less motion of the internal particles, if the wave a a is drawn out sideways from a block of water of the form A A, than if drawn out from a rectangular, triangular, or any other form of block. The dotted lines indicate the directions in which the pressure will be transmitted, and if we suppose these lines to be tubes, it will be obvious that the surface of the water in these tubes will only conform to the outline of a wave, if the side of the vessel has that outline. If we suppose the portions of those tubes rising above the water-line to be very much en- larged, then the height of the outline will fall from A A to a a, but the same order of figure will still be preserved, as it involves less expenditure of power to give this form at once than to give some other form which is afterwards reduced by the action of gravity to this one, so on this ground it is preferable to make the cross-section of the vessel of the form suggested. Taking all things into account, a curve of the same kind that has been shown to be the best for the water-lines, appears to be also the best for the cross-section ; and the same ordinates which answer for the water-lines will answer for the cross-section, only in the latter case the ordinates must be placed closer together. If, for example, we have a vessel 200 feet long, and if the ordinates of the water-lines be 5 feet apart, there will be 40 ordinates ; and if the vessel be supposed to draw 20 feet of water, the same ordinates placed 6 inches apart will give the proper form of the cross-section below the load water-line. The nearer the form of the cross-section approaches to a semicircle the less friction there will be in the vessel ; and the proportions of the cross- section should in alt cases, where practicable, approach to the proportions of a semicircle, or in other words the depth below 414 STEAM NAVIGATION. the water should be a little more than half the breadth at the water-line. The ascending water will move more and more rapidly as it comes nearer to the surface, like the motion of a falling body in- verted ; and its momentum will carry it above the surface to a height equal to that which would generate the velocity. This motion of the water above the surface constitutes the second half of the beat of the pendulum which each ascending particle may be supposed to be the motion of the particle from the keel to the water-line being the first half of such beat. But as, after passing the surface of the water, the particle has to encounter more of the power of gravity, whereas below the water line it is floated by the other contiguous particles, it will follow that the Fig. 49. motion of the particle above the surface will be smaller in the proportion of the greater retarding force it there has to encoun- ter. This action will be better understood by a reference to fig. 48, where the parallelogram A B o D is supposed to be the side of a ship, w L is the surface of the water in which the ship swims, and the vertical dotted line at a shows the position of the midship frame. If we suppose a particle of water to be situated at a; a little below the water-level at the bow, then as the vessel moves onward in the direction of the arrow, such particle will be moved upwards faster and faster, until midway between the bow and the midship frame, where its velocity upwards is great- est, it will rise above the surface of the water w L, and its own momentum and that of other ascending particles will carry it upwards until it reaches the position of the midship frame, when it will begin to sink, until at y it reaches the same level from MEASURE OF THE HYDROSTATIC RESISTANCE. 415 which it rose. The surface particles, no doubt, which terminate their motion at y, hegin it at w and not at z, and to this circum- stance we may trace the origin of the hydrostatic resistance of the bow. The depression at y will be as great below the mean water-level w L as the elevation at a is above it ; and if the sur- face of the water at the stem stood at x instead of at w, the fore- body would be in equilibrium, seeing that the depression tx~w would suck the vessel forward as much, or nearly so, as the pro- tuberance from t to a would impede it. As the hydrostatic pressure from a to s pushes the vessel forward as much as the depression from s to y holds it back, the two portions of the after body will be in equilibrium ; and the whole moving vessel would be in equilibrium if the surface of the water at the stem stood at x instead of at w. As, however, the water stands higher at the stem than at the stern, there will be a hydrostatic resistance to be encountered which is equal to the height of the wave midway between a and w, which will be a, acting against the breadth of the ship. This will readily be understood by a reference to fig. 49^, which represents a horizontal slice of a floating body of the height of the wave which the body raises in passing through the water, and the form of the wave is represented by the trian- gular figure w a o, which is delineated on the plane surface formed by cutting away one-quarter of the model so as to clear the problem of the complication involved by the introduction of the curved form of the side. A transverse ordinate is drawn at &, and at the point & 5, where this ordinate meets the side, a line is drawn parallel to the axis, intersecting the line c e. From the point of intersection a vertical line 5 is raised, on which is set off the height of the wave at & &, and by drawing any desired number of similar lines the wave w a c will be set off on the midship section in the form ce d, which figure represents the hy- drostatic resistance of half the vessel. The area of the figure c e d is manifestly half the area of the parallelogram a c e d ; and as there is a similar figure on the other side of the vessel, the total area representing the hydrostatic resistance will be equal to half the height of the wave acting against the breadth of the ship. Supposing that no disturbing forces were in existence in in- 416 STEAM NAVIGATION. terfering with the upward and downward motion of the water, a particle of water at the forefoot B, fig. 48, would, as the vessel moved forward, follow the curved line B A; and if on rising above the lino w L it had not to encounter more of the force of gravity, it would pursue its course along the dotted line a D. Fig. 49J. As, however, as soon as the particle passes ahove w L, it has to encounter nearly the whole force of gravity, its momentum will not suffice to carry it up far, and it will proceed ahove the wa- ter level only to some such point as a, and wiU then immedi- ately pass downward and astern in the track of the curved line a o. The whole of the ascending and descending particles will pursue courses nearly parallel to these tracks ; and such lines might be drawn mechanically by a tracing point attached to a HOW TO REDUCE LOSS OF MOMENTUM. 417 pendulum in the manner already described, only that the half of the beat answering to the motion of the particle above the water-line, would be reduced in length by the ball being made in this part of its motion to compress a spring representing the increased power of gravity to which the particle is subjected during this part of its course. Hitherto we have discovered no source of loss of mechanical power in the movement of the water by a vessel passing through it, except that involved by the necessity of overcoming a con- stant hydrostatic resistance in consequence of the difference in the level of the water at the bow and stern. There will, how- Fig. 50. ever, be the loss of the momentum left in the undulating mass of water. But this last loss will be diminished, if we shift the midship frame further forward, as say to a, fig. 50, which is one- third of the length from the bow, instead of half the length. For, although we have still the hydrostatic resistance equal to half the height of a above w L multiplied by the breadth of the vessel to encounter, yet if the after-body of the vessel be prop- erly formed with diverging sides, the undulating mass of water will have surrendered most of its power to the vessel in aid of her propulsion before it leaves the stern at y. If we snppose the vessel to be cut off" at the water line, we shall get rid of the question