FUNDAMENTAL IDEAS MECHANICS AND EXPERIMENTAL DATA. A. MORIN. REVISED, TRANSLATED, AND REDUCED TO ENGLISH UNITS OF MEASURE BY JOSEPH BENNETT, NEW YORK: D.APPLETON AND COMPANY, 346 & 348 BROADWAY. LONDON: 16 LITTLE BRITAIN. 1860. ENTERED, according to Act of Congress, in the year I860, by D. APPLETON & CO., In the Clerk's Office of the District Court of the Southern District of New York. DEDICATORY PREFACE. To GENERAL J. G. SWIFT: In the summer of 1857, 1 was engaged upon a survey of Lakes Squam and Newfound, with a view of making some alterations in the delivery of those great reservoirs of the Lowell water power ; on completing the work, my employer, James B. Francis, made me a present of his only copy of " Arthur Morin's Legons de Mecanique Pratique ;" speaking of its value in terms of the highest commendation, but expressly stating that the gift was not to be considered as involving the expectation of a translation, or as anywise im- posing upon me an obligation to attempt it. The temptation proved too strong to be resisted ; though, could I have foreseen the labor required in the reductions, I should hardly have ventured upon the undertaking. It will be seen that I have not, in all cases, adhered to the unit of the foot. In the matter of the " Draught of Vehicles " and " Resistance of Fluids," the near approach of the yard to the metre, and a fidelity of translation, seemed to call for its substitution in place of the foot. The tables were not merely transferred to our units, but were calculated from the data, as a double check upon the correctness of the original, and its translation, and whatever errors may have crept in the latter, it is certain that some grave errors of the former have been discovered and corrected. In the calculations, I have been helped by many friends, and I appreciate their valuable assistance in clearing the way- through this forest of figures. We all acknowledge the advantage of well established practical formulae ; to them the mechanic and engineer must IV DEDICATORY PREFACE. look for ready aid in producing harmonious combinations of strength and dimensions, ensuring to their mechanical devices and structures, agreeable forms, convenience, security, and economy : it is a matter of regret that so little has been done in our country towards establishing them. It is not every one like Brunei can congratulate his employers upon the falling of a bridge, on the score of its preventing the erection of a hundred more on the same plan. With us, the fall of one would be " the hoisting of the engi- neer with his own petard." The recent calamity at Lawrence (Mass.) cries out in thunder tones against the merciless destruction of life, and most painfully shows that too much care or skill can not be exacted of our constructors. The frequent record of loss of life or property, arising from a want of skill in those intrusted with the management of our most vital interests, has been creating a wide-felt disgust for the too prevalent system of placing them in hands who have no other claim but that of political or partisan preference, and its evil influences have been operating upon a profession which, in point of attainment or utility, should stand second to none in the country. It is to be hoped that our government may yet take in hand a matter that cannot well be done at individual cost, and thus institute a series of experiments, so that for the strains of wood, of iron, for the properties of materials, and general experimental results, there may be found many an able native Barlow, Fairbairn, or Morin, to elicit valuable information, and supply the great void existing in the testing of our own materials. In dedicating this translation to one of your great ability and experience, I express the hope that I have done justice to the gifted author, and that in presenting the results of his ingenious experiments, I may have done the profession some service. Most respectfully, Jos. BENNETT. BROOKLYN, JAN. 13, 1860. CONTENTS. PRELIMINARY IDEAS. PAGE Extension 1 Simpson's formula 2 Quantity infinitely divisible 5 FORCES AND THE MEASURE OF THEIR WORK. Inertia of matter. Definition offeree 8 Action offeree 10 Measure offeree. The unit of measure 11 Different names of forces. Constitution of bodies 12 Action andreaction, equal and opposite 13 Point of application of forces. Effect and work of forces 14 Measure of work of a constant force, where the path described by its point of application is in its own direction 15 Representation of this work by the surface of a triangle. Measure of the work of a variable force 16 Mean effort of a variable force 19 Mode of calculation in English practice. Case where the arithmetical mean of variable values may be taken for the mean effort. Applica- tions 20 Idea of work independent of time 21 Denominations and unit of mechanical work 22 Conditions of mechanical work 23 Horizontal transportation of loads 24 Case where the force does not act in the direction of the path described.... 25 The work of gravity upon a body describing any curve 26 The crank and its connecting rod. Direction of effort in its relation to the path described 27 Springs. Expansion and contraction 28 Proper limits to variations of temperature to be used 29 VI CONTENTS. DYNAMOMETERS. PAGE Conditions to be fulfilled by these instruments 33 Rules for proportioning spring-plates 34 Ratio between the different proportions 35 Longitudinal profile of plates. Disposition of plates 36 Permanent trace of spring flexures 38 Motion of paper receiving the trace of the style 38 Quadrature of traced curves 39 The Planimeter 41 Dynamometer, showing the whole quantity of action for a considerable interval of time and space 45 Indications of the number of turns made by small wheel dynamometer, with chronometer motors 48 Rotating dynamometer, with styles 49 Transmission of the motion of shaft to the band of paper 51 Results of experiments made with the rotating dynamometer 51 Rotating dynamometer with counter. 52 Watt's gauge, perfected by MacNaught 53 New style indicator 55 TRANSMISSION OF MOTION BY FORCES. General remarks upon the laws of motion 58 Vertical motion of heavy bodies 59 Successive fall of heavy bodies 60 Forces proportional to their velocities 61 Measure of motive forces and of inertia 62 Case where the force is constant. Relation offerees to accelerations 64 Quantity of motion 65 Equal forces acting during equal times 66 Proof of preceding considerations by direct experiment 69 Shock of two elastic bodies 73 Observations upon the preceding results , 74 Quantity of motion imparted bya constant force 75 Observations upon the use of quantity of motion 78 OBSERVATION OF THE LAWS OF MOTION. Determination of the intensity of forces by observing the laws of the motions they produce 80 Means of determining the laws of motion. Colonel Beaufoy's and Eytele- wein's apparatus 81 New apparatus 82 Zinc plates. Contrivance for tabulating the curves , 88 Description of a chronometric apparatus with cylinder and style, for observ- ing the laws of motion 89 Discussion of results furnished bythis apparatus 91 CONTENTS. Vil PAGE Determination of the velocity 92 Experimental demonstration of the principle of the proportionality of forces to the velocities 93 PRINCIPLE OF VIS VIVA. Mechanical work developed byforces, in variable motion 97 Vis viva 98 Effects of powder in fire-arms. 99 Application of the principle of vis viva 101 Relation between charges and velocities. 102 Initial velocities and vis viva imparted to balls by different charges of gunpowder 104: Vis viva imparted by different powders. Mean efforts 106 Effects of powder and pyroxile on fire-arms 107 Consumption and restoration of work by inertia 113 Rams, punching-machines, &c 113 Work expended in the shocks of two non-elastic bodies 114 Work due to compression, and the return to the primitive form in the case of elastic bodies. 116 Work lost in the shock of bodies imperfectly elastic 116 Masses in motion reservoirs of work 117 Periodical motion 118 COMPOSITION OF MOTION, VELOCITIES, AND FOECES. Composition and resolution of simultaneous motions 119 Simultaneous motions in same direction 120 Composition of motions directed in any manner 121 Variable motion 124 Components at right angles 125 Resultant of three simultaneous motions or velocities in space 127 Resultant of any number of simultaneous motions or velocities 128 Varignon's theorem of moments 129 Extension of these theorems to systems impressed with a common motion of translation. Independence of the simultaneous action of forces upon the same point 132 Forces acting in different directions 134 Quantity of work of a force whose point of application does not move in the same direction as the force 135 Application of Varignon's theorem to forces. Resultant work of forces equal to algebraic sum of its component works ... 136 Forces acting in any direction 137 Case of the point turning around a fixed axis 138 Conditions of uniform motion or equilibrium, the forces being in the same plane. 139 Forces acting in any manner in space 140 Ylll CONTENTS. PAGE Parallel forces 142 Consequence of the composition of parallel forces ; 143 Point of application of resultant of parallel forces 143 Work of resultant of parallel forces 148 Centre of parallel forces 148 Use of moments in determining position of resultant 149 Condition of uniform motion or of equilibrium 149 The balance 150 Proof of balances 154 Double weighing. The steelyard 155 Steelyard with a fixed weight (Peson) 157 Quinteux's platform balance. 159 Theory of the lever 162 THE CENTEE OF GEAVITY AND EQUILIBEIUM OF TENSIONS IN JOINTED SYSTEMS. Application of preceding theorems to gravity 165 Determination of the centre of gravity 165 Geometrical method. Triangle 166 Quadrilaterals. Polygons . Triangular pyramid 167 Centre of gravity of a body of any form 168 The stability of equilibrium 169 Application of the principles of composition and resolution of forces 170 Equilibrium of cords 171 Equilibrium of efforts transmitted by cords or rods meeting in the same point 172 Movable pulley. Towers 172 Funicular polygon 173 Weights acting upon the funicular polygon 174 Determination of the tensions by a graphical construction , 175 Suspension bridges 176 Application....... 180 COMPOSITION AND EQUILIBEIUM OF FOECES APPLIED TO A SOLID BODY. Forces applied to solid bodies. Motion of translation of a system of bodies parallel to itself. 182 Case of variable motion ...; 183 Quantity of motion and vis viva of a body 184 Work of gravity in compound systems 185 A system of forces, acting upon a solid body, may always be reduced to two equivalent forces, applied to two of its points, one of which may be chosen at will 186 Condition of uniformity of motion or of equilibrium 187 CONTENTS. IX MOTION OF ROTATION. PAGE Work and equilibrium of forces, in the motion of rotation around a fixed axis 189 General conditions of the uniformity of motion or of equilibrium of a solid body, free in space, and subjected to any forces 191 Centrifugal force its measure 193 "Work developed by centrifugal force 195 Action of centrifugal force upon wagons 197 Action of centrifugal force in fly-wheels 198 Application to the motion of water in a vase turning round a vertical axis 199 Surface of water in the bucket of a hydraulic wheel with a horizontal axle 201 Regulators with centrifugal force 202 Distribution of a regulator with centrifugal force 207 Results of observations upon the effect of this regulator 210 Comparison of the data of experiment with the formula. Modification of the balls for obtaining a greater regularity 212 Transmission of motion by the endless screw 213 Indispensable disposition in the use of these regulators 214: Modification of apparatus. Other regulators. Variable motion around an axis 215 Observations upon the moment of inertia 217 Principle of vis viva in the motion of rotation about an axis 219 Theory of the pendulum 221 Time of oscillations of a pendulum with small vibrations 224 Compound pendulum 226 Length of the simple pendulum which makes its oscillations in the same time as the compound pendulum 228 Moment of inertia of a compound pendulum 228 Centre of gravity of compound pendulums 230 Centre of percussion 231 Theory of the ballistic pendulum 234 APPLICATION OF THE PRINCIPLE OF VIS VIVA TO MACHINES. Application of " vis viva " to machines 240 Maximum effect of machines. Work of powers and of useful resistances 242 Work of passive resistances. Pieces with alternating motion 243 Influence of vis viva acquired at each period 244 Periodical motion 245 Advantages and conditions of uniform motion. 245 Means of diminishing variable motion 246 Observations upon the starting of machines, and the variations in velocity which then take place 246 Perpetual motion. Periodical motion 248 Limitations of the deviation of velocity. Theory of fly-wheels 249 X CONTENTS. PACK High-pressure steam-engines 253 Fly-wheels for expansion engines, and for forge-hammers 254 German hammers geared 255 Geared tilt-hammers. Necessity of using fly-wheels when there are- shocks 256 Proportions of fly-wheels for powder-mills with twenty stamps. Rolling- mill for great plates 257 Use of fly- wheels 258 FBICTIOK Ancient experiments 260 Experiments at Metz and description of apparatus 262 Graphic results of experiments 223 Formulae for calculating results of experiments 264 Relations between the tensions of the cord and friction of the sled 268 Friction of oak upon oak, without unguents .- 270 Friction of elm upon oak, without unguents 271 Friction of soft limestone upon soft limestone, without unguent 272 Friction of strong leather, placed flatwise upon cast-iron 273 Friction of brass upon oak, without unguent 274 Friction of cast-iron upon cast-iron 275 Experiments upon friction at starting, and results of experiments 276 Friction of oak upon oak, the fibres of the sliding pieces being perpendicu- lar to those of sleeper k 277 Friction of oak on oak, the sliding pieces having their fibres vertical, while those of the fixed pieces are horizontal and parallel to the direc- tion of motion 278 Friction of limestone upon limestone, when the surfaces have been some time in contact 279 Friction of limestone upon limestone, the surfaces having been in contact with a bed of mortar 280 Expulsion of unguents and influence of vibrations upon friction at starting 281 Influence of unguents. Adhesion of mortar 282 Experiments upon friction during a shock 283 Apparatus employed in the experiments 284 General circumstances of the experiments 285 Formula for calculating the results of experiments 286 Acceleration of motion of sledge during fall of shell may be neglected.... 287 Results of experiments 289 Friction of cast-iron upon cast-iron, with an unguent of lard, during the shock 290, 291 Transmission of motion by belts. Slipping of belts upon cylinders 292 Slipping of cords and belts upon wooden drums and cast-iron pulleys 294 Friction of belts upon wood drums 296 CONTENTS. XI PAGE Friction of belts of curried leather upon cast-iron pulleys 29 7 Conclusions 298 Variation of the tension of belts in transmitting motion 299 Remarks upon preceding results 306 Friction.of journals 307 Friction of cast-iron journals upon cast-iron bearings 309 Advantage of granulated metals 310 Light mechanisms. Use of experiments 311 Friction of plane surfaces which have been some time in contact 312 Friction of plane surfaces in motion upon each other 313 Friction of journals in motion upon their pillows. 314 Application to gates 315 Application to saw-frames 316 Application to journals 317 Axles of wagons 320 RIGIDITY OF COEDS. Rigidity of cords. Experiments of Coulomb with apparatus of Amontons 321 Results of Coulomb's experiments 323 General expression of resistance to rolling 325 Other experiments of Coulomb 328 Rigidity of cords with movable rollers upon a horizontal plane 329 Extension of Coulomb's experiments to those of different diameters 330 Rigidity of cords in function of number of strands 331 Remarks on cords that have been used 332 Tarred cords 333 Rigidity of cords of different diameters rolling upon a drum one foot in diameter 334 Moistened cords. Use of preceding tables 336 DRAUGHT OF VEHICLES. Draught -of vehicles 338 Experiments on oak rollers, rolling upon poplar. Vehicles moving upon common roads 341 Ratio of the draught to the load 343 Influence of the pressure 345 Experiments upon the influence of pressure upon the draught of vehicles 346 Influence of the diameter of the wheels 347 Experiments upon the influence of the diameter of wheels upon the resist- ance to the draught of vehicles 348 Influence of the width of the rims 349 Influence of the velocity. 350 Approximate expression for the increase of resistance with the velocity... 351 Xll CONTENTS. PAGE Practical consequences of these experiments. Comparison of paved and metalled roads , 354 Influence of the inclination of the traces 356 Application of the general results of experiments 359 Draught and load of carriages for different soils and vehicles 360, 361 Consequences relative to the construction of vehicles 364 Destructive effects of vehicles upon roads 365 Influence of great diameters of wheels 366 Direct experiments upon the destructive effects of wagons upon roads 367 Influence of the width of tires. 367 Experiments with the same carriages under equal loads 369 Experiments made upon the influence of the diameter of wheels, in their destructive effects upon roads ^ 369 .Influence of velocity upon the destructive effects 370 Comparative experiments upon the wear produced by carriages, carts, and wagons without springs 371 Experiments to determine the loads of equal wear and tear 372 EESISTANCE OF FLUIDS. Eesistance of fluids 373 Work developed per second by the resistance of a medium. Equivalent expressions of the resistance 376 The body at rest in a fluid in motion 377 Experiments upon the resistance of water to the motion of variously formed bodies ,. 377 Mode of observation 378 Observations upon the results 379 Influence of the acuteness of the angles of cones upon the resistance 380 Resistance of water to the motion of projectiles 381 Resistance of water to the motion of floating bodies. Influence of the form of floating bodies 382 Flat-bottomed boats with raised fronts 383 Velocity of waves 385 Experiments upon the velocity of solitary waves produced by boats 388 Results of experiments upon the resistance of boats to towing. Fast boats 389 Consequences of the experiments 393 Comparison of the resistance to towing of mail-boats, when the wave is spread along the sides, and when it is towards the bow 394 Work developed by horses in hauling fast boats 395 Days work developed by horses in different modes of transportation 397 Observation upon the daily work of horses 398 Resistance of water to the motion of wheels with plane paddles 398 Causes which alter the law of resistance 401 Proper distance of paddles apart. Value of the second term of the resist- ance K,.. .. 403 CONTENTS. PAGE Influence of the presence of a boat near the wheels 405 Application to the wheels of steamboats 406 Resistance of air. Results of experiments 408 Thibault's experiments upon bodies inmotion in air 410 Remarks upon regulators and wind-mills. Experiments upon different formed surfaces 413 Influence of the inclination of the wings 414 Approximation of surfaces exposed to the resistance of air. Influence of the form of surfaces. Resistance of air to the motion of spherical bodies 415 Experiments atMetz upon bodies moving in air 417 Mode of reckoning the effects of acceleration 419 Proof of the exactness of the formula i 421 Influence of the extent of surfaces 422 Experiments upon parachutes 424 Case where the parachute presents its convexity to the air, and where its motion was accelerated 425 Resistance to the motion of inclined planes in air. 426 General conclusions from the experiments at Metz 427 Effort exerted by the wind upon immovable surfaces exposed to its direc- tion 428 Observation upon the velocity of the wind 429 Means of measuring the velocity of air 430 Anemometer of M. Combes 431 Remarks upon the use of the instrument 432 New anemometer 433 Testing of the instrument 435 Remarks upon the test of the instrument, and its extension to great veloci- ties 436 M. Thibault's experiments upon the effort of wind upon immovable sur- faces, exposed to its action, perpendicular to its direction 439 Accordance of these results with those of Professor Rouse, cited by Smeaton. 440 Influence of the curvature of surfaces, and of their inclination to the wind 441 Difficulties in the directing of balloons 442 \ FUNDAMENTAL IDEAS OF MECHANICS AND EXPERIMENTAL DATA. PEELIMIJSTAEY IDEAS. 1. Extension. Extension has three dimensions : length, breadth, and depth. In its measurement we call that a line, or linear dimension, which has length without breadth ; that a surface which has length and breadth ; and that a solid, or volume, which combines the three di- mensions. The measurement of extension constitutes the science of Geometry, the use of which in this treatise will be its application to Mechanics. Lengths are measured by a comparison with a conven- tional unit adopted in any country, which with us is the foot, subdivided into tenths, hundredths, &c. To appre- ciate any thing smaller than hundredths, we make use of the vernier and other contrivances, such as micrometer screws, compensators, &c., whose description is within the province of Industrial Geometry. Surfaces are measured by the rules of Geometry, and are expressed in square feet. But it is often the case that they are bounded by lines and contours, not con- forming to any known geometrical law, when we must have recourse to approximate modes of quadrature, or some mechanical means. The use of these being a matter of daily occurrence, in tabular abstracts, and in the dis- PEELIMINAEY IDEAS. cussion of experimental results, to provide for any future reference to them, we will speak of them somewhat mi- nutely. One of the most sim- pie and exact methods for determining approximate- ly by calculation, a sur- face bounded by curved lines, or partly composed of curved and straight lines, is the following : Draw across the surface a line AB, and divide the distance between the points of its intersection with the contour, into an even number of equal parts, numbered 1, 2, 3, 4, .... 7, 8, 9, for example. At the points of division raise perpendiculars to the line AB, (called the axis of abscissa,) giving II", 2'2", 3'3" .... 8'8", 9'9" for the lengths of ordinates. This done, the surface S, terminated by the curved line, will have the value nearly S=|(l, 2)|"l / l // + 9 / 9 // +4(2 / 2 // +4: / 4 // + . . . 8'8")+2(3'3"+5'5" . . . 7'7")1 that is to say, the third of the space between two consecutive, equidistant ordinates, multiplied ~by the sum of the extreme ordinates, plus four times the sum of ordinates of an even order, plus twice the sum of the ordinates of an uneven order. M. Poncelet gives the following demonstration of this rule, page 187 of " L'introduction a la mecanique indns- trielle." (Second Edition.) 2. Demonstration of Simpson's formula. The area to be measured being limited by the contour line PRELIMINARY IDEAS. 3 . . . g'g . . . ba, if we divide the line ag into six equal parts, we shall have at once a first approximation by taking the sum of the right lined trapeziums ', &c., which gives , &c., which is equal to This method is usually followed. But it is manifest that for curves always concave towards the Jine ag of abscissa, this formula will give too small a result ; and otherwise it will give too much for curves convex toward the line ag : so that the only approximate compensation, will be for the case of curves alternately concave and convex. But if we consider the space between two odd consecutive ordinates cc f and ee\ and di- vide ce into three equal parts, cm=mn=ne, we have imme- diately a nearer approximation to the mixtilinear area cc'd'e'e ; by substituting the three right lined trapeziums cc'm'm, mm'n'n^ nn'e'e, in place of the two trapeziums cc'd'd and dd'e'e : the sum of the area of these three trapeziums is - o since cm=mn=ne=-ab. o In drawing the line m'n 1 which meets dd! in 0, we have if the body is held by obstacles and superior resistances, the action of the force is cancelled, since it has produced no motion. Such is the case with a sup- port, a column, a man sustaining a burden, horses which cannot start a mired wagon, and with overpressed rollers which cannot overcome the resistance of iron. For these forces to produce a mechanical or industrial effect, or useful work, they must have passed through a certain path in their own direction, at their point of ap- plication. Tims, a fundamental' condition of the mechan- ical or industrial work of forces, is, that there must be, at the same time, an effort exerted, and a path described in virtue of this effort. 16. Measure of work of a constant force, where the path described ~by its point of application is in its own direction. It is evident that the effect, the work produced by force, is proportioned : 1st. To the intensity of the effort : 2d. To the space described, and consequently to the product of these two factors. Thus in raising burdens, or minerals ; in the draught of wagons and ploughs ; in towing boats, and drawing water, it is evident that for the same weight or effort, the effect is doubled when the space is doubled ; and that for the same space the effect is double or triple, if the resistance is double or threefold Comparing efforts with weights, whose action will produce the same effect, and the spaces described being expressed in feet, we see that the work of a constant force may be represented by the product of its intensity, (ex- pressed in units of weight, or in pounds,) into the space described in its own direction, expressed in units of length, or in feet. If we take for the unit of work, the pound raised one foot, then the work of a force F, whose point of application has described the path S, will be ex- pressed by FS pounds raised one foot, which we may designate by Ib. ft., written a little above the right of the product FS. Thus FS lb8 - ft - 16 FORCES AND THE MEASURE OF THEIR WORK. IT. Representation of this work ly the surface of a rectangle. If we take the space S for the base of a rec- tangle, whose height at a certain scale shall be the effect F, it is evident that the product FS will be the measure of the surface of the rectangle ; or that reciprocally this surface may be taken to represent the work FS. 18. Measure of the work of a variable force* When a force is variable, we may apply the same method of measurement to each of the small elementary spaces, s, in which we may consider the force as constant. The work corresponding to each of these elementary spaces is repre- sented then by the product FS. If we place upon the straight line AB taken for the axis of abscissae, the spaces described, and at each point of division raise a perpendicu- ^ n ~^_^ lar representing at a certain ^"l""" scale, the effort exerted, we shall i I then have a curved surface limi- -^j j ted by the line of abscissa, the extreme ordinates, and the curve -J b passing through the extremities -^ s " of all the ordinates. If we consider the small elementary trapezium as corresponding with any effort F, and with an element of the space, s, it is clear that. the surface of this small trapezium will be Fs, and that it will represent the elementary work corresponding to the small space s. The whole work for a space s being composed of the sum of all the elementary quantities of work Fs, it is evi- dently represented by the whole surface limited by the curve. We have only then to find this surface, or the sum of all the elementary products Fs. Calculation in certain cases affords direct methods of obtaining it t>ut in many others, and in practice, it is best to employ the method of quadrature, particularly that of Simpson, which has been already explained. Moreover, it is absolutely indispensable to recur to FORCES AND THE MEASTTKE OF THEIR WORK. 17 these methods when we wish to estimate work transmitted by animal motors, and by many machines, in which the efforts transmitted are constantly varying, according to laws impossible to be found. 19. Application to work developed by horses hauling a mail-boat on the canal de rOurcq. By means of appa- ratus, to be described hereafter, we have obtained an ex- perimental curve, or graphic relation between the space described and the efforts exerted. It would be impossi- ble by any direct method of calculation, to obtain the relation existing between the efforts and spaces described, for deducing the work ; but the rule of quadrature fur- nishes us the means. Operating, for example, over a space of 157.48 ft> in length, which we divide into twelve equal parts, we find for the ordinates F 15 F a , . . . . F 12 , F, 8 the following values, according to the scale of flexures of the spring : FFFFFFFFFF F FF x - 1 - A A ^ - LJ -- L - t - L - L - L F, =191.9 F 2 =275.08 F 13 =283.4 F 4 =266.92 475.3 F 6 =200.08 F 8 =241.55 F 10 =158.38 F 12 =187.51 F 3 =258.10 S, F 6 =216.84 I2~ 12 F 7 =208.46 F 9 =208.46 F n =158.38 1050.24x2=2100.48 1329.52x4=5318.08 18 FOECES AND THE MEASURE OF THEIK WOEK. The whole work for this space is then I x 13.12(475.3+5318.08+2100.48')=34522.4 lbs - ft - This experiment was made with a boat weighing, in- clusive of load, 15766 lbs - moving at the rate of 15.45 ft - per second, or 10^ miles per hour. 20. Indretfs Steam Engine. The diameter of the piston being 1.18112 " its surface =1.181 1 2 x 0.7854= 1.0958 sq - ft -=157.766 s<1 - ins - The whole stroke is 3.0184 ft - Dividing it into 16 1 S equal parts we have - =0.062883 ft - 3 lo The abstract of the curve of pressure furnished by the index, gives the following result for the pressiire upon each 0.00107643 sq - ft - of surface of piston. FFFFFFFF i * a J- a J-4 J-s-^e x 7 x lbs. lbs. 148.35165 F, =.425657 F a =2.43043 F 3 =3.49567 F a -}-F n = 0.88219 F 17 =. 456534 F 4 =4.25657 F 6 =4.33375 4(F 2 ..+F 16 )=99.35212 .882191 F 6 =4.37786 F 7 =4.37786 2(F 3 ..+F 15 )=48.11734 F 8 =4.37786 F 9 =4.37786 F 10 =4.25657 F n =3.79702 F 12 =2.50982 F 13 =2.27825 F 14 =1.59676 F 15 =1.39826 F 16 =1.03216 24.05867 x 2=48.11734 24.83803x4=99.35212 FORCES AND THE MEASURE OF THEIR WORK. 19 And for the work developed by the steam in one stroke, .062883 ft - x 1018 x 148.3516 lbs ~ 9498.4 lbs - ft - 21. Mean effort of a variable force. It is often useful, and even necessary to ascertain the mean effect of a va- riable force ; that is, the constant effort which would pro- duce the same work, causing it to pass through the same space at its point of application. From this definition and the preceding remarks, if we call W the work devel- oped by the variable force, and S the whole space described by the point of application, we shall have W W=FS, whence F=- 5 -. Thus we shall obtain the mean b effort of a variable force in dividing the total work by the space described. It follows from this that the work of the va- riable force being repre- sented (Fig. 8) by the area AabcdefM, the work of the corresponding con- stant mean force will be represented by the sur- face of the rectangle FIG. s. AA'M'M, having the same area as the curve. We may here remark that the points 5, c, and 0, where the curve of variable effort cuts the line A'M' of the mean constant effort, correspond to the positions where these two efforts, as well as the elementary work developed by them are equal. Moreover, the areas A!ab and cde above the straight line A'M 7 , represent the excess of the work of the variable effort above that of the constant effort while the body passes through the space AB and CE; in the same way the areas contained between the line A'M' and the curve below it, represent the excess of work of the constant effort, above that of the variable force. The 20 FOKCES AND THE MEASURE OF THEIR WORK. sum of these first excesses should evidently equal the sum of the second. 22. Observations upon the mode of calculation adopted in English practice. Some authors, and particularly the practical English engineers, in calculating the effect of steam engines, often take, for the mean effort, the arith- metical mean of the extreme pressures or forces, and mul- tiply it by the space described. If, for example, the work developed by the steam during its expansion were required, the curve which gives the effort corresponding to each stroke of the piston, as we shall see, and as the figure indicates, would be convex to- ward the line of abscissae ; and taking the arithmetical mean between the extreme ordinates v and multiplying it by ac, we have the area of the trapezium abdc, much greater than that Fl0t 9 . of the curve. 23. The case where the arithmetical mean of a certain number of variable values may be taken for the mean effort. When the values of the effort oscillate periodi- cally around some determinate effect, or between certain limits very numerous, taken independently of the irregu- larity of their periods of oscillations, the arithmetical mean of a great number of these values may be taken for the mean effort, with sufficient accuracy for common practice. This is particularly the case in experiments upon the action of animal motors, and in the efforts trans- mitted by various manufacturing machines, as will be seen in some following examples. 24. Applications. We have seen in an experiment cited in (19), that the work developed by three horses harnessed to a mail boat, was 34522.4 lbs - ft - for a space of FORCES AND THE MEASURE OF THEIR WORK. 21 157.48 ft - The mean effort to produce the same work would be MS^-". 157.48"- or for each horse - =73.06 lbs - 3 So, also, in the example relating to work developed upon the piston of the engine of the millwright's shop at In- dret, we have found, for a stroke of 3.0184 ft - the total work to be 9498 lbs> The corresponding mean effort would be 9498.4 lbs - ft - lbs - 3.0184 ft - =3146 The surface of the piston being 1 57.76 sq> In " this mean effort corresponds to a pressure of 3146 lbs * sq . in. = 19 - 9 lbs * P er s a u are inch. 25. The idea of work is independent of time. We see from what precedes, that in the measure of work, we have only regarded the effort exerted, and the space de- scribed in the direction peculiar to this effort. It is therefore independent of time. Thus in raising goods, the effect is not measured by the duration of labor, but by the product of the load into the height of its elevation. Still, when the work is of long duration, and is peri- odically repeated in the same manner, the measure of one determined period is evidently sufficient to ascertain any other. It is thus, in the periodic action of steam engine work, of hydraulic wheels, of animal motors, that we re- fer the work to a unit of time, which we usually make equal to a day, an hour, a minute or a second. (This last unit is most frequently used.) For animal motors, whose work is limited by fatigue, 22 FOECES AND THE MEASURE OF THEIR WORK. and the need of rest, in our estimate of its value per sec- ond, we must regard the whole duration of the work, since it has great influence upon its value in each unit of time. Thus a strong wagon-horse may travel from 8 to 10 hours per day, developing in each step at a velocity of 3.28 f * per second, from 434 to 480 lbs - ft - of work; while horses employed in hauling mail boats, which, in the case we have cited, (19,) developed a mean effort of 73.06 lb % run- ning at the rate of 15.45 ft - per second, and giving the value of work as 73.06 lbs -xl5.45=1129 lbs - f % can only travel two hours at most, with four relays per day, resting one day in four, and yet rapidly wearing out in this hard service. 26. Various denominations of mechanical work. The mechanical effect of forces, which we shall measure by the product of the effort into the space traversed in its peculiar direction, has received different names, which it is well to know. Smeaton, the English engineer, to whom we are in- debted for useful experiments on water-wheels, and wind- mills, called it mechanical power ; Carnot, the moment of activity ; Monge and Hachette, dynamic effect ; Coulomb and M. Navier, quantity of action MM. Coriolis and Poncelet, quantity of work. We shall adopt the last ex- pression, as most appropriate to the industrial view which we shall take of mechanics. 27. Unit of mechanical work. As to the value of the unit of work, we have said that we shall adopt the pound raised one foot. Some French authors have proposed for a unit of work, 1000 kil s 2205 lbs - raised l metre =3.28 ft - in height, giving" it the name of dyname or dynamode. Another unit which has come into use, notwithstanding its faulty denomination, is what is termed horse-power. This expression, introduced by Watt, at a time when the FORCES AND THE MEASURE OF THEIR WORK. 23 steain-engine had been successfully substituted for horse- power, expresses a work equivalent to 33000 lbs - avoirdu- pois raised l ft - per minute, equal to 550 lbs - per second. The value generally adopted in France is 542.7 lbs> Though this estimate of the horse-power is at present used as a conventional unit, it has no legal value, though it is very desirable that a legislative act may give it this character, for it is the money of industrial work. "We need hardly add, that this expression has no direct relation to the work actually developed by horses tackled to gins, which seldom exceeds a mean of from 289 to 325 ibs. ft. per second. Example. In the experiment relative to the steam- engine at Indret, where we found the work developed by the steam, in a stroke of the piston equal to 9496 11)S - f % there were 28 double strokes per minute, so that the work per second would be v Kfi 6 =8863 lbs - ft -, 60 and the force in horse-power would be 8863 lbs.ft. 550 =16 horse-power. 28. Observations upon the conditions of mechanical work. "We have said, that the work of a force will be measured by the product of its intensity, into the space traversed in its proper direction, but it should be under- stood that this space is described by the effect of the force itself. Thus a man in a boat or rail-car, who exerts a force, in the direction of motion, upon an object which receives from it no relative motion, will not produce any useful work, although the body may move in the direction of the effort, by the general effect of the motion trans- mitted. 