of the hydrostatic resistance, as the water rising above the water-level will in such case run over the deck ; but the momentum of the undulating mass will remain, and the ob- ject to be attained is so to form the stern part of the vessel that the upward motion of the water above the water-line at the stern shall be resisted, whereby the mechanical power resident 18* 418 STEAM NAVIGATION. in the heaving water will be communicated to the vessel. This is done at present practically by causing the stern part of the vessel to spread outwards near the load water-line, so that the ascending column of water is intercepted by it and gradually brought to rest. The rise of water at the bow, it will be observed, increases not merely the hydrostatic pressure against which the vessel has to force her way, but also the opposing area against which the pressure acts. In like manner the deficient height of water at the stern diminishes both the pressure and the pressed area. It is very important, therefore, that the difference of level at the bow and stern should be as small as possible. And although we have supposed that the height of the wave a, fig. 50, would only be the same if we shifted forward the centre frame, it would in point of fact be higher if the same speed of vessel were maintained. On this ground, therefore, it appears prefera- ble to maintain the midship frame near the position shown in fig. 48, the more especially as the forward and ascending cur- rent due to the friction of the bottom of the vessel on the water has a tendency to bring the surface of the water relatively with the ship into the condition represented by the waving-line x t a s y. Before entering upon the consideration of the friction of the bottom, however, it may be stated that the hydrostatic resistance consequent on the increased elevation beginning at w instead of at x is not all loss. For while the height of the wave increases the pressure of the water beneath, it also helps to sep- arate the water ; and if the vessel be made without any straight part between the fore and after-bodies, a portion of the increased elevation which the mean water-line w L receives at the bow, will be retained to increase the elevation of the water at the stern, so that under certain conditions nearly the whole of the power expended in moving the water would be theoretically re- coverable. In practice, however, such a result is never reached ; and however perfect the arrangements for recovering the power may be made, yet a certain percentage of it is lost at every step ; and the safest indication is to employ such a form of vessel as will disturb the water as little as possible. This will be a body of the form which I have indicated with a considerable BEST MODE OF SHAPING VESSELS. 419 proportion of length to breadth, so that the vessel may be sharp at the ends. A length of 7 times the breadth is found to be a good proportion for such speeds as 15 or 16 miles an hour. But the proportionate length that is advisable, will increase with the intended speed. It is not difficult when the intended speed of the vessel and also its length and breadth are determined, to find what the proper form of the vessel will be, and also the height of the wave which the vessel will raise at the midship frame by her passage through the water, one-half of which height multiplied by the breadth of the vessel will be the measure of the hydro- static resistance. For as each particle of water at the stern has to describe the motion described by the ball of a pendulum which makes a double beat during the time that the vessel passes through her own length, the breadth of the arc will an- swer to half the breadth of the vessel, and the vertical height of the arc or the vertical distance fallen by the ball in passing from the highest to the lowest part of the arc, will be the height of the wave raised at the midship frame that being the height neces- sary to give the velocity of motion, with which the particles of water must be moved sideways through half the breadth of the vessel, to enable the vessel to pass through in the prescribed time. If we suppose the ball of the pendulum to be replaced by a mass of liquid moving in a circular arc, the motion of this liquid will be the same except in so far as it is affected by friction as if it were frozen and suspended by a rod of the same radius as the arc ; but if the mass of liquid be large so as to occupy any considerable part of the length of the arc, the motion will not be the same as that of a suspended point, as the whole of the particles will no longer rise and fall through the same height, while all of them will have still to be moved with the same velocity. So also if we have a tube open and turned up at both ends, and if we pour water into it and depress the water in one leg so as to disturb the equilibrium, the water when released will vibrate upward and downward like a pendu- lum. Such a tube is represented in fig. 51, where E A B n is the tube which is filled with water to the level of x. If the level in one leg be depressed from o to G, it will rise in the other leg 420 STEAM NAVIGATION. from D to H ; and if the depressing force be now withdrawn, the water will fall from H with a velocity corresponding to its height above G, and will be carried by its momentum above o to E, just as the ball of a pendulum ascends in '} ' its arc by the momentum it possesses and the water will continue to oscillate np and down ** like the ball of a pendulum, until it is finally D brought to rest by friction. If the tube be of equal bore throughout and be bisected in o, then as the accelerating force is the difference in the masses of the two unequal columns di- vided by their sum, the accelerating force will be represented by E G divided by o A B D, or what is the same thing, by E A B F ; or it will be proportional to the half of this, or to E o divided by o A o. The time of the oscillation or the time in which the surface of the water will fall from the highest to the lowest point, is equal to that in which a pendulum of the length o A o makes one vibration. Hence the time in which the surface will pass from the highest point to the lowest, and to the highest again, will be that in which a pendulum of the length o A o will make two vibrations, or it will be that in which a pendulum of four times that length makes one vibra- tion, or a centrifugal pendulum of the height equal to o A o makes one revolution. These relations equally hold, if we sup- pose the same kind of motion which exists in the water to be produced by a piston at o ; and the side of the ship may be supposed to be such a piston, and if properly formed, the ship will impart sideways to the water precisely the same kind of motion which exists in the case here illustrated. If a sheet of paper be drawn vertically behind a pendulum furnished with a tracing point, then I'ig- 52. if the pendulum be stationary, the (2.) tracing point will draw a straight line represented by the dotted line fig. 52. But if the pendulum be put into motion, then the tracer will describe the waving line A B o D SHARPNESS SHOULD VARY WITH SPEED. 421 Fig. 53. where the point A answers to the stem of a ship, the point B to the midship frame, and the point o to the stern ; and the paper will pass from A to o during the time the pendulum makes two oscillations. Since the pendulum has to make two oscilla- tions while the vessel passes through a distance equal to her own length, the combined motions of the tracer and pencil will delineate the proper form for the side of the vessel ; and if made in this form the particles of water will have the same motion as the ball of a pendulum, which motion enables the water to be moved with the minimum of loss. It will be useful, however, to take a particular case to show in what manner the proper form may be practically determined. Suppose A c, fig. 53, to represent the keel of a vessel which we may take at 200 feet long and 40 feet wide and which is intended to maintain a speed of 10 statute miles per hour, or 880 feet per minute. Now as the vessel has to pass through her length, or from A to c, during the time that the pendulum p makes a double beat, or to pass from A to B, which is 100 feet, during the tune the pendulum make a single beat, there will be 880 divided by 100, or 8*8 vibrations of the pendulum per minute ; and the rod of the pendulum must be of such " length as to produce that number of vibra- tions. Now to determine the length of the rod of a pendulum which shall perform any given number of vibrations per minute, we divide the constant number 375-36 by the number of vibrations per minute, and the square of the quotient is the length in inches. Hence 375-36 divided by 8'8 = 42-6, the square of which is 1814*76 inches or 151-23 feet, and apendulum 151-23 feet long beating in an arc 20 feet long with the paper travelling at a speed of 880 feet per minute, will describe the line ABO, which will be the proper water-line for the side of a ship if the cross-section be rectangular; and whatever the form of cross- 422 STEAM NAVIGATION. section this figure will equally determine the proper area of cross-section at each successive frame. If instead of moving at 10 miles an hour, the vessel has only to move at the rate of 5 miles an hour, the figure described will be that represented by D a E, and the breadths & & in the longer figure and V V in the shorter are the same, both being equal to half the breadth at a a. The rod of the pendulum p p passes through the point 5, and the pendulum vibrates from the plane of the keel to the plane of the side, so that the chord of the arc in which the vibration is per- formed is equal to half the breadth of the vessel, while the versed sine or height through which the pendulum falls at each beat, will be equal to the height of the wave at the midship frame. To find the versed sine of the arc, we divide the square of half the chord by twice the length of the pendulum. The chord being 20 feet the half of it is 10 feet ; and the pendulum being 151-23 feet long the double of it is 302'46 feet, and 100 divided by 302-46 = '33 feet or 3'96 inches. The height of the wave at the midship frame, in a vessel formed in the manner in- dicated, will accordingly be 3'96 inches, or rather this would be the height if the water were moved without friction, so that practically the height will be somewhat greater than is here in- dicated. If we increase the speed of the vessel, or increase the breadth, the hydrostatic resistance will increase very rapidly. Thus, if the speed of the vessel be increased to 20 miles an hour, or 1,760 feet per minute, the pendulum will require to make 1T6 beats per minute, and its length will be 375'36 divided by 1T6 = 21'3, the square of which is 453'69 inches, or S'T'S feet. Now, 100 divided by 37'8 = 2'6 feet, which will be the height of the wave at the midship frame in this case, and the hydrostatic pressure will be the half of this, or equivalent to 1'3 feet of water acting on the breadth of the vessel. In like manner, successive addi- tions to the breath of the vessel without increasing the length add rapidly to the hydrostatic resistance, as they involve the ne- cessity of the oscillating particles ascending higher and higher in the arc to enable the vessel to pass. FRICTION OF WATER. 423 FRICTION OF WATER. It remains to consider the friction of water upon the bottom of the vessel, and this is by much the most important part of the resistance which ships have to encounter. Beaufoy made a number of experiments to ascertain the amount of this resistance by drawing a long and a short plank through the water : and, by taking the difference of their resistances and the difference of their surfaces, he concluded that the friction per square foot of plank was, at one nautical mile per hour, '014 Ibs. ; at two nautical miles per hour, -0472 Ibs. ; at three, -0948 Ibs. ; four, 153 Ibs. ; five, -2264 Ibs. ; six, -3086 Ibs. ; seven, -4002 Ibs. ; and eight, '5008 Ibs. At two nautical miles an hour, the force re- quired to overcome the friction was found to vary as the 1*825 power of the velocity, and at eight nautical miles an hour as the 1'713 power. Other experimentalists have deduced the amount of friction from the diminished discharge of water flowing through pipes. If there were no friction in a pipe, the velocity of the issuing water should be equal to the ultimate velocity of a body falling by gravity from the level of the head to the level of the orifice.* But as the velocity is found by the diminished discharge to be only that due to a much smaller height, the dif- ference is set down as the measure of the power consumed by friction. This mode of estimating the friction is not applicable to the determination of the friction of a ship ; for, in the first place, the discharge is a measure not of the maximum, but of the mean velocity ; and, in the second place, there is every reason to believe that the friction per square foot on the bottom of the ship is quite different near the bow from what it is near the stern. As the water adheres to the bottom there will be a film of water in contact with the ship, which will be gradually pat * There is sometimes misconception on this subject, arising from a neglect of the difference between the ultimate and mean velocities of a falling body. Thus, if water flows from a small hole in the side of a cistern, the water will issue with tho ultimate velocity which a heavy body would acquire by felling from the level of the head to the level of the orifice, which, if the height be IS^j feet, will be 82J feet per second. The mean velocity of falling, however, is only 16^ feet per second, BO that the ultimate velocity is twice the mean velocity. 424 STEAM NAVIGATION. into motion by the friction ; and the longer the vessel is the less will be the friction upon a square foot of surface at the stern seeing that such square foot of surface has not to encounter sta- tionary water, but water which is moving with a certain velocity in the direction of the vessel. The film of water moving with the ves- sel will become thicker and thicker as it passes towards the stern, and it will rise towards the surface by reason of the virtual re- duction of weight consequent upon the motion. The whole of the power, therefore, expended in friction is not lost, as the power expended in the front part of the vessel will reduce the friction of the after part ; added to which, the rising current which the friction produces may be made to aid the progress of the ship, if we give to the after-body of the ship such a con- figuration as to be propelled onward by this rising current. Finally, when the screw is the propelling instrument, the slip of the screw will be reduced, and may even in some cases be rendered negative, by the circumstance of the screw working in this current ; and whatever brings this current to rest will use up the power in it, and so far recover the power which has been expended in overcoming the friction. In my investigations respecting the physical phenomena of the river Indus in India, I observed that the water not only ran faster in the middle of the stream, but that it also stood higher in the middle, so that a transverse section of the river would exhibit the surface as a convex line. At the centre of the river the stream is very rapid, but it is slow at the sides, so that boats ascending the river keep as close as possible to either bank ; and in some parts at the side there is an ascending current forming an eddy. I further observed, that not merely were there rapid and considerable changes in the velocity, which I imputed partly to the agency of the wind in deflecting the most rapid part of the current to the one side or the other of the river, but there were also diurnal tides ; or, in other words, the stream ran more swiftly in the afternoon than in the early morning. -This had been long before observed, and was imputed to the heat of the sun melting the snows in the mountains more during the day than during the night. But although such an effect might be INFLUENCE OF HEAT ON VELOCITY OF KIVEES. 425 observable in a single feeder, the river is supplied from so many sources at different distances tbat such intermittent accessions would equalise one another. Moreover, the effect of the sun in the daytime in swelling the volume of the river, if acting without any equalising influence, could only produce a wave like a tidal wave in the river ; and the increase of velocity would at some points take place at night and at some in the morning, whereas I found it to take place eceryichere at the same time. I finally came to the conclusion that the phenomenon is caused by the in- fluence of the sun in heating the water of the river, and thereby increasing its liquidity and its velocity throughout the whole length of the river. The temperature of the water in the river is commonly about 94 Fahr., but as the river is wide and shal- low, it is rapidly heated and cooled, and there are several de- grees difference between the temperature of the day and the night. In the early morning the river is coldest, and at that time also other things being equal its velocity is least. It may hence be concluded that any thing which gives more mo- bility to the particles of the water in which a vessel floats will diminish the friction of the bottom ; and this end seems likely to be attained by the injection of air into the water at the stem and forefoot or front part of the keel. It is not difficult to understand how it comes that the water hi a river should stand higher at the middle than at the sides, as shown in fig. 54. If we hang a weight upon a spring balance we shall find the amount of the weight to be indicated on the scale or index ; and Fig. 54 this weight will continue to be shown so long as we hold the spring balance sta- tionary. But if we allow it to move tow- ards the earth with the velocity which a heavy body would acquire in falling by gravity, the index of the spring will show no tension at all proving that with this amount of downward motion the body imparts no weight. If the spring is moved downward slower than a body falls by grav- ity, the spring will show that it is sustaining some weight ; but at any velocity downward there will be a diminution in the 426 STEAM NAVIGATION. Fig. 55. weight of the body answerable to that velocity. In two columns of water, therefore, moving at different velocities, the slower will exert most hydrostatic pressure on the pipe or channel con- taining it ; and where two such columns are connected together sideways, as in a river, the faster must rise to a greater height to be in hydrostatic equilibrium sideways with the slower. The surface of the water consequently becomes convex, as shown at M in fig. 54, where H is the water and A B o D the bed. It will be seen from these observations that there is a hy- draulic as well as a hydrostatic head of water ; and the hydraulic head is equal to the hydrostatic head, diminished by the height due to the velocity with which the water flows. This law is further illustrated by fig. 55, which represents a bulging vessel in which the water is maintained at a uniform height by water flowing into it at the top, while it runs out at E at the bottom. The velocity with which the water flows downward from A to E, varies with the amount of enlarge- ment or contraction of the vessel; and the height of water which will be supported in the small pipes J, c and y the area in square feet of the immersed transverse section of the vessel, and divide the product ~by the indicated horse-power. The quotient will 5e the coefficient of that particular type of vessel. Example. The steamer Fairy, with an area of immersed section of 11^ square feet, and 363 indicated horse-power, at- tained a speed of 13'3 knots an hour. "What is the coefficient of that vessel? Here 13'3 cubed = 2352-637, which multiplied by 71 '5 and divided by 363 horse-power = 465, which is the coefficient of this vessel according to Boulton and Watt's rule. A good num- ber of coefficients for different vessels is given at page 77. TO DETERMINE WHAT SPEED WILL BE ATTAINED BY A STEAM VESSEL OF A GIVEN TYPE WITH A GIVEN AMOUNT OF ENGINE POWER, BY BOULTON AND WATT'S EULE. RULE. Multiply the indicated horse-power "by the coefficient proper for that particular type of vessel, and divide the product J)y the area of the immersed transverse section in square feet. Extract the cube root of the quotient, which, will l)e the speed that will lie obtained in knots per hour. Example. What speed will be obtained in a steamer of which the coefficient is 465, and which has an immersed section of 71 J square feet, and is propelled by engines exerting 363 horse-power. Here 363 x 465=168795, which divided by 71 '5=2360. The logarithm of this is 3-372912, which divided by 3=1-124304, the natural number answering to which is 1331. Now the in- dex of the divided logarithm being 1, there will be two integers in the natural number answering to it, which will consequently be 13-31, and this will be the speed of the vessel in knots per hour. MEAN VELOCITY OF WATEB IN CANALS, ETC. 433 The coefficient of a steamer sometimes varies with the speed with which the vessel is propelled. If the vessel is properly formed for the speed at which she is driven, then her coefficient will not become greater at a lower speed ; and if it becomes greater, the circumstance shows that the vessel is too blunt. When the ' Fairy' was sunk to a draught of 5 feet 10 inches, her speed was reduced to 11-89 knots, and her coefficient was re- duced from 465 to 429, showing that she worked more advan- tageously at the higher speed and lighter draught. The ' War- rior,' which when exerting 5,469 horse-power attained a speed of 14.356 knots with a coefficient of 659, attained when exerting 2,867 horse-power a speed of 12-174 knots with a coefficient of 767 ; and when exerting 1,988 horse-power a speed of 11-040 knots with a coefficient of 825. This shows that the 'Warrior' is too blunt a vessel for a high rate of speed. It will be satisfactory to ascertain the comparative eligibility of the forms of the 'Fairy' and the "Warrior,' which we may easily do by comparing the speed attained by each, with the speed which would be attained by an equal weight of water running in a river or canal, and impelled by an equal motive force. The rule for determining the speed of water flowing in rivers or canals of any given declivity is as follows : TO DETERMINE THE MEAN VELOCITY WITH WHICH WATEE WILL PLOW THROUGH CANALS, AETEKIAL DRAINS, OE PIPES, SUNNING PARTLY OE WHOLLY FILLED. RULE. Multiply the hydraulic mean depth in feet "by twice the fall in feet per mile. Extract the square root of the product, which i* the mean velocity of the stream in feet per minute. Now the 'Fairy,' when realizing a speed of 13'3 knots per hour with 363 horse-power, had a draught of water of 4-8 feet ; a sectional area of 71'5 feet; a wetted perimeter of 24'7 feet, and a displacement of 168 tons. The hydraulic mean depth be- ing the sectional area in square feet, divided by the length of the wetted perimeter in feet, the hydraulic mean depth will in this case be 71 -5, divided by 24-7=2-9. The engine made 51-6 revolutions per minute, and the screw 19 434 STEAM NAVIGATION 258 revolutions per minute, being five times the number of rev- olutions of the engines. The stroke is 3 feet, and the pitch of the screw 8 feet. Now a horse-power being 33,000 Ibs., raised 1 foot per min- ute, and as there were 363 horse-power exerted, the total effort of the engines will be 363 times 33,000, or 11,979,000 Ibs., raised through 1 foot each minute. But the engine makes 51 '6 revolu- tione each minute, and the length of the double stroke is 6 feet, so that the piston moves through 309 '6 feet per minute; and the power being the product of the velocity and the pressure, the power 11,979,000 Ibs. divided by the velocity of the piston, 309'6 feet per minute, will give the mean pressure urging the pistons, which will be 38,691 Ibs. But the speed of the screw- shaft being five times greater than that of the engine-shaft, the pressure urging it into revolution must, in order that there may be an equality of power in each, be five times less ; or it will be 7,538 Ibs. moving through 6 feet at each revolution. Then the pitch of the screw being 8 feet, the thrust of the screw will be less than 7,538, in the proportion in which 6 is less than 8, or it will be 5,653, supposing that there is no loss of power by slip and friction. It is found on an average in practice, that about one-third of the power is lost in slip and friction ; and the actual thrust of the screw-shaft will be about one-third less than the theoretical thrust, or in this case it will be 3,769 Ibs. or 1*68 ton. Now, in order that 168 tons of water may gravitate down an inclined channel with a weight of 1*68 ton, the declivity of the channel must be 1 in 100. In 1 mile, therefore, it will be 52-80 feet. A cubic foot of salt-water weighs 64 Ibs., so that there are 35 cubic feet in the ton, and in 168 tons there are 5,880 cubic feet. Dividing this by the sectional area 71 '5 feet, we get a block of water 82'2 feet long, and with a cross-section of 71 '5 feet, weighing 168 tons; and the wetted perimeter being 24*7 feet, and the length 82'2 feet, we get a rubbing area of 2020*34 feet; and as the friction on this surface balances the weight of 3,769 Ibs., there will be a friction of 1*8 Ib. on each square foot. If this block of water be supposed to be let down a channel falling 1 in 100, its velocity will go on increasing until SHIPS COMPARED WITH RIVERS. 