24 FORCES AND THE MEASURE OF THEIR WORK. It is so in case the effort is perpendicular to the path described ; there will then be a pressure, an effort, but no work produced by the effort. From this cause we have disturbances, and friction, causing, as we shall see, losses of work, but not of immediate useful effect. "We would also remark here, that the definition of the work of any force applies as well when the path described by the point of application of the force, is in an opposite di- rection to that of the force, as when it is in the same direction. We speak of the latter case, the point of ap- plication following the direction of the force, as develop- ing a motive work ; and of the former, the point of application moving in a direction opposite to that of the force, as developing a resisting work. 29. Horizontal transportation of loads. This kind of work is not measured by the method we have adopted, since it produces certain effects, and expenditures of work, depending less upon the weight transported, than upon the mode of transportation. Thus, the transportation of a weight of 2205 lbs - by means of a sledge slipping along the ground, where the friction equals T \ of the pressure, would require per yard passed over, a work of 661.5 lbs - x 1 yd - ; while that by a wagon of common dimensions, where the draught is V the load, would require a work of 76.83 lbs> xl yd - ; and that by a railway car at a small ve- locity the resistance being but ^ of the load would 2205 require a draught of :=7.35 lb % and the w r ork per yard would be 7.35 lbs -y ds -. We see, then, that the work relative to the horizontal transportation of loads, cannot be measured, as we have hitherto done, by the product of the weight into the path described ; but rather by a comparison of results made for the particular service, and kind of transportation. FORCES AND THE MEASURE OF THEIR WORK. 25 30. Case where the force does not act in the same di- rection as the path described. If the path described is Aa, _F while the direction of the force is AF, it is clear that the path described in the direction of the force will be determined by the _ perpendicular a5, let fall from a upon AF, and equal to AJ). The work developed by the force F will then be, according to the definition, F x A5. This may be otherwise readily understood from a con- sideration of the adjoining figure. Let AB be the direction of a force P, acting, at a certain in- stant, upon a body describing the curve LM, when the body is sup- posed to have arrived at A. If we conceive the line AB to be an inextensible and perfectly flexi- ble thread, and the action of the force P to be replaced by a weight Q, acting at the end of this thread, which rolls over a pulley o, perfectly movable around its axis ; it is clear, that in the elementary displacement of the body from A to a, the work of the force P, will be measured by the product of the weight Q into the quantity W which it will have fallen. Now this quantity W is equal to the difference of length of the lines AB and aB, where the point of . intersection B, may be regarded as the point of instantaneous contact of the directions AB and aB with the periphery of the pulley. But on rolling the thread aB upon the periphery, its extremity a will describe an elementary arc of the involute aa 7 perpendicular to AB, and the length Ka' will measure precisely the difference sought. The arc aa' being merged at the smallest limit 26 FORCES AND THE MEASURE OF THEIR WORK. into a perpendicular let fall from the point a upon AB, we see clearly that Aa' is what is geometrically termed, the projection of the path Aa really described upon the direction of the force : and thus it is evident from this figure, that the elementary work of the force P is meas- ured by the product P x Aa', of the intensity into the projection of the infinitely small path Aa, upon its own direction. When the force is not in the direction of the path de- scribed, the work due to the elementary displacement Aa of its point of application, is the product of the intensity of the force by the projection of the displacement Aa upon its own direction. This product is what is termed, in rational mechanics, the virtual moment, though we shall give it the name of elementary work. This identity will lead us to many analogies with the results of rational mechanics, and this natural expression of work will help us toward an easier appreciation of its demonstrations. 31. The work of weight upon a body describing any curve. If we consider the body as arrived at A, and then describing the small elementary path Aa, the correspond- ing elementary work, developed by gravity, whose direction is always vertical, will be the product of the weight of the body, P, by the height ] Ab, which it has described in the di- rection of this force. Weight being constant, for the same place, and for heights varying but little at the earth's surface, the total work developed after the body has de- scended from L to M will be the product of P by the sum of the projections analogous to Ab, or by the total height of the descent II, and will consequently be equal to PH. Whatever, then, may be the curve of descent, it is the FORCES AND THE MEASURE OF THEIR WORK. 27 same, and depends only upon the difference of level of the extremities of this curve. 32. The crank and its connecting-rod. When the arm of a crank is sufficiently long for us to disregard its obliquities, it is clear, if the effort exerted in its direction is constant, that the total work de- veloped during a semi-revolution will be the product of the constant effort F, into the sum of the pro- jections of its elementary arcs Aa upon its direction, a sum evidently equal to the diameter 2R. Consequently, the work developed in a semi-revolu- tion is F x 2E. 33. Observation respecting the direction of the effort, in its relation to that of the path described. If the path described is in a direction contrary to that of the effort, it is evident that the body is impelled by another force, in relation to which the effort F is a resistance overcoriie ; we say, then, that the work of a force F is resistant, sub- tractive, or negative, that is, it must be deducted from the motive work, a part of which it has consumed and absorbed. Thus, when a body descends by the action of gravity, its path being in the direction of the force, it acts as a power, and its work is positive ; but when the body as- cends, the path is in an opposite direction to that of the force, which acts as a resistance, and the work is negative. If the body descends and ascends alternately the same height, the motive work developed during the descent is equal to the resistant work consumed during the ascent, and the total work is nought. There is then an alternate production and consumption of work, in all cases where the bodies periodically ascend and descend, as in cranks, pistons, pendulums, &c. 28 FOECES AND THE MEASURE OF THEIK WORK. 34. Springs. A consumption of work is produced also in the flexure of springs, and a restitution is made in their return to their primitive form. It is complete, if the spring, in unbending, recovers exactly the form it had before : it is incomplete, and a consumption of work occurs, when-' ever it but partially returns to its primitive form. 35. Expansion and Contraction. It is the same also with a body dilated by the action of heat, and the enor- mous efforts developed in this case are similar to those produced by other causes. Indeed, we know by expe- rience, that bodies expand or contract between certain limits, by quantities proportional to the efforts to which they are submitted. Thus, for example, a bar of iron ex- pands or contracts by a quantity I, which expressed in feet, is given in the formula Plbs. , 28457800 calling P the load per square inch of section, and I the expansion per running foot. Reciprocally, when a bar contracts, it exerts an effort equal to the force required to produce the same contrac- tion, and this effort will depend upon the expansion per running foot. If, for example, a bar of iron 1.1811 ins - square expands a quantity 1= 0.0005 ft - per foot, the effort capable of pro- ducing this elongation will be P=284r5r800x 0.0005 ft -=14228.9 lbs - per square inch, or in all 1.1811 2 x 14228.9=19844 lbs - Observing now that between 32 and 212 a bar of iron expands 0.0012205 ft - per foot, it follows that the FORCES AND THE MEASURE OF THEIR WORK. 29 quantity required to expand it 0.0005 ft - per foot will be found by the proportion 0.0012205"- : 180 : : 0.0005"- : a ?= Thus by an increase solely of the temperature of the bar by about 74, we may bring to bear against obstacles opposing its expansion, an effort of 19844- pounds. Keciprocally, if this bar, after heating and expansion, is cooled, it exerts efforts of traction depending upon the degree of cooling. In case of the reduction of the tem- perature of a bar 1.18 ins - square, by about 74, there would be an effort of contraction exerted equal to 19844 pounds. This important property of bodies exerting considera- ble efforts of expansion, and of contraction or shrinkage, is often advantageously used in the Arts. The tires of wheels, naves, and the shafts of water-wheels ; the gird- ling of domes, particularly that of the cupola of St. Peter's at Rome, are examples of its use. It is said that the righting of the walls of the ancient Conservatory Library was effected by similar means with great success. The bars used were 2.3622 x 0.86615 in % having a section of 2.04:6 square inches. They were heated by means of suspended gridirons, and as they ex- panded, were held in place by strong screw nuts with cast-iron washers ; they were then left to cool. If, for example, their temperature was reduced 73.74r, the contraction would be 0.0005 ft - per foot, and the cor- responding effort would be 14:228 pounds per square inch ; the effort exerted by each bar would therefore be , U228.90 x 2.046=29112 lbs -. As to the work developed by this force, it is easily calculated. In fact, from 32 to 212, and even beyond 30 FOECES AND THE MEASUEE OF THEIE WOEK. this, experiments prove that the elongations are propor- tional to the temperatures, so that If representing the ex- pansion up to 212, and 1 that relative to T, we have T : 180 : : I : T; whence T rT_0.0012205T ~180~ ""180 If we call L the length of the bar at the temperature of 32, this length will increase per lineal foot, in passing to the temperature T, by the quantity I=KT and will become Also, in passing from the temperature 32 to the tem- perature T', the length of the bar will become The expansion of the bar, in passing from the temper- ature T to that of T will then be L'L=L 1 K(T'--T), and the expansion per lineal foot will be whence we see that the expansion per lineal foot depends upon the difference of the temperatures, and not upon their particular elevations. Consequently, it is the same with the force, P=I x 28457800 lbB -=284:57800K(T / -- T) ; which increases proportionally with the differences of temperatures, and is the same for equal differences. This granted, if we place upon a line of abscis&a, start- ing from 32, the expansions L x L=L 1 K(T'-T), which at first are nothing, for T'=T 15 and at the resulting points of division erect perpendiculars or ordinates equal to the FORCES AND THE MEASURE OF THEIR WORK. 31 efforts of expansion or contraction, which have for their values those of the force P=284:57800. I=28457800K(T / -T), it is clear that the ordinates be- ing proportional to the abscissa, the points thus determined will be in a straight line, and thus will form a triangle, whose sur- face expresses the work devel- FIG. 14. oped by the efforts of expansion or contraction, corre- sponding to the different temperatures T' T. The surface of the triangle is moreover i ft. So that the work developed by the efforts of expansion or contraction has definitely for its value 14228900K 2 L 1 (T / -T) 2 lbs - ft -, A iLilLi f TT * i 0.0012205 4 , . and substituting for K its value - - this expression lot) of work becomes 0.0006541 eLXT'-T) 2 lbs - ft -. It shows that this work is proportional to the length of the bar at the temperature of 32, and to the square of the difference of temperatures. We also see, that it does not depend upon the temper- atures themselves, but rather upon their differences, so that for the same variation we have always a correspond- ing work. If we suppose T=68, T'=141.8, we have T 7 T=T3.8 and consequently the work of the bar, per running foot and per square inch of section, is 0.00065416 x 73.8 a =3.5629 lbs - f 32 FORCES AND THE MEASURE OF THEIR WOKK. and for the 2.046 square inches for a length of 32.808 ft - it will be 3.5629 lbs - ft - x 2.046 x 32.808=239.17 lbs - ft - 36. The proper limit to variations of temperature to be used. We have confined ourselves in the preceding calculations to a variation of temperature, because, as we have seen, it corresponds to an expansion or a shrinkage of .0005 per foot, and to an effort of 14228.5 lbs - of exten- sion or compression, which, from observations of good constructions, is the highest limit of effort which forged iron can support per square inch of section, without fear of deranging its elasticity, as we shall hereafter see. It is important, therefore, that we should confine ourselves to limits of extension or contraction between which elas- ticity is not altered. DYNAMOMETEES, OK, THE DESCRIPTION AND CONSTRUCTION OF INSTRUMENTS ADAPTED TO THE MEASUREMENT OF WORK DEVELOPED BY ANIMATE OR INANIMATE MOTORS. 37. General and particular conditions which these in- struments should fulfil. We have seen in the preceding lessons, that the work developed by a constant force F, (whose point of application has described the path S in its own direction,) had for its measure the product FS, and that if the effort F is variable, the total work developed after the body has passed through any path S, was the sum of all the elementary quantities of work Fs, succes- sively developed along the elements s of the path described. In this last case, we have shown, either by calculation or Simpson's quadrature, how we obtain the sum of products analogous to Fs, for the total given path S described in the direction of the effort. Finally, we have defined the mean effort of a variable force, and shown that the total work is deduced, by dividing the former by the total space described. The instruments designed to measure work developed by animate or inanimate motors, should, then, afford us by their indications the product of the effort into the space described, whatever may be their simultaneous va- riations. Such is the general condition to be fulfilled in all cases not involving impossibilities. The illustrious Watt was the first to satisfy this con- 3 34: DYNAMOMETEES. dition, in the construction of a dynamometric apparatus, to which he gave the name of " Pressure Indicator" a description of which will be given hereafter. "We will now consider the particular conditions to be fulfilled by these instruments : 1st. The sensibility of the instrument should be pro- portioned to the intensity of efforts to be measured, and should not be liable to alterations by use. 2d. The indications of flexures should be obtained by methods independent of the attendance, fancies, or pre- possessions of the observer, and should consequently be furnished by the instrument itself, by means of tracings or material results, remaining after the experiments. 3d. "We should be able to ascertain the effort exerted, at each point of the path, described by the point of appli- cation of the effort, or, in certain cases, at each instant in the period of observations. 4th. If the experiment from its nature must be con- tinued a long time, the apparatus should be such as can easily render the total quantity of work expended by the motor. To meet the first condition, we use plates, which bend in proportion to efforts exerted, and which have the form of solids of equal resistance. This affords much assist- ance in making tabular statements, while it gives great sensibility to the instrument. 38. Eules for proportioning spring-plates. The theory of the resistance of materials to flexure, according with the known results of experiment, shows that, when a me- tallic plate of a constant rectangular section is fastened at one end, and subjected at the other to the action of an effort P, perpendicular to its length or primitive direc- tion ; or when an elastic plate of the same form is placed freely upon two supports, and subjected in the middle to an effort P, directed in the manner.described, its flexure DYNAMOMETERS. 35 F, so long as it does not exceed the limits of elasticity, will be : 1st. Proportional to the effort P ; 2d. Proportional to the cube of the arm of the lever c of this effort ; 3d. In an inverse ratio of the width a of the plate, in a direction perpendicular to the plane of flexure ; 4th. In an inverse ratio of the cube of the depth b of the plate, at the fixed point for the first case, and at its mid- dle for the second ; 5th. In an inverse ratio of a number E constant for each body, called the coefficient or modulus of elasticity, and which expresses in pounds the weight required to extend a prismatic bar of the same material, with a unit of surface for its transverse section, to double its primitive length, if such change in its dimensions may be made, without varying the value of E. Furthermore, if the longitudinal profile of the plate presents the parabolic form of solids of equal resistance, the flexure will be double that of a plate of uniform thick- ness throughout its length, subjected to the same efforts; while the resistance to rupture is the same in both. Hence, we have for springs of equal resistance, con- formably to theory and experience, the relation a formula, by means of which we can calculate any one of the quantities composing it when the others are known. I have found in the construction of many spring-plates, that if made of good German steel, properly tempered and annealed, the value of the coefficient of elasticity to be used will be E =4273700000 pounds per square foot. 38. jRatio to be established between the different pro- 36 DYNAMOMETERS. portions. The width a of the plate should, at most, not exceed the limit of from .1312 ft - to .164 ft -, since the warp- ing produced by tempering increases with its width and creates difficulties in its adjustment. An examination of the springs which I have made, shows that the flexures of springs remain proportional to their efforts, when for the strongest they do not exceed T V of their length, and for the weakest , the measure being taken from the place where they are embedded. With these data it will be easy to calculate the thick- ness I to be given to the plate at the place of its setting, so that under a determinate effort it may take a known flexure. It is derived from the following formula : V- 70 ' - 39. Longitudinal Profile of the plates. The above dimensions being obtained, we determine the form of the longitudinal profile by means of the formula s V y=-x in which 5 and c being the quantities already designated, A. e _ i #: _Z? FIG. 15. a; will represent the abscissa of the curve measured from its origin B, and y will be the corresponding ordinate. 40. Disposition of the plates of springs.* The plates of springs designed to measure the traction of animal mo- tors upon wagons, ploughs, boats, &c., are disposed as shown in Fig. 16. * For further details, seethe description d&Apparrih dynam&metriques, etc. Chez L. Matluas. DYNAMOMETERS. 37 Two plates aa! and W exactly alike, with the inner faces plane, and the outer parabolic, are terminated at FIG. 16. the ends by a knuckle joint of the same width, pierced with a drilled hole. Small steel bolts traverse these holes, with moderate friction, and are secured in the straps/y to which they are fastened by screw nuts. A posterior catch c, is pierced with an opening for receiving the plate, which is passed through it lengthwise ; a shoulder with its length equal to the width of the catch, is prepared midway the plate, and fits this opening accu- rately. Adjusting screws , and a constant flexure or rectangular height of 0.2296 ft - answers to an effort of 1234 lbs - ; calling P the weight of the strip 0.2296 ft - high, p the weight of the part contained between the curve and the zero line, DYNAMOMETERS. 41 E the length of the space described, and F the mean effect developed by the motor, we shall have F=1234.| pounds, and the whole work of the variable effort would have for its value the product FE. 45. Use of the Planimeier. In employing Ernst's Planimeter, furnished with a wooden cone, we have a second method of obtaining the quadrature of curves rap- idly and without calculation. This instrument is composed of a cone l>cb (Figs. 18 and 19), with its axis inclined to the plane of the table Plan. Fio. 18. 4:2 DYNAMOMETEES. which supports the instrument, so that its upper edge is parallel to this plane. This cone rests by its points upon two supports fastened to the plate X, and upon its pro- longed axis is a small wheel aa, which presses against a strip LL parallel to the guides along which the plate XX slides; so that when this plate is pushed forward or back, in the direction LL, the small wheel of the cone makes a number of turns proportional to the space de- scribed by the plate. A counter, the principal piece of which is a wheel dd, vertical and perpendicular to the upper edge of the cone, turning around an axis parallel to this edge, is mounted by points upon a piece with slides ff, which is moved with the plate XX, but which, moreover, may have a motion perpendicular to the band LL, so that the wheel Section on A B. Scale i FIG. 19. DYNAMOMETERS. 4:3 can at pleasure be brought near or removed from the apex of the cone. The counter resting upon the surface of the cone by its own weight, when this cone turns the wheel turns with it; evidently, the number of turns it makes will always be proportional : 1st, to the number of turns of the cones, or to the space described in the direction LL ; and 2d, to the distance of the wheel from the apex of the cone, or to the product of these two quantities. Suppose, now, the wheel being at the summit of the cone, that a pointer g placed upon a slide ff^ corresponds with a line RS parallel to the guide LL, and is over the point R, it is evident that if we push the plate XX, so that this point will follow exactly the line RS, the wheel will not turn, since the velocity at the summit of the cone is nought ; but if the point g is at M, and the wheel at a distance from the apex of the cone equal to MR=NS, when the point is pushed from M to N", the number of turns of the wheel will be proportional to the length RS, which is the base of the rectangle MNSR, and to the height of the same rectangle. It will consequently be pro- portional.to the surface of this rectangle. So also, if we cause the point to follow the line OP, the number of turns of the wheel will be proportional to the surface of the rectangle ORSP. But in using the instrument we cannot bring the wheel to the summit of the cone, which is truncated, and we must somewhat modify the method of obtaining the surface of the rectangle to be measured. Suppose, for example, it were required to calculate the surface of the rectangle OM1STP. We first bring the point g over the line MN, being careful that it conforms exactly to the motion of the plate XX. We then push the instrument so that the point g shall pass from M to !N". The counter wheel makes then a number of turns, proportional to the surface of the rectangle RMNS. We then draw the slide ff and bring the point g over the point P, and then draw 44 DYNAMOMETERS. back the plate XX, so as to have the point g follow the linePO. In this retrograde movement the wheel turns in a con- trary direction, and makes a number of turns proportional to the surface of the rectangle ORSP, and as in these two consecutive movements it has passed in two opposite directions, it is evident that the definite number of turns made is proportional to the difference of the two rectan- gles OBSP and MUSE", or to the surface of the rectangle OMXP. The motion of the wheel is transmitted by gearing to indicators with two limbs, one giving the units, tenths, and hundredths of square millimetres, and the other the thousandths, the square millimetre being =.00152 square inch. What we have said respecting the rectangle applies exactly to the quadrature of a surface terminated, as in the curves traced by the style of dynamometers, on one side by a straight line, and on the other by a curved line op, for each element of this surface uvxy may be regarded as a small rectangle, whose base is ux, and the height the arithmetical mean between uv and xy. To get the abstract of the curve, or the quadrature of the surface MN/>0, we operate as follows : We place the sheet of paper under the plane table of the planimeter, so that the point g being but little removed from the table may follow exactly the line MK of zero of efforts, when we push the plate XX from M to N. This done, we bring back the point g over M, raise up the counter and place by hand the indices at zero. We .then place gently the wheel upon the cone, and push the plate XX, so that the point g shall go from M to N". We then draw the slide ff, so as to bring the point g over the point p ; then, by means of the double motion we can impart to it, we fol- low exactly with this point all the sinuosities of this curve, until it arrives at o. We then read upon the two limbs the number of square millimetres contained in the sur- DYNAMOMETEKS. face, and dividing it by the length of the base MN", ex- pressed in millimetres, we have the mean ordinate, or the height of the rectangle of the same surface, which is the mean effort exerted. But that these operations j ust indi- cated should lead to an exact result, we must be sure, in the forward or back movement, that the wheel does not slide along without turning. We secure this result by substituting for the polished metallic cone of common planirneters one of unpolished wood. 46. Dynamometer for showing the whole quantity of actions, for a considerable interval of time and space. When we would observe the work developed by animate or other motors, for a long distance, the dynamometer with the style, whose band of paper cannot serve but for a distance of from 2,000 to 3,000 ft -, will not answer our purpose. Besides, it is quite often more convenient to obtain at once the quantity of work developed at any given distance, and it is essential that we have an appa- ratus which of itself shall record a total of the successive elementary quantities of work, and thus dispense with the quadrature, whose use we have just explained. Such is the design of the following modifications attached to the dynamometer described in the preceding sections : The posterior catch bar c (Fig. 20) is traversed by an axis of rotation, upon which is screwed a plate B with a radius 0.26 f % placed above the springs, and which has at its lower end a pulley to which the motion of the wheel is transmitted by means of an endless cord passing over pulleys. A support E embodied in the ante- Qd O 1 L FIG. 20. 46 DYNAMOMETERS. rior catch bar d bears a counter, which follows all the motions of flexure of the front spring. The principal piece of the counter is a wheel mounted upon an axis parallel to the plate and to the direction of the efforts of traction. This wheel acts as that of the counter of the planimeter, but, since instead of the cone we have here a plane, it will be at the centre of the circle when the instrument is at rest. From what has been said in No. 45, it is needless to describe the action of this instrument, and we see that the number of turns of the wheel is proportional to the sum of elementary products of the efforts exerted, and of the elements of the path de- scribed, or to the total work. Calling T the distance in feet of the wheel from the centre of the plate, under the effort of a traction expressed in pounds, or the flexure of the spring under this effort, provided the instrument is arranged so that the wheel rests upon the centre of the plate when the effort is zero ; r' the radius of the small wheel ; e the space described in one second by the wagon in the direction of its draught, if the effort is constant, and in an infinitely small period, if the effort is variable ; E the radius of the wheel from which the motion is derived ; p 71=: the number of turns of the wheel correspond- 27TXV ing to the space e ; -p K=- the ratio of efforts to the measured flexures ; T !N" the number of turns of the small wheel answering to the space e ; R/ the radius of the hub of the wheel, by which the motion of the plate is produced ; T' the radius of the pulley of the plate ; It is evident that this plate will make a number of -p/ turns equal - r for one turn of the wheel, or, rather, to DYNAMOMETERS. 47 T>/ ., for the space e described in the direction of the 27T.K T draught. rip The small wheel will make turns for one turn of the rl plate ; we shall have then N= ?_._.-,, for the number of turns of the small wheel, corresponding to a space , described under the effort of a traction F. The number !N" is finite or infinitely small, according as we deal with a constant effort, and a finite space, or with a variable effort, and an element of space. But we TT TT have by definition K=-, whence r=^, and consequently Thus, whether for a constant effort and a finite work, or for a variable effort and an elementary work, we see that the work developed by the motor is measured by the product of the constant factor - -^57- > and of the num- ber !N" of turns, or elementary fractions of turns made by the small wheel, so that the total work, at the end of any interval, being the sum of the elementary quantities of work successively developed, will be equal to the same product in taking the number N equal to the total number of turns of the small wheel during the observed interval. Instruments of this kind have been successfully em- ployed, and have afforded great facilities in prolonged experiments upon the draught of carriages, and have ena- bled us to determine the total quantities of work devel- oped by six horse teams, during their entire daily trips, and for routes from Paris to Amiens, and from Nantz to Mans. 47. Arrangement to obtain the indications of the num- ber of turns made by the small wheel. We may easily conceive, that when the axis of the wheel has an endless 48 DYNAMOMETERS. screw, its motion may be easily communicated by properly proportioned gearing to two limbs, one of which will give the units and tenths of turns, and the other the hundredths and thousandths of turns of the small wheel. But further, in order to be able to observe the divisions of these limbs without stopping the instrument or the trip, two styles are so arranged that traversing two cups filled with thick ink, they may deposit upon the enamelled limbs a black dot, by placing the finger upon a button. Observations can thus be made and multiplied, without confounding the results. 48. Dynamometer with chronometer motors. When we wish to experiment upon the resistance of tow-boats and foot-swing ploughs, it would be at least difficult, and in some cases impossible, to put the motion of the paper in constant ratio with the space described. In this case, it is very convenient to employ a chronometer motor, which communicates to the paper a uniform motion. Then the developed lengths of the paper represent the time, and the quadrature of the curve of flexures gives the sum of the product F x t of each effort by its elementary dura- tion, or what we call, as we shall see hereafter, the total quantity of motion developed in the observed interval of time, or by the length of the developed paper, we have the mean effort of the motive power. In the towing of boats, and in all cases where the ve- locity influences the results, we provide two auxiliary brushes, one of which serves to mark upon the paper intervals of time, 15", 30", &c., and the other the dis- tances described -in the passing of mile posts, or of objects whose distances are known. 49. Rotating dynamometer. The instruments we have been describing are constructed for measuring the effort or work developed by motors whose action takes place in straight or circular lines, but it is easy to modify them so DYNAMOMETERS. as to obtain the work transmitted by an axis of rotation, to any machine, in applying the principle of styles, or that of the counters. 50. Description of a rotating dynamometer with styles. Upon a shaft resting on two cast iron supports fastened to a wooden platform, are placed three pulleys of the same diameter (figs. 21 and 22) ; the one A is FIG. 21. Fia. 22. 50 DYNAMOMETERS. fixed, the other 0, near the first, is loose, and the last, B, is movable around the shaft, between limits which we shall indicate. This apparatus being placed between the motor shaft and a machine whose resistance is to be measured, the loose pulley C receives the transmission belt of the motor shaft, and when this belt is passed over the fixed pulley A, the shaft is set in motion and acquires a velocity de- pending upon the ratio of the diameter of the pulley to that of the drum of the motor shaft. The pulley B receives a belt which serves to transmit motion to the machine, and to overcome its resistance, and as it has but a slight friction upon the shaft, it would not be impressed with the motion imparted to the shaft by the fixed pulley, unless a stop embodied in it were pressed by the extremity of a spring-plate planted upon the shaft in the direction of one of its spokes. This spring turning with the shaft, acts upon the stop, whose resist- ance bends it, and when the resistance to flexure is able to overcome that opposed by the machine, motion com- mences, and is thus found to be transmitted from the mo- tor shaft to the machine experimented upon, through the agency of a spring, whose flexures are the immediate measure of the resistance to be overcome. A style adjusted upon an arm of the pulley can be brought to any desired proximity with the paper, endowed with a motion of its own, in a constant ratio with that of the pulley or the shaft, and then traces a curve of flexures of the spring exactly in the same manner as in dynamom- eters employed on wagons. Another style, immovable relatively to the first, traces at the same time a line corresponding to a flexure zero, or in the position occupied by the movable style when the effort is zero. This line of zero will be found in the mid- dle of the width of the paper, so that the effort may be measured in either direction. The springs used have a parabolic section, and may be DYNAMOMETEKS. 51 multiplied at pleasure, according to the intensity of efforts to be measured by the instrument. A catch placed upon the shaft limits the displacement of the pulley, and consequently the flexure of the springs, so as to prevent their being strained in case of any con- siderable accidental efforts. 51. Transmission of the motion of the shaft to the land of paper. A toothed ring is adjusted with gentle friction upon the shaft, and its helicoidal teeth-range is geared with a pinion, whose axis, being in a plane perpendicular to that of the shaft, does not come in its way. The axle of this pinion has an endless screw driving another pinion mounted upon the prolongation of the axle of the small cylinder, on which is rolled the silk which drives the fusee. When we wish to set the band of paper in mo- tion, we make the toothed ring immovable by means of a stay, upon which a catch fastened to this ring impinges, when it is properly turned. Then the toothed ring being fixed in space, while this pinion driven by the shaft rolls around it, this pinion will acquire a relative motion which is transmitted to the screw, to the fusee, and to the band of paper. This apparatus is provided with a conical fusee to con- trol the motion of the bobbin which carries the paper ; by means of this we compensate the relative increase in the velocity of translation of the band of paper, which, without this precaution, would ensue from the increase of diameter of the motor wheel, upon which the succes- sive layers of paper accumulate according as the trace of the dynamometric curve is effected. 52. Results of experiments made with the rotation dy- namometer. As examples of the results derived from the rotation dynamometer, we will report some that were 52 DYNAMOMETEKS. obtained at the saw-mills and wheelwright's machines of the imperial coach establishment at Chaillot : is 1 C * * JS Kind of Machine. Condition and kind of Wood. 5j E i I S g, It r i ir f feet. sq. ft. horse power. Ibs. Oak cut 8 years, 1.164 7.6803 2.82 163944 Vertical saw, with one blade, " Ash " 2 " Soft Elm cut 4 years, Aspen " 4 " 0.970 1.570 1.279 6.4801 12.5425 6.6759 2.45 4.60 2.67 162470 198682 150414 Twisted Elm cut 1 year, Ash cut 8 years, 0.566 5.8224 1.5124 3.48 2.80 260040 312400 Circular saw, 8 " 0.402 1.6910 2.375 176990 2.03 ft in " 8 " 0.284 1.3455 1.665 166986 diameter. 3 0.187 0.6480 1.910 156772 " 3 " 0.068 0.3240 1.775 126368 WHEELWRIGHT MACHINES. Kind of Machine. Condition and kind of Wood, or nature of Work. Mean work in Horse Powers. For sawing felloes, Diagonal cut saw, Elm cut 2 years, Ash " 3 1.390 1.225 Machine for spoke tenons, Oak " 2 " 0.460 " piercing Felloes, Elm cut 2 years, holes for spokes, 0.253 4t 16 U 1C " " pins, 0.125 " " making pins, Oak, pins of 0.118 feet, 0.390 " piercing iron, Holes 0.115 feet, 0.551 Blower forcing air upon -] 19 fires making 1296 "I t 13 " 1816 1 tu s A (t -1 KQ/ ?~ <* 2.860 2.750 I 1327 J minute > 1.920 53. The counter of the rotation dynamometer. The movable pulley and the mounting of the plate springs is precisely the same as in the dynamometer with styles. A bevel-toothed ring with gentle friction is geared to a con- ical pinion whose axis stands at right angles with that of the shaft. The axis of this pinion is terminated by an endless screw, which drives a toothed-wheel, whose axis parallel to that of the shaft, carries at the other end a brass plate, with its plane perpendicular to the axle. The movable pulley carries a wheel counter similar to that described in No. 46, which is displaced with this pulley a quantity proportional to the flexure of the springs. By DYNAMOMETEKS. 53 endless screws we can place the small wheel in the centre of the plate when the machine is at rest. The theory and action of this instrument is also analogous to those of dynamometers with counters for wagons. This instrument can easily be proportioned so as to obtain the total quantity of work transmitted by a rotating axle, during a day, a week, or a month, and in this regard will be very useful in observations relative to the distri- bution of motive force among different work-shops, or the consumption of fuel by steam-engines. 54. Gauge of pressure of steam in the cylinders of engines. Watt's gauge per- fected ly Mac-Naught. It is of the greatest utility, for appreciating the effects of the distribution of steam in the interior of cylinders of steam-engines, to have the means of measuring the pressure of the steam at different points of the stroke of the piston. Watt gave his attention to the con- struction of a small instru- ment for this purpose, which he named Indicator of Pres- sure, and which since his time has received many improvements in its details. It is composed of a free piston, with moderate fric- tion, and without packing, fig. 23,) contained in a small cylinder, terminated at its lower end by a tube, provid- Flo> 28 . 