435 the friction balances the gravity, which, according to the rule given above, "will be when the water attains a speed of 11 miles an hour, from whence we conclude that the sum of the resist- ances of a well-formed skip are less than the friction alone of an equal weight of water of the same hydraulic depth, moved in a pipe or canal by an equal impelling force. If instead of taking the declivity in 2 miles, as the rule prescribes, to ascertain the velocity of the water, we take the declivity in twice 2, or 4 miles, we shall arrive at a pretty exact expression of the speed of the vessel in this particular case. Taking the knot at 6,101 feet, 13'3 knots will be equal to 15'3 statute miles, and the de- clivity in 1 mile being 52 - 8 feet, the declivity in 4 miles will be 211'2 feet. Multiplying this by 2'9, the hydraulic mean depth, we get 612'48, the square root of which is 24*7, which multi- plied by 55, gives the speed of the water in feet per minute = 1,358-5. This, multiplied by 60, gives 81,510 feet as the speed per hour, and this divided by 5,280, the number of feet in a statute mile, gives 15 '4 as the speed in statute miles per hour. The 'Warrior,' with a displacement of 8,852 tons, a draught of water of 25 J feet, an immersed midship section of 1,219 square feet, and 5,469 horse-power, attained a speed of 14*356 knots, or 16'6 statute miles. The number of strokes per minute was 34J, and the length of the double stroke 8 feet, while the pitch of the screw was 30 feet. The wetted perimeter is 88 feet, which makes the mean hydraulic depth 13 '8 feet. The power being 5,469 horses, 33,000 times this, or 180,477,000 Ibs., will be lifted 1 foot high per minute. But as the piston travels 54'25 times 8 feet, or 434 feet each minute, the load upon the pistons will be 415,845 Ibs. The pitch of the screw, however, being 30 feet, while the length of a double stroke is 8 feet, the theoretical thrust of the screw will be reduced in the proportion in which 30 exceeds 8, or it will be 110,892 Ibs. If from this we take one-third, on account of losses from slip and friction, we get 73,928 Ibs., or 33 tons, as the actual thrust of the screw. Now 8,852, which is the displacement in tons, divided by 83 tons, which is the motive force in tons, gives 268, or, in other words, the declivity of the channel must be 1 in 268, in order 46b STEAM NAVIGATION. that 8,852 tons may press down the inclined plane with a force of 33 tons. This is a declivity of very nearly 20 feet in the mile, or 40 feet in two miles, or 80 feet in twice two miles. The mean hydraulic depth being 13'8 feet, 80 times this is 1,104, the square root of which is 33'2, which multiplied by 55 = 1,826 feet per minute, or multiplying by 60=109,500 feet per hour. Dividing by 5,280, we get the speed of 20 miles per hour, which ought to be the speed of the ' Warrior ' if her form were as eligible as that of the 'Fairy.' The speed falls 3'4 miles an hour short of this, which defect must be mainly imputed to the de- ficient sharpness of the ends for such a speed and draught, and the increased resistance consequent on the greater depth. In a paper by Mr. Phipps, on the ' Eesistances of Bodies pass- ing through "Water,' read before the Institution of Civil En- gineers in 1864, it was stated that these resistances comprised the Plus Resistance, or that concerned in moving out of the way the fluid in advance of the body ; the Minus Eesistance, or the diminution of the statical pressure behind any body when put into a state of motion in a fluid ; and the Frictional Resistance of the surface of the body in contact with the water. The Plus Resistance of a plane surface one foot area, moving at right angles to itself in sea water, was considered to be 64'2 x 2 It = -, and the Minus Resistance was one half the Plus 2? Resistance. For planes moving in directions not at right angles to them- selves, the theoretical resistances were, for the Plus Pressure a , #64-2fl 2 8 = -, and B = , r 8 ' 2g ' the Minus Pressure being one-half the above ; where JK was the resistance of the inclined plane ; , the area of the projection of the inclined plane upon a plane at right angles to the direc- tion of motion ; r, the ratio of the areas of the projected and the inclined planes ; and $, the area of a square-acting plane of equivalent resistance with the inclined plane. But, besides these theoretical resistances, the experiments of Beaufoy showed, that when the inclined planes were of moderate RECENT COMPUTATIONS OF RESISTANCE. 437 length only, the Plus Resistance was considerably in excess of the above ; so that when the slant lengths of the planes were to their bases in the proportion of 2 to 1, 3 to 1, 4 to 1, and 6 to 1, the actual resistances exceeded the theoretical, as 1-1 to 1, 1-98 to 1, 3-24 to 1, and 6'95 to 1. Mr. Phipps proposed a method of approximating to these ad- ditional resistances, by adding the constant fraction of |th of a square foot for every foot in depth of the plane to the quantity /S previously determined, which empirical method he found to agree nearly with the results of Beaufoy's experiments. The resistances of curved surfaces, such as the bows of ships, were adverted to, the method of treating them being to divide the depth of immersion into several horizontal layers, and then again into a number of straight portions, and to deal with each portion as a separate detached plane, according to the preceding rules. The question of friction was then considered. The experi- ments of Beaufoy were referred to, giving 0*339 Ib. per square foot as the co-efficient of friction for a plained and painted sur- face of fir, moved through the water at 10 feet per second, the law of increase being nearly as the squares of the velocities, viz., the 1-949 power. Mr. Phipps was, however, of. opinion, that a surer practical guide for determining the coefficient of friction would be, by considering all the data and circumstances of a steam-ship of modern construction, moving through the water at any given speed. The actual indicated horse-power of the engines being given, the slip of the paddles being known, and the friction and other losses of power approximated to, it was clear that the portion of the power necessary to overcome the resistance of the vessel might be easily deduced. By determining approxi- mately, by the preceding rules, the amounts of the Plus, the Minus, and the Additional Head resistances, and deducting them from the total resistance, the remainder would be the resistance 438 STEAM NAVIGATION. due to the friction of the surface. By this process, and taking as an example, the iron steam-ship ' Leinster,' when perfectly clean, and going on her trial trip 30 feet per second in sea-water, her im- mersed surface being 13,000 square feet, the coefficient of fric- tion came out at 4*34 Ibs. per square foot. Beaufoy's coefficient of 0*339 Ib. per square foot at 10 foot per second would, according to the square of the velocities, amount to 3'051 Ibs. at 30 feet per second. The difference between this amount and the above 4*34 Ibs. might be accounted for by a difference in the degree of roughness of the surfaces. Other methods for the determination of the coefficient of friction were then discussed. One, derived from the known friction of water running along pipes, or water-courses, was shown to be considerably in excess of the truth. It was founded upon the observed fact, that at a velocity of 15 feet per second, the friction of fresh water on the interior of a pipe was 25 oz.* per square foot. Apply ing this to the ship ' Leinster,' and increasing the friction as the square of the velocities up to 30 feet per second, the above friction would become 100 oz., or 6 Ibs., per square foot, which, acting upon 13,000 square feet of surface, would absorb, at the above speed, no less than 4,395 H.P., whilst the total available power of the engines (after deducting from the indicated 4,751 H.P. ^th for friction, working air-pumps, and other losses, and |th of the remainder for the observed slip), was only 3,421 H.P. ; thus showing an excess of resistance equal to 974 H.P., without allowing any power to overcome the other re- sistances. The assumption of 25 oz. being the proper measure of the friction per square foot, at a velocity of 15 feet per second, upon the clean surface of an iron ship, seemed to have arisen from the opinion, very generally entertained, that there was no difference in the amount of friction in pipes and water-courses, whether internally smooth like glass, or moderately rough like cast-iron, and that the surfaces of ships were subject to the same action. The comparatively recent experiments, in France, of the late M. Henry Darcy were in opposition to the above view, and * For sea -water this quantity must be increased as the specific gravity, or as 62-5 to M-2. EFFECT OF SMOOTHNESS OF SUKFACE. 439 showed that the condition as to roughness of the interior of a pipe modified the friction considerably. Thus, with three differ- ent conditions of surface, the coefficients were : A. Iron plate covered with bitumen made very smooth, Q'000432 B. New cast-iron 0-000584 C. Oast-iron covered with deposits . . . .0-001167 The friction was, therefore, nearly as 1, 1, and 3. As there appeared no reason to doubt the correctness of M. Darcy's experiments, even in pipes the notion of the friction being uninfluenced by the state of roughness of the interior could no longer be entertained. The 25 oz., previously mentioned as the measure of friction per square foot for the interior of pipes and water-courses, could not, therefore, be regarded as a constant quantity, applicable to all kinds of surfaces; but from Mr. Phipps' calculations, it appeared to come intermediately between the coefficients of the surface B and 0, given in the above scale ; as at 15 per second, A would give 13^ oz. per square foot B " 20 " " and " 40 " " Besides, there was another cause for an excess of friction in pipes and water-courses, over that upon ships, even when the surfaces were equally smooth. It arose from the circumstance, that where the velocity of the water hi a pipe, or open water- course, was spoken of, the meaning was, its average velocity; whilst the velocity of a vessel through still water meant what the words implied, namely, the relation of the vessel's motion to the fluid at rest. If the case were taken of a water-course of such width, that the friction of the. bottom only need be considered, with an average velocity of flow of 15 feet per second, the friction upon the bottom would be equal to 25 oz. per square foot ; but according to the rules generally used, an average velocity of 15 feet per second corresponded to a surface velocity of 16-66 feet 440 STEAM NAVIGATION. per second, which was the velocity with which a vessel should pass through still water, to give an equal friction upon its sides. According to Beaufoy, the velocity of 16-66 feet per second would produce a friction of -932 Ibs. or 14-91 oz., where 15 feet would only give 12'2 oz. The difference between 14*91 oz. and 25 oz. (equal to 10*09 oz.) must, therefore, Mr. Phipps thought, be set down to the different degree of roughness of the surfaces in the water-course and the vessel. Taking then 4*34 Ibs. as the friction per square foot of a new iron ship, moving through the water at a speed of 30 feet per second, it would be found, Mr. Phipps considered, that this was equal to the o-g4-erfttures, 159 Elbow-jointed lever, 89 Elliot, Brothers, indicator by, 835 Energy, conservation of, 78 Engines, if perfect, power producible by, 181 Equations, nature of, T4 Equation for determining the speed of steamers, 76 Equivalent, mechanical, of heat, 91 Ericsson the designer of the American monitors, 461 Evaporation, latent heat of, 162 in locomotives, 331 Evaporative powers of combustibles, 175 power of coal, 812 468 INDEX. EXH Exhaustion of chimneys, 304 Expansion of air by heat, 146 of gases, 166 by link, diagrams showing, 355, 356 of steam, 182; measure of benefit from, 183 ; mean pressure of expanding steam, loO producible by a given proportion of lap to stroke of valve, 1ST, 188 Expansion producible by throttling the steam, 198 Exponents, fractional, 51 Eyes of cranks of wrought-iron, 271, 272 Eye of air-puinp crosshead", 283 TUCTOES denned, 29 J? Fahrenheit's thermometer, 133 'Fairy' steamer, lines of, 454 Falling bodies, laws of, 90, 97 Fans, power required to drive, 391 Feed pipe, rule for proportioning, 221 pump, to find the proper capacity of, 224 Feeding boilers by Giffard's injector, 384 Film of water moving with a ship, 428 Fire bars of locomotives, 314 Fishes, shape oij translatable into that of ships, 407 Flaud, pumps by, 385 Flax mills, 395 Floating bridge, diagrams from, 855, 356 Flour mill, 387 Flues, proper sectional area of, 310, 811 boilers, proper proportions of, 811 Flues, sectional area of, required to evaporate a cubic foot of water per hour, 314 collapsing pressure of, 32T Fluids, motion of, 100 Fly-wheels, momentum of, 106 ; burst- ing velocity of, 110 should have power equal to six half strokes, 216 Fly-wheel, Boulton and Watt's rule for the, 228 shaft, 239, 240 Force, conservation of, 78 centrifugal, 107 ; how to measure, 108 ; bursting velocity, 110 of dilatation, 143 elastic, of steam at different temper- atures, by M. Kegnault, 159 Form of least resistance in ships, 402 Formula for determining the speed of steamers, 76 Foot valve, passages to find the proper area of, 228 Fractions, nature of, 5 ; vulgar, 5 ; deci- mal, 6 multiplication by, 9 nature and properties of, 31 ; how to GRA reduce a fraction to its lowest terms. 33 addition and subtraction of, 34 how to reduce a common denominator, 35 multiplication and division of, 38 squares and square roots of, 45 cubes and cube roots of, 48 resolvable into infinite series, 66 Fractional exponents, 52 Frame, midship, of ships, best position of, 417 Franklin Institute, experiments on steam by, 157 French Academy, experiments on steam by, 157 Friction, 118; coefficient of, 119; experi- ments on, by Morin and Bochet, 118 of crank pins, 120 bearings varies with the pressure, 121 ; relations of pressure and velocity, 122 Friction of flowing water, 199 engines, 367 water in pipes does not vary with the pressure, 429 bodies moving in water varies with nature of surface, 439 the bottom the main source of re- sistance in ships, 423 bottom of steamer, ' Leinster,' 438 Fuel, different kinds of, heating power, 175 consumed per indicated horse powei per hour at Chelsea Water-works, 868 Fulling mills, 394 Furnaces, temperatures of, 178 rates of combustion, 179 importance of high temperature of, 377 GAS into a vacuum, velocity of, 102 Gases, dilatation and compression of, 143, 144, 145 and vapours, difference between, 158 liquefied by cold and pressure, 153 specific heats of, 164, 165; densities, volumes, and rates of expansions of, 166 Gearing, differential, 86 Gearing, proportions proper for, 231 Gibs and cutters of crosshead, 258 side rods. 260 through crosstail, 266 air-pump crosshead, 280 air-pump side rods, 286 Giffard's injector, 383 Glass works, 897 Governor for steam engines, 117 to determine the right proportions of the, 281 Grate coal burned on each square foot in different boilers, 314 INDEX. 469 GEA Grate surface to evaporate a cubic foot per hour, 314 bars per nominal horse-power in steamers, 315 Gravity, nature of, 93 Gudgeons in side lever, 269 Guns, piston, 460 Gyration , centre of, 112 ; to find the posi- tion of, 113 Gyroscope, phenomena of the, 93 Gwynn's centrifugal pump, 386 'TTAJfSA' steamer, proportions of 11 machinery of, 314 Haystack boiler, 316 Heat, motive power of, 90 mechanical equivalent of, 91, 167 power producible by, 131 sensible, defined, 135 latent, defined, 135 specific, defined, 135 dilatation by, 140 specific, 162 unit of, 162 effect of, in accelerating the velocity of rivers, 425 Heating surface of boiler per square foot of fire grate, 314 to evaporate a cubic foot of water per hour, 314 and cooling surface of con- denser, 815 in modern boilers, 375 Height from which bodies have fallen determinable from their velocity, 98 Height from which bodies have fallen determinable from their time of fall- ing, 98 of chimney proper for different boilers, 805 Hodgkinson, strength of woods accord- ing to, 128 ; law of strength of pillars by, 128, 131 ; of cast-iron beams, 183 Horse-power, nominal, definition of, 79 actual, definition of, 79 Hot well, Indicator diagrams from, 359, 860 Hydraulic press, pressure producible by, 81,84 head of water different from hydro- static head, 426 mean depth of a ship, 481 Hydrostatic resistance of vessels increas- es with speed and with breadth, 422 head of water different from hydraulic head, 426 ICE, weight of at 82% 140 made in a red-hot crucible, 170 Improvements required in boilers and condensers, 879 LES Inches, square and circular, spherical, cylindrical, and conical, 9 Incommensurables, nature of, 46 Indian system of numeration, 3 Indicator, construction of the, 833; Kichards', 334 ; method of applying the, 335,870 diagrams, how to read 835 ; how to take, 370 ; various examples of, 838 ; from air-pump, 344, 351, 357 ; from hot well, 359 ; from water pump, 361 ; from double cylinder engine, 365, 869 Indicator diagrams, showing momentum of indicator piston, 347 Inertia defined, 105 Infinite series, how to resolve fractions into, 66 strains from crank and elbow-jointed lever, 89 Injection pipe, to find the proper area of, 222 Injector, Giffard's, 551 Invisible light, 96 Iron, steel, and other metals, strength of, 125 fusible at low temperatures, 151 works, 897 Iron-clad steamers penetrable by shot, 457 Irrational numbers defined, 46 1 Island Queen, 1 indicator diagram from, JET, composite, in chimney, 819 tl Joule's experiments on the conden- sation of steam, 173 Journals of crosshead, proper dimensions of, 257 air-pump crosshead, 284 T AMBETH "Water-works, engines at, JL 862; diagrams from, 865; duty of, 868 Latent heat defined, 135 of liquefaction, 151 heats of steams from water, alcohol, ether, and sulphuret of carbon, 154 Lap of valve proper for a given amount of expansion, 187, 189. 190 on eduction side, effects of. 187, 193 Lead plug, 830 'Leinster,' steamer, computation of fric- tion of, 438 Length of pendulum to vibrate at any given speed, 115 vessels should vary with intended speed, 421 Leslie's explanation of the strength of iron, 126 470 LET Lctcstu, pumps by, 385 Lever, action of the, 83 elbow-jointed, 89 Levers of Stanhope press, 89 Light, invisible, 94 Lineal measure explained, 7 Lines of ships, 400 ; illustrated by shape of fishes, 40T Link motion, 198 expansion by, diagrams showing, 355, 356 Liquefaction, 150 ; latent heat of, 151 ; of gases, 153 Liquids, dilatation of, by beat, 143 specific heats of, 164 Locomotive engines, proper proportions of, 301 boiler, example of, 329 efficiency of steam vessels, 814 Logarithms, nature of, 52 ; mode of using, 56 Lowest terms, how to reduce a fraction to, 33 TITADAGASCAE, mode of numeration M used in, 2 Machines, strains and strengths of, 81, 87 how to determine power of, 82 Magnitude, standards of, 7 Magnus, his experiments upon ebulli- tion, 168 Main links, 232 centre of land engines, 232 centre of marine engines, 268 beam of land engines, how to propor- tion, 233 Maize mill, 3S7 Marine engines, proportions of the parts of, 254-301 boilers, proportions of, 314 Marquis do I'lftipital, his rule for finding the centrifugal force, 108 Materials, strength of, 124 Maudslay and Co.'s side lever engines, dimensions of, 290 Maximum density of water, 139 Midship section of ships, best form of, 409 frame of ships, best position of, 417 Mill gearing, proportions proper for, 246 Mills: flour, 387; barley, 387; rye, 387; maize, 887 ; bean, 388 ; oil, 388 ; saw, 888 ; sugar, 390 ; cotton, 391 ; weaving, 393 ; wool, 393 ; fueling, 394 ; flax, 394 ; paper, 396 ; rolling, 397 Millwall Ironworks, engines by, 882 Mechanical power from the Bun, 79 nature of, 90 of the universe constant, 92 equivalent of heat, 91 Melting points of solids, 148 Membrane pump, by Brule, PER Mercury, relative density of, 100 into a vacuum, velocity of, 101 Merry weather, pumps by, 385 Metals, strengths of, 125 conducting powers of, 172 Molecular attraction of water retards boiling, 168 Momentum defined, 105 ; of rams, 105 ; of cannon balls, 105 of heavy moving bodies, how meas- ured, 106; of a revolving disc, 111 indicator piston, 347 Monitors, features of their construction, 461 ; weak points of, 462 ; mode of de- stroying, 463 Moors brought decimal system into Eu- rope, 3 Morin's experiments on friction, 118 Morin, General, his experiments on va- rious machines, 387-397 Motive power of heat, 90 Motion of fluids, 100 power required to produce, 106 in a circle, 107 Multiplication, nature of, 16; multipli- cation table, 19, 23 ; examples of, 20 ; mode of performing, 22 Multiplier defined, 20 Multiplicand defined, 20 Multiplication by fractions, 9 of fractions, 88 ' Munster,' indicator diagrams from, 340 Mylne, his constant for velocity of water in pipes, 206 ATAPIEE, DAVID, his haystack boil- ll ers, 173, 316 Numerator of fractions defined, 5 OAK posts, proper load for, 180 Oil mill, 888 Ordnance, increased power of, attainable, 459 ' Orontcs' indicator, diagrams from, 348, 349 Oscillation, centre of, 114 Oxygen required for combustion, 175 207 T)ADDLE shaft, 294 I Paper mill, 896 Parallel motion, how to describe the, Parallelepiped, circumscribing, 406 Peclet's rule for proportions of chim- neys, 306 Pendulum, action of the, 95 laws of the, 114 centrifugal, 116 Percussion, centre of, 112 Perrin, pumps by, 387 Perry, pumps by, 887 471 PEE Persian wheel, 386 ' Persia,' steamer, 8S9 Phipps. on resistances of bodies by, 436 Pillars, law of strength of, 131 Pipes, velocity of water flowing in, 199, 433 and passages, proper proportions of, for different powers, 300 Piston rod for land engines. 231 of marine engines, 261 table of proportions of, 299 valves, by D. Thomson, 363 guns, 409 Plates of boilers, proper thickness of, 322 Plus, the sign of addition, 10 Pneumatic Despatch Company's engine, indicator diagrams from. 354, 355 Portsmouth floating bridge, diagrams from, 355, 356 Posts of oak, proper load for, 130 Powers and roots of numbers, 49 Power, m chanical, from the sun, 79 mechanical, nature of, 90 motive, of heat, 90 required to produce motion, 106 resident in a revolving disc, 111 producible by a given quantity of heat, 181 in a perfect engine, 181 cheapest source of, 181 nominal, how to determine, 208; Ad- miralty rule for, 211 of boilers an indefinite expression, 809 and performance of engines, 333 loom weaving, 393 required to produce a given speed in steam vessels, 430, 432, 443 Press, Stanhope, 89 Pressure, atmospheric, how produced, 100 permissible on bearings moving with a given speed, 121 strength of boiler to withstand, 823 safe, in a cylindrical boiler, 325, 326 collapsing of flues, 327 Pressures and volumes of gas, 147 Printing machines, 396 Product defined, 20 Projectiles should contain rocket com- position, 460 ; and have spiral feathers to put them into revolution, 460 Proportion, nature of, 42 Proportions of steam-engines, 208, 214 engines laid down to curves, 288 locomotive engines, 801 boilers, 304 wagon boilers, 310 ; of flue boilers, 811 Pump, combined plunger and bucket, SAW Pumps, relative efficiency of different kinds, 337 by various makers, 3S7 Pumping engine at St. Katherine's docks, diagram from, 346 engines, friction of, 867 ; duty of, 368 AUOTIEXT defined, 24 pADIATIOX of heat, 171 It Eankine. his method of computing speed of steam vessels, 443 Ratio, or Proportion, nature of, 42 Reaumur's thermometer, 138 Red-hot crucible, ice made in. 170 Reduction. 67 Rcgnault, his experiments on dilatation of gases, 145 Regnault's formulae for the elastic force of steam, 158 Relative bulks of water and steam at at- mospheric pressure, 102 Rennie, tensile strength of metals ac- cording to. 126 ' Research,' indicator diagram from, 349 Resistance of vessels, 399 mainly caused by friction, 423, 442 at bow and stern, 436 hydrostatic, of vessels, increases with speed and with breadth, 422 ' Rhone' steamer, proportions of engines and boilers of. 382 Richards' Indicator, 334 Rivers, velocity of, 199 have water highest where stream is fastest, 424; effect of temperature on velocity, 425 Riveted joints, best proportions of, 820 ; strength of, 320 Revolving bodies, centrifugal force of; 109 ; bursting velocity, 110 Rocket vessels propelled by rockets, a new expedient of warfare, 464 Roman method of numeration, 3 Roots, square, 44 ; cube, 48 Ropes tightened by pulling sideways, 84 Rule of three, 42 Rye mill, 387 OAFETT valves, rule for proportion- ij ing, 219 Saw mill, 388 ; for veneers, 889 472 INDEX. SAW Saw circular, 389 for stones, 889 Screw, pressure producible by, 81 differential, pressure producible by, 81,85 of Archimedes, 380 ' Scud,' diagram from hot well of, 360 Seaward and Co.'s side lever engines, di- mensions of, 292 Sectional area of boiler flues or tubes, 314 ; of chimney, 314 Sensible heat defined, 135 Side lever, proper proportions of; 267; studs of, 269 ; thickness of eye round, 271 engines, dimensions of, by Caird and Co., 2S7 ; by Maudslay, 290 ; by Seaward, 292 rods of marine engines, proper pro- portions of, 258 rods of air-pump in marine engines, 284 Solid measure explained, 8 Solids, melting points of, 148 Specific heat denned, 135 162; of different bodies, 1G3, 165, 166 heats under constant pressure and under constant volume, 164, 167 gravities, tables of, 165 of oxygen and carbonic acid, 175 Speed of steamers, rule for determining, 77 steam vessels, how to determine, 430, 432, 443 vessels a main condition of success in war, 460 common steamers may be increas- ed by rocket composition, 464 Shafts, strength of, 133 of fly- wheel, 238. 239 for paddles, 294 ; sizes of wronght-iron shafts for different powers, 294 Ships, maximum breadth of, best posi- tion of, 417 length of, should vary with intended speed, 421 resistance of, mainly caused by fric- tion, 423, 442 Spherical measure, 9 Spheroidal condition of water, 109 Square measure explained, 8 and circular inches, 9 roots, nature of, 44 of fractions, 45 root, method of extracting, 47 Squares and square roots, 44 of fractions, 45 St. Katharine's Dock, diagram of engine at, 346 Standards of magnitude, 7 Stanhope press, levers of, 89 SUN Stays of boilers, 321 Steam-engines, great waste of heat in. 91 Steam-engine, theory of the, 134 Steams, latent heats of, from water, al- cohol, ether, and sulphur of carbon, 154 Steam and water, relative bulks of, at at- mospheric pressure, 102 of atmospheric pressure, density of, 102 rushing into a vacuum, velocity of, 102 ; velocity the same at all pressures, 102; velocity into the atmosphere, 103 sensible and latent heat of, by M. Regnault, 155; elastic force of, 155- 161 expanding, mean pressure of, 285 ports, 216 pipes, proper size of, 218 boilers, proportions of, 304 room, 315 ports of locomotives, 330 pipes of locomotives, 331 navigation, 399 vessels, locomotive efficiency of, 313 Steamers, equation for determining speed of, 77 Steamer ' Fairy,' body plan of, 454 ; ' Bat- tler,' 455; 'Bremen,' 456; 'Persia,' 457; ' Warrior,' 458 Stones, strength of, 125 machine for sawing, 889 Strains of machines, how measured, 81, 86 infinite, how produced, 89 Strap of side rod, proper dimensions of, 259 connecting rod, proper dimensions of, 265 Straps of air-pump side rods, 285 Strengths of machines, how determined, 81, S6 Strength of main beam of an engine, 87 of materials, 124; elastic strength, 124 cast-iron columns, 129, 131 ; of cast- iron beams, 133 ; of shafts, 133 boiler to withstand any given pres- sure, 322 Studs of the beams of land engines, 232 in side lever, 269 ; metal round studs, 271 Subtraction, nature of, 13 ; indicated by or minus, 14; method of perform- ing, 15; examples of, 16 of fractions, 84 Sugar mill, 890 Sun the source of mechanical power, 79 INDEX. 473 SUP Superficial measure explained, 7 Superheater, proportions of, in steamer 'Khone,'3S3 Surds or incommensurables, 46 Surface of boiler required to evaporate a cubic foot of water per hour, 309 condensers, proportions of, in steamer ' Hansa,' 814 condensers, 315 heating, of modern boilers, 375 condensers cause internal corrosion in boilers, 331 : proportions of, in steamer ' Khone,' 3S8 rp ABLE of addition, 11 1 Tables, multiplication, 19, 28 'Tay 1 steamer, dimensions of, 2S7 Temperature defined, 135 Temperatures of liquefaction and ebulli- tion constant, 137 steam at different pressures, 159 Tensile strengths of metals, 126 ; of woods, 127; crushing strengths of woods, 128 ; iron, 129 strength of boiler plates, 321 ' Teviof steamer, dimensions of, 287 Theory of the steam-engine, 134 Thermo-dynamics, 134 Thermometers, 187; Centigrade, Eeau- mur's. and Fahrenheit's compared, 139 Thermal unit, 162 Thomson, D., rotative pumping engines by, 862; double cylinder engines by, 862; combined plunger and bucket pump by, 363 Throttling the steam, effect of, 198 Time during which bodies have fallen de terminable from their velocity, 99 by height fallen through, 98 Toothed wheels, proportions proper for, 246 . Torsion, strength to resist, of different metals, 133 Transverse section of ships, best form of, 409 Tubes of locomotive boilers, 830 'Tweed 1 steamer, dimensions of, 287 Tylor, pumps by, 885 'TTLSTER,' Indicator diagrams from, U 842, 345, 852 Unit, meaning of the term, 5 of heat, 162 Uptake of boilers, sectional area of, 876 TACTJUM, velocity of air, water, and mercury into. 101: of steam and gas, 102 WAG Values of different coals in generating steam, 177 Yalve piston, by D. Thomson, 363 Vaporisation, 152 ; latent heat of, 154 Vapours and gases, difference between Velocities, virtual, law of, 79 Velocity of falling bodies, 95 determinate from height fallen, 97 ; from time of falling, 97 air, water, and mercury into a va- cuum. 101 ; of steam and gas, 102 rotation that will burst by centrifu- gal force, 110 permissible in bearings moving un- der a given pressure, 123 water in rivers, canals, and pipes, 199 water flowing in pipes and canals, 433 Veneer saw, 389 Vermicelli machine, 888 Vertical tubes, advantages of, 377 Vessels, resistance of, 399 ; proper shape of, 401 maximum breadth of, best position of, 417 length of, should vary with intended speed, 421 resistance of, mainly caused by fric- tion. 423, 442 Vibrations of pendulums, rule for deter- mining, 115 ' Victoria and Albert,' indicator diagram from, 352 Virtual velocities, law of, 79 Viscosity or molecular attraction, 163 Vis ulna, nature of, 90 Volumes, relative, of water and steam at atmospheric pressure, 102 and pressures of gases, 146 of gases, 166 Vulgar fractions, nature of, 5 "ITTAGON boilers, proportions of. 810 VV "Water, relative density of, 100 into a vacuum, velocity of, 101 and steam, relative bulks of, at atmos phcric pressure. 102 maximum density of, 189 weight of, at 82% 189 velocity of, In rivers, canals, and pipes, 433 works, indicator diagram from pump, 861 lines of ships, 400 ; illustrated by phape of fishes, 408 in pipes, friction of the same at all pressures, 42d velocity of, in pipes, 434 ; In canals, 434 474 INDEX. WAE "War, maritime, new resources available for, 401-464 'Warrior' steamer, body plan of, 453; transverse section of, 459 "Waste water pipe, to find the proper di- ameter of, 224 Wave raised by a vessel, 414, 419 ; mo- tion of, 420 Weaving by steam, 393; by compressed air, 398 Weights lifted by machines, 82 Wenhanrs double cylinder engine, 308 ZIIR Wheels, teeth of, 24T Winch wcijrhts lifted by, SI Wirtx.'s Zurich machine, 3S6 "Woods, strength of, 125 Wool-spinning mill, 8!)3 Working beam of land engines, how to proportion, 233 ZEKO, absolute, 1ST Zurich machine, 3S6 THE EKD. Any of these Books sent ly Mail, free of postage, on receipt of the price. SCIENTIFIC WORKS PUBLISHED BY D. APPLETON & CO., 443 & 445 Broadway. ALLEN'S Philosophy of the Mechanics of Nature. Illustrated. 8vo. ANTISELL, on the Manufacture of Photogenic or Hy- dro-Carbon Oils. 8vo, cloth. APPLETONS' Cyclopaedia of Drawing a Text-Book for the Mechanic, Architect, Engineer, and Surveyor. 1 hand- some vol. royal 8vo. 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