54: DYNAMOMETERS. ed with a stop-cock screwed on to the head of the cylin- der. When the cock is open, the steam rushing into the cylinder tends to drive the piston upwards, but the stem of it being connected with a spiral spring, this spring is compressed and serves to measure the effort exerted. With this kind of spring, and with cards, we may ob- tain flexures proportioned to the efforts, and need only a trace of these flexures. For this purpose, the stem of the small piston bears a pointed arm or lever, furnished with a crayon, which is brought in contact with a sheet of paper rolled upon a copper cylinder, whose axle is parallel to the stem of the piston; a groove is made upon the lower part of this cylinder, in which winds round a thread, the end of which is fastened to a small winch. The number of turns of the thread around this winch has a development a little less than that of the cylinder, and upon its axle is a pulley receiving many turns of a thread, whose development is equal or superior to the stroke of the piston. Within the cylinder is a spiral spring which forces it back to its first position in the return stroke of the piston. It follows from this disposition, that during the intro- duction and expansion of the steam, the style will trace upon the sheet of paper a curve giving the excess of the internal over the external pressure ; then that, in the pe- riod of its escape, the cylinder turning back, the style will trace another curve giving the pressure during the escape, and that the second branch on the following stroke closes in upon the first. The length of the paper developed being proportional to the stroke of the piston, and the ordinates bounded by the two 'curves being in all cases proportioned to the motive pressure of the steam, it is evident that the area of surfaces comprised within these curves represent the work developed upon the small pis- ton, and consequently that upon the great. The use and application of this instrument is easy, and may give good indications, even though somewhat worn, "DYNAMOMETKRS. 55 but we would remark that when the crayon has traced many successive curves, they become confounded or over- lay each other, so as sometimes to create confusion ; still, the facility of its establishment causes it to be in great demand with constructors of steam-engines. 55. New style indicator. To avoid the confusion of curves, I have proposed to adapt to the indicator the arrangement used for common dynamometers. (Figs. 24:, 25, and 26.) Instead of acting upon a spiral spring, FIG. 25. the piston of the instrument has a square head d pierced with an opening, in which is fastened the end of a parabolic spring plate, the other end being secured to a support /. the plate has such a length that on either side it may bend several centimetres, (cent. =.0328"-) and as we may 56 DYNAMOMETERS. use plates more or less rigid, the instrument may serve to measure pressures, comprised between one and ten atmos- pheres. Thus, for example, for a high-pressure engine working at four atmospheres above the atmospheric pres- sure, each atmosphere may correspond with a flexure of from .0328 ft - to .0361 ft - of the spring, which is exact enough in practice. The head of the piston carries in front of the spring blade a style g, which traces upon a sheet of paper the curve of flexures, or the tensions of FIG. 26. the steam. Another fixed style A, adjusted so as to trace the same right line with the movable style when the spring is at rest, indicates the zero of pressures. When the steam is let in upon the piston of the machine, it drives that of the instrument outward, and the curve traced is beyond the line of zero : when, on the other hand, the DYNAMOMETERS. 57 steam expands and escapes, whether in the air or the con- denser, the curve approaches the line of zero, and may pass by it. In either case we have upon the band of paper a trace of all the variations of pressure. A third fixed style k marks at each stroke a point which serves to connect the curves with the commence- ment of the stroke of the piston. Notwithstanding the advantages possessed by this instrument, for the study of the effects of steam-engines, by the multiplicity and dis- junction of its curves, we must admit, that for common use, "Watt's improved indicator, in its greater portability, and convenience of establishment, answers quite as well for ascertaining the condition of a steam-engine. We see that the two principles upon which are founded all the instruments we have described, to wit : 1st. The use of a style tracing a curve of efforts upon a sheet of paper set in motion by direct means ; and 2d. The use of a small wheel counter, to totalize the quantity of work, readily applies to All kinds of observations we may have to make ; and finally, I would bear in mind that the main idea of these two solutions of the questions we have dis- cussed, were pointed out to me by M. Poncelet, my friend and teacher, and whatever I may claim in the construc- tion of these instruments is only relative to the realization of this pregnant and ingenious thought. THE TRANSMISSION OF MOTION BY FORCES. 56. General remark relative to the laws of motion. We have derived from mechanical geometry a knowledge of the laws of uniform motion, as of those of motion uni- formly accelerated or retarded. Experience also shows us, that there exists motions subjected to these laws. Thus, for example, we admit, by means of various chro- nometric contrivances adapted to these observations, that the motion of descent of different formed bodies in air or water, quickly becomes uniform when they present sur- faces sufficient for the resistance of tl^e air to acquire a suitable intensity. We also admit that heavy bodies with small surfaces fall to the earth with a uniformly accelerated velocity. These facts established, it is proper to deduce their consequences. We know, (No. 3) according to the fundamental prop- erty of matter called inertia, " that all bodies in a state of uniform motion proceed in the same straight line, unless some obstacle constrains them to change that state." If, then, a body is impressed with a uniform motion so that no foreign cause or force operates to change this state of motion, or if many forces solicit it to equal changes, their action will be counterbalanced, neutralized, and will be in equilibrium. Such is the case with parachutes descending with a uniform motion. The action of gravity, and that of the resistance of the air, compensate and destroy each other. TRANSMISSION OF MOTION BY FORCES. 59 57. Consequences relative to the causes producing accel- erated or retarded motion. In motion uniformly acceler- ated or retarded, the increase or diminution of velocity being always the same for equal times, the force producing this modification of motion is then constant, since it pro- duces constant effects. Thus, when observation has shown us that the motion is uniformly accelerated or retarded, we are justified in the conclusion, that the force which accelerates or retards is constant. 58. Vertical motion of heavy bodies. Experiment proves that in a vacuum, all bodies subjected to the action of gravity fall from the same height in the same time, whatever their density. It follows from this, that gravity operates in the same manner upon all the material mole- cules. In air and other resisting mediums, the resistance experienced by bodies depends on the extent and form of their surfaces, and the nature of their motion is notably modified, when the velocities are great, and the bodies have very great bulk relative to their weight. But for bodies such as stones, wood, metal, used in construction and for common heights of fall, the resistance of the air is so small, that we may usually leave it out of account. Galileo, in observing the times employed by bodies rolling upon inclined planes or falling vertically, was the first to observe the fact, that the spaces described in a vertical direction, and in that along planes, were to each other as the squares of the times employed ; whence he concluded that, "for the same place upon the surface of the earth gravity was uniform and constant" It is thus that, from experiment, has been derived this important mechanical law. Applying to this case the laws which we have found for all accelerated or retarded motions, we shall have for the velocity imparted or destroyed at the first second, and which is usually designated by the letter ^, V t =#=32.1817 ft - The space described in the vertical 60 TRANSMISSION OF MOTION BY FORCES. direction, or the height is designated by the letter H. We have then for the formula of motion of heavy bodies TT 59. Use of this formula. The first formula serves to determine approximately the height of a tower, or depth of a pit, by a simple observation of the duration of the fall of a body. If, for example, we have found that a body (for which, in case we try a pit, we make use of a light) has taken 2.5" to pass from the curb to the bottom of a pit, we shall have for its depth 11=16.0908 x (2.5 // ) 2 =100.57 fu The third is of frequent use, especially in calculating the gauging of the discharge of water, and gives the velocity corresponding to a known height. Thus, for a height H=3.9371 ft - we find Y= 1/64363 x3.9371=15.91 ft - It has been reduced in tables, which may be found in most of the works on mechanics ; but the rule for calcula- tion is a substitute for these tables, when they are not at hand. Bringing a pointer under the number 64.363, read at the upper scale, we find in the lower scale the veloci- ties corresponding to all the heights read upon the reglet, or, reciprocally, reading the velocities at the lower scale, we find upon the reglet their corresponding heights. 60. Successive fall of heavy lodies. The laws of the motion of descent of heavy bodies serves to explain, among other phenomena, that of the increasing separation TRANSMISSION OF MOTION BY FORCES. 61 of bodies ; of water-drops, for example, which raised together and contiguously in a jet of water, fall in a shower of separate drops. In fact, it is easy to see that the drops starting from the summit of the curve, one after the other, must separate more and more. Suppose, for instance, that a drop of water commences its descent 0.01" before the following : V after the starting of the second, the first drop will have fallen during 1.01", and through a height H=16.0908 ft - x (1.01) 2 =16.4:l ffc - while the next, which has been only 1" in motion, will have fallen only H=16.0908xl' /2 =16.091 ft - Already the first is in advance of the second by 0.31 ft - and the separation constantly increasing, the jet falls back in rain. 61. Principle of the proportionality of forces to their velocities. The observation of facts shows, and it seems quite natural to admit, ih&t forces are really proportional to the degrees of velocity which they impress in equal infi- nitely small times, upon the same ~body, yielding freely to their action and in the proper direction of this action. This is one of the fundamental axioms admitted by all geometricians, and is proved in the exactitude of conse- quences deduced from it. If, then, we call F and F' two forces which, acting successively upon the same body, impress it with or de- prive it of infinitely small degrees of velocity, v and v' 9 in an element of time , we shall have from this principle the proportion F : F : : v : v r . To get the expression and measure of the force F, we may compare it with another force, whose effect upon the 62 TRANSMISSION OF MOTION BY FORCES. body is known ; with gravity, for example, and as we know that the velocity imparted to heavy bodies in an element of time is v f =gt^ and as we designate by P, the weight of the body, or the force exerted by gravity, the above pro- portion will then become F : P : : v : gt ; Pv Before proceeding farther, we remark that the same principle applied to actions exerted by gravity upon the same body in different places, where the weight of the body is respectively P and P', gives us the proportion P : P' : : gt : g't : : g : g' , p 9 p p' whence it follows that the ratio r is constant for all places upon the earth. This constant ratio of the weight of a body to the velocity communicated to it by gravity, in the first second of its action, is what we term its mass, and is designated by the letter M. 62. The measure of motive forces and of inertia. "We have, then, for the expression of the force F capable of imparting to or taking from a body of the weight P or mass M an element of velocity v, in an element of time t We see, by this expression, that when the weight of a body is given, or its mass, we shall have the value and measure of its force in pounds, when we know the ratio -. If, for example, this ratio is constant, which is the t TRANSMISSION OF MOTION BY FORCES. 63 case with motion uniformly accelerated or retarded, the force F is constant. But, since to communicate to a body of the weight P, a variation of velocity v 9 in an element of time , there Pv must be developed an effort , then there is a resistance gt to be surmounted, of which this effort is the measure. This resistance is the force of inertia, the reaction which takes place every time that a variation of motion is produced. Thus the preceding expression will be at once the measure of the motive force, which produces the change of motion and that of the force with which the body, by virtue of its inertia, opposes or resists this change. Yv An examination of the formula F shows that, for gt a weight P, or a given mass M, the magnitude of the force F will increase as the change of motion becomes rt* more rapid, or the ratio - becomes greater. It is thus we t account for the magnitude of efforts and reactions devel- oped in the transmission of motion by the shocks expe rienced between hard bodies, in very short intervals of time, when the velocity varies or is destroyed stiddenty. r\\ This ratio - of the increase or diminution of velocity t in the element of time during which this change is pro- duced, is that to which for many years past geometricians have given the name of acceleration. Thus, in treating upon the action of gravity, the con- stant acceleration produced by it is represented by the number g=-. t It follows from this definition and the preceding gen- eral principles, that the force which produces an elemen- tary change in the motion of a body, is proportional to 64: TRANSMISSION OF MOTION BY FORCES. the weight P, or to its mass -, and to the acceleration - 9 * which it produces. "We may make sensible the increase of effort F to be exerted, with the rapidity of the communication of mo- tion, by means of a spring balance, or any kind of spring whose flexure, indicated by a style or a scale, is so much the greater as the transmission of motion is more rapid. If, for example, we suspend to a spring balance a weight of 10 lb % in which case a pasteboard scale placed opposite the upper part may stop at the fifth division, and then raise the balance and weight with an accelerated motion, the spring will bend still more, and so much the more, as the acceleration of motion is the more rapid. The in- crease of flexure indicated by the displacement of the scale will measure the effort, the resistance opposed by inertia to the acceleration of motion. 63. Case when the force is constant. If the force F, or the ratio j is constant, we have then at the end of a cer- 5 tain time T, when the force has communicated or destroyed a velocity Y, the equality 7= , and consequently F=M =M- ; o J. JL t whence FT=MY and Ft=M.v. 64. delation offerees to accelerations. If two forces F and F' act in succession upon the same body, and im- n\ ny part to it different accelerations - and - , we see that they t t will be proportional to these accelerations, and that we shall have F : F 2 v - t'f TRANSMISSION OF MOTION BY FORCES. 65 It is by reason of this proportionality that the acceler- ations are sometimes taken for the measure of forces. But these quantities cannot be an exact measure of forces, inasmuch as they only express a ratio. Thus, when we say absolutely and without other ex- planation that the quantity ^, which expresses the accel- eration produced by gravity, is the measure of this force, we give to students an incorrect idea, since g is in reality only the velocity imparted to or taken from a body by gravity during each second of its action, and the velocity which is expressed in feet cannot measure a force which should be compared with pounds. 65. Quantity of motion. The products MY, Mv, P P equal to Y or v, have received the name of quantity & \J of motion : it is a conventional phrase to which we attach no other signification than that of the product of a mass into the velocity imparted to or taken from it. "We would further observe, that this product MY, M-y, is equal to FT or F, of the force and time during which it has acted. If we consider two forces as acting for dif- ferent times upon two bodies of unequal mass, we shall have F$=M*>, FY=MV; and consequently F* : FY : : Mu : M V ; whence it follows that the quantities of motion M-y, MV, imparted to or taken from different bodies in unequal times, are as the product of the forces to which they are due, into the time during which these forces have acted. It is only when the times are equal that the quantities of motion impressed or destroyed are proportional to the forces, and can serve for their measure. From the preceding remarks it follows, as we shall explain in the following section, that in shocks there is no 5 66 TRANSMISSION OF MOTION BY FORCES. loss of quantity of motion, which is expressed in saying that there is a preservation of the quantities of motion. But we shall see hereafter that shocks occasion a loss of work. 66. Equal forces acting during equal times. If the forces are equal and act during the same time, the quan- tities of motion imparted or destroyed in the two bodies with masses M and M' are equal. This occurs in the re- action of two bodies which press, push, or impinge upon each other. The efforts of compression and resistance being equal, opposed and developed during the same time, it follows that the quantity of motion imparted in the reaction, to one of the bodies, is equal to that which is lost by the other. Here is a fact which is a necessary con- sequence of the theory of the shocks of bodies. Thus, for example, when a body with a mass M im- pressed with a velocity V, impinges on a body with a mass M x animated with a velocity V, in the same line, whether in the same or opposite directions, it develops at the point of contact equal and opposite efforts of com- pression, in an element of time , taking from the imping- ing body -a small degree of velocity -y, and consequently a quantity of motion Mv, and imparting to the body shocked, if it moves in the same direction as the first, an increase of velocity v' and a quantity of motion MV. These quantities being equal, we have then, at each instant of the mutual shock or compression of bodies, M.v=~M. f v'. In this case one of the bodies loses a quantity of motion equal to that gained by the other, and the sum of their two quantities of motion remains the same. The same thing transpiring at each instant of the shock, it follows then that the total quantity of motion lost by a body is equal to that gained by the other during the compression, and that at each end of this period, the sum of their quantities of motion is the same after the shock as before. This consequence constitutes the principle of TRANSMISSION OF MOTION BY FORCES. 67 the conservation of the quantities of motion, otherwise termed the principle of the conservation of motion of the centre of gravity. If we are dealing with soft bodies, whose elasticity is completely impaired by the shock, and which after com- pression unite and travel together with a common velocity U, the quantity of motion after the shock is (M+M)TJ, and from what proceeds we should have whence we derive for the common velocity after the shock TT _MY+M / Y / M+M~ If the body shocked was at rest, we should have V=0, and the above expression is reduced to MV U= M+M'' If, in the first of these two expressions, we divide the two terms of the fraction by the mass M' of the body shocked, the common velocity after the shock becomes -V+V M' Under this form we see that the common velocity of motion of two soft bodies will differ so much the less from the velocity V of the body shocked as the mass M of the impinging body is smaller compared with the body shocked. At the limit, or when the impinging body is infinitely small compared with the body shocked, the ra- tio ^ vanishes, and we have U= V, that is to say, the velocity of the mass shocked will not be changed. This 68 TRANSMISSION OF MOTION BY FORCES. case occurs in the motion of liquids and elastic fluids, when infinitely thin edges impinge successively upon finite masses endowed with a less velocity in the same direction. If bodies strike against each other in opposite direc- tions, a similarity of action exists ; but then, at the end of the compression, either the bodies are both brought to a state of rest, and we have MV=M'V and U=O, or one of the two goes backwards, and they proceed with a common velocity U. If it is, for example, the body M' which goes backwards, the quantity of motion lost by the body M is M(V U), and the quantity of motion devel- oped during the period of compression, by the forces of reaction upon the body M', is composed of that which has been destroyed, or M'V, plus that imparted in an op- posite direction M'U, and since the quantities of motion developed on both sides upon each of the bodies should be equal, we have whence we deduce for the common velocity, after the shock or compression, M+M' a formula in which we also see that the velocity of the impinging body will be so much the less changed, as its mass M is greater in its ratio with that of the body shocked for, dividing both parts of the fraction by the mass M of the impinging body, we have TRANSMISSION OF MOTION BY FORCES. 69 This shows that in machines working by shocks we must increase the weight, the mass of the impinging pieces, in their ratio to the pieces shocked, in a ratio so much the greater, as it is desired to maintain a greater regularity of motion. If the body shocked is at rest, such as a pile driven by a ram, we have V=O, the common velocity of the de- scent of the pile and ram after the shock is _ ,M / + Which shows that this velocity will differ so much the less from that of the arrival of the ram upon the head of the pile, as the mass M of the rani is greater in its ratio with that of the pile. It is best in this case, then, to increase the mass of the ram rather than its velocity, for the work employed to raise it increases only with its weight, while its work will be increased proportionally to the height of elevation, or to the square of velocity of its descent. 67. Proof of the preceding considerations ly direct experiment. The results which we have recorded relative to the shock of soft bodies have been verified by direct experiments, made by me at Metz in 1833,* with the fol- lowing apparatus : A wooden box (Fig. 27), in which was placed successively clay, more or less soft, sand, pieces of wood, &c., was suspended to a dynamometer having a style and turning plate. The plate was impressed with a uniform motion, which was transmitted by a weight, and regulated by a fan fly-wheel. "When the box was im- movable, the resistance of the dynamometer was in equi- * New experiments upon friction, and upon the transmission of motion by shock, Iz.^o 147.97 1.64 0.65 10.28 6.50 0.92 1.08 0.93 1.08 .020 .019 Experiments made with clay, whose resistance to penetration of pro- 132.84 26.64 159.28 0.98 7.96 1.40 1.81 .021 [ jectiles with small velocity was 1.81 9.19 1.52 1.51 .024 6080 Ibs. per square foot. 182.84 44.78 177.57 0.65 6.50 1.64 1.61 .020 147.82 147.82 12.23 44.73 161.05 192.54 0.65 0.65 6.50 6.50 0.53 1.51 0.54 1.44 .063 .072 1 Experiments made with clay, whose V resistance to penetration was 34G ) Ibs. to sq. ft. 48.45 26.64 74.85 0.65 6.50 2.27 2.16 .0075 48.45 26.64 74.85 0.98 7.96 2.79 2.75 .0074 48.45 44.73 98.14 0.38 4.59 2.20 2.26 .0060 We see by the results entered in the above table, that the velocities, so far as we are able to verify them with TRANSMISSION OF MOTION BY FORCES. 73 such means, are the same as those deduced from the pre- ceding theoretic considerations. 68. Shock of two elastic bodies. If we suppose that the two bodies in consideration are perfectly elastic, the effects of compression will be at first the same as in the preceding case, and at the end of this time the body M will have lost a velocity Y U, or a quantity of motion M(Y U) and the body M' will have gained a quantity of motion M'(U Y'), and the quantities being then equal, we have for the common velocity at the end of the com- pression V __MY+M / V / M+M' But, after the instant of greatest compression, the elastic bodies regain their primitive form, and in the return to it develop, if the elasticity is powerful, efforts equal to their resistance or compression, and consequently destroy or impart quantities of motion equal to those which they have previously destroyed or imparted. It follows from this that in the unbending of the molecular springs the body M will further lose a velocity =Y U, and that its final velocity will be V_2(Y-U)=2U-Y, and that the body M' will receive a new increase of ve- locity equal to U Y', and will then have a final velocity equal to If the body were at rest at the beginning, in supposing it to be perfectly elastic, it will then receive a velocity ._ ~M+M'* That is to say, twice that imparted to a soft body in the same circumstances. TRANSMISSION OF MOTION BY FOKCES. 69. Observations upon the preceding results. The foregoing reasonings relative to soft or elastic bodies pre- supposes the existence of bodies deprived of all elasticity, and of others endowed with perfect elasticity. Now, neither of these hypotheses is exact, and according to the circumstances in which they are placed, a body may act as if deprived of all elasticity, or as if possessed of only a partial elasticity. So also a body which, in certain con- ditions, acts as if it were perfectly elastic, will only appear as if but partially so in other cases. I will cite as examples the results of some experiments analogous to the preceding, and which were effected by placing at the bottom of a movable box a plate of cast- iron, upon which fell a spherical body. Experiments upon the transmission of motion l>y the shock of a spherical projectile falling upon a cast- iron plate. Velocity Weight of Velocity imparted to the Approximate the box Weight of Height of due to Box. duration and the Total the fall of the this By By of the its load. Iron ball. Weight. ball. height. Theory. Experi- transmission. P P P+P h 2 U ment. Ibs. Ibi. Ibs. ft. ft. ft. ft. seconds. 185 13.23 148.24 1.31 9.19 1.64 1.64 0.0085 136 13.23 148.24 1.64 10.28 1.84 1.87 0.0081 135 13.23 148.24 1.9T 11.26 2.07 2.05 0.0080 185 26.44 161.45 1.31 9.19 3.01 2.98 0.0065 185 26.44 161.45 1.64 10.28 3.36 3.44 0.0075 The results recorded in this table show that the cast- iron plate shocked has acted as a body perfectly elastic. But it is proper here to make some important remarks. The projectile which, (had it been in the condition of a perfectly elastic body,) as well as the parts of the plate with which it came in immediate contact, would have risen a height corresponding to the velocity 2U Y, did not by any means attain this height. This proves that the intensity of the shock in these experiments had changed TRANSMISSION OF MOTION BY FOECES. 75 in a great measure the elasticity of the molecular springs of the parts in contact, while the elasticity of flexure or of the general form of the plate had not been altered. "We see by this, that although bodies endowed with a cer- tain elasticity apparently resume their primitive form, there is nearly in every case a notable loss of work pro- duced by the shock, by reason of the more or less com- plete alteration of its elasticity. We shall see this more explicitly stated hereafter in JSTo. 95. 70. Quantity of motion imparted by a constant force. When the force is constant we have FT=MY, whence MY Y=-= r . This expression shows that the effort required to impart or destroy a given quantity of motion MY is so much the greater as the time employed is less, and since the reciprocal action of bodies is more rapid as the spaces described, their compressions, flexures, and penetrations are less for the same quantity of motion destroyed. We have here explained why it is that the shock of hard bodies, the transmission or destruction of motion by bodies slightly flexible, compressible or extensible, occasion such great efforts and such ruptures and accidents, and how it is, on the other hand, by the interposition of soft and com- pressible bodies that the intensity of efforts and their con- sequences is so much diminished. We see by the expression F=-=- that a finite velocity Y could never be imparted in an infinitely small time (nul) to a mass M except by an infinite effort, which shows the error in the hypothesis of the instantaneous transmission of motion by forces, to which we are then compelled to give a special name, and thus suppose a special nature in calling them forces of percussion : this error is often too explicitly admitted in the teachings of rational mechanics. Nothing like an instantaneous operation really occurs in nature ; quantities of motion are imparted and destroyed 76 TRANSMISSION OF MOTION BY FOKCES. in greater or less periods of time, sometimes, indeed, im- perceptible to our senses and means of observation, but never instantaneous. The idea of percussion is then erroneous in itself, if regarded in the sense just indicated. Examples will enable us to better appreciate this mat- ter. In case we require the quantity of motion imparted to a ball weighing 26.46 lb % and upon which gunpowder has impressed a velocity of 1 640.4"- in 1", we have FT=0.8222 x If we suppose successively T=1.00", 0.50", 0.10", 0.01", we have F=1348.7 lb % 2697.4 lb % 13487. lbs -, 134870 lb8 - The velocity being communicated in less than T i of a second, gives us an idea of the enormous efforts de- veloped by powder, though we have regarded it but as a mean constant effort, and consequently far inferior to the maximum value of the real effort. When horses impress upon a coach weighing 9924 lbs - a velocity of 32808 ft - per hour, or the quantity of motion to be imparted is WQ have, then, FT=2809.2. TRANSMISSION OF MOTION BY FOECES. 77 If we suppose that each one of the five horses exerts in any time a mean effort of 220.5 lb % we shall have neglecting the resistance of the ground, and the friction of the wheel-boxes, which in common cases would require an effort of 9924 30 =330.8 lb % or of 66.2 lb8 - per horse. "We see, in this case, that to impart this velocity in 2. 55", each horse must develop a mean effort of about 286.7 lb % which is more than four times the mean effort to be exerted after the velocity has been once acquired. It is proper to observe here, that the breaking of traces, of swing bars, wounds upon the breasts, and straining of hams, arises from the great rapidity of the destruction of the quantity of motion impressed by the horses upon their own mass by the resistance and reaction of the inertia of the vehicle : whence the necessity of starting with slack traces, and of warning and urging the horses gently with the voice. Similar effects are produced in starting and stopping railroad trains ; and in seeking the means of promptly checking these enormous masses, we must bear in mind that too sudden changes of velocity are dangerous for the passengers. Finally, the means adopted for the connection of ma- chines, or for a rapid transmission of motion to them, should be disposed or proportioned agreeably to these ideas. Jugglers, clowns, herculean fellows, in their feats of skill or strength, are led by observation to a practice con- forming to that above indicated, and they are never seen to raise, hurl, or arrest very heavy weights, or make their 78 TRANSMISSION OF MOTION BY FORCES. jumps suddenly, but always gradually increasing the time and the spaces described, so as to dimmish the ef- forts. 71. Observations upon the use of quantity of motion. When we know the product of the mass of a body and the velocity imparted or taken from it, we have the meas- ure of effect produced by the force during the period of its action ; but we see that this measure cannot be taken as a term of comparison except for analogous cases, where the velocities are really imparted or destroyed by the force, and it does not follow that the product FT of the force, by its period of action, (equal, when there is a change in the state of motion, to the quantity of motion imparted or destroyed,) should always serve as a measure of the effect of forces, as is sometimes admitted for certain in- struments and for certain kinds of work. Indeed, it is readily seen that an effort may continue a long time with- out producing a mechanical effect. Thus, horses pulling upon a mired wagon without starting it develop consider- able efforts, which, multiplied by the period of their action, would give an enormous product without any use- ful effort resulting, any mechanical work, and nothing but the fatigue and exhaustion of the motors. Take, for example, the draught of a plough, which in strong earth requires a mean total force of 794 pounds. We suppose the furrow to be 393. 7 ft long, the horses in one take 100" and in the other 200" to plough it. We shall have for the first case FT=794 lb8 -xlOO"=79400, and for the second FT=794 lbs -x 200" =158800, and yet in both cases they have accomplished the same work. An instrument giving the product of efforts, by the times or periods of duration, would by no means lead to an exact appreciation of the mechanical effects produced. The true measure of these effects is, as we have said, the product of the effort exerted by the path described in it8 direction. TRANSMISSION OF MOTION BY FORCES. 79 72. Important observation. We should here observe, that it is only in the case of a constant effort acting during a time T=l f/ that we can take the product MY for the measure of exerted effort F, and then we have F=MV=? V, or F : P : : Y : g, y a proposition resulting directly from the general principle enunciated in No. 61. But in the case of variable efforts, the same mode of measurement does not apply for finite times, for forces varying according to very different laws may in the same time impart equal quantities of motion to the same body or to different bodies. The formula F=MV will only give then the value of a mean con- stant effort capable of imparting in the same time the same quantity of motion. OBSEKVATION OF THE LAWS OF MOTION. 73. Determination of the intensity of forces ty observ- ing the laws of the motions they produce. The formula shows that if by observation of the laws of motion we ft\ know for each instant the value of the ratio -, we shall t then have that of the corresponding effort F. If, for exam- ple, we know, by experiment, that the motion is uniformly accelerated, we have whence Vi . v , = ss consequently If, for instance, a wagon weighing 2205 pounds runs with a uniformly accelerated motion a distance of 32.8 ft - in 2", we have - 2205 2x32.8 OBSERVATION OF THE LAWS OF MOTION- 81 . for the value of the force capable of imparting this accel- erated motion, deduction being made for friction. 74. Means employed for determining the laws of motion of bodies. According to the nature of the case, we make use of different contrivances or instruments for observing the laws of motion of bodies. For a slow motion, we employ watches, pendulums, and note the time corre- sponding with the given spaces. But for rapid motion these methods do not afford the requisite precision. 75. Colonel Beaufoy's apparatus. (Fig. 30.) This v U ;:, El m FIG. 30. experimenter, in his researches upon the resistance of water, was provided with a pendulum which traced at each oscillation a mark upon a rule, whose motion was in a known ratio with that to be observed, and as in his ex- periments the motion soon became uniform, the velocity of the motion was easily obtained. 76. Eytelweirfs apparatus. This learned engineer seems to have been the first to entertain the idea of com- bining a known uniform motion with that to be deter- mined, so as to obtain a trace of simultaneous motions, from which might be deduced the condition of the un- known motion. For his experiments upon the hydraulic ram, he used an endless band of paper, (Figs. 31 and 32,) rolled upon two cylinders, to which a regular motion was imparted OBSERVATION OF THE LAWS OF MOTION. by the hand. The lengths of the paper passed were then nearly proportional to the time. A style fixed to the J_L II , r^ =L L Fig. 31. stem of the valves traced upon this band a curve whose ordinates were the spaces described. M. Eytelwein could, by means of this imperfect con- trivance, nearly determine the intervals of time between the opening and shutting of the valves. But we know that the motion imparted by the hand could not have been uniform, and that this disposition could not give very exact results. 77. New apparatus. For experiments made at Metz, upon friction, and for other researches, at M. Poncelet's suggestion, I made use of a combination of a known mo- tion with one whose law was to be determined. I have since modified the arrangement of this machine, and that deposited in the repository of arts and machines was made in the following manner, (Figs. 33 and 34 :) a plate 1.05 ft - in diameter, perfectly plane, receives a uniform motion by means of a weight hung to a first axle. The motion of this axle is transmitted to a second axle by a wheel and pinion, whose number of teeth are to each other as 6 : 1. A wheel mounted upon the second axle drives a second pinion fixed upon the axle of the plate. This wheel and its pinion have also their number of teeth OBSERVATION OF THE LAWS OF MOTION. 83 in the ratio of 6 : 1, so that the plate makes 36 turns for one of the first axle. Elevation. FIG. 38. Fio. 34. Upon the axle of the plate is a fly-wheel with 4 wings, 0.33 ft - at the sides, which serves to regulate the motion by the resistance of the air, which is, as we know, nearly proportional to the square of the velocity. It follows from this arrangement, that in a short time the motion of the plate becomes uniform, its centre of gravity, as well as that of the other pieces, being upon the axis of rotation. This uniformity may be readily ascertained by taking the number of turns of the second axle, as indicated by the pointer. Just opposite the plate, and parallel to its surface, is a pulley, whose motion is in a known ratio with that to be observed. The axle of this pulley bears a small arm, upon which is placed a style, formed usually of a brush filled with India ink. By simple means we test the parallelism of the circle described by the point of the style with the surface of the OBSERVATION OF THE LAWS OF MOTION. plate, upon which is fastened a sheet of paper. The style may, before the experiment, be kept at a short distance from the paper, and be brought in contact with it the in- stant the motion begins. We easily conceive, after this description,* that the plate turning with a uniform motion, and the style with an unknown, there follows from these simultaneous mo- tions a trace left upon the paper, which, depending upon the synchronous motion, should give in the tabulations the relation of angles described by the pulley, or of the spaces described by the observed body, with the angles described by the plate, or their corresponding times. This readily appears in observing that, if the plate were at rest, the style would de- scribe upon its surface a circle with a radius equal to its distance from the axis A of the pulley. While, on the other hand, if Jhe style were at rest, and the plate in motion, the latter would have for the trace of its contact with the brush, a circle whose radius is that of the plate, and its radius, the distance of the style from its centre. This granted, let o be the origin of the curve traced during the experiment. Through this point describe a circle with a radius Ao, equal to the distance of the style from the axis of the pulley, with its centre at a known distance Ac from that of the plate, and divide this circle into ten equal parts at the points 0, 1, 2, 3 ... 9. FIG. 86. * For further details, see the description " des Appareils chronometriquesj inserted in the journal of the scientific convention held at Metz in 1837. OBSERVATION OF THE LAWS OF MOTION. 85 Through each of these points we draw circumferences of circles, with their centres at c, radii CO, 01, 02, &c. These circles will cut the curve in the points 1', 2', 3', &c. ]STow, it is evident that the point V results from the simultaneous motions of the style from to 1, and of the plate describing the angle 101' ; consequently, the arc 01 gives the angle described by the pulley, or the space de- scribed by the body, and the angle 101' furnishes the measure of the corresponding time. We may then suc- cessively observe these spaces, and from them make a table representing the law of motion. Taking, then, the spaces described for the abscissae, and the times for the ordinates, we shall have a curve with rectangular co-ordinates, the nature of which must be studied to derive the law of the observed motion. If, for example, the abscissae or the spaces described are proportional to the squares of the times, or the ordi- nates, the curve will be a parabola, and the motion will be uniformly accelerated. If the curve, either at its ori- gin, or after a certain time, should change into a straight line, from that instant the motion will be uniform. When the motion is very rapid, for which styles charged with ink are not suitable, we may use metallic styles to trace upon soft materials, such as wax mixed with tallow. Thus we may easily determine the law of motion in the cock of a musket, though this motion is made in nearly T ^ F of a second. We begin by tracing upon the plate at rest the arc of the circle described by the style fixed in the head of the cock, and formed of a light steel pointer, which serves to determine upon the plate a circle of a radius CA, upon which is projected at A the centre of the tumbler. This done, we set the plate in motion, and when by direct ob- servation we have obtained the uniform velocity of its motion, we let go the pointer and the cock, and obtain a curve 0, 1, 2, 3, 4. The origin, o of this curve can be nearly determined at first sight by examining its point of 86 OBSERVATION OF THE LAWS OF MOTION. tangency with the arc of the circle traced by the style before the starting of the cock. We trace the circle de- scribed by the point and passing through the origin o. This arc terminates at the circumference which the style had traced when the cock was arrested. "We divide it into any number of parts, or rather start- ing from the point 0, we take arcs 01, 02, equal to a given number of degrees, and corresponding consequently to known angles described by the cock. FIG. 80. Then from the point C as a centre, and with radii Cl, C2, C3, etc., we describe arcs of circles, meeting the curve in 1', 2', 3', etc. Finally, the angles 101', 2C2', 3C3', give the corresponding times. . We may then compare the an- gles described by the cock with the time employed. We find thus, for example, for the infantry percussion mus- kets, modelled in 1840, the following results : Arcs described by the centre of the coun- tersink Corresponding ft. 0.01T38 sec. 00406 ft. 0.0347 sec. 00604 ft. 0.0525 sec. 000754 ft. 0.0698 sec. * OOS86 ft. 0.0741 sec. 000936 ft. 0.1046 sec. 01050 ft. 0.1230 sec. 01126 ft. 0.1397 sec. 01190 ft. 0.1571 sec. 01372 ft. 0.1706 sec. 01318 Ratios of the squares of the time to the arcs described. 0.00094 0.00105 0.00108 0.00112 0.00118 0.00105 0.00103 0.00101 0.00120 0.00103 ** Mean. .0.00107 * In ray edition it is written 0.08864 probably a misprint. TRANSLATOR. ** In the last column of ratios, Morin has apparently made some error ; the mean, as he gives it, for metres, is .00341. It should be .00351. OBSERVATION OF THE LAWS OF MOTION. 87 ^Repeating the experiment thrice, a mean ratio was found rp r> | =.0010759=^-, and the total arc being 0.1706 ft -, we find T=0.01355", and so __ _ .0010759' and finally for the velocity of the style. This style was 0.2001 ft - from the axis of the tumbler, while the centre of the counter- sink was but 0.1975 ft -, consequently the velocity of the cock at this centre was r x25.1S ft - in 1 second. 0.2001 We see by this example that we can determine a great many points of the curve which represent the laws of motion, and as the ratio of the spaces described to the squares of the time is constant, it follows that this curve must be a parabola, or that the motion of the cock is uni- formly accelerated. Thus the force which produced it is constant, and the form of the curve of the tumbler, as well as the resistance of the spring to flexure, are so com- bined that the effort of the thumb to cock the gun is con- stant. This shows how the art of the mechanic may sometimes prove the solution of quite difficult mechanical problems. We may extend the use of these instruments to the observation of still more rapid motions ; for, in the ex periments upon the cock of the gun, the plate only made 6 turns in 1", and we could easily obtain 10. Then the circumference of this plate being 3.281"-, would run 88 OBSERVATION OF THE LAWS OF MOTION. 32.81 ft - in 1", and as by the instrument for observing curves we can appreciate 0.00065"-, we may then obtain the time to nearly ^7^^: of a second. "We might go still further by increasing the dimensions of the plate. If we should furthermore combine the movement of the plate with electricity, we might perhaps determine the duration of certain phenomena so rapid in their dura- tion as thus far to have eluded all our attempts at meas- urement. Among these, for example, would be the law of mo- tion of projectiles in the air, for the solution of which attempts have already been made, but with small success, since instead of an apparatus with continuous motion, we have used chronometric instruments, with an oscillating or intermittent motion. 78. Zinc plates. The sheet of paper which usually receives the traces of the style is wet and pasted at its edges upon a zinc plate, which is fastened to the plate. This avoids the inconvenience of unequal shrinkage of the paper, and greatly facilitates the tabulations. 79. Contrivance for tabulating the curves. The sheet of zinc, when taken from the plate, being exactly centred, is placed upon an instrument for making abstracts of the curve. The circumference of this instrument is divided into a thousand parts. An alidade movable around its axis, bears a disc provided with ten pointers, whose ex- tremities divide into ten equal parts the circumference described by the style upon the plate, while at rest. We loosen the thumb-screw which fastens this disc, and then turn it round so that one of these points shall correspond with the origin of the curve. This done, we fasten the disc firmly to the alidade with the thumb-screw. Since, then, it is evident that each of these ten points, in the OBSERVATION OF THE LAWS OF MOTION. 89 movement of the alidade, will describe the auxiliary circle 11', 22', 33', 99', etc., by turning the alidade so that all the points may successively meet the curve, and reading the angles described by the alidade corresponding to each point, we shall have the times which are proportional to these angles, and the angles described by the pulley of the style, by the number in order of the points. This instrument, for which we are indebted to M. Didion, Captain of Artillery, combines great precision and utility in all operations of this kind. These contrivances just described and constructed by me, are the first of this kind, and are conveniently used in various experiments, but the results require an elabo- rate abstract, and do not address the eye quickly enough for the purposes of instruction. It has for some time seemed desirable to apply the principle of this construc- tion in a more simple form. It was thus that I was led to the construction of the following apparatus : 80. Description of a chronometric apparatus with cyl- inder and style, for observing the laws of motion. The principal piece of this apparatus, in the model adopted for the Lyceums by the minister of public instruction, is a vertical cylinder AB, (fig. 37), 6.89 ft - high, covered with common white paper slightly wet, and pasted at its edges. This cylinder is 0.41 ft - in diameter, answering to a circum- ference of 1.2S6 ft> . It rests upon a pivot, and is put in motion by means of a contrivance similar to a kitchen- jack. A weight is suspended to a cord rolled round the surface of a drum C. At one end of this drum is a wheel D with teeth, inclined 45, which drives an endless screw, whose vertical axis carries at its upper end a fly-wheel with wings, which serves to regulate the motion by means of the resistance of the air. We may incline these wings to increase or diminish the regulating action of the air, 90 OBSERVATION OF THE LAWS OF MOTION. and as we may also vary the motive weight, we may ob- tain a uniform motion of the cylinder at a velocity of one turn per second, or a more rapid motion if desired. The motion being regulat- ed gradually, it is well not to make the projected ex- periment before the motive weight has run through from | to | of its fall; there remains more than time enough for ordinary cases. There is at the Con- servatory of Arts and Ma- chines a much larger and more complete model of this instrument, whose cyl- inder is 10.17 ft - high and 3.2S ft - in circumference. If the circumference of the upper and lower base of the cylinder is di- vided into 100 equal parts, each of these points wilt correspond to - f i of a revolution or of a second, and as each of these may be divided into 10 parts of 0.032 f % we see that each of them aifords the means of measuring the time with the precision of T i- - or T oV o of a second, and even less. Here, then, we have FIG. ST. a very delicate chronometer. The division of which we OBSERVATION OF THE LAWS OF MOTION. 91 have spoken is easily effected, by means of a wooden rule placed upon a post parallel to the edge of the cylinder and near its surface. The base of the cylinder has a cir- cle, with 100 equidistant ratchet teeth, in each of which falls a catch to fasten the cylinder while the generatrix is traced with the rule. The rule is also divided at spaces of 0.16 ft - into notches, in which is placed a crayon, which is held fast while the cylinder is turned, so as to trace parallels to the base, whose developments give the ordi- nates of the curve of motion. Now, suppose a body M, hung near the summit of the cylinder upon a bent lever ab, is left to fall to the earth, guided vertically, by means of two metallic threads, well stretched by the screws v v' parallel to the edge of the cylinder, and that the body carries a style, formed of a brush soaked in ink, or rather of a crayon pencil pressed against the surface of the cylinder by a spring, we see that if during the descent of the body the cylinder is at rest, the style will leave a trace of the generatrix of the cylinder, or of a straight line. But as the cylinder moves at the same time the weight falls, the style traces upon the sheet of paper a curved line depending upon the two simultaneous motions. 81. Discussion of the results furnished ~by this appa- ratus. When the curve has been obtained, and the gen- eratrix of the cylinder has been traced, corresponding to the origin, it is easy to recognize its nature, and to prove that the motion of the style which traced it, and conse- quently that of the body which bore the style, was uni- formly accelerated. Indeed, if we cut the paper and take it from the cyl- inder, the generatrix drawn through the origin of the curve may be taken for the axis of the abscissae, and the lengths upon this line, measured from the origin, will be of the same magnitude as the spaces described by the 92 OBSEKVATION OF THE LAWS OF MOTION. body in its fall. The ordinates of this curve will be the developments of so many arcs of the circle of the cylin- drical surface, and each millimetre =0.003 ft - of these or- dinates will represent a given fraction of the second. We shall then have obtained with this contrivance a curve whose abscissae are the spaces described by the body, and its ordinates are the corresponding times. Now, comparing directly the spaces described with the times measured upon the curve, we see at once that the first arc is in a constant ratio with the squares of the seconds, which shows that the motion of descent of the body was uniformly accelerated. By a simple graphic construction, which consists in drawing at sight with a ruler a series of tangents to the curve, to determine the point where they cut the axis OY of ordinates, and in raising at these points perpendiculars to each of these tangents, we find that all these perpen- diculars intersect in the same point, a property of the parabola : a curve whose abscissae are in a constant ratio with the square of its ordinates. The point thus determined is the focus of the parabola, and furnishes the true position of the axis of the abscissae and of the origin of the curve, positions thus far supposed to be determined by the eye, and naturally attended with some uncertainty. 82. Determination of the velocity. Since we have determined, directly, and by the instrument, the curve representing the law of motion uniformly accelerated, which we have recognized as a parabola, we shall have the velocities of motion changing at each instant, by the inclination of the tangents of this curve with its ordinates. Now, this inclination in the parabola is equal to double 2S the abscissae S divided by the time T, or to -^ ; we have T7 2S , Y 2S then Y^-^, whence = , and as, by comparison of OBSERVATION OF THE LAWS OF MOTION. 93 the abscissae S with the squares of the'ordinates T, we have S Y found that ^ is constant, we see also that the ratio - is constant, which shows that, agreeably to the definition of uniformly variable motion, the velocities here are propor- tional to the times. If we seek the velocity Y x ac- quired at the end of the first second of the fall, we find ,=28^ S x being the space described after the first second, or the abscissae, cor- responding to the ordinate T=l". We see, then, that this new appa- ratus enables us to determine directly by observation all the circumstances Fia 8g of the descent of bodies falling freely . to the surface of the earth, to prove that this motion is uniformly accelerated, and to obtain also, with exactness, the value of the velocity acquired by bodies in the first second of their fall, which at Paris is equal nearly to 32.1817 ft -* 83. Experimental demonstration of the principle of the proportionality of forces to the velocities. This prin- ciple which we have admitted, in Art. 61, as the resultant of all. the observed phenomena, and of which, till the present, no direct demonstration has been given, may be easily verified, by means of an apparatus made jointly by M. Tresca and myself, for the course of Mechanics of the Institute. Imagine a movable weight P, subject to the action of gravity, to be connected by a thread with another movable weight p, free to move upon a horizontal plate, placed firmly near the apparatus. The action of gravity upon the second body will be destroyed by the plate, and * The models of this cnronometric apparatus, made for the Lyceums, came from the workshops of M. Glair, machinist, rue du Cherche-Midi, 93. 94: OBSERVATION OF THE LAWS OF MOTION. when the body P descends, it is solely by the action of gravity upon this body that the two bodies are put in motion, and so from the force P alone, that motion is ini- parted to the mass - . In these conditions, the fall g will take place with a uniformly accelerated motion, but evidently less rapid than when the body P is entirely free. The curve traced by the style is a parabola, more open than in the case of a free fall, and serves to demon- strate, as we have before observed, the velocity Y of a system, at the end of any time, a second, for example. The continuous action of a force P imparts, then, in a second, to the mass --- |r a velocity Y measured experi- mentally. P'-fl/ We may also observe the velocity V of a mass -- =- j/ in similar circumstances, by substituting a weight P' for the weight P, and for the body drawn a weighty/ instead of p. Now, in this twofold substitution we may make P'+2/, in which case the mass put in motion in P+z? P'-h0' both cases is identical : -- being equal to S- ; we t/ l t^QOOOOOOOOOOOOOO OO COO VAIA iOCOOCO'* i i , whose decreasing lengths were regulated as follows : 3.55 ft - 2.T3 ft - 2.12 ft - 1.62 ft - 1.23 ft - O.S9 ft - 0.61 ft - 0.36 ft - 0.2S ft - 0.22 ft -; which corresponded to the numbers of calibres respect- ively equal to 64 49 38 29 22 16 11 7 5 4 calibres. VIS VIVA. 109 The charges of 4.5 drachms of powder and of 1.69 drachms of pyroxile, had from previous experiments been adopted as nearly equivalent, but it was found in the course of the experiments that 1.6 drachms of pyroxile sufficed to impress upon the same ball of 0.0635 lbs> weight a veloc- ity of 1235. 9 ft> , equal to that imparted by 4.5 drachms of powder. Future comparisons will be based upon these charges. The velocities imparted to balls were measured with the ballistic pendulum, by placing the gun-barrel upon a frame, so that the face of the muzzle was 6.56 ft> from the ballistic receiver. Bearing in mind that the vis viva imparted to the ball is, by the principle of vis viva, equal to double the quan- tity of work developed by the gas, and that the mean effort of this gas, or the constant effort which would in each case impress the ball with the same vis viva, is equal to half of the vis viva divided by the length of path de- scribed by the projectile in the barrel, we see that from observation of the velocity of the latter, which is termed the initial velocity, we may easily deduce the value of the mean effort. It is also evident, that the value so determined will always be below the maximum effort, and will decrease with the. length of the barrel: so that the conclusions, from a comparison of the mean efforts of the gas of pow- der and of pyroxile, will approach more nearly the truth, as the path described by the projectile in the barrel shall be less, and will very nearly approach the truth, in the first moments of its displacement, which are precisely those in which the efforts should be studied. The length of the chamber occupied by the charge was the same for the powder as for the gun-cotton, and was 0.157 ft< ; and subtracting this from the interior length of the barrel, we have the space described by the hind part of the ball in the barrel, and dividing the half of the 110 VIS VIVA. imparted vis viva by this length, we obtain the mean effort sought. It is proper to remark that this estimate of the space described by the ball, while subject to the action of the gas, is that usually adopted in calculations of this kind, but it is not wholly exact. In fact, when the centre of the ball has passed the face of the muzzle, a portion of the gas escapes around it ; still, these gases issuing with great velocity, their impulse is partly continued outwards. However this may be, the value abov adopted for the space described by the ball under the action of the gas is too great rather than too small ; consequently, the mean effects deduced from it are too small, and our conclusions err on the safe side. We have represented the results of experiments and of calculations in two different ways. In the first (Fig. 40) we have taken the lengths of the barrel described by the ball for abscissae, and the vis viva for ordinates ; in the second (Fig. 41) we have also taken the lengths of the barrel described by the ball for the abscissa, and for ordi- nates the corresponding mean efforts of the gas of powder and of pyroxile. We thus have a graphic expression of results contained in the following table : Results of comparative experiments upon the velocities, the vis viva, and the mean efforts developed by the gas of war-powder and that of pyroxile. Lengths of barrel. Velocities imparted. Vis Viva imparted. Mean efforts exerted. Total Described By 4.5 drms. By 1.61 drms By By By gas By gas by the ball. of powder. of pyroxile. powder. pyroxile. of powder. of pyroxile. a ft. ft. ft. Ibg. ft. IbB. ft. Ibs. Ibs. 3.55 3.39 1235.9 1235.5 3015.1 3013. 447.4 444.3 2.73 2.57 1234.2 1270.7 3006.5 3187.3 584.9 620.1 2.11 1.96 1146.7 1245.4 2595.6 3061.7 662.1 781. 1.61 1.46 1039.6 1176.2 2133.2 2730.8 730.6 935.2 1.22 1.06 938.6 1182.3 1788.6 2759.2 | 820. 1301. 0.89 0.73 856.9 1348.6 1449.5 2264.9 992. 1541. 0.61 0.45 724.9 965.8 1037.2 1841.1 1152. 2045. 0.39 0.28 530.4 821.9 555.2 1333.6 1207. 2599. 0.27 0.12 378.2 577.3 282.3 657.6 1176. 2740. 0.22 0.06 293.1 341.2 169.5 302.7 1412. '2521. VIS VIVA. Ill An examination of Fig. 40 shows : 1st. That for powder, the vis viva and consequently the velocity of the ball was not sensibly increased beyond A length of barrel 2.62 f % answering to 49 calibres. 2d. That with pyroxile, the vis viva and maximum velocity seems to correspond with the same length, and to decrease with greater lengths. 3d. That finally, the v-is viva imparted by charges of 4.5 drachms of powder, and of 1.6 drachms of pyroxile, are equal, for lengths of 3.55 ft - or 64 calibres, but that for greater lengths the pyroxile lost the advantage it pos- sessed in shorter lengths. Fig. 41 shows that starting with a length of 3.55 ft> , for which the charge of 4.5 drachms of powder and '1.6 drachms of pyroxile have given the same vis viva, and so the same mean effects, the effort exerted by the gas of pyroxile prevails always over that of powder, in the pro- portion of the diminution of the length, and that for small lengths of barrel, or in the first displacements of the pro- jectile, the mean maximum tension of the gas seems to correspond with the instant of 0.246 ft - displacement, and 9ftfi9 4- was then 2862.4 lbs - or ^^=1044821 lbs -* per sq. ft, 1044821 or finally, 493.4 atmospheres, while the mean -2116.4 maximum tension of the gas of powder was not over 1 9QO S 1290.8 lbs - or -f^=471170 lbs - per foot square, or .002 1 o9 222.6 atmospheres, in taking even its value an- swering to the smallest length, which seems to depart somewhat from the law followed by the other lengths. It follows, therefore, that the mean maximum pressure produced by the gas of powder will not attain the half of ft. 05905 2 * The surface of the great circle of the ball is ' =.002739 >. ft. 112 VIS VIVA. that produced by the gas of pyroxile, for charges pro- ducing the same ballistic effect. The dimensions of infantry muskets are such, that when the projectile is displaced 0.246 ft - it is found in a part of the barrel having a thickness T 0.017716 ft> , and it is readily seen, from the formula of the resistance of a cylinder to rupture, supposing the rnetal to be of a me- dium quality, that the interior pressure capable of pro- ducing rupture will be for the unit of surface 2T E .03543 8195580 and when the metal is impaired by the firing, or is of an inferior quality, it is .03543 5122210 Thus, in th'e last case, the maximum pressure of the 222 5 1 gas of the powder would only be ' = - of the mod- ulus of rupture, while that of pyroxile would be 493.4 1 1451.4 2.94 If we refer to the comparative results previously re- ported upon the vis viva imparted by explosive materials, according to which we have seen that the charge of pyroxile was to the equivalent charge of fine sporting powder as 72.83 : 159.25, we see that the charge of pyrox- ile equivalent to that of 15.5 drachms of sporting powder used in the tests of guns, would be 7 drachms. ISTow it sometimes happens that guns burst with a charge of 15.5 drachms of sporting powder, and since, with the same ballistic effect, the pyroxile develops, at the first instant, much greater pressure than the powder, it would seem to VIS VIVA. 113 follow that guns could not generally resist a charge of 6.77 drachms of pyroxile. Without attaching an undue amount of precision to these calculations, we may yet have in them a confirma- tion of the fears first entertained as to the effects of the rapid combustion of pyroxile. Later experiments have confirmed these deductions, and when it was wished to determine the velocities im- parted by increased charges, it was constantly the case that new guns burst with charges of 3.95 to 4.23 drachms, and sometimes with less. If we bear in mind that infantry guns are made of a choice quality of iron, submitted to a close inspection and even severe tests before reception, and that the thickness of the metal is much greater than that of fowling-pieces, we cannot doubt that the use of pyroxile in portable fire- arms is far from affording the same security as powder. 93. Consumption and restoration of work ly inertia. It follows, from the above, that to impart to a body a certain velocity, answering to a certain vis viva, a quan- tity of work must be developed which is expressed by the half of the vis viva, and reciprocally, if the body lose a part or the whole of its vis viva, a work is developed in virtue of its inertia, which is also expressed by half of the vis viva destroyed. In the first case the motive work is transformed into an imparted vis viva, and in the second the vis viva is transformed into a resistant work. 94. Rams for pile-driving, punching machines, c&c. In driving piles, the work employed to raise to a height H, a ram of the weight P, is transformed in its descent P Y 2 into a vis viva - PH ; for when the ram reaches the 9 * head of the pile, it develops by its inertia efforts which 8 114 VIS VIVA. compress the head, overcoming its resistance, sinking, and so producing a corresponding useful work. In boring and punching metals with a ram, the resist- ance overcome is that opposed by the metal to the sepa- ration of its molecules, and the thickness of the piece is the space described. The example already cited, of the action of powder upon balls, shows us, at first, the work transformed into a vis viva, then, during the penetration of the balls into any medium, the vis viva is employed in overcoming the resistance of the medium. 96. Work expended during the period of compression from the shock of two non-elastic lodies. Calling M and Y the mass and velocity of the impinging body, and M' and Y' the mass and velocity of the body shocked, we would remark that the total vis viva of these two bodies before the shock was MY 2 +M'Y /2 , and that after the shock, since they move with a common velocity, TT _MY+M'Y' "M+M 7 "' their vis viva will be (M+M')!! 2 . Consequently, the vis viva destroyed during the compression, and employed in producing it will be and the work consumed by this compression is MW_ fA If the body shocked had before the shock a motion against the impinging body, we have seen when the body VIS VIVA. 115 M' after the shock recedes, and takes the direction of the body M, that we have for the common velocity after the shock TT _MV-M / Y / M+M 7 "' and then the loss of vis viva producing the compression is and the work consumed by this compression is If, after the shock in the last case, the velocity II were zero, which happens when MY^M'Y', the work con- sumed by the compression is reduced to ^(MY'+M'Y' 2 ), which is quite evident, since the two bodies are brought to rest by the shock. If the mass of the impinging body is very great com- pared with that of the body shocked, the loss of work M jyr/ is reduced by reason of the smallness of the ratio -^ to iM'(V=FV')*; in the first case iM'(Y-Y') 2 is the work answering to the vis viva gained by the body shocked, and in the second M'(Y-f Y') 2 is the work answering to the vis viva, due to the sum of the velocity Y' which the body shocked has lost in one direction, and of Y which it has received in an opposite direction, because then U is reduced to - MV-M'V'_ _M__ V M+M' M' ' 116 vis VIVA. in consideration of the smallness of in its ratio with M unity. If the body shocked were at rest before the shock, we have V=0, and the loss of work due to the compression is so long as the velocity of the impinging body is not sen- sibly altered, and U=V as above. If, on the other hand, the mass of the body shocked is very great compared with that of the impinging body, we have for the loss of work relative to the first case, where the bodies move in the same direction, on account of the smallness of the ratio =-=. M 97. Of the work due to compression, and the return to the primitive form in the case of elastic bodies. If the bodies are perfectly elastic in their return to the primi- tive form, the molecular springs must develop the same efforts in returning to the same degrees of tension, and the points of application describing the same paths as in the compression, the total work developed by these efforts, varying in the same manner in both cases, will be the same, and the work due to the unbending of the molecular springs will be the same as the work consumed in their compression ; so that in reality the consumption of the work due to the shock is nothing. 98. The work lost in the shock of bodies imperfectly elastic. If the bodies are imperfectly elastic, as is gener- ally the case ; or rather, if the flexures and changes they experience during the shock exceed the limits of those VIS VIVA. 117 which can be produced without an alteration of the elas- ticity, then the parts shocked remain more or less changed in form, and only a part of the work consumed in pro- ducing it is restored. There is then a loss of work. Now, in machines imparting shocks it nearly always happens, either in their first use, or after a period of ser- vice, that the elasticity of the parts in contact is more or less changed, and that the loss of work by the shock is very nearly the same as that which takes place in the shock of soft bodies. This last quantity is, moreover, the greatest limit which this loss can attain. In recapitulating, we see that in shocks there is in practice always a more or less great loss of work, due to the disturbance of the parts in contact, and that it is well to substitute, as far as possible, parts with a continuous motion, for those working with shocks, intermissions, or sudden changes of velocity. 99. Masses in motion may be regarded as reservoirs of work. It follows, from the preceding remarks, that bodies, in virtue of their inertia, absorb and store up mechanical work, when the forces are employed in communicating to them velocity and vis viva, and on the contrary, trans- mit and restore the work when their motion is retarded. In this view we may regard them as reservoirs of mechan- ical work, which are filled during acceleration, and emp- tied in the retardation, absolutely in the same manner as the reservoirs of hydraulic motors. 100. EXAMPLE. We have already seen, in Art. 94, that it was in virtue of the work thus stored up that the pile- driving rams produced their effects ; it will be the same whatever the number of intermediate parts of the ma- chine : we find a striking example of a similar application in the walking-beam employed in many mechanical de- partments. 118 VIS VIVA. If the fly-wheel of a machine is set in motion with a certain velocity, by any motive force, and then left to itself, it will continue to move until the frictions and other resistances have entirely expended the work which was accumulated under the action of the motor when this work is consumed it will stop. But if, while animated with a certain velocity testify- ing to the accumulation of a certain work, we oppose to the machine a useful resistance, we see that it then devel- ops a useful mechanical action, such, for example, as the coining of money, the stamping a metal plate into a given form, the piercing of holes, &c. 101. Periodical motion. If the motion of the body varies periodically, that is to say, if its velocity increases or decreases successively in equal quantities, it is evident that the work consumed in the period of acceleration is equal to the resistant work during retardation, and that then the total work developed by inertia is nothing. If we watch what passes in these successive periods, where the velocity and vis viva become without ceasing the same, at the end of each period, it is not necessary to take any account of the vis viva. We shall see hereafter the great importance of inertia in the work of machines. COMPOSITION OF MOTIONS, VELOCITIES, AND FOKCES. 102. Composition and resolution of simultaneous mo- tions. We have thus far considered material points as animated by a single motion, or solicited by a single force, and before extending the preceding theorems, it would be well to examine what passes when a body or material point is impressed simultaneously with many motions, or solicited by many forces. Observation affords frequent examples of its occur- rence. Thus, when a traveller promenades the deck of a steamboat under way, he is impressed with the motion of the vessel as well as that of his own will. If, while walking, he throws a body from him, this body already partaking of the two motions of the traveller, takes a third in falling upon the deck ; besides, the vessel partakes of the daily motion of rotation of the earth, and also of its motion about the sun. All these motions are simultaneous and are independ- ent of each other, since the causes which produce them are. By a very simple experiment of M. Tresca, sub-director of the Conservatoire, this independence of simultaneous motion is made very apparent. In placing the chronom- etric cylindrical apparatus, described in No. 80, upon a truck impressed with a uniform, or even variable motion, and in repeating during this motion the experiment of the 120 COMPOSITION OF MOTIONS, VELOCITIES, AND FORCES. fall of bodies, left to the action of gravity, it was seen that the parabola traced by the style was exactly the same as was obtained when the apparatus was immovable. The vertical motion of the heavy body, and the rotary motion of the cylinder, were wholly independent of the motion of the apparatus. From this principle of the independence of simultane- ous motions follow rules which enable us to determine the real motion resulting from many simultaneous known motions, and which is called the resultant motion. 103. Case of the simultaneous motions having the same direction. The first and most self-evident case, is that of a material point impressed with simultaneous motions acting in the same straight line. They are added if in the same direction, and subtracted if in opposite direc- tions, in order to obtain the resultant. In the case of a person walking upon the deck of a vessel, it is evident that if he walks in the direction the vessel is going, his motion and displacement in respect to a fixed point on the shore, supposed to be parallel to these motions, will be equal to the sum of the displacements of the vessel and of his path described upon the deck. If the boat has advanced 26 feet while the traveller has passed 9 feet forward, the displacement of the traveller in respect to the banks will be 26 ft -+9 ft 35 ft - If he walked 15 feet in an opposite direction to the boat, while the boat advances 26 feet, his displacement, or the space described by him in respect to the banks, will then be 26 ft --15 ft -=ll ft - If the traveller walks towards the stern of the vessel a distance equal to that which the latter has advanced in the same time, his displacement in respect to the banks is nothing, and though impressed with two simultaneous motions, he is at rest in respect to the banks. COMPOSITION OF MOTIONS, VELOCITIES, AND FORCES. 121 Finally, if the traveller walks towards the stern a dis- tance of 30 feet, or 4 feet greater than that described by the boat, his displacement in respect to the banks will be negative, an expression indicating that he has receded in- stead of advancing in respect to the banks. It would be the same for any number of simultaneous motions directed in the same straight line ; calling : S, S', S", &c., the paths directed from left to right, and regarding them as positive : Si, S/, S/', &c., the paths directed from right to left, and regarding them as negative or subtractive, the total resultant path of these simultaneous motions will be equal to ,-^-^/ S/' &c., which is expressed in saying that the resultant path is the algebraic sum of all the simultaneous or component paths : understanding here by the word sum the result of the operation of adding all the paths from left to right, and subtracting all those from right to left. 104. Composition of several simultaneous velocities di- rected in the same straight line. All that we have said upon the composition of spaces simultaneously described by a material point in the same direction, applies to the simultaneous velocities impressed upon a point, since in uniform motion the velocities are proportional to the spaces described in the same time, and since in variable motions the velocities are those of uniform motions which the bodies would possess if these motions ceased to be variable. 105. Composition of two motions directed in any man- ner. Let us consider now the point A, which may be the point of a pencil placed upon the inclined rulers MA1ST. If the rulers are moved uniformly a quantity AB, its side AM will be displaced parallel to itself the same quantity, 122 COMPOSITION OF MOTIONS, VELOCITIES, AND FORCES. moving also uniformly, and with it the point of the pen- cil to which it is attached. But if in the same time T, the pencil moves upon the side AM, uniformly a quan- tity AD, it is easy to see that at the end of the time T, the point of the pencil will have arrived at the point C, and at the summit of the parallelo- gram constructed upon AB and AD as sides. In fact, this point constantly resting upon the side AD being displaced with it parallel to its primitive position a quantity equal to AB, would be found upon the line BC, parallel to AD, and as it is also displaced in the direction AM, by the quantity AD, it would likewise be found upon the line DC, drawn parallel to AB. The in- tersection of the two lines BC and DC determines the direction of the diagonal of the parallelogram constructed upon the simultaneous paths. Whence it follows that, when a material point is ani- mated ~by two simultaneous motions in two given direc- tions, the position of the point at the end of these two motions will be at the extremity of the diagonal of the parallelogram constructed upon the two paths as sides. The distance AC, at which the point is found from its first position A, is called the resultant path, and the two simultaneous paths AB and AD are called the component paths or the relative paths passed over in the direction of the lines AN and AM. For any two other but also simultaneous paths AB' and AD' passed over by the point A, in another time T', the point A will arrive at a position A' determined by the extremity of the diagonal AC' of the parallelogram con- structed upon the paths AB 7 and AD'. !N~ow, as these second simultaneous paths are by hy- 123 pothesis described with a uniform motion as well as the first, we have AB : AB' : : T : T and AD : AD' : : T : T, whence AB : AB' : : AD : AW. The angles at A being moreover equal, it follows that the triangles ABC and AB'C', ADC and AD'C' are simi- lar, and that the diagonals AC and AC' are in the same direction. Moreover, the diagonals AC and AC 7 are also propor- tional to the times T and T' employed in reaching the points C and C'. When a material point moves simultaneously and uni- formly in two given directions, the path really described in virtue of these two motions, and called the resultant path, is represented in magnitude and direction ~by the diagonal of a parallelogram constructed upon the two paths simultaneously described, and its motion in this direction is uniform with a velocity represented by the ra- ,. AC AC 7 /o/) ~f ~~ T'' The first proposition of No. 105 enables us to determine the position of the point in consequence of its two simul- taneous displacements, the second gives us its real motion. Reciprocally, when a material point moves in a right line AC, uniformly or not, we may always find its simul- taneous displacements, as referred to any two given direc- tions. It suffices for this to construct a parallelogram whose diagonal is AC, and whose sides AB and AC are parallel to the given directions. We see that a path or a given motion may then be resolved or decomposed in an infinity of ways into two others with given directions, so that the two paths or mo- tions shall answer to but one path or resultant motion. 124: COMPOSITION OF MOTIONS, VELOCITIES, AND FORCES. O We might demonstrate also that if the two simulta- neous motions of the point A in the directions AIST and AM are uniformly accelerated, the resultant motion along the diagonal AC will be so likewise. 106. Variable, motion. All that has been, said being independent of the absolute magnitude of the paths and velocities, will hold good for two infinitely small compo- nent paths. Thus in curvilinear motion, an element of the infinitely small path AC may be decomposed into two other infinitely small paths described parallel to any two given axes in the same plane; and reciprocally, if we know the relative elementary paths AB and AD described in an element of time in the direc- tion of the axes Ox and Oy, we may deduce from them the absolute elementary path AC described by the body. We would remark that this absolute elementary path AC is the element of a curve, whose prolongation gives the tangent AT, at the point A, and as its direction de- pends upon the ratio of the relative paths AB and AD, and not upon their magnitude, it follows, that if this ratio is known we can determine this diagonal or tangent by constructing upon the directions of AB and AD, a paral- lelogram whose sides have to each other the same ratio, and tracing its diagonal. This principle is often advan- tageously applied in tracing tangents of curves. Moreover, in variable motion, we see, if the ratio of the elementary path AC to the element of the time t em- ployed in describing it, is given, the construction of the parallelogram ABCD will give the ratios - , , or the t t FIG. 43. 125 relative paths AB and AD, described in the same time, which will be the values of the relative velocities in the direction of the axes, and that if the absolute velocity is proportional to AC, the relative velocities will be propor- tional to AB and AD. Then, in variable motion, the velocity at any instant can be decomposed into two others, in any two given direc- tions, and represented in magnitude l>y the sides of a paral- lelogram constructed upon this velocity and the diagonal. Reciprocally, the resultant velocity is the diagonal of a parallelogram constructed upon the relative velocities. 107. Case where the directions of the compo- nents are at right angles. In this case the paral- lelogram is a rectangle, the diagonal the hypothe- nuse of a right angled triangle, and its square is equal to the sum of the squares of the sides. We have, then, the simple relation : =V cos CAD. Y'=V.:|?=VcosCAB, 108. Composition of any number of simultaneous mo- tions or velocities in the same plane. We see by the pre- ceding that the path or resultant velocity of two simulta- neous motions in any two directions will be determined in constructing the triangle ABC, and drawing the side AC, and taking in the given directions AB and BC=AD respectively equal to the spaces, or the relative and simul- taneous velocities. If the body is also impressed with a third motion, or a third velocity AE, we construct the triangle ACF, in which AC is the motion or resultant 126 COMPOSITION OF MOTIONS, VELOCITIES, AND FOECES. velocity of the two preceding, and OF is equal and par- allel to AE. AF will consequently be the motion and resultant velocity jgr of the two simulta- neous motions AC and AE, or the three motions or velocities AB, AD and AE. So for a fourth motion or velocity AG, the motion or resultant velocity will be giv- en by the side AH of the triangle AFH in which AF is the preceding resultant, and FH is equal and parallel to AG. Then, in general, the motion or the resultant velocity of many simultaneous motions or velocities in the same plane will be given in magnitude and direction by the last side of the polygon ABCFH, &c., constructed from the origin A, with sides equal and parallel to the given simultaneous motions or velocities. If we project the last side of the polygon thus con- structed, upon any line, by perpendiculars or by parallel lines in any direction, a simple inspection of the figure shows that A'H'^A'B'+B'C'+C'F'-F'H', &c., which signifies that the projection of the last side, or the resultant path or velocity, is equal to the algebraic sum of the projections of the sides, or simultaneous paths orveloci- We understand here by the algebraic sum, the result obtained by adding or taking as positive, the sides, paths, or velocities, in the real direction of the motion, and by COMPOSITION OF MOTIONS, VELOCITIES, AND FOKCES. 127 subtracting or taking negatively the sides, paths or veloci- ties in the opposite direction. It follows from this, that if the last side is zero, and the polygon returns upon itself, the resultant path or velocity is zero, and the body is not displaced, and has no velocity, notwithstanding the relative motions imparted to it. It is also the case, when the algebraic sum of the paths or the velocities projected upon the same straight line is zero. 109. Resultant of three simultaneous motions or veloci- ties in space. If the body is impressed with three simul- taneous motions or velocities AB, AD, AF, in any three directions in space, it is evident that if we at first com- pound AB and AD, then their resultant AC with AF, or AB and AF, and their resultant AE with AD, or AD and AF and their resultant AG with AB, we shall find in all cases for the final resultant the diag- onal AH of the paral- lelopipedon constructed upon the given motions or velocities. Then, the resultant of three simultaneous motions or velocities in space, is represented in magnitude and direction lyy the diagonal of the parallelopipedon con- structed upon these three' FIG 4G motions. 110. Reciprocally, any motion or velocity may he de- composed into three motions or velocities according with 128 COMPOSITION OF MOTIONS, VELOCITIES, AND FORCES. three given directions. Let AH be the path described or the velocity : we may decompose it into two others, one according with one of the given directions, the other following AC in the plane of the other two directions, and regard the body as impressed with these two simul- taneous motions or velocities. Then we may decompose the motion or velocity AC into two others AB and AD, according with the two other given directions. The motion or velocity All will then be replaced by the three motions or velocities AF, AB, and AX), in the three given directions. Case where the components are at right angles. If the three directions are at right angles, putting AB=Y', AD=Y", AF=V", and AH=Y, we have Y= v/Y /2 +Y //2 +v //a , and Y'=YcosBAH, Y"=YcosDAH, V'"=VcosFAH. 111. Resultant of any nuinber of simultaneous motions or velocities. If, instead of being impressed with three simultaneous motions or velocities AB, AD, AF, the. body had a fourth, it is readily seen, that the final motion or velocity would be represented in magnitude and direction by the diagonal of the parallelogram constructed upon the resultant of the first three motions, and upon the fourth as sides ; now, this line is the last side of the poly- hedron formed on the supposition, that the body receives these simultaneous motions or velocities. Then, in general, the resultant 'motion or the velocity of any number of simultaneous motions or velocities, di- rected in any manner in space, is represented in magni- tude and direction ~by the last side of the polyhedric poly- gon, formed on the supposition that the body was succes- sively impressed with these simultaneous motions. COMPOSITION OF MOTIONS, VELOCITIES, AND FORCES. 129 But we arrive more simply at the determination of the motion or resultant velocity by recalling our previous statement, that any motion of translation may be decom- posed into three other simultaneous motions, in any three given directions, which are the sides of a parallelopipedon ? whose diagonal is the motion, and whose sides follow the given directions. This being established, if we conceive each of the mo- tions, or each of the simultaneous velocities, impressed upon the body to be thus decomposed, the motion or the final velocity will not be altered. But as all motions or velocities along the same axes have partial resultants equal to the sum of the components, in these directions, it follows that the movement or resultant velocity will be represented in magnitude and direction ~by the diagonal of the parallelopiped constructed upon the sums of compo- nents of partial motions in any three directions. Following, then, the reasoning of Art. 108, and sup- posing that after having compounded into a single motion all the simultaneous motions impressed upon the same material point, we project these motions, or the resultant motion or the corresponding velocities upon any axis, by as many planes perpendicular to this axis, we shall see that the projection of the resultant motion or velocity, which is the diagonal of the polygon already mentioned, is equal to the algebraic sum of projections of the component mo- tions or velocities. 112. Case where the resultant is zero. When the line joining the extremities of the first and last side of the plane or polyhedric polygon, formed upon the directions of the component paths or velocities is zero, which hap- pens when the polygon returns upon itself, the resultant motion or velocity is naught. 113. Varignon's theorem of moments. If from any point O, taken in the plane of the parallelogram ABCD 9 130 COMPOSITION OF MOTIONS, VELOCITIES, AND FORCES. of velocities, and outside of the angle BAD, we draw the straight lines OA, OD, and OC, the quadrilateral OADC being the sum of the triangles OAD and ODC, we shall have OAC=OAD+ODC-ADC. If, then, we let fall from the point O perpendiculars Oa or Ocj Ol and O<#, upon the sides AB, AC, and AD, we have for the surfaces of the triangles The above relation becomes, then, AC x 05= AD x Od+ AB x Oa. The products AC x O5, AD x Od, AB x Oa of the sides AC, AD, AB, by the perpendiculars O5, Od, Oa, let fall from the point O, upon their respective directions, are COMPOSITION OF MOTIONS, VELOCITIES, AND FORCES. 131 called the moments, and the above relation shows that if we apply the preceding remarks to the component and resultant motions of the point A, we may enunciate the theorem in saying, that the moment of the diagonal or of the resultant is equal to the sum of the moments of the sides or components. In the preceding figure, the two motions or velocities tend to turn the body in the same direction around the point O, placed outside of the angle BAD. If the point O is within this angle we shall then have OAC=OAD-{-ODC ADC, then ODC^DCxOc, 2i and therefore AC x O5= AD x Od AB x Qa. And, as in this case, the body, in virtue of its two motions, is urged in opposite directions around the point O, Yarignon's theorem may be enunciated in general, being thus extended for any number of simultaneous 132 motions or velocities, in saying that the moment of the re- sultant is equal to the sum of the moments of the compo- nents, which tend to turn the body in one direction, minus the sum of the moments of components tending to turn it in an opposite direction, or more generally, that the mo- ment of the resultant is equal to the sum of the moments of the components, provided that, taking as positive the moments relative to a certain direction of motion, we agree to adopt as negative those which belong to an op- posite direction. 114. Extension of these theorems to bodies or systems impressed with a common motion of translation. All that has been said in relation to a material point applies to bodies or material systems impressed with a common translation, since a determination of the resultant motion or velocity of one of the points will give us that of the others. For if all the points are impressed with one or many common velocities in given directions, the resultant velocity will be the same for all. 115. Independence of the simultaneous action of many forces upon the same point. From observations which show that a material point may be impressed with many simultaneous and independent motions or velocities, it fol- lows quite naturally that the causes or forces which produce these motions or impart these velocities exert actions independent of each other. Thus experience shows that when a body is subjected to the action of many forces, each one of them communicates, in an element of time t, and in its own direction, a small velocity v, proportional to its intensity, which is the same as if it acted alone, whatever may have been the previous motion of the body. 116. Case of the forces acting in the same direction. When all the forces act in the same direction, the veloci- ties imparted by them being in the same direction are COMPOSITION OF MOTIONS, VELOCITIES, AND FORCES. 133 added, and the body is impressed with a resultant velocity equal to the sum of the component velocities. Now, these forces being proportional to the velocities with which they impress the same body in the same time, it follows also that all the forces acting upon the material point in question, have a resultant equal to their alge- braic sum. In fact, calling F, F', F" the forces acting in the same direction upon a material point with a mass M ; v, v', v", the finite or elementary velocities imparted by them in the same time, we have P_ _ = ~T' ~T> : and as the resultant velocity is we have, calling R the resultant of the forces, or K=F+F'+F"+&c. Moreover, if we multiply this last relation by the space s described by the material point in the common direction of the forces and their resultant, we have Ifc=F+F'+F"+&c., an expression showing that the work of the resultant is equal to the algebraic sum of the works of the compo- nents, acting either as motive or resistant works. (No. 93.) Finally, in order that the motion of the body may be uniform, it is necessary that the sum of the motive works 134: COMPOSITION OF MOTIONS, VELOCITIES, AND FORCES. should equal the sum of the resistant works, which leads to the relation K=F+F'+F"+&c.=0, or which expresses that the result is nothing, or that the work of the motive forces is equal to that of the resisting forces. Equilibrium is but a particular case of uniform motion, and when the velocity is zero the preceding condition is also that of equilibrium. 117. Case where the forces acting upon the body have different directions. We have seen by the examples of No. 100, relative to the fall of bodies impressed at the same time with a horizontal motion, that the velocities imparted in different directions were wholly independent of each other. In obedience then to the simultaneous action of these forces, the body will receive the velocities A, A^, pro- portional to their intensities, and in the direction of the forces, and these com- ponent velocities will have a resultant which will be the diagonal of the parallelogram Abed. If we take AB and AD proportional to the velocities Ab J[ jo >jp and Ad to represent Fie. 49. the forces P and Q producing these small velocities, the resultant of these forces to which the resultant velocity is due, will be pro- portional to the velocity imparted in the same time and in the direction of its action, or to Ac ; we have, then, COMPOSITION OF MOTIONS, VELOCITIES, AND FORCES. 135 Then the resultant R will be represented in magnitude and direction by the diagonal AC of the parallelogram ABCD. Then the resultant of two forces acting simultaneously upon the same body is represented in magnitude and di- rection by the diagonal of Q a parallelogram construct- ed upon these two forces. Reciprocally, every force can be resolved into two others, in any two arbitrary directions, equal to the sides of the parallelogram FIG. 50. whose diagonal is the given . force, and whose sides are parallel to the given directions. If the two directions are perpendicular to each other we have =E cos CAB, Q=R=K cos CAD. 118. Quantity of work of a force whose point of application does not move in the same direction as the force. When a force R does not act in the same direction of a, the path described by its point of application, it can be resolved into two ; the one P represented by AB in the direction of this path; the other Q, repre- Fro - 6L sented by AD, perpendicular to it. The work of P will be P x Aa. Designating by Aa the path really described, the work Q will be zero, since it has no motion in its own direction. Then the work of the force R will be 136 COMPOSITION OF MOTIONS, VELOCITIES, A3TD FORCES. measured by that of its component P. But in dropping the perpendicular db upon AC, we have by the similar triangles ACB and Aab. K : P : : Aa : AJ, whence ~R.Ab=P.Aa. Consequently, the work of the force H may be measured by that of its component P in the direction of the path described, or by the product of its intensity K into the projection Ab of the path Aa upon its own direction. 119. Application of Varignorfs theorem to forces. Since the resultant of two forces is represented in magni- tude and direction by the diagonal of the parallelogram constructed upon these forces, as sides, it follows that the purely geometrical theorem of Yarignon applies to forces as well as lines, and that consequently, The resultant of two or any number of forces acting in the same plane has for its moment, in relation to any point of this plane, the sum of the moments of the forces which tend to turn it one direction, minus the sum of the forces tending to turn it in the other direction. "Which is expressed by the relation &c. In calling P, P', .... the forces tending to turn the body in one direction, and p,p r the respective lever arms of these forces ; Q, Q', .... the forces tending to turn the body in the other direction, and q, q f , ---- the respective lever arms of these forces ; R the resultant and r its lever arm. 120. The resultant work of any number of forces is equal to the algebraic sum of its component works. In the most simple case, when the forces all act in the di- COMPOSITION OF MOTIONS, VELOCITIES, AND FORCES. 137 rection of the path described, the resultant of all the forces is evidently equal to the sum of those acting in one direction minus the sum of those acting in an opposite direction, and as the path described by their points of application is the same, the proposition is evident. 121. Forces acting in any direction. If we first consider the forces P and Q and. their re- sultant R as respective- ly proportional to 'the lengths AB, AD, and Jr AC, and AM, the direc- tion of the path describ- ed, and project P, Q and II upon this direc- tion, we shall have AB'=F, AD'=:Q' and AC'=R' for the components in the direction of any path described, A#, for example, and the work of these components, which is equal to that of the primitive forces P, Q and R, will be respectively P'.Aa, Q'.A&, R'.A&. Now it is evident, according to Fig. 52, that FIG. 52. ^R'^AB' x B'C'=P'+Q'. Thus in the case of this figure, In the case of Fig. 53 we have ^R'^AB' AD^P' Q', and consequently 138 COMPOSITION OF MOTIONS, VELOCITIES, AND FORCES. The difference of these two results arises from the fact that in the first, the two forces P and Q act both in the direction of the path described, while in the second, the force Q acts in an opposite direction and occasions a re- sistant work. Further proof of this result is derived from the fact that the projection of the resultant is equal to the alge- Draic sum of the projections of the components upon any line in the direction of the path really described, and the multiplication of the two members of this equality by the space described, is an expression of the following general theorem. When a material point is acted upon ~by any number of forces, situated in the * / same- plane, which tend to impart a motion of trans- lation, the work developed ~by the resultant is equal to the sum of the works of the forces which urge the body in the direction of the path described, minus the sum of the works of the forces which urge it in an opposite direction. Without entering into theoretic developments which are foreign to the special purpose of this treatise, we re- mark that analogous reasonings apply to the case of many forces acting upon the same body, in any direction in space. The elementary work being termed the virtual mo- ment, the above enunciation may be thus stated, that the sum of the virtual moments of the components, taken with the proper sign, is equal to the virtual moment of the re- sultant : which is the principle Jcnown as that of virtual velocities. FIG. 53. 122. Case where the point tends to turn around a COMPOSITION OF MOTIONS, VELOCITIES, AND FORCES. 139 point or a fixed axis. If the point O, from which is let fall the perpendicular upon the direction of the two forces P and Q, Fig. 54, is the projection of the axis of rotation, or the point around which the plane of the forces and the body tend to turn, the relation of the moments (No. 113), K x 0&=P x 00 Q x Oc or "Rr= making OJ=r, Oa=j> and Oc=([, becomes by multiplying all the terms by the arc described at a unit of distance Rra Now ra l} pa ga^ are respectively the elementary or finite arcs described by the foot of the perpendicular or the paths described by the point of application of the forces K, P and Q, in their proper directions, and con- sequently Kra^ Pj?^, and Q^, are the works re- spectively developed by these forces, and the above relation demonstrates for motion of rotation the proposition already established for motions of translation. FIG. 54. 123. Conditions of uniform motion or equilibrium. Case where all the forces are contained in the same plane. If the material point considered is acted upon by forces in ^the same plane, it must remain in this plane, and at any instant it can only act in obedience to a motion of translation, or to one of rotation, or to these two com- bined. Since every motion of translation may be resolved into two others in the same plane, the real motion of the 140 COMPOSITION OF MOTIONS, VELOCITIES, AND FORCES. material point will be uniform if its two components are so. Then the condition of uniform motion of translation is the same as that of uniform motion in any two given di- rections. This latter will be fulfilled if the forces or their components urging the material point in the direction of said axis, while acting for the acceleration of its motion, develop a work equal to that of the force which retards it ; or, in other words, the sum of components acting in one direction must be equal to the sum of those acting in an opposite direction, or their algebraic sum must equal zero, according to the previously established condition. Then the motion of translation of a material point will be uniform when the respective sums of the component forces soliciting it in any two directions within the plane shall ~be separately equal to zero. Equilibrium being but a particular case of uniform motion where the velocity is zero, the same conditions exist for it as for uniform motion of translation. In rotation, it is evident, that if all the accelerating forces in the direction of the motion develop a work equal to that of the retarding forces iii an opposite direction, the motion will be uniform, that is, for uniform motion of ro- tation, the sum of the moments of the forces tending to produce motion in one direction must be equal to the sum of the moments tending to produce rotation in the opposite direction. Equilibrium being but a particular case of uniform motion where the velocity is zero, the same conditions exist for the equilibrium of any number of forces situated in the same plane. 124. Case where the forces act in any manner in space. The motion of bodies is generally composed of one of translation and one of rotation round a certain point, and since the motion of translation must be uniform so long as the three motions resolved parallel to three COMPOSITION OF MOTIONS, VELOCITIES, AND FOKCES. 141 perpendicular axes are uniform, we are led to the condi- tion, that the sums of the works developed in the direction of each axis must separately be zero. In respect to motion of rotation we remark, that generally the point around which a body rotates varies at each instant : on this account, we give it the name of the centre of instantaneous rotation. This being the case, it is evident that the rotation around any centre may be re- solved into rotations around the three preceding axes, or axes parallel to them, through the centre of instantaneous rotation at the moment of its consideration. Moreover, the resultant motion of rotation will be uniform, if the components are so. The rotation around these axes being due to the components perpendicular to each axis, uni- formity of motion will take place if the sum of the mo- ments of the component forces, respectively parallel to the two axes taken in their relation to the third are sepa- rately equal to zero : this leads to the relation between the moments which must be taken successively in their relation to each of these axes. The general condition of uniform motion of a material point solicited by any two exterior forces is then reduced to the following : 1. The sum of the component works in the direction of any three rectangular axes must equal zero. 2. The sum of the moments of the given forces in rela- tion to these three axes must be separately equal to zero. Equilibrium being but a particular case of uniform motion where the velocity equals zero, these conditions exist also for the equilibrium of forces. The preceding discussion shows that the study of mo- tions produced by any forces may always be reduced to that of translation in the direction of the forces or their components and of the motion of rotation around a given axis. We have already examined the first of these mo- tions, and will now investigate the second, but first it is 142 COMPOSITION OF MOTIONS, VELOCITIES, AND FOECES. proposed to extend the theorem to the case of parallel forces. 125. Parallel forces -We have seen (No. 108) that the projection of the resultant of any number of forces applied to the same material point, upon any right line, is equal to the algebraic sum of the projections of these same forces upon the same straight line. The demon- stration of this proposition being entirely independent of the direction of the forces, and the angles contained be- tween them, and with their / Q ^^ resultant, it must be true A" : -^7?^ also when we make the projection upon the re- sultant itself, whence it - ' follows, that the resultant of any number of forces applied to the same point is equal to the algebraic sum of these forces acting in its own direction. This may also be shown from Fig. 55, from which we have AC or R=CD'+D'A=:AB'+AD', or R=P'+Q', in calling P' the projection AB' of AB or P upon AC and Q' the projection AD' of AB or Q upon AC. The pro- jections P' and Q' of the forces P and Q upon the direc- tion of the resultant are moreover evidently the compo- nents of these forces, in the direction of their resultant. In the case where the angle formed by the direction of the forces P and Q is obtuse, it is easy to see that the proposition of No. 180 holds good, and the one just estab- Flo< 56> lished is so modified that the re- COMPOSITION OP MOTIONS, VELOCITIES, AND FOECES. 143 sultant is equal to the difference of the projections of the components. In fact, we see by the figure that AC or K^AB'-CB'^AB'-AD'^P'-Q'. 126. Consequences of the composition of parallel forces. The preceding propositions are wholly independ- ent of the magnitude of the angles BAG and DAC, or the direction of the forces P and Q ; they hold good also when the point A of meeting of the forces becomes more and more distant, until these forces becoming parallel, it is found at an infinite distance. We have then for two par- allel forces, in the first case, when they act in the same direction, E=P+Q, and for the second case, when they act in opposite direc- tions, K=P-Q. 127. Point of application of the resultant of parallel forces. The theorem of moments (No. 119) demonstrated for any point around which forces tend to produce rota- tion, being also independent of the direction of the forces, must be equally true for parallel forces. Whence it fol- lows, that the moment of the resultant of any number of parallel forces, situated in the same plane, in relation to any point in this plane, is equal to the algebraic sum of the moments of these forces, and thus we are enabled to determine the position of the resultant. Let there be, for example, two parallel forces acting in the same directions. The preceding proposition becomes and moreover we have 14:4: COMPOSITION OF MOTIONS, VELOCITIES, AND FORCES. If we take for the centre of moments a point of the resultant itself, we have evidently r=o and Br=o, and consequently PpQ^=o, which can only be the case when we have P#> Q^, and when the forces P and Q having the same direction tend to produce rotation in opposite directions around the centre of moments. Then this point and the resultant itself are comprised within the directions of the forces P and Q, and all the perpen- diculars to the resultant and to the two forces are divided into parts reciprocally proportional to these forces ; which is expressed hv the relation -^=~. Q P It is the same for every secant drawn between the directions of the forces A CO JB ^ an( ^ Q 5 wherever may be the points of application A and B of these forces, we see that their resultant cuts this line parts reciprocally proportional to n a their intensities. We have moreover from the figure, in calling d the dis- tance between the directions P and Q, =dp and Pjp=Q(d m p), Qd whence consequently whence In the case where the forces P and Q are in opposite directions, B=P-Q, and we have the relation Now in order that, in this case, the forces P and Q, which are in opposite directions, may produce rotation in COMPOSITION OF MOTIONS, VELOCITIES, AND FOECES. 145 " ..... ~~ opposite directions in respect to a point of the resultant, the point and the resultant itself must be outside of the two direc- tions of the forces. If we call d the distance of these two direc- tions, we have by the above re- lations d=qj} ; whence Q whence or -?= which gives the distance of the resultant from the direc- tion of the force P, and consequently its position. If the points of application of the forces P and Q are at A and B, the resultant cuts the line AB produced in C, so that these distances p and q from the directions of the forces P and Q are reciprocally proportional to these forces. 128. Reciprocally, every given force may le resolved into two other parallel forces acting at given points. If a force E acts at a given point C, of a right line supposed to be rigid and inflexible, it will always be easy to find the values of two other forces P and Q which acting at given points A and B shall produce the same effect as this force. It will be sufficient if we have at the same time P-t-Q=E if they act, the one at the right and the other at the left of E, P Q=E if the points A and B are both on the same side of E, and in both cases 10 whence = 9 146 COMPOSITION OF MOTIONS, VELOCITIES, AND FORCES. These two relations will give whence P== 87 p+q d being the distance of the two given directions ; or K, whence P=-2-. p-q Which indicates that the force P acts in an opposite direc- tion to K. The two forces P and Q thus determined have a single resultant precisely equal to K, and may be substituted for this force, since they develop the same work. This decomposition of a single force into two others act- ing parallel and at given points has frequent applications in practice. When, for example, we wish to determine the pressure that a beam or a shaft of a hydraulic wheel, of a known weight, or loaded with a given weight, exerts upon its supports, we are led to a resolution of this kind. Suppose a beam loaded at a , f ...... t'_ ________ point C of its length 20, with a j4 ........... \&_ _ j^ weight 2P, and resting upon the two points of support A and B, situated at distances I' and I" from the point C. Calling P' and P" the two pressures or components sought, observing that P / +P // =2P, and taking the mo- ments of the resultant and of the components alternately in relation to the points of support A and B, we have in the first case P7' P"x2c=2PxZ', whence P"= , o in the second P'x2q f . g the arm of the lever of the weight M of the beam. In this new position of equilibrium, the condition of equality of moments will be expressed by the relation make We have then from which we deduce the value of the weight which in- clines the beam and maintains it in this new position, In order that m may be equal to zero, we must have p^o, or p'=q'i and go. The first of these conditions can only be satisfied when the two blades are in the same plane, since we see by the figure that then we always have pq and p'=q f COMPOSITION OF MOTIONS, VELOCITIES, AND FORCES. 153 in all positions, which shows the necessity of conforming to this rule of construction. ivlSJ We cannot make =o, unless we make g=o, so that the centre of gravity of the beam will be found upon the edge of the blades ; but then the beam and scales will be in equilibrium in all inclinations, and for all cases where P=Q. The balance would be indifferent, and would have no marked or determinate position of equilibrium. We then relinquish the attempt to satisfy wholly this condition ; but, in order that the balance may incline un- der the smallest additional weight m, when P=Q, we make the distance of the centre of gravity from the axis so small, that under a given inclination of the limb, of half a degree, for example, the arm of the lever g of the weight of the beam, shall be such as that the weight m producing this inclination may be .0154 or .0077 grains Troy. This result is independent of the magnitude of the weights P and Q, so that the balance appreciates the ad- dition of .007 grains, for all weights from the smallest to the greatest, designed to be measured. Artisans have attained this perfection by dint of great care. M. Fortin has made balances for weighing from 2.2 pounds to near 0.015 grains. The Conservatory of Arts and Manufac- tures possesses a valuable collection of balances, among which are two, one Fortin's, the other Gambey's, which gives the weight of from 2.2 pounds to nearly 0.015 grains, and one of M. Deleuil, which can weigh from 22 pounds to nearly 0.007 grains, and one presented by the United States government which can weigh from 110 pounds to .007 grains nearly. For assaying balances, such precision has been attained as to appreciate a weight of 0.0007 grains. As for balances used in trade, we do not require such delicacy, and well made balances, for weights of 110 154: COMPOSITION OF MOTIONS, VELOCITIES, AND FORCES. pounds, for example, appreciate a difference of 0.15 grains only. The general conditions of stability of equilibrium show, moreover, that if the centre of gravity is above the edge of contact with the blades of the axle, equilibrium cannot subsist, and the balance will be useless. The great sensibility of balances, due to the conditions just indicated, as well as to finish in execution, and to the polish of surfaces, has this inconvenience, that their oscil- lations are slow and take considerable time to arrive at the position of equilibrium. This defect is remedied by various contrivances, having for their object the stopping of the scales before being left to the action of the weights, the placing of them gently upon the edges, the stopping their oscillations at will, and finally the limitation of their amplitude and duration. For the preservation of the form of the knife edges and their cushions, we should protect good balances by some disposition which enables us to raise up the beam and the plate while we are loading the scales, or when they are not in use. 136. Proof of balances. Being assured that the knife edges are well made, and contained in the same plane, that their cushions are well planed and polished, we put the beam in place, and prove whether it is in equilibrium when the index is at zero or is vertical. We turn the beam, end for end, to see if it is the same in both direc- tions ; we test the equality of the arms of the beam, by hanging the scales upon the beam, and making sure that the latter retains its position of equilibrium when we change their sides. "We load the scales with weights graduated from the smallest to the greatest, which the balance is designed to weigh, and see if its sensibility re- mains the same throughout. After one weighing, we change the weights of the plates, and see if the results are the same. COMPOSITION OF MOTIONS, VELOCITIES, AND FORCES. 155 The sensibility of balances for the' trades is fixed by the statutes at ^oVo- of the weights to be tried, from the smallest up to the greatest. When the addition of this fractional weight does not incline the beam to the side on which it is placed, the balance is imperfect. 137. Method of double weighing. Notwithstanding all the pains taken in their construction, when it is desired to operate very exactly, to be free from all liability to error, we use a very simple method of the illustrious Borda, called double weighings. After placing the body to be weighed in one scale we bring the balance in equilibrium by loading the other with weights, or scraps of lead, iron, &c. When the equi- librium is well established, we take away the body to be weighed, and substitute for it a number of units of weight, required to re-establish equilibrium ; we obtain the weight exactly, wholly disregarding the inaccuracies of the bal- ance. 138. The steelyard. In this system of balances (well known to the ancients, and of which many light and sim- ple Chinese models are to be found in the Conservatory) the arms of the lever of the load P, and of the weight Q are unequal,^?, that of the load remaining constant, while q that of the weight Q varies; but the counterpoise is always the same. The condition of equilibrium is then satisfied, in varying the arm of the lever q of the constant weight Q, so that we always have The ratio being constant, the arm of the lever of 156 COMPOSITION OF MOTIONS, VELOCITIES, AND FORCES. the constant weight, must vary proportionally with the weight of the body to be weighed. But in the graduation FIG. 62. of the long arm of the lever, we must take account of the weight of the lever, and of the scale ; which is done in determining first the position of the sliding weight Q, when the beam is horizontal or in equilibrium, under its own weight, and that of the hooks or scales. Let q' be .the distance of this weight from the axis ; the moment Qq' will be equal to the excess of that of the empty scale, and its arm of lever, above that of the other arm. If to make equilibrium with the weight P in the scale, whose moment is P>, it is necessary to put the running weight Q, at a distance ^, we shall have the relation ~Pp+Qq'=Qq, whence p= or Thus in taking account of the weight of the apparatus, we see that the distances of the running weight from zero increases proportionally with the weights to be weighed. The lengths p and q', as well as the running weight Q, being known when the balance is made, we may calculate COMPOSITION OF MOTIONS, VELOCITIES, AND FORCES. 157 the value of the distance q, corresponding to the greatest weight to be determined by the formula but it would be better, after having thus calculated for the maximum weight Q', to determine it exactly by ex- periment. This done, we divide the interval q q' into as many equal parts as we wish to have subdivisions of the weight Q'. Steelyards for weighing from 40 to 50 pounds usually have divisions corresponding to the pound, and the frac- tions are read at sight, according to the position of the running weight. It is moreover evident, that in this balance, as well as the preceding, the blades of the axis of suspension, those of the hook bearing the weight, and those of the running weight, as well as the notches in which it is arrested, should be in the same plane, in order that the ratio of the levers may be independent of the inclination of the beam ; the centre of gravity of the latter should also be a little below the axis of suspension. "With these conditions the balance oscillates freely. The use of non-oscillating (folles) steelyards is forbid The degree of exactitude of steelyards is fixed by statute a * 5^0- f the weight to be determined from the smallest to the greatest. 139. Steelyard with a fixed weight. (Peson.) "We sometimes nse for the determination of small weights, a balance having a single plate and a fixed weight, arranged as follows : a beam with two unequal arms, resting upon a cylindrical steel axle, receives upon its longest arm the scale ; the shortest arm is terminated by a fixed weight, usually of a lens-like form. An index OGr, fixed perpen- dicularly to the length of the lever, passes through the 158 COMPOSITION OF MOTIONS, VELOCITIES, AND FORCES. FIG. 63. axis of rotation, bearing at its end a weight q. The different weights, the lever, the scale, the index, are so proportioned, that when the plate is empty and the beam horizontal, the centre of grav- ity of the apparatus, composed of the beam, the counterpoise and the index, is found at a point G of the index, upon the vertical passing through the axis. In this position the point of the index corresponds with zero of the graduated limb of the instrument. A weight P put in the scale inclines the lever, and we are to determine the position which the end of the index shall take upon the limb. Call : Q the total weight of the lever, the counterpoise and the scale, &c., and consider it as acting at the centre of gravity G of the system, (see 138,) which will then have taken the position G'. The condition of equilibrium of the index will give us P.B'C^Q.G'H, whence P=Q.52*; _D L> now the triangles OB'C and OG'H are similar, and we have OB' or OB : B'C : : OG' or OG : OH, whence fc^-~ RT OHxOB -OG ' and consequently -TJ .~ OG GH r . OG The weight Q of the beam and pieces connected with it is known and constant, the invariable distance of the centre of gravity may be determined by experiment, the COMPOSITION OF MOTIONS, VELOCITIES, AND FORCES. 159 length of OB the long arm of the lever is known and in- variable ; the constant factor Q . - is then determined, O-t> and we see that the weights P are proportional to the tan- gents of the inclination of the lever, or to the arcs de- scribed by the end of the index. The division of the limb is then easy, since we shall have O~R P tangGOG'=g|.| and making successively P=l lb> , l.o lb % 2 lb % &c., we may calculate the values of tangents of the angles GOG 7 , an- swering to the positions of equilibrium, and consequently the angles described by the index. In practice, we determine these angles by experiment ; we see, however, that this kind of balance, though not susceptible of the same precision as the common balance, is yet quite handy for certain purposes. 140. Quintenz's Platform Balance. This contri- vance, bearing the name of its inventor, serves, accord- ing to its proportions, for weighing common bales, or the FIG. 64 heaviest loads. Its form varies with its destination, but its general arrangement is nearly always the same. 160 COMPOSITION OF MOTIONS, VELOCITIES, AND FOECES. It is composed of a horizontal platform AB, resting at one end upon a triangular blade C, placed on the lever DI, which is supported at D, upon a fixed point, and is sustained at I by a vertical rod HL. The other end of the platform AB is sustained at F, by a rod FG, by means of the strut EF. The two rods HL and FG are sustained by blades, forming a part of the lever HOK, resting at O, upon the support of the balance, and upholding at K the platform on which the weights are put. This apparatus is usually borne upon a movable frame, but is established solid upon masonry, when we make use of weighing bridges, designed for heavy wagons. In all cases the frame and platform should be level in all direc- tions. The first condition to be fulfilled, is to maintain a horizontal position when the platform is loaded, which is attained by giving proper proportions to the different arms of levers. Indeed, if the point C of the platform, or the blade sustaining it is lowered the height A, the point I at the end of the lever AI, and consequently the point H, the upper end of the rod III, will descend the height k x =r-~, ; but, at the same time, the point G of the lever DO OH, and so the end F of the rod GF, which sustains the other end of the platform, will be lowered the height DI OG X DC ?X OH' That the two supports of the platform C and F may descend the same quantity, the factor - x ^-^ must be DO OH equal to unity, which leads to the condition, that DC 7 , OG , , -=-=- and . shall be equal. This condition being satisfied, we see that in whatever part of the platform the load is placed, its action upon the COMPOSITION OF MOTIONS, VELOCITIES, AND FORCES. 161 lever OH will be the same as if it were suspended by the rod GF, upon the blade G. In fact, suppose the centre of gravity of the load rests upon O', if we resolve it into two, the one acting at C, and the other at F, and call L the vertical distance between C and F : P.# The component of P at F will be -^ , and will act _L directly upon G. Pv The component of P at C will be -^, and this will act JL at I, and consequently on H with an effort -x--^-^, .L DL , , . Py DC' HO which occasions an effort at G equal to -- . x-^- ; L Ul (J(JT DC' HO ,. now -x-=lj according to the preceding remarks. Then the two components of the load P occasion an effort atG since a?+y=L, according to the figure. Thus, in whatever part of the platform the load is placed, it is found in virtue of the connections of the sys- tem, referred to the point G, with its integral value, and we have for the equilibrium between the load P and the weight Q the relation P.OG=QxOK. Of^ The ratio -^ is usually equal to T V for the common UJY portable balances of commerce, and to T ~ for those de- signed for the weighing of heavy loads. All the fulcrums and joints of the system are formed of steel blades, resting upon well prepared seats. The blades corresponding to the larger parts of the platform are double, and exactly parallel to each other. Above 11 162 COMPOSITION OF MOTIONS, VELOCITIES, AND FOECES. the platform with the weight is a small basin 1ST, in which we place weights or scraps, to establish the equilibrium of the platform when unloaded, and of the levers of the apparatus, when by use or accidental causes, their primi- tive condition has been changed. To save calculation, the weights used are generally marked in figures, indicating the decuple or centuple of their real weights, according to the proportions of the balance. A stop-lever with a handle serves to raise the lever HOK, to relieve the blades, when the balance is not loaded, and to avoid shocks upon their edges at the time of loading. This system of balance has been modified in form, but not in principle ; it has been ingeniously attached to cranes, which weigh the goods while being raised for loading. Yarious arrangements have been applied which have furnished some improvements, and have even regis- tered the weighings, but we cannot give a detailed account of them. 141. Theory of the Lever. From what has been said in respect to a body urged by parallel or concurrent forces, around a point or fixed axis, it follows, that the moment of the resultant is equal to the sum of the mo- ments of the components of the forces, producing rotation in the same direction, or to their difference, when acting in opposite directions. The perpendiculars r, p, q, being let fall from the centre of rotation, upon the direction of the forces, are called the arms of lever of the forces, and we have the relation In case of equilibrium, we have Tt,r=o, and conse- quently Pj}=Qq, if one of the forces P, is the power, and the other, Q, a resistance to be overcome ; we see then, COMPOSITION OF MOTIONS, VELOCITIES, AND FORCES. 163 that, for equilibrium, the moment of the power must be equal to that of the resistance. If the resistance and its moment are given, the effort to be developed by the power which is given by the formula will be so much the smaller, as the arm of the lever p is greater. This relation contains the theory of the simple contrivance of the lever, employ- ed for the moving of heavy weights by the long, with 32 suspension rods, 3.937 ft> apart, except at the sides, which are 4.59 ftt from the vertical supports. The width of the platform is to be 16.4 ft -, and there are four chains. In conformity with the usual constructions the plat- form weighs 672.21 lbs - per running foot. The test load being 40.977 lbs - per square foot, this amounts to 672.21 lbs - per running foot of the four chains, or in all per chain and running foot 336.1 lbs - ; and the spaces of the suspension rods being 3.937 ft - we have P=336.1 x3.937=1323.4 lbs The height of the towers is 16.404: ft - above the lower horizontal side. EQUILIBRIUM OF TENSIONS. 181 * The vertical tension at each point of suspension is V= weight = 22054 lb % and the horizontal tension is H=Y cotang angle of suspension = 44109 lb % and the whole at the end is - } /V f +R t = ^22054? +4:4:109*= 49314 pounds ; we have, then, T, = V(44109) a + 1985 3 = 44153 lbs - T 2 = V 44109 2 + 3308 3 = 44233 T 3 = -v/ 44109 2 + 4631* = 44352 T 4 = V 441Q9 2 + 595? = 44508 T 5 = V 44109 2 + 7277 2 = M705 T 6 = V 4109* + 8601 2 = 44938 T 7 = V 44109 2 + 9924 2 = 45212 T 8 = V 44109 2 + 11247 2 = 45521 T 9 = V 44109 2 + 12571 2 = 45865 T 10 = V 44109 2 + 13894 3 = 46244 T n = |/44109 2 + 15218 2 = 46659 T 13 = V 44109 2 + 17864 2 = 47590 T I4 = y 44109 2 + 19187* = 48101 T 15 = V 44109 s + 20511* = 48644 T 16 = |/ 44109 2 + 22054 2 = 49312 * Morin has made a marked error throughout this calculation, arising from taking the number of metres in the span, instead of the number of panncls, nor in the example has he conformed to the statement of Art. 159, where, in the case of several chains being used, it is recommended to bring the summit to the lowest part of the polygon. The common formula for the horizontal tensions of the whole chain is : H = G cotang a? = and for the tension at the ends is : Where G is the weight of the loaded half of chain, & = half span ; a = versed sine, or height of arc ; a? = anglo of suspension. I have not followed literally the steps of Morin, and have used these formulae, as more direct and less complicated than those given by hi na. Translator. GENERAL COMPOSITION AND EQUILIBRIUM OF FORCES APPLIED TO A SOLID BODY.* 161. Forces applied to solid ~bodies. If we refer to what has been said in Arts. 11 to 14, and the following, upon the constitution of bodies, the mode of action of forces, and their point of application, we readily perceive that all the propositions relating to work, and the compo- sition of forces, acting upon a material point, and to the conditions of uniform motion and of equilibrium, may be extended to solid bodies, composed of molecules or mate- rial points so strongly united by the molecular attractive forces, that their form may be regarded as invariable. And first, let us examine what occurs when a body is impressed with a motion of translation. 162. Motion of translation of a l^ody or system of ~bodies parallel to itself. The motion of a body or system of bodies, is called parallel translation, when all its points or parts describe simultaneously equal and parallel paths, whether in a finite time, or one of infinitely small duration. In the motion of translation, the elementary spaces described by all the points of a body being equal, the sum of the elementary works of forces soliciting the body * We borrow the demonstration of principles, recited in Art. 161, from the course of M. Poncelet, at Sorbonne, and from the work of M. Reisal, en- titled Elements of Mechanics. COMPOSITION AND EQUILIBRIUM OF FORCES. 183 in this direction, or the total elementary work developed upon the body, is equal to the algebraic sum of the pro- jections of forces upon the common direction of the path described, multiplied by the elementary path. In order that the total elementary work may be zero, or that the motion of the body may not be changed, it is only requisite that the sum of these projections of forces upon the path described shall be zero. This sum is equal to the resultant of all the forces tending to produce the translation. This resultant should then be zero, in order that the motion of the body may remain uniform in this direction, or that the body may maintain an equilibrium in this direction. 163. Case of variable motion. "When external forces produce a variation in the motion of translation, the forces of inertia are developed, and react against them. If we call jp the weight of one of the elementary masses composing a body, the motive force and the inertia cor- responding to an elementary change of its velocity will be and all the similar forces will be parallel, and in the direc- tion of the common velocity of translation. Their result- ant F will be equal to their sum, and we shall have F= g ~/t~~g't~ *? P and M being respectively the total weight and mass. As to the point of application, all the partial forces of inertia /',/ y ,/ r/ , are proportional to the weights p,p' ->$'-> &c., of the different parts of the body, and the point of application of the^r resultant will be the same as that of the total weight, or as the centre of gravity. 184: GENERAL COMPOSITION AND EQUILIBRIUM Then, in the motion of parallel translation the total force of inertia is 9 * * and its point of application is the centre of gravity of the lody. This consequence being independent of the amount of motion of translation, holds good for finite motions, and for any instant of motion in a curved line. But the resultant of the forces of inertia developed in the variation of motion, is by virtue of the principle of action and of reaction, precisely equal and opposite to that of the external forces producing this variation, so that F really expresses this resultant, and the relation be- tween the external forces, and the forces of inertia in the motion of the translation is . g t t The acceleration - produced by the force is given by the t formula which shows that it is proportional to the force F, and inversely proportional to the mass of the body. It is for this reason that we give considerable weight to anvils, and place them on large blocks of wood, to diminish the shock, and to render the velocity imparted by the hammer nearly imperceptible. 164. Quantity of motion, and vis viva of a l)ody. We see, also, that in parallel motion, the total quantity of motion of a l)ody has for its value OF FORCES APPLIED TO A SOLID BODY. 185 From this it follows that, yew a body to receive a motion of parallel translation, it is only requisite that the result- ant of the applied forces shall pass through its centre of gravity : for, if it passes, otherwise, the body solicited on one side by this force, and on the other, in an opposite direction, by the resultant of the forces of inertia, which passes through the centre of gravity, must necessarily take a motion of rotation, at the same time that it does a motion of translation. It is further evident, that the total vis viva imparted to a ~body in parallel motion is equal to -V a =MV, g and is equal to double the quantity of work developed in producing it. 165. The work of gravity in jointed or compound sys- tems. The work of all the components being equal to that of the resultant, it follows that in the motion of bodies, or of heavy jointed systems, we may substitute the total work of the resultant or of the total weight, for the work of all the partial weights, and as the work of the resultant is measured by the product of the total weight, and of the space described by its point of applica- tion, it follows that, in machines or systems, with pieces which ascend or descend under the action of weight, the total work developed by the weight is measured by the pro- duct of the total weight, and the height, that the general centre of gravity is raised or depressed. Then, also, the condition requisite for the descending weights to be in constant equilibrium with the ascending, or that the work developed by one shall be equal to that of the other, is that the general centre of gravity must re- main always at the same height. Such is the condition for the equilibrium of weigh-bridges, balance machines, &c. 186 GENERAL COMPOSITION AND EQUILIBRIUM The principle set forth in Art. 31, on the measure of work developed by weight upon a body describing any curve, ascending or descending, applies also to any sys- tem of heavy material points, since, in this case, the work developed by weight, upon all the points, is equal to that corresponding to the elevation of the centre of gravity, which is in itself but a heavy material point. 166. A system of any forces, acting upon a solid body, may always be red^lced to two equivalent forces, applied to two of its points, one of which may le chosen at will. It is readily seen that the proposed forces may be resolved into three others, applied at any three points within the body. Let F be one of these forces, and its point of application ; we may resolve it into three others F a , F 6 , F c in the directions AC, BO, CO, and suppose these com- ponents to be transferred to the points of application A, B, and C. These three components will de- velop the same work as their resultant. Operating in the same way upon all the other forces acting on the body, we shall have at the points A, B, and C, three groups of concurrent forces, which each have a single resultant, and the work of these three resultants R a , K fc , K c will be equal to the sum of works of all the forces applied to the body. Moreover, the system of three forces K a , Rj, E c , may be reduced to two forces, equivalent to the proposed, one of which may be ap- plied at a chosen point A. Let us conceive, through the point A, and the directions of the forces R ft and R c , two distinct planes to be passed, Upon this line take a point FIG. 81. intersecting at the line AD. OF FOKCES APPLIED TO A SOLID BODY. 187 A', and draw AB, resolving the force R 6 into two others, one in the direction BA', which we transfer to A', the other in the direction BA, which we transfer to A. We do the same for the force R c , resolving it into two others, one in the direction CA', which we transfer to A', the other in CA, which we transfer to A. The three forces K a , R fi , R c , may thus be represented by two groups of forces, acting, the one in A 7 with a sin- gle resultant R', the others in A, the point chosen at ran- dom, and having a single resultant R. Finally, the system of all the forces acting upon a body may be replaced by two equivalent forces, one of which passes through an arbitrary point within the body. It is evident, that the work of these two forces will be equal to the total work of all the forces applied to the body. 167. Condition of uniformity of motion, or of equili- brium. The motion of a body will not be disturbed, or modified, by the action of these two forces, or of those which they replace, if the quantity of work developed by one is equal and opposite to that developed by the other ; which requires these two forces to be equal and directly opposite to each other in all possible displacements of the body. Such is also the condition of equilibrium, which is but a particular case of uniform motion. Reciprocally, where, for all possible displacements of a body, the sum of the works of the forces soliciting it is zero, these forces will not modify the motion of the body, and are in equilibrium. If among all the possible displacements we conceive an elementary displacement of the body, for which the point of application A of the force R remains fixed, we may regard this point as the centre of rotation, and the point of application A' of the force R' will describe 188 COMPOSITION AND EQUILIBRIUM OF FOKCES around A, with the radius AA' of a circle, an elementary path A V, perpendicular to AA'. But, since by hypothe- sis, the work of the force K is zero, its point of application not being displaced, in order that the sum of the elementary works of E and E' may be zero, it is necessary that the work of E' should be so; which requires the path AV to be perpendicular to the force E', or that the latter shall have the direction of the line AA'. The force R must also be directed in the same line. In order that the sum of the elementary works of the two forces E and E', in opposite directions, may be zero, as supposed, these forces must be equal and opposite, and therefore will not change the state of motion of the body, and will be in equilibrium. Then the motion remains uniform, and the body is in equilibrium. It follows that three forces which are not in the same plane^ cannot be in equilibrium ; for the resultant of any two cannot take the same direction as the third. MOTION OF KOTATION. 168. Work and equilibrium of forces in the motion of rotation around a fixed axis. We have seen, (No. 122) when a material point is subjected to the action of many forces, contained in the same plane, and tending to turn it around an axis perpendicular to this plane, that the work of the resultant of these forces is equal to the sum of the works of the components. We arrive at a similar result when the forces have any direction whatever in relation to the axis of rotation, for if we decompose each of these forces into two others, the one in the direction of the axis, the other in the plane perpendicular to the axis, and passing through the mate- rial point, it is evident that the component in the direc- tion of this axis supposed to be fixed, can only produce a motion of translation, destroyed by the support of the axis, in consequence of which its work will be nought ; the only work developed then will be that of the component con- tained in the plane perpendicular to the axis. It will be the same not only for all the forces applied to one of the material points of the body, but also for those acting upon the other parts. We have then only to consider the forces comprised in the plane perpendicular to the axis, and as the body is supposed to be rigid and inflexible, we are at liberty, so far as concerns rotation, to suppose all the 190 MOTION OF ROTATION. forces comprised in one and the same plane perpendicular to this axis. If we call a the elementary arc described by a point of the body situated at a unit of distance from the axis O, in an ele- ment of time, r the distance of any point m from this axis, The arc described in the same time by the point m will be ra. If the force F, acting upon this point, is not perpendicu- lar to the radius Om, we may decompose it into two others, one in the direction of this radius, which will be destroyed by the resistance of the solid, since it only has a motion of rotation around the axis, the other F', perpendicular to the radius, will produce the only work due to the force F, which will have for its expression Fra. It will be the same for all the other forces F 1? F 2 , acting at distances 7\, r# from the axis, in planes perpen- dicular to it. Their respective work will be due to their components F' 15 F',, perpendicular to the radii. It results from this that the total work of all the forces acting upon a body and corresponding with an angular displacement a measured with the unit of distance, will have for expression which is expressed in saying that the work is equal to the sum of the moments of the exterior forces mul- tiplied by the elementary arc meas- uring the displacement of points situated at the unit of distance. In fact, it is evident that the product FV is equal to the moment J?op of the force F, as may be easily proved MOTION OF ROTATION. 191 from the figure, and we have mF or F : raF' or F' : : om or r : op, whence The elementary work of all the external forces can not then be zero, for any angular displacement whatever, ex- cept that the sum of the moments of the forces in relation to the axis of rotation shall be zero, which requires the moment of their resultant to be so likewise, or that it shall pass through the axis of rotation. What we have said in respect to this isolated case ap- plies to every other axis of rotation, and consequently, in order that the forces applied to an invariable body shall mutually produce equilibrium, or no motion of rotation, the above condition must be satisfied for any axis of rota- tion whatever. 169. General conditions of the uniformity of motion or of equilibrium of a solid body, free in space, and sub- jected to any forces. It is evident that a solid body, entirely free, can receive and take but one of the three following motions : A motion of translation without rotation, a motion of rotation without translation, and a simultaneous motion of translation and rotation. Every motion of translation may be resolved into three other motions similar in relation to any three rectangular axes drawn in space, and it is evident that if each of these component motions is separately zero, the resultant motion of translation will be so likewise, since it will be represented by the diagonal of a parallelepiped, whose sides are zero. This condition is moreover necessary and sufficient. Now, in order that these three motions shall be zero for each of these axes, the sums of the components parallel 192 MOTION OF ROTATION. to the axes should separately be zero, (No. 124.) Then, if we call X, Y, and Z the sums of the components of the exterior forces applied to the invariable solid considered, these forces cannot impart a motion of translation if we have at the same time X=O, Y=O, Z=0, and the motion will remain uniform or the body be in equilibrium as to translation. So also, every motion of rotation of a body or of mate- rial points composing it, may be resolved into three mo- tions of rotation around three rectangular axes drawn through any point. In order that the body shall receive no motion of rotation, it is only requisite that the rotations around each of the three axes shall be separately zero, which requires the sums of the moments of forces in rela- tion to each of the three axes to be separately zero, so that if we call L, M, and N these three sums, we must have at the same time L=O, M=O, N=0. When these conditions are satisfied, the work developed in imparting a motion of rotation will be zero, and it will continue to move uniformly or will rest in equilibrium. In order that the body receive no motion of transla- tion, nor of rotation, or that its motion be in no wise altered, all that is requisite is, 1st. That the sum of all the components of the forces soliciting the body, in relation to any three rectangular axes, shall be separately zero. This is expressed by the relations X=0, Y=0, Z=0, L=O, M=O, N=0, which we call the six equations of uniform motion, or MOTION OF KOTATION. 193 the equilibrium of an invariable body, free and solicited by any forces. 170. Centrifugal force. Every one knows, that if we tie a stone or other heavy body to a cord, impress it with a circular motion of which the hand is the centre, the cord will experience a tension, the greater as the motion is more rapid. From observation of this fact came the use of the sling as an implement of war among the ancients, and which is but a boy's play. Similar effects are seen in wagons running swiftly in short curves, in circuses, when the horses and riders are naturally induced to lean towards the centre of the curves they describe to prevent being overthrown. The reader may readily find other effects from the same cause : all of them prove that in curvilinear motion the bodies are subjected to a peculiar force tend- ing to drive them from the centre, which force is called the Centrifugal force. 171. Measure of the centrifugal force. To understand what takes place when a material point is submitted to the action of the centrifugal force, let us examine first how this force is developed in circular motions. When a material point or an elementary mass m passes from one element of a curve which it describes to another, it tends by virtue of its inertia to continue its motion in the direction of the prolongation of this element, or of the tangent fid of the curve, and is what is termed flying off at a tangent, as is the case with the sling at the moment one suddenly lets go his hold upon the cord. If the mass m takes the direction of the next element, it is then retained upon the curve, either by the resistance of the curve itself, upon which it then exerts a pressure, or by the tension which it develops in the cord. This pressure or tension is itself the measure of the centrifugal force, in contradistinction to which it is sometimes called the centripetal force. 13 194: MOTION OF ROTATION. This force is in the direction of the radius of the curve or of the corresponding circle, and if we call Y the ve- locity with which the mass in is impressed in the direction of ab, and take the length T)d to represent it, it is clear that the velocity destroyed by the resistance of the ci)rd or the centripetal force, will be represented by the side de of the parallelogram Icdf, whose side do is parallel to the radius ob, in the direction of which this force is ex- erted. Now, an inspection of the figure shows that the angles dbO and ~bdc are equal as internal and external, and the angles deb and cbO as alternate and internal, and as moreover the angles cbO and aOl) being formed on both sides of the radius by two equal and consecutive ele- ments of the circle or of the polygon whose infinite num- ber of sides replace it, it follows that the angles ~bdc and deb are equal, and the triangle T)dc is isosceles. Then the velocity ~ba with which the mass m is moved in the direction of the following elements It, is the same as "that it had in the direction of the preced- ing element. Thus in circular motion, the centrifugal force does not alter the velocity of rotation : which is conformable with the principles upon work, which we have already recited, since this force, in the direction of the radius, or normal to the path described, produces no work in the direction of motion, so long as there is no path described in its own direction and by its action. This being settled, the velocity destroyed in the ele- ment of time t by the centripetal force has, according to the figure, dc for its measure, and the centripetal and cen- trifugal forces, which are equal and directly opposite, have for a common measure t MOTION OF ROTATION. 195 Now, the triangle fide and Obt having equal angles, are similar ; we have then lOilti-.M: dc, whence , Idxlt Vs -^- In calling R the radius of the circle described, and s the elementary arc run over in the element of time t ; and as we have V=? or s=Vt, t it follows that , _YxY*__V 2 K "E"' and finally, that the centrifugal force has for its measure Y a if, moreover, we call V\ the angular velocity, or that at the unit of distance, we have Y= V^, and the expression for the centrifugal force becomes . ~ JLV What we have said of the centrifugal force applies to a material point describing any curved line, since in each of its positions, an osculating circle may be substituted for the curve ; the only difference being in the fact that the radius K of this circle varies for each position of the mov- ing body, while that in the circle is constant. 172. Work developed ly the centrifugal force. "When instead of being retained by a circular curve or at a con- stant distance from the centre of rotation, the material 196 MOTION OF KOTATION. point is removed farther from it, the centrifugal force will cause it to describe a certain path in the direction of the radius ; it develops upon this body a work easily appre- ciated. In fact, if in an element of time the material point is displaced in the direction of the radius by a certain ele- mentary quantity Y, the corresponding work of the cen- trifugal force will be and the total work due to this force when the material point shall have passed from R" to R' at a greater dis- tance from the centre, will be given by the sum of all the analogous elementary works taken from R=R" to R=R'. Now we have seen by the preceding examples that this sum is equal to --R //2 )=w (V /2 -V" 2 ) if we call V'^VJR/ and V'^VJR", the velocities of ro- tation of the point around the centre. We have then for the work of the centrifugal force We remark that the second member of this relation is no other than the variation of the vis viva of rotation, ex- perienced by the material point while partaking of this motion in its removal from the centre of rotation, what- ever may be the curve or path described in this removal. This expression then could be directly deduced from the principle of vis viva. In the case just considered, the centrifugal force tends to increase the absolute velocity of the body moved, and MOTION OF EOTATION. 197 acts thus as a motive force which is developed in the rno tion of rotation. When, on the other hand, the body approaches the centre, the centrifugal force is opposed to it, and acts as a resistance in developing a work having indeed the same expression, but which is resistant, since the path described is in a direction contrary to the action of the force. The preceding considerations will find their application in the study of the effects of certain hydraulic receivers. 173. Action of the centrifugal force upon wagons. When a coach with great speed turns upon a short curve, the effects of the centrifugal force is felt by the passengers who are driven towards the outer curve with an intensity often dangerous for those placed on the outside, and which may even disturb the stability of the coach itself. There is often a prejudice against the effects of this force upon railways, w r hen it is pro- posed to use curves of small radius ; but it is easily shown by figures, that in this regard the greatest velocities with the common radii of curves pro- duce no danger. In fact, calling P the weight of the car or any carriage, h the height of its centre of gravity above the plane of the track, F^-V/R the centrifugal force, y %c the width of the track. It is evident that when the car passes around the cen- tre O of the curve, and is arrested by some obstacle, such as the falling or rising of the rail, it tends to upset out- wards, in turning around the point a of instantaneous sup- 198 MOTION OF ROTATION. port. This motion is counterbalanced by the weight P of the carriage, and at the moment when the weight and centrifugal force are in equilibrium as to the point, we have between the moments of the two forces P and F^-V^K the relation 9 which shows that, with equal velocities and weights, the stability of the car will be so much the greater, and the equilibrium better secured, as the width 2 with cars whose centre of gravity when loaded is 3.28 ft> in height, and with curves 1312 ft - radius will "be given by the relation ^K, whence YJl^K^ a velocity beyond the greatest speed of railroads. This shows that in this regard, the centrifugal force occasions no danger. But we should not forget that it brings the flanges of the outer wheels to bear against the rails, pro- ducing a cutting away which wears them out and greatly contributes to their running off the track. 174. Action of the centrifugal force, in fly-wheels. For regulating the irregularities of machines, we make use of rotating pieces of considerable weight and diameter, impressed with quite a great velocity, upon which the mo- *Morin has 47^.5; it should be 5S W .24. MOTION OF ROTATION. 199 tion of rotation developes a centrifugal'force of considerable intensity. Thus, for example, the fly-wheel of an iron rolling mill, established at the iron works of Fourchambault, weighs 13232 lb % its radius is 9.58 f % the number of turns it makes is 60 in 1', or 1 per second. "We have thus V,=6.28 ft ' in 1", and consequently, ^=6.28x9.58. If we consider a segment of the ring equal to of its circumference, corresponding to a single arm, its weight will be 2205 pounds ; and if its connection with the adjoin- ing segment is broken, the arm will experience, in the direction of its length, a traction expressed by x 6.28 2 x 9.58=25887 pounds, 32.1817 which shows that in fly-wheels the centrifugal force ac- quires a dangerous intensity, and that it is well to give great solidity to their connections. The velocity of rota- tion of these machines should be confined within certain limits. If, for example, we were to impart to the above fly a double velocity, or 120 turns in 1', the centrifugal force of the segment just considered would be four-fold, or equal to 103548 pounds. 175. Application to the motion of water contained in a vase turning round a vertical axis. In this case the liquid molecules are simultaneously subjected to the ver- tical action of their own weight, and to a centrifugal force developed horizontally; in order that they shall be in equilibrium under the action of these two forces, it is requisite that the resultant of these two forces should be normal to the surface assumed by the fluid mass, for if 200 MOTION OF ROTATION. this resultant was inclined to the surface, the molecules would yield to its oblique action. Let us consider a molecule m with the weighty and mass -, sit- y uated at the distance mp=E, from the axis of rotation AC. In a horizontal direction and perpen- dicular to the axis, it will be im- pressed with a centrifugal force expressed by Let us take mDp, and construct the parallelogram mBED, whose diagonal normal to the surface assumed by the fluid intersects the axis at i. The similar triangles mpi and wBE give us whence mB or : BE OY p : : mp or E :pi, . g Thus the distance^', which is called the subnormal, depends only upon the constant number ^, and the angular velocity supposed also to be constant. Consequently this distance is constant, which, according to the known prop- erties of the parabola, shows that the generating curve of the surface of the level is a parabola whose summit is at the point O, and whose axis is that of the rotation, and MOTION OP ROTATION. 201 we readily see that its parameter is ~-^ so long as we have pp' or 2x : mp or y : : mp or y : pi or =^- a . whence 1Y6. Surface of water contained in a bucket of a hy- draulic wheel with a horizontal axle. In following the reasoning of the preceding case, it is easy to see that, if we represent by ab the centrifugal force -^V^E, and by ad the weighty of any molecule situated on the surface, we shall have the proportion whence ah or^V x a E : be orp : : E : 01, which shows that the distance OI is constant for all points of the surface of the liquid, and that consequently this FiS. 86. 202 MOTION OF ROTATION. surface is that of a cylinder, with a circular base of radius al, whose axis is parallel to that of the wheel. This the- orem, for which we are indebted to M. Poncelet, serves as the basis of the theory which this illustrious engineer has given upon the effects of water in bucket wheels with great velocities.'* 177. Regulators with centrifugal force. The action of centrifugal force is utilised in the construction of an apparatus called a Governor. It consists principally of a vertical spindle AH, (Fig. 87,) which receives from the machine to be regulated a motion of rotation. Upon this spindle are suspended two rods AP and AP', jointed at A and terminated by the equal weights or bobs P and P'. At the two joints B and B' of the rods AP and AP 7 are jointed two other equal rods BC and B'O', forming with the first a lozenge, and which at their ends C and C' are also jointed with a collar traversed by the vertical spin- * See Lemons sur 1'hydraulique. MOTION OF KOTATION. die with which it turns, having at the same time a motion of translation, in the direction of the length of this spindle. This collar has a yoke in which is fastened the fork of a lever DE, which acts upon the throttle valves for steam, or upon any other piece. The working of this contrivance is readily understood. By the effect of the rotary motion of the vertical spindle, the balls of the regulator are thrown outwards from the axis, and so raise the collar a certain height. If the ma- chine has attained and preserves its normal velocity, the balls and the collar are held in the same position, because there is established a state of equilibrium between the centrifugal force and the weights of the different parts of the apparatus. "When the velocity increases, the centri- fugal force increases, tending to spread outwards the balls and to raise the collar, and consequently the lever DE. Inversely, if the velocity diminishes, the balls approach the spindle ; the collar and the end of the lever DE are lowered. Let us examine the mechanical conditions of the ac- tion of this apparatus, and first suppose the collar CC', as well as the rods BC and B'C', to be in equilibrium with the lever DE, so that, neglecting friction, we may regard the rods AB and AB' as free to yield to the centrifugal force which tends to separate them, and to the weight of the balls which tends to bring them nearer to the spindle. p The centrifugal force of each ball is V* x OP, and its y moment in relation to the axis of joints A is ?Y 2 1 xOPxAO. 9 The moment of the weight P of each ball in respect to the same axis is PxOP. 204 MOTION OF EOTATION. Consequently, the condition of equilibrium of each is -V* x AO=P, whence =^ 5 y if ^ which shows that the distance of the balls' separation from the spindle depends not upon their weight, but solely upon the angular velocity of rotation, and enables us to so dispose of the weight of the balls as to satisfy other conditions. If we call T the time of the revolutions of the balls around the vertical spindle, we have Y 1 T=27r=6.28, whence ,= > and consequently ^ whence which is double the duration of oscillations of a pendulum having for its height the height AO, at which the balls would be raised to the normal velocity. The above formula enables us to determine approxi- mately the height AO at which the balls are raised with a given velocity, and thus to establish their mean position. It gives, in fact, Thus for T=r AO=0.81517 ft - AO=3.2606 ft - In this calculation we have neglected the weight and the centrifugal force of the rods AB and AB'. The preceding remarks are not sufficient to insure the action of the pendulum as a regulating apparatus, since it is a requisite that it should be able to move the lever DE and the parts for the distribution of the steam or MOTION OF ROTATION. 205 water, upon which this lever operates, or in other terms, it should be able to overcome the resistances experienced in the motion of the collar, when the balls are separated or brought nearer to each other. These resistances can be estimated or measured when the apparatus is con- structed, and if we call 2Q the vertical force applied to the collar in the direc- tion of the vertical spindle, Y / 1 =(l+7z- / ) V t another angular velocity, greater, for example, than the mean velocity Y t by a fraction n' of the latter. It is easily seen that the force 2Q can be re- solved into two other forces parallel and equal to Q, applied at each of the joints B and B 7 , and that then we shall have for the equilibrium corresponding to these new conditions, at the instant of its being broken, the relation ?V'\ x OP x AO=P x OP+Q x BO 7 . g Calling a the distance AB=AB 7 and 5 the length AP AP' of the rods to the centre of the balls, we remark that I : a : : OP : BO 7 , whence B0 7 =|. OP, and consequently PV /2 n >.AO=P+Q. 9 l We have previously found that the value of AO cor- responding to the mean position of the balls was the above relation becomes, then, 206 MOTION OF ROTATION. whence we derive a so long as n'* is very small compared with n'. We also see, then, from these considerations, due to M. Poncelet, that there exists a necessary relation between the ratio of the weight of the balls to the resistance and the degree of regularity of which the apparatus is suscep- tible. We see, also, that for a degree of regularity desired or considered as necessary in the operation of the machine, the weight of the balls increases proportionally with the resistance which the collar opposes or experiences. Then for example, if we have the proportions a= 0.666, and if we have n'= =0.02, we find oU P_ 0.66 Q~2^002~ so that, if the resistance of the collar was only 22.05 pounds, the weight of each of the balls should be This result shows that this apparatus cannot give a great degree of regularity to machines, without great dimensions and weights, if we would overcome, directly by the collar, considerable resistances. It is from a disregard of these circumstances, that many constructors have failed in the establishment of this kind of regulators, made for the purpose of raising sluice gates, or in fixtures for the distribution of steam. This serious inconvenience may be avoided ; and, with this sim- ple and solid apparatus, we may obtain a proper regulation by arranging it in the following manner, which I will de- scribe for the case of a hydraulic wheel. MOTION OF ROTATION. 207 178. Distribution of a Regulator with centrifugal force. The vertical spindle of the regulator bears a coni- cal wheel aa', geared with a wheel W of the same diame- ter, fixed upon a horizontal axle cc'^ which consequently makes the same number of turns as the spindle AH. This axle carries two wheels with gentle friction, toothed with FIG. 88. conical gearings ee' and ff provided with a shield with catches, which can engage with the collar dd r movable upon the axle cc f in the direction of its length. It follows, from this disposition, that these two wheels ee' andj^' do not partake of the motion of rotation of the axle the action of the regulator must have a certain duration. In closing suddenly all the gates of the mills upon the same canals, for obtaining a rapid rise of the levels, the velocity may be increased to 11.1 turns in 1 minute. The maximum departure above the mean velocity thus produced has been as high as 1.3 turns in 1 minute, or - of the mean T .5 velocity, and it requires from 50 to 60 seconds' action of the collar to regain the velocity of 10.1 turns in 1 minute, at which it is disengaged. When, by reason of an increased resistance, or a low- ering of the level, the velocity of the wheel has reached 9.5 turns in 1 minute, the collar engages with the wheel ee f , which then drives the wheel gg' and the screw in an opposite direction, and lowers the gate, and so restores the velocity to its mean value. On lowering suddenly the level in the upper reach, the velocity of the wheel may be as low as 8 turns per minute, which corresponds to a departure of 1.8 turns per minute of - of the mean value. But by the action of 5.5 the regulator, the wheel is restored to the velocity of 9.5 turns per minute at the end of 30 seconds. We see by these details that this regulator, by the delicacy of its operations, may be employed very usefully in many cases. 212 MOTION OF ROTATION. 180. Comparison of the data of experiment with the formula. The results of direct observations made upon the governors at Bouchet's powder-mills, give us the means of verifying the exactness of the formula of No. 1YT. We have seen, beyond a velocity of 10.1 turns in 1 minute, exceeding the mean velocity of , that the collar 32 engages with the wheel ff, to check the speed, and is afterwards disengaged, when from a gain of speed it tends to return to its normal value. Now, the normal velocity of the regulator is 48 turns of its vertical spindle, and for an increase of =n', itbe- 32 comes 49 turns per minute, which corresponds to an angu- lar velocity of fi 28 '- x49.5=5.18 ft - per second. We have then OP=1.062 ft - nearly. Consequently, the centrifugal force of each ball be- comes p 4.fi n** - Y: x OP=-^?- x (5.18) 8 x 1.062 ft -=41.49 lb % which, for the balls, gives us an effort of 82.98 lb % very nearly equal to that indicated by direct observations for raising the collar and the gearing lever. 181. Modification of the balls for obtaining a greater regularity. If we wish to impart a greater sensibility to the machinery, it will suffice to increase the balls, know- ing that for the regulator in consideration, the effort to be exerted by the balls is 82.98 lbs - or 41.49 lbs - for each. If the degree of regularity n' is to be - of the mean veloci- 50 MOTION OF ROTATION. 213 ty, the number of turns of the axle, at the commencement of the action of the balls, should be 48+g=48.96; consequently, V^ x 48.96 =5.1244 ft - Admitting that OP=1.062 f % we shall have for the de- termination of the new weight P of the balls whence x (5.124) 2 x 1.062=41.49% .. = g> (5.124) 2 x 1.062 182. Observations upon the transmission of motion ty the endless screw to the gate. In the first applications of this kind of regulators, it was observed that their play was in- cessant, and that the gate was raised or lowered continu- ally, so that the action of the wheel was quite irregular, and was far beyond or short of its mean velocity. This inconvenience, experienced also in many other applica- tions, has caused constructors to regard the governor as defective, and more injurious than useful for regularity of motion. But on examining more carefully its action, and that of the gates, I discovered that the ceaseless vari- ations arose from imparting too great motion to the gate, so that it rose and fell too suddenly, and always beyond the adopted limit of mean action ; and thus irregularities, inherent upon the mode of action of the machine upon variations in the level of the water, &c., were constantly added to the already exaggerated motion of the gates. I changed the motion of transmission to the gates so as to reduce considerably the motion of the latter, and thus 214 MOTION OF ROTATION. confined these variations of velocity within convenient limits. Similar observations were made at one of the great sharpening establishments at Chatellerault, upon the effect of a regulator applied to a turbine ; and the working of this motor has been very well regulated by establishing similar proportions for the transmission of motion of the screw to the gate. According to these observations, it was admitted, that when the vertical spindle of the regulator, and that of the endless screw, were regulated to a mean velocity of 48 turns in 1 minute, the gates of the weirs for the side wheels should not be raised or lowered more than from .033 to .049 ft - for 10 turns of the endless screw, and that for turbines, Fontaine's, for example, the run of the gates should not exceed from .005 to.006 ft - for 10 turns of the screw. In satisfying these conditions, by suitable proportions in the transmission of motion from the screw to the gate, the deviations of velocity were confined within sufficient limits. 183. Indispensable disposition in the use of these regu- lators. The gate of the hydraulic motor to which is ap- plied this kind of regulator being driven by the motor itself, we see that for gates in weirs, which are to be low- ered when the motion slackens, and for gates with water upon their summits, which are to be raised when the resistance increases, the run of these organs being neces- sarily limited, it is important to stop the action of the regulator before these limits are attained, else some rup- ture might occur. It is necessary then to arrange some disengaging con- trivance, to interrupt the action of the endless screw upon the gate, as soon as it is upon the point of attaining the limit of its course. MOTION OF ROTATION. 215 184. Modification of the apparatus just described. To render this apparatus more sensible to the small variations in velocity, M. Delongchamp, Civil Engineer, proposed to transmit the motion to conical wheels, by means of a belt, passing from one pulley, always loose, over two others placed on the right and left of it, which were fastened upon the axle of bevelled wheels. The action of the pen- dulum was then reduced to passing the belt from one pulley to the other, which only required a small effort, and avoided the shock of the, clutches of the collar, in the other disposition. Thus modified, one regulator might answer for a great number of different cases. 185. Other regulators. There are other contrivances constructed for the same purpose as the preceding, which we have mentioned only to show an example of the cen- trifugal force. In passing, we will only allude to the Molinie regulator, that of M. Lariviere, and that of Siemens, based upon the use of the conical pendulum of Huyghens, &c. Good results may be obtained by them, but this is not the place to discuss them. 186. Variable motion around an axis. We have seen from what precedes, that in the motion of rotation about an axis, the work of all the external forces soliciting the body is equal to the work of their resultant. We may then consider all these forces as replaced by this resultant. If the motion is uniform, it is evident that the work of all the forces tending to accelerate the motion will be equal to that of the forces tending to retard it, or that the work of the resultant will be zero. But if the work of this resultant is not zero, there must necessarily be pro- duced a certain variation in the velocity of the body, and then the inertia of each of the elementary masses com- posing it develops in an opposite direction, efforts propor- tioned to the degrees of velocity either imparted to or taken from it. 216 MOTION OF ROTATION. Let us call V, the angular velocity, or the circular space described by a point at a unit of distance from the axes, during a unit of time, so that at the instant consid- ered the motion may be uniform, and v r the elementary variation which this velocity expe- riences in the element of time t, any elementary mass m situated at a dis- tance r will be impressed with the velocity Y^, and the elementary va- riation of this velocity will be v^r. Consequently, inertia, by its reaction, will develop in a direction opposite to the work of the resultant of the exterior forces, an effort m directed tangential to the t circumference described by the mass m. Now, if to each of these elementary masses m is applied, in a direction opposite to the variation of motion, a force equal to 0) />* m , and in the direction of the reaction of inertia, this t force will be able to destroy the variation of velocity sy, and consequently the effect of the general resultant of the external forces upon the mass m ; then these forces combined will destroy the effects of the general resultant of the ex- terior forces, and consequently they will be in equilibrium. Now, these forces which we have supposed applied to each of the molecules of the body, are precisely equal to the reactions developed by inertia, and have the same direction. There is then also at each instant of the varia- tion of motion, an equilibrium between the reactions and the resultants of the exterior forces, or, what amounts to the same, the work developed by all the reactions must be equal to the work developed by the resultant. The elementary arc described by the mass m being a^r, calling a v the elementary arc at the unit of distance, MOTION OF KOTATION. 217 the work developed by the force of inertia during the variation of the velocity will be for the mass m v.r m.-L.a?. Now, ^Yj? 1 , or the velocity possessed by the mass t m at the instant considered ; then the elementary work of the force of inertia of the mass m has for its expression mv.r x V 1 r=mr i V l v l . The product of the mass, by the square of its distance r from the axis of rotation, entering in this expression, is called the moment of inertia of this mass. For another elementary mass w', situated at a distance /, we shall have for the work of inertia mV^V^, and for any similar number of masses of elementary work, the sum of the quantities developed by their inertia will be V " + &c. 187. Important observations upon the moments of iner- tia. Geometry teaches us how to calculate the sums of the moments of inertia of the elements of different formed bodies, as we shall show further on ; but, for the present, it is well to explain an im- portant theorem as to the moment of inertia of a body in its relation to any axis when the moment of its in- ertia with respect to a paral- lel axis passing through the centre of gravity of the body is known. Let us consider an ele- mentary mass m of a body whose centre of gravity is G, FIG. 91. 218 MOTION OF ROTATION. and which turns around an axis A. The moment of iner- tia of this element, in respect to the axis A, will be But if we call AG^^Z the distance of the centre of grav- ity from the centre of rotation, and Gm=r 1 the distance of the molecule from the centre of gravity, we have by the triangle Aaw, found by letting fall ma upon AG pro- duced, Am 8 = K.c?-\- f mc< ) or since Aa a =AG 2 +2AG x or r* The moments of the mass m is then % .md. a formula in which we remark that m . aG is the moment of the mass m, in respect to a plane perpendicular to the line AG, and passing through the centre of gravity. So for the other masses m', m", &c., situated at the distances r', r", r" f from the axis A, and at distances r/, 7 1 /', //", from the centre of gravity, we shall have 'd. a'G, And consequently, calling I the total amount of inertia in respect to the axis A, and I 1 =m/* 1 8 +mV/ 2 4-m / V/ /2 +&c., the moment of inertia in relation to the parallel axis pass- ing through the centre of gravity of G and M, the total mass of the body = m-\-m f +m"+&G. ) we have MOTION OF EOTATION. 219 Now, the term in the parenthesis is the sum of moments of the elementary masses composing the body, in respect to a plane passing through the centre of gravity ; it is then zero, and the above relation is reduced to which expresses that the moment of inertia of a body, in respect to any axis, is equal to the moment of inertia of the same ~body in relation to an axis parallel to the first, and passing through the centre of gravity of the body ', plus the product of the mass of the body, by the square of the distance of the two axes. 188. Principle of vis viva in the mot^on of rotation about an axis. It follows, from what has been said in 'No. 186, that in calling I the moment of the total inertia of the body in consideration, the work developed by inertia during the elementary valuation v l of the angular velocity will be 1^^ and we have seen this quantity should be equal to the work developed in the same time, by the re- sultant of the exterior forces ; which otherwise may be apparent, in observing that, if the work of forces of inertia was inferior to that of the resultant, in taking the first from the second, the excess would produce an acceleration or reduction of the velocity other than that which really takes place. We have, then, at any instant, I.V^^K.a^, in calling E the resultant of all the external forces, and r, its arm of lever. At the end of a certain time, the work of this variable K, variable or constant, may be obtained, either directly or by Simpson's method, and may be represented by W. As for the total work of the forces of inertia, the factor I depending solely upon geometrical dimensions, and the material of which the body is composed, the sum of all the similar quantities of work, from the instant or the po- 220 MOTION OF ROTATION. sition where the angular velocity is Y 1? to that where it has reached Y/, will be, from what we have previously seen, represented by if the motion is accelerated, R being a motive force, or by if the motion is retarded, R being a resistant. Consequently, at the end of any time, when the angu- lar velocity shall have passed from the velocity Yj to the value Y/, we shall have between the quantities of work developed by the external forces or their resultant, and by the forces of inertia, the relation We would remark that, I being the sum of the elementary products rar 2 , mV a , &c., we have an expression in which mr*V*, mVYj 2 , are evidently what we have hitherto called the vis viva of the masses m, m f , &c. ; then lY^, IY/ 2 , are the sums of the vis viva of the body, and the above relation shows us that, in the motion of rotation, as well as in that of translation^ the work developed by the exterior forces at the end of a cer- tain time, is equal to the half of the vis viva acquired or lost ~by the body during the same time. We see, then, that the principle of vis viva, previously demonstrated for the parallel motions of translation, is also true for the motions of rotation about an axis. Now, as any elementary motion, velocity or work can always be decomposed into two elementary motions, velocities or works, the one of translation in the direction MOTION OF ROTATION. 221 of a certain axis, the other of rotation, perpendicular to this same axis, and as in this decomposition the square of the resultant velocity is equal to the sum of the squares of the component velocities, and as the sum of the com- ponent living forces is equal to the resultant vis viva, and as the resultant work is equal to the sum of the compo- nent works, it follows, evidently, that in any motion the work developed at the end of a certain time by exterior forces is equal to half of the variation of the correspond- ing vis viva during the same interval. Such is the enun- ciation of the princi- ple of the vis viva in its most general form, and it serves as a base for the general theory of machines and of the motions of bodies. Before applying this principle to the motion of machines, we will make use of it in the study of the motion of pendulums and of ballistic pen- dulums in particular. 189. Theory of the Pendulum. For a first application of the preceding principles, we will attend to the theory of the pendu- lum; and first suppose that we consider the FIG. 92. motion of an elementary mass suspended to an infinitely 222 MOTION OF ROTATION. thin wire, and search out the various conditions of the motion of this contrivance, which we term a simple pen- dulum, and which would also be found very nearly in the same circumstances with a lead ball suspended upon a very fine silk thread supposed to be rigid. Suppose the pendulum, starting from the point B, has arrived to M, and has consequently fallen a height MP=H, the work developed by gravity will be mgH 9 and if we call Y the velocity of the mass m, in the direc- tion of the tangent to the circle described, its vis viva will be mY 3 , and, according to the principle of the vis viva, we shall have or v =: The velocity Y of this variable motion has also for its o expression the ratio - of the elementary arc described in the element of time i; the above relation is then re- duced to s a a 2^H; whence f= or 't= If we compare this pendulum, whose length is AB=/*, with another whose length is AB'=/, which describes an equal angle, and is placed, when the velocity imparted to heavy bodies in the first second of their fall is g ', we shall also have We shall have, then, for these two pendulums the pro- portion S* '* fit":: ^ MOTION OF ROTATION. 223 But the condition that the angle described by two pendulums may be equal give us for the same elementary angular displacement, s i s' : :r:r', or 8* : s'* : : r* : / 2 , and, moreover, we have H : H' : : r : /, whence consequently, * . JL r r f H'H" ' . - ..-.- whence ct w* We would observe that the ratios y - and -, being giv- t/ en and independent of the described angles, it follows that the elementary times, employed to describe the ele- mentary arcs s and s', are in constant ratio, and that con- sequently it is the same for the sum of the elementary times of the total times T and T' employed. in describing an entire oscillation. We have then, also, g Such is the relation between the times of the oscilla- tions of simple pendulums in different places, and for different lengths. If we compare pendulums of the same length, we have r=r f , and then 224 MOTION OF ROTATION. which shows that the duration of oscillations of pendu- lums of the same length, at different places of the earth, are to each other in the inverse ratio of the square roots of the values of g, and may serve to determine the latter. At the same place we have g=g', and then whence it follows that then the times of oscillations are to each other as the square roots of the lengths of pendu- lums, as Galileo had discovered, by direct observation, before the making of the theory. 190. Time of oscillations of a pendulum with small vibrations. If, in the relation t _ _ _ we seek to introduce the value of the elementary arc $, described in the instant t, as a function of the data of the figure, we have by the similar triangles MQ1ST and MAF, (Fig. 92,) MN : QN : : AM : ME, or * : QN : : r : ME; whence 'ME" Now, ME is a mean proportional between CE and 2r CE, 2r being the diameter of the circle described by the pen- dulum ; we have, then, E= V2r x CE-CE 2 . But when the amplitude of oscillation is very small, we may neglect the square of CE, or the sagitta of the de- scribed arc, in its relation to the product 2r x CE, which reduces the above value to MOTION OF ROTATION. 225 We have then t _ s QN.r whence ' CE But if we describe upon CD as a diameter a circle, and if we draw the parallels Mm and !NV& to the chord BD, we shall have ^E 2 =CE xDE=CE xH, which gives E Now the similar triangles mOE and mqn give 7i or QN : mE : : mn : mO ; whence QN_ mn and consequently whence We see, then, that the infinitely small time employed by the pendulum in describing the elementary arc MN is equal to the product of the constant factor _ __ g'mtf by the element mn of the circumference of the circle de- scribed upon CD as a diameter. Then the sum of all the elements of time successively employed in describing the arc BC will be equal to the same factor multiplied by the 15 226 MOTION OF ROTATION. semicircle of which DC is the diameter or mO the radius, which is equal to 7r7^O=3.14. mO ; we shall have then, for the total duration of the semi- oscillation, g' mO and for the entire oscillation Such is the formula which gives the time of oscillation of the simple pendulum. From this we deduce the ve- locity imparted to heavy bodies by gravity in the first second of their fall, which shows how the knowledge of the duration of small oscillations of a simple pendulum of known length may serve to determine the value of the number g. But the simple pendulum is only an abstraction, and is only used as an approximate method for measuring time by the duration of the oscillations of lead balls or other heavy bodies suspended upon a thread, which appa- ratus is considered as a simple pendulum, the mass of which is collected at the centre of its figure. In ordinary cases, for the pendulums of clocks, and with still better reasons for those employed in the determi- nation of the velocities impressed by powder upon pro- jectiles, which are therefore called ballistic pendulums, we must take into account the distribution of the mass. 191. The compound pendulum. Let us consider, then, a solid body turning or oscillating about a fixed axis, and regard the various conditions of its motion. MOTION OF ROTATION. 227 Calling, as we have already done, I the moment of inertia of the body in respect to its axis of rotation, and V 1 the angular velocity at an instant when its centre of gravity has fallen the height H, we shall then have, by the principle of vis viva, and if we call d the distance of the centre of gravity of the pendulum from the axis, and Hj the height which a point at the unit of distance has fallen, the proportion H, i l ft - : : H : d, whence 11=11^, and the above relation becomes This relation is of the same form as that presented by the simple pendulum, and only differs in the factors -y- which depends solely upon the dimensions and the nature of the body. We have also for the angular velocity Y a = -, which t leads to the relation Keasoning here precisely as we have done for the sim- ple pendulum, and supposing the amplitude of oscillation very small, which admits of our neglecting the influence of the resistance of 'the air, we shall see that the fraction mo mo 228 MOTION OF ROTATION. by reason of ^=1"- ; whence it follows that the duration of an elementary fraction of an oscillation has for ex- pression 2 ~M.dg mo ' and that the total duration of the oscillation is T=' . .dg 192. Length of 'the simple pendulum which makes its oscillations in the same time as the compound pendulum. If we compare the formula of the simple pendulum with that of the compound, we see that, in order that the times of oscillations may be equal, we must have p which gives for the length sought of the simple pendulum I 193. Determination of the moment of inertia of a compound pendulum. When, in the formula we know the total mass of the pendulum, and the distance d of its centre of gravity from the axis of knife blades, or the suspension, observation of the duration T of the oscillations will give for the moment of inertia in respect to the axis MOTION OF ROTATION. 229 in which we may dispense with the calculation (quite laborious in many cases) of the moment of inertia. This formula will be found peculiarly applicable in the determination of the moments of inertia of fly-wheels, ballistic pendulums, &c. It will suffice to make them oscillate around any axis, placed at a known distance from their centre of gravity, in drawing them slightly from the vertical, and observing the duration of their oscillations by counting their number. Let us farther bear in mind (No. 187) that in calling I, the moment of inertia in respect to an axis passing through the centre of gravity, and parallel to the axis of suspension, we have the relation, which gives, at our need, the moment of inertia in respect to an axis passing through the centre of gravity. It follows, also, that the length of the simple pendulum which makes its oscillations in the same time as the compound pendulum, and which we will designate by &, has for ex- pression ,_ I , ~~ + -Hi and that it is always greater than that of the centre of gravity from the axis. In placing upon the line AG, which joins the axis to the centre n of gravity, a length -^i. FIG. 93. all the points which can be found upon the line parallel 230 MOTION OF KOTATION. to the axis, drawn through the point O, may be regarded as the centres of so many simple pendulums, whose oscil- lations are made in the same time with those of the com- pound pendulums. This point O, thus determined, is called the centre of oscillation of the pendulum. It is well to remark that the point A will be recipro- cally the centre of oscillation of the same pendulum if the centre O becomes the point of suspension. In fact, if we call df the distance OGr of the centre of gravity from the point O, we shall have for the distance W of the new cen- tre of oscillation from the axis O, But whence we derive and consequently Jc'=7c d+d=L 194. Determination of the centre of gravity of corn- pound pendulums. This operation is done by calculation, or by the means pointed out in No. 14:2, or by a combi- nation of the two methods. Sometimes, for ballistic pen- dulums whose weight may reach several thousands of pounds, we adopt the following method : We fasten to the cannon or receiver, at any point of their suspension, a cord passing over a pulley, on which we hang a weight, which holds the pendulum at a de- terminate inclination. The pulley should be large, its axle small and well oiled, so that friction may be disre- garded, and the certainty insured against the commission of any palpable error in the calculation. The friction of the knife blades, which only roll upon their cushions, MOTION OF ROTATION. 231 may be neglected. We know, moreover, and can de- termine at the start, the position and the traces of the vertical plane, which contains the centre of gravity, when the ap- paratus is free. It is easy, then, to measure the inclination which this plane takes under the action of a given counterpoise. This done, call p the weight of the pendulum ; d the distance sought of its centre of gravity from the axis of the knife blades ; a the inclination of the plane which passes through the centre of gravity, and through the axis of the blades, with the vertical ; L the perpendicular let fall from this axis upon the direction of the cord ; T the tension of this cord, and we have the relation whence t =p . d sin a ; -&; ^ . sin a 195. Centre of percussion. When a body (Fig. 95) re- ceives a motion of rotation around the axis A, which we suppose here as perpendicular to the plane of the table, each elementary muss of this body develops partial forces of inertia, perpendicular to the respective distances of 232 MOTION OF EOTATION. each of them from the axis whose intensity is measured by rar-, according to the notation adopted in No. 186. t The moment of each of these forces in respect to the axis of rotation is mr* . - -, and the sum of all the similar moments /y has for its value I . - 1 . t If we decompose each partial force m . r- into t two others, the one hori- zontal and the other vertical, and call x and y the abscissa and ordi- nate of m in relation to a vertical plane and a horizontal plane passing through the axis A, the first component will evidently be FlG - 95 - and the second will be v. y mv, v, x mv, -> -= l t T t If we call a?! and y 1 the co-ordinates of the centre of gravity of the body, we shall have, in making separately the sum of all the horizontal and vertical components, according to the theory of parallel forces, MOTION OF ROTATION. 233 and [mse+m'a/+ . . . ]= t t whence it follows that the resultant of these two groups of rectangular forces is and that it makes with the horizontal axis and the verti- cal axis of co-ordinates, angles whose cosines are respect- *?/ * ively ~ and -^, so that it is perpendicular to the distance du cL d of the centre of gravity from the axis. This granted, if we designate by O the point of appli- cation of this resultant F= M . d, its moment will be t equal to the sum of those of all the forces of inertia of the body, and we shall have T' whence A0=~ M..d The point thus determined is called the centre of per cussion. It is such that a force capable of producing in an element of time, the variation of angular velocity v l9 and which, when applied to this point, shall be precisely equal to the resultant of all the forces of inertia, of the different elements of the body. Thus the pressure, or, as we commonly say, the percussion upon the axis, of this force and this resultant, being equal and directly opposite, will be zero. Then, reciprocally, that this pressure may be zero, the exterior force producing the variation of motion, must pass through the centre of percussion, so that no shock 234 MOTION OF ROTATION. may occur upon the knife blades or upon the axis of rotation. We would observe that the distance of the centre of percussion from the axis is the same as that of the centre of oscillation, and that these two points merge into each other. This is the reason that, in ballistic pendulums, they are so arranged as to receive the action of the pow- der, or the shock of the projectile, precisely at the height of the centre of oscillation. 196. Theory of the ballistic pendulum. In the recep- FIG. 96. MOTION OF ROTATION. 235 tion and testing of powder, we generally make us of a contrivance known by the name of ballistic pendulum, (Fig. 96,) the inventor of which was Kobins, a celebrated English Professor of Artillery, and which has lately,- in France, received material improvement. The ballistic pendulums used in the French powder magazines, whether for trials of guns or cannons, are com- posed of a cast iron receiver, suspended in an iron frame. This receiver contains soft or compressible matter, capa- ble of receiving and deadening the shock and the velocity of a projectile without any rupture of the receiver. The firing takes place at the height of the axis of the receiver, which is horizontal. We will here, as in the " Aide-Memoire des officiers d'artillerie," call B. the radius of the arc described by the index along the graduated limb, showing the angles of the recoil : i the distance of the point shocked, or point of impact from the horizontal plane of the knife blades ; k the distance of the centre of oscillation from the horizontal plane of the knife blades ; p the total weight of the loaded pendulum, that is to say, including the buffers pr barrels full of sand, for can- nons, or the block of lead or wood for guns ; d the distance of the centre of gravity of the loaded pendulums from the line of the blades ; 5 the weight of the projectile ; c the chord of the arc of recoil ; Y the velocity of the projectile at the instant of con- tact with the receiver ; Y! the angular velocity imparted to the pendulum after the shock. We must first remark that during the shock there is developed, at the point of contact of the projectiles and receiver, efforts of action and reaction, equal and directly opposite. 236 MOTION OF EOTATION. The action exerted upon this receiver accelerates its motion, and, from what precedes, the moment of this force in relation to the axis of rotation should be equal to that of all the forces of inertia of the material molecules com- posing the pendulum. In continuing to call v 1 the small increase of angular velocity imparted to the pendulum during the element of time t) the resistance of an elementary mass m, situated at the distance r from the axis will be expressed by mr. ] its moment in relation to the axis will be wr 2 . '; t t AJ the sum of all the similar moments will be I . - 1 , and should t ", be equal to the moment of the effort exerted at the same instant by the projectile. But, on the other hand, the projectile, acting perpen- dicularly at its distance i from the horizontal planes of the blades, loses in an element of time a small degree of velocity v, and its inertia, which is the same for all the points which are impressed with velocities very nearly equal and parallel, occasions a motive effort expressed by -. -, the moment of which in relation to the axis of the ff * , , , . ~b . v blades is -&.-. g t Thus, at any instant of the shock, we must have, be- tween the actions developed by the projectile and the reaction of the pendulum, the relation or I. T -^.v=Iv l . 9 In establishing analogous relations between all the ele- mentary degrees of velocities, lost successively by the MOTION OF KOTATIOST. 237 projectile and gained by the pendulum, we shall have, in adding them, Now, the sum v+v'+v" + &c., is evidently equal to the total velocity lost by the projectile, from the moment it struck the receiver with the velocity Y, to that when, having lost all relative velocity in respect to the receiver, it partook of a motion in common with it equal to Y^', in calling Y! the angular velocity imparted to this body ; we have then On the other hand, the receiver starting from repose, and acquiring by the shock the final angular velocity Y 1? we have The above relation becomes then v y j snce =*- 9 "We deduce from this expression w&Sf and we have elsewhere seen that we must have i=k in order that no shock should be produced. But on the other hand, when the pendulum recoils, its centre of gravity is raised, and its vis viva, as well as that received by the projectile, being soon extin- guished, should be equal to double the work developed 238 MOTION OF KOTATION. by gravity and by the friction of the rolling of the blades, which we have neglected. The angle described by the pendulum being a, it is clear that its centre of gravity is raised by the quantity d <#cos a=d (1 cos a) = The projectile was at rest at the distance i from the axis of rotation, and has been raised the height ^ ^ cos a= v ' then the work developed by gravity upon the pendulum and the ball has for expression (pd+li) 2 sin 2 -#. The vis viva possessed by these two bodies at the end of the shock, or of their reciprocal reaction, is \7 V We have then L* 9 whence -rr /( Vd+bi) Q c. . V = A / -^- ! 2. . 2 sin , 2 Making the value of 'V l equal to the preceding we have Kir from which we deduce li 2 MOTION OF ROTATION. 239 Such is the formula which serves to calculate the initial velocities of projectiles by means of the data within it and of the angle of recoil. We would remark, that in calling C the chord of the arcs of recoil, whose radius is R, we have 2sin-a-- which gives y = V(pdk+W)(pd+li)g C U ' R' This is the form given in " Aide-Memoire d'artillerie" We have seen that the conditions of having no shock upon the blades led us to that of i=Jc. If it was com- pletely satisfied, the above formula would be reduced to which shows that then the measured velocities would be proportional to the chords of the arcs of recoil. But this condition which has been nearly attained in the construction of the new pendulums for cannons at Metz, and Yincennes, and of Bouchet, is by no means satisfied in the pendulums for guns, and this accounts for their so sensible vibrations. Officers of artillery will find farther details upon these contrivances in the instructions for semi-annual trials of powder with ballistic pendulums. GENEEAL APPLICATION OF THE PKESTCIPLE OF VIS VIVA TO MACHINES. 197. Application of the principle of vis viva to ma- chines. In applying this principle to the motion of ma- chines, we must examine separately the circumstances and conditions of action of the different forces to which they are subjected. These forces may be classified as follows : 1st. The powers which produce, maintain, or acceler- ate motion, and whose work, which we shall designate by F . S, is always developed in the direction of the motion, and is consequently positive; we designate by F the mean effort of the resultant. 2d. The useful resistances which must be overcome or destroyed to produce the effect proposed or the work which the machine is to do, and which destroy, retard, or moderate the motion. The work of these resistances, which we shall designate by Q . S', is always developed in an opposite direction to that of the powers and must be subtracted. 3d. The prejudicial or passive resistances, existing in motion such as frictions, the resistance to rolling, that of the air, of water, &c., which absorb unprofitably a portion of the motive work, and retard, moderate, or destroy the motion, and whose work we shall represent by E . S", is always to be subtracted from that of the powers. 4th. The action of gravity, which should be regarded PRINCIPLE OF VIS VIVA APPLIED TO MACHINES. 24:1 separately whenever it acts, sometimes as a power, some- times a resistance, and its work, represented by P . H, will be positive or additive in regard to that of the pow- ers in the first case, and negative or subtractive in the second. But, when gravity acts always as a power, as in hydraulic wheels, clock-weights, &c., it should be reck- oned among the powers ; and inversely when it acts as a useful resistance, as in machines for raising weights, &c., it should be joined with the useful resistances. With this classification of forces, the principle of vis viva will be represented by the equation [V/ a - Y, 2 ] :=FS-QS'-K . S"'PH, a in case the machinery is composed of rotating pieces ; or in general hi [V V]=FS-QS'-K . S"PH ; the expressions IY /2 , MY /a , etc., representing the sum of all the analogous vis vivas of the parts of the machine. This relation refers to a finite interval of time, and we have seen that for an element of time, or an infinitely small displacement, we have also The aim in the establishment of every machine, being to overcome a useful resistance, or to do a certain work, it is evident that it is the work QS' or Qs' of these useful resistances, which should be rendered the greatest possi- ble, or a maximum ; if we deduce from the above relation the value of QS' we have 16 242 PRINCIPLE OF VIS VIVA APPLIED TO MACHINES. or for the element of time 198. Conditions of the 'maximum of effect of ma- chines. Let us examine successively the conditions which should be satisfied, for a maximum of useful work. 199. Work of powers, W& remark, first, that for each kind of motor or of power, there is a maximum effort cor- responding to a velocity zero, for which the work is zero, and a maximum velocity corresponding to an effort zero, where also the work is nothing. Thus for animal motors the effort and the velocity have absolute limits, for which the one is zero, when the other is a maximum. It is the same for hydraulic wheels, for which the effort is a maxi- mum when the velocity is zero, and at its minimum when the velocity is the greatest which the water can impart to its course in open space. It is also the same for steam machines, wind-mills, &c. Between these limits there is a certain velocity which, for each motive power, according to its nature and its combination of mechanical parts, corresponds to a maxi- mum quantity of work, developed by the power, and as it often happens that, for the greatest or smallest veloci- ties, the work diminishes rapidly, it follows that it is very important to preserve, at the points of application of the motive power, the velocity which corresponds to its maxi- mum of effect, and therefore a uniform motion for the recipient of the power. 200. Work, of useful resistances. We make the same observations for the work of useful resistances, for, accord- ing to the nature of the tools and the products, there is a certain velocity which answers to the best quality of pro- ducts, the best effect, or the longest duration of the tools ; thus for the grinding of corn, the rolling of iron, for the PRINCIPLE OF VIS VIVA APPLIED TO MACHINES. 243 drawing and spinning of cotton, of wool, &c., there is a velocity suited to the quality and nature of the products to be obtained ; in saw-mills, the turning of metals, pumps, &c., the preservation of the tools, or the economy of work, exacts a velocity within certain limits, &c. Then, also, it is best that there should be a uniform motion for useful resistances, as well as for motive powers. 201. The work of prejudicial or passive resistances. As to prejudicial or passive resistances, the work which they consume being always expended at a loss, we must evidently seek to render them the smallest possible. It is necessary, then, to diminish the friction, and conse- quently the weight of the pieces which slide upon each other ; to polish their surfaces, to keep them well oiled, and to diminish the spaces described by the rubbing parts. For the resistance of the air or of water, we should limit the velocities, and give to the bodies the forms best adapted to lessen these resistances, etc. 202. Pieces with alternating motion. The work due to the weight of pieces alternately and periodically as- cending and descending the same height, being zero for each period, we see that there is no occasion to concern ourselves with machines whose motion embraces a great number of similar periods, if these alternatings, while in- creasing or diminishing periodically the motive work, do not produce corresponding variations in the motion, and so alter the uniformity of motion, the necessity of which has been recognized. If, then, we cannot wholly suppress the pieces which ascend or descend periodically, it would be well to limit their number and influence as much as possible, and the general condition will be to employ only pieces well cen- tred in relation to their axis of rotation, or whose centre of gravity remains at the same height. On this subject, we could show a dynamometric ex- 244 PRINCIPLE OF VIS VIVA APPLIED TO MACHINES. perimental curve, obtained upon a ventilator which, by the nature of the resistance to be overcome, should have been a uniform motion, but which, by reason of a defect in centring, presented, on the contrary, very considera- ble periodical variations. "What we have said of pieces which rise and fall peri- odically under the action of gravity refers also to pieces with alternating motion, such as the horizontal frames of saws, etc., whose variable vis viva is opposed to the uni- formity of motion. 203. Influence of the vis viva possessed or acquired at each period. We see by the equation of the principle of vis viva, that if the vis viva has diminished during the period considered, the half of this diminution represents a work, which is added to that of the motor, and that if, on the contrary, the vis viva has increased, the half of its variation represents the portion of the motive work ab- sorbed to produce it. If, then, the motion is periodical, we see that in the accelerations of motion, the inertia of the masses absorbs and stores up a portion of the motive work, which it restores in its retardations. Inertia, then, performs here truly the duty of a reservoir of work, abso- lutely like the pond, the reservoir of the hydraulic wheel, which receives and preserves the water of a stream, when the wheel does not consume all the flowing water, and, on the other hand, furnishes it, in emptying the water con- sumed by the wheel, when this wheel expends more than the supply of the source. Examples of these effects are as numerous as remarka- ble in the working of machines : thus, in the working of rolling-mills, which are set in motion before passing the iron between the cylinders, all the pieces of the machine receive an accelerated motion, and absorb a considerable portion of the motive work ; thus, when we pass the metal to be drawn, the work of resistance prevails over that of PBINCIPLE OF VIS VIVA APPLIED TO MACHINES. 245 the power, the motion is retarded, and the inertia of the masses restored, develops in favor of the motor the quan- tity of work which it had previously absorbed. It is the same in the action of walking-beams, of hammers, of the treadles of knife-grinders, &c. But as these variations of vis viva correspond with variations of velocity, it becomes us to restrain them and to limit them as far as possible, so as to obtain as near an approach to uniformity as may be. 204. Case of periodical motion. There are many ma- chines which, by their constitution,, or by the nature of their work, cannot be impressed with uniform motion. Of this number are all those where the motors or the tools act intermittently or in alternate directions, as steam en- gines, or a column of water on one side, and on the other saws, pumps, hammers, &c. In all such cases it is neces- sary to reduce the number of pieces impressed with alternating motion to what is strictly necessary, and to distribute the variations of resistance or of work among equal spaces. When, by these means, we have attained an exactly periodical motion, and when the vis viva absorbed in the accelerations is restored in the retardations, we may, in calculating the effect of an entire period, dispense with the reckoning of the vis viva, which will be zero for the entire duration of this period. 205. Advantages and conditions of uniform 'motion. But, in general, uniform motion being the most favorable to the action of motors and of tools, and occasioning less loss of work by the effect of the passive resistance, since it admits of giving to all the pieces of machines smaller dimensions, and accordingly less weight, it follows that we should try all possible means to obtain it, or at least to approximate to it. 24:6 PRINCIPLE OF VIS VIVA APPLIED TO MACHINES. It becomes us, then, to use, if possible, as organs for the transmission of motion, parts with a continuous mo- tion, with their centres of gravity resting at the same height, with wheels exactly centred, &c., to distribute the materials to be worked in a continuous manner, or at least at equal intervals, as is done with the "babillard des moulins " of mills, the claws of saw-mills, etc. 206. Inconvenience of variable motion and means of diminishing it. Besides the inconveniences which we have pointed out, relative to the irregular action of the motive power and of the useful resistance, there is another which obliges us to give to the parts subjected to it larger dimensions than those required by uniform motion for the same work, since the efforts which these pieces have to resist are, at certain instants, much greater than the con- stant effort corresponding to uniform motion. From this results an excess of weight and an increase of friction, be- sides the shocks or the more or less sensible alterations of forms produced by changes of velocity. All these inconveniences being greater as the vis viva of the pieces with alternating motion are more considera- ble, it will be requisite, after having limited their dimen- sions to what may be necessary, to make their velocities as small as possible in relation to those of the pieces en- dowed with a uniform motion, or one approximating to it. 207. Observations upon the starting of machines and the variations in velocity which then take place. The relation gives us for the elementary variation of velocity PKINCIPLE OF VIS VIVA APPLIED TO MACHINES. 247 We see that the velocity will increase when the elemen- tary motive work Fs is greater than the sum of the quan- tities of work of all the resistances ; but that, for a given excess of work, the variation, or the increase of velocity, will be so much the less as the velocity V a possessed by the body is greater, and as the moment of inertia I of masses in motion is the more considerable. Also, when the elementary motive work is inferior to the work of the resistances, the velocity decreases, but so much the less as the velocity and the moment of inertia are the greater. Rapid motions, and those in which the moments of inertia are considerable, are then the more stable , and ex- perience less alteration from the action of given causes. When a machine starts from rest, its velocity, at first zero, increases gradually, since the work of the motor prevails at each instant over that of the resistance. But, on the one hand, the motive work attains its maximum value at a certain velocity, having passed which it de- creases ; and, on the other hand, the work of the resist- ance increases often with the velocity, so that soon we have the equality ' At this instant the variation or the increase i> r of the velocity is zero, and the velocity has attained its maxi- mum. If this equality of motive work and of resistant work subsists, the nlotion becomes uniform ; but this can- not happen except the term =pPA is zero ; that is to say, that the centre of gravity of all the pieces remains always at the same height. This condition of uniform motion is in some sort self- evident, since it amounts to saying that the work of pow- ers tending to accelerate or maintain motion should be equal to that of the resistances which tend to retard or destroy it. The elementary work being, as we have already re- marked in No. 120, what is termed, in rational mechanics, 24:8 PKINCTPLE OF VIS VIVA APPLIED TO MACHINES. the virtual moment, we see that the preceding statement amounts to saying that, for uniform motion, or for equi- librium, which is but a particular case of it, the virtual moment of powers must be equal to that of resistances, or their sum equal to zero. 208. Observation relative to perpetual motion. The velocity only remaining the same when the elementary variation ^=0, we should then have ISTow, in supposing, even, the work of useful resistance Qs f to be zero, in which case the machine serves no useful purpose, that of the prejudicial resistances Ks" can never be zero, since we cannot have machines without weight, and consequently without friction. "We must then always have a certain motive work Fs to maintain the motion, which shows the absurdity of all the attempts to obtain what is called perpetual motion, or a motion self sus- taining, without the aid of any exterior motive force. 209. Periodical motion. It seldom happens that the motive work remains always equal to that of the resist- ances starting from the instant when the velocity has ac- quired its maximum value ; most usually, on the contrary, the resistant work begins then to prevail over the motive work, the variation in the velocity becomes negative, and the motion slackens. But as the work of useful or passive resistances may diminish, while, at the same time, that of the power increases, the excess of the first above the second diminishes, the motion is retarded less and less, and we have again The velocity ceasing then to diminish, it attains its mini- mum. If the diminution of the velocity does not cancel the PRINCIPLE OF VIS VIVA APPLIED TO MACHINES. 249 motion, there follows then another period of acceleration, limited by a second maximum, and so on. Machines then work, for the most of the time, with a periodical motion, sometimes accelerated, sometimes re- tarded, in which the velocity attains successively and alternately maxima and minima*; but,-these periods being accomplished usually in equal times, we substitute, as we have said, for this variable velocity, quite difficult to be determined, the consideration of a mean velocity. 210. Manner of limiting the deviation of velocity Theory of fly-wheels. After having used all ordinary means to regulate the play of machines, there remains still another to restrain the variations of the velocity, be- tween suitable limits for each case, under the action of given and alternating excesses of the motive or resistant work. In fact, if we consider the equation by means of which we express the principle of vis viva. I [Y/ 2 Y, 2 ] =2 [FS-QS / -ES // PH]=2W, we see that, for a determinate period, in which the velocity shall have varied from Y x to Y/, under the influ- ence of a given excess "W of motive work above the resultant work, the variation of the squares of the veloci- ties V /2 V 3 V 1 " V 1 ~~ "r"" will be so much the smaller as the moment of inertia of the pieces endowed with the motion of rotation, or the mass of the pieces impressed with the motion of transla- tion, are more considerable. Thus, after having by a good disposition of machines, by a symmetrical distribu- tion of the resistances, etc., diminished, as far as possible, the alternating excess of work causing the irregularity, 250 PRINCIPLE OF VIS VIVA APPLIED TO MACHINES. we may check, as far as we wish, the variation of velocity, by increasing the moment of inertia or the mass of the movable pieces, or more simply, the moment of inertia of one of them specially appointed for this end. This piece is called the fly-wheel, and is usually com- posed of a cast-iron ring of great diameter, with cast-iron arms, and is placed as near as possible to the parts of the machine impressed with variable motion, in order that their irregularities may be lessened in the transmission to the other parts. In the establishment of the fly-wheel, we usually neglect the regulating influence of the other masses which nevertheless contribute towards insuring a greater regu- larity than could be attained by the fly-wheel alone. We would h'rst remark, that the difference of the squares of the velocities, which is evident by an examination of the figure, where AB=Y/ and EF=Y,. In fact, we see that =AIHK=AIxIH=(Y 1 / +Y 1 ) (VY V,). E\ If, further, we call U the arithmetical mean FIG. 97. jffOO - 3 - between the velocities Y/ and Y 15 we remark that U will differ but very little from the mean velocity of the machine PRINCIPLE OF VIS VIVA APPLIED TO MACHINES. 251 derived from the number of turns which it makes, and which is usually given beforehand, according to the ar- rangement of the machine ; we shall have then and consequently whence If now, to obtain a given degree of regularity, we im- pose the condition that the angular velocity shall not vary over a fraction - of the mean velocity IT, we shall have and consequently U_W 7*~UI ; from which we deduce nW 1= IP We see, then, that when the excess of the motive work above the resistant work, or vice versa, is given, as well as the mean angular velocity of rotation of the shaft of the fly-wheel, and the regulator number n, we may de- duce from this simple expression the moment of inertia of the fly-wheel. We observe that the moment of inertia will be so much the smaller as the mean angular velocity is the greater, and that consequently it is proper to place the fly-wheel upon the axle whose motion is the most rapid. 252 PRINCIPLE OF VIS VIVA APPLIED TO MACHINES. In calculating the moment of inertia of a fly-wheel, we usually neglect the influence of the arms, and we have then very nearly V I= E 2 , g P' being the weight of the ring, and R, its mean radius. Moreover, we know that a and 5 being the width and thickness of the ring, R, its mean radius, and walking-beam. In these formulae, N is the nominal force in horse-pow- ers, m the number of turns of the fly-wheel in 1", Y the mean velocity of the mean circumference of the ring. The number n, according to the common practice of Watt, is usually equal to 32, for all cases not requiring extraordinary regularity. For flour-mills, saw-mills, &c., it may be diminished a little, while for spinning it may be increased as high as from 50 to 60. The fly-wheel for the spinning-mill of Logelbach affords us the following data : Diameter of fly-wheel 20.01 ft - w=19, j =35 horae power . 60 254 PRINCIPLE OF VIS VIVA APPLIED TO MACHINES. If we make 7i=40, 124091.5 40 x 35 _ -D . :_____ __ .23067 pounds. For n=35, we have P=20183 pounds. The con- structors have set P=20555 pounds. 212. flywheels for expansion engines. The irregu- larity of action of steam being very great, the fly-wheel should be increased, and I give here, for examples, for- mulae for high-pressure engines with expansion, without condensation, at five atmospheres of pressure in the boiler. nN Expansion commencing at | the stroke, PY 2 = 168089 - . PV 2 =193864 . m PV 2 =218849 in EXAMPLE. Let N=40 horse powers, expansion com- mencing at a half of the stroke, ,=32, m=16, D=26.739, y^Ux 26. and consequently 168089x32x40 213. Fly-wheels for forge hammers. In machines which work by shocks, such as trip-hammers, the irregu- larity of motion arises from the intermittent action of the resistance, and the losses of work produced by the shock. "We can submit these effects to calculation, to determine directly the loss of vis viva, and consequently limit the variations of the velocity, to a given fraction of the mean PRINCIPLE OF VIS VIVA APPLIED TO MACHINES- 255 velocity ; but this is no place to unfold the theory, and we confine ourselves to saying that it has led to the fol- lowing formulae : 214. " Frontaux " or Tennant helves hammers. From 6616 to 7718 lb % striking 70 to 80 blows, p_ 474890 From 8822 to 10S07 lb % striking 72 to 80 blows, 712213 The regulating number has been taken in these cir- cumstances equal to from 50 to 55 nearly. EXAMPLE : The " frontal " hammer of the forge at Framont, weighing 6616 lbs - and upwards, has a fly-wheel with ra- dius K=7.054 ft -, whose ring weighs 9327 lbs -, and has worked for nearly 12 years. 215. German hammers geared, Weighing from 1323 to 1764 lbs> , including " manche et hurasse," beating 100 to 110 strokes in V. At the works of the new mill connected with the foun dries at Hyange, K=5.413 ft ; P=11358 lbs - nearly. OF THE 256 PRINCIPLE OF 'VIS VIVA APPLIED TO MACHINES. 216. Geared tilt-hammers. Beating from 150 to 200 blows in 1', 794 lbs - in weight, including all, p= 142442 E a Beating from 150 to 200 blows in 1', and weighing 1103 lb % all told, -p_213664 217. Vertical saws for cutting large timber. Obser- vations show that it suffices to take p _712213 EXAMPLE Saw mill at Mete. The radius K=2.49 f % the number of turns of the saw is 88 in 1', whence we conclude CO V=|?6.28 x 2.49=22.93"- ^ The formula gives p _ 712213 lt , ~(2O8)'~ We usually place two fly-wheels, each one-half of the above weight. At the saw mills of Metz, the two fly- wheels weighed together but 1129 pounds. 218. Necessity of using fly-wheels in machines when there are shocks. A striking example of the necessity for using fly-wheels where shocks are produced, was observed in 1845, at the powder-mills of Youges, and at that of St. PRINCIPLE OF VIS VIVA APPLIED TO MACHINES. 257 Ponce, in four stamping-mills. In substituting for these new constructions cast-iron gearings, in place of the old wooden wheels, care had been taken to increase in the ratio of 2 to 3, the dimensions of the teeth and of the wheels, furnished by the ordinary rules of practice. Not- withstanding this precaution, these mills having been set at work, the wheels of the gearing could not resist the vibrations produced by the shocks, and were broken at the rims after a short service. To remedy this evil, two methods presented themselves; one, which consisted in increasing considerably the dimensions of the wheels was adopted from the necessity of the case, for the broken wheels. The other, the most rational, was to place fly- wheels upon axles with cams, to diminish the vibrations of the velocity, and consequently the shocks between the wheels and pinions. It was perfectly successful, and the gearing of the fourth mill, exactly similar to those which had been broken when there was no fly-wheel, had resisted well, with the employment of this means of regulation. 219. Proportions of fly-wheels for powder-mills with twenty stamps. The stamps of powder-mills weigh from 88 to 92.4 pounds, and beat 56 blows per minute, there being two for each turn of the shaft with cams. Experi- ence has proved that fly-wheels of 8.2 ft - diameter, .557 ft - of width at the crown in the direction of the axle, and ,59 ft - in that of the radius, were sufficient. 220. Rolling-mill for great plates and l)ulky iron. In these machines, observation shows that we may calcu- late the fly-wheel by the following formula : p_ 3086258KK . wV a N being the force in horse powers transmitted to the shaft of the fly-wheels ; 17 258 PRINCIPLE OF VIS VIVA APPLIED TO MACHINES. Y the mean velocity of the middle circumference of the ring. m the number of turns of the fly-wheel (usually placed upon the same axis as the cylinder), in V ; K a constant numerical co-efficient which we may take equal to ; K=20 for machines from 80 to 100 horse powers, and with 6 to 8 equipments of cylinders ; K=25 for machines of 60 horse powers, and from 4 to 6 fixtures of cylinders ; K=80 for machines from 30 to 40 horse powers, with one or two cylinders for great sheets of iron. EXAMPLE. D =9.3177"-, w=60, V=60.368 ft - for six cylinders working together, p _3086258 x 60 x 25^ lbs 60 (60.368) 2 The manufactory at Fourchambault, placed in these circumstances, has a fly-wheel of 17643 pounds only. When the machines to be regulated have for a motor hydraulic wheels with rapid motion, such as wheels with plane and curved floats, the moment of inertia being usually considerable, it may be added to that of the fly- wheel, which may then be somewhat diminished, espe- cially if the motor is near the resistance. 221. Use of fly-wheels, It follows from what has been said, that fly-wheels have for their object the confining of the velocity within given limits, when there is in the course of th'e pieces, or in the action of the motors or of the resistances, inequalities or inevitable alternations, or, in certain cases, to accumulate, during a portion of the periods of motion, a quantity of motive work, to be re- stored when the work of resistance prevails over that of the motor. It is, then, only momentarily that the use of PRINCIPLE OF VIS VIVA APPLIED TO MACHINES. 259 the fly-wheel can, in the last case, increase the power of the machine. But, the fly-wheel being always a heavy piece, caus- ing a useless consumption of work by its friction and the resistance of the air, we must restrain its use to cases of absolute necessity, and give it a suitable limit of weight. FKICTION. 222. We usually distinguish two kinds of friction? One, called friction of sliding, is produced when bodies slide one upon the other, whence it results that the primi- tive points of contact are found ceaselessly at distances respectively different from new points of contact, which is expressed in saying that they have experienced dis- placements, relatively unequal, and in opposite directions. The second kind of friction, improperly called rolling friction, takes place when bodies roll one upon the other, when the distances of the new points of contact from the old are the same upon both bodies, and when the relative displacements are equal. As the word friction implies, generally, the idea of sliding, and not that of rolling, it will be proper to admit only one kind of friction, that of sliding, and to designate the other by the name of resist- ance to rolling. 223. JKeview of ancient experiments. The first experi- ments known upon the friction of sliding, are due to Amontons, and are inserted in the Memoirs of the An- cient Academy of Sciences, 1699. This philosopher knew that friction was independent of the extent of surfaces, but he estimates its value at a third of the pressure for wood, iron, brass, lead, etc., coated with lard, which is far too much. FRICTION. 261 Coulomb, officer of the military engineers, and some years later a member of the Institute, presented, in 1781, to the Academy of Sciences, experiments made at Boche- fort, and much more complete than those of Amontons'. The apparatus he used consisted of a bench, formed of two horizontal timbers 6 feet long, upon which a sledge loaded with weights slid by the action of a weight sus- pended to a cord, which, passing over a fixed pulley, was attached horizontally to the sled. By means of this disposition, Coulomb at once de- termined the effort necessary to produce motion after the bodies had remained some time in contact. This is what he called the resistance or friction of departure. He saw that this friction was proportional to the pressure, and he expected to find it composed of one part proportioned to the extent of the surface of contact, which he termed ad- hesion and of another part independent of this surface. He then sought the value of friction during motion, and for this effect he observed, with a stop-watch of half seconds, the time employed by the sled in running suc- cessively the first three feet and the next three feet of its course. But as in these durations, sometimes equal to V or 2", he might be mistaken by a half second at the end, and also at the commencement of the experiment, there re- sulted very great uncertainties which did not admit of establishing his conclusions in a positive manner, and we may say he rather conjectured than observed the laws which he inferred from his experiments. Still he admit- ted, that generally, friction during motion is : 1st. Proportional to the pressure. 2d. That it is independent of the extent of the surfaces of contact. 3d. That it is independent of the velocity of motion, with some restrictions, which subsequent experiments did not confirm. 262 FRICTION. Coulomb also first established the fact, that for com- pressible bodies, the friction at starting, or after a contact of some duration, was greater than it was after the first displacement. 224. Experiments at Metz. The uncertain observa- tions, and the restrictions adduced by Coulomb, and above all the more general use of inetals in the construction of machines, called for a new series of experiments, which I made at Metz, in 1831, '32, '33, and '34:, by means of new processes. 225. Summary description of the apparatus used. In the smelting yards of this ancient foundry, upon a flag- stone foundation, and at the side of a trench, (Fig. 98,) FIG. 93. was established a horizontal bed, composed of two parallel oak beams AA, 0.98 ft - square, and 26.24"- long, connected and supported by sleepers 3.28 ft - apart. These beams, which jutted about 4.26 ft< beyond the edge of the trench, were connected with four uprights BB, between which was placed a platform FF, which bore the pulley for pass- ing the cord, to which was suspended the motive weight, FRICTION. 263 placed in a box K. This cord was fixed horizontally to a sled D, charged with weights, under which was placed the body to be experimented upon. The cord, instead of being attached directly to the sled, was fastened to the front plate of a dynamometer with a style, whose flexure measured the tension of the cord, both at its starting and during its motion. The axis of the pulley had a copper plate H, perfectly smooth and covered with a sheet of paper. Opposite this plate, clock-work communicated uniform motion to a style, formed of a brush filled with India ink, whose point described a circle 0.459 ft - in diameter. The parallelism of the plane of the circle, and that of the plate, was also perfectly established by precise methods, and the contact of the brush was produced or interrupted at will. Upon the box K may be placed two others for holding weights, which, after producing the motion, may at a cer- tain height be stopped by cleats, so that the motion con- tinues only in virtue of the load and weight of the box Q. By this means, we may at will obtain, with the box Q alone, an accelerated motion, and with the three boxes, a motion at first accelerated, then uniform or retarded, ac- cording as the weight of the box is sufficient to overcome the friction or is inferior to this resistance. Further details of these experiments may be found in the " Recueil des savants etrangers, tomes IV and V," as well as in a memoir published in 1838, by M. Carilian Gosury. 226. Examination of the graphic results of experi- ments. We may conceive, from what has been said upon a similar apparatus now at the Conservatory of Arts and Manufactures, that from the synchronism of the two mo- tions, the one of the style being uniform and with a known velocity, and the other unknown, corresponding in a con- stant ratio with the spaces described by the sled, there 264: FRICTION. must result a curve whose abstract will give us the law of the motion of the sled. "We may then, by this ab- stract, form a table of spaces described, and of the corre- sponding times, and construct a curve whose abscissae are the spaces and whose ordinates are the times. The curves thus constructed are perfectly continuous, and we see, as has been indicated in No. 81, that they are parabolas, that is to say, their abscissae are proportional to the square of their ordinates. From the fact of this curve being a parabola, we are justified in the inference that the motion is uniformly accelerated. Now the motive weight being constant, the motive force producing the acceleration of motion is the excess of this weight above the friction, and since this ex- cess is constant, it follows, necessarily, that the friction is constant and independent of the velocity. Experiments, repeated with all the bodies used in the construction of machines, with or without unguents, having always led to the same consequences, we are au- thorized in regarding this law as general, at least within the limits of the velocity of observation ; that is to say, of about 11.5 ft -, and in the assumption that the restrictions which Coulomb anticipated have no existence in reality. 227. Formulae employed in calculating the results of experiments* The apparatus which we have just described affords a simple example of a machine in which the mo- tion is variable, and enables us to apply the general prin- ciples which we have previously pointed out. We take advantage of it to show the method of procedure in similar cases. We call P the weight of the descending box, including its load and that portion of the cord which hangs always under the pulley, neglecting, however, the quantity by which it is increased in its descent, which seldom exceeds 2 lb8 - ; T the tension of the horizontal strip ; ^=13.T9 lb9 - the weight of the pulley. FRICTION. 265 V, the angular velocity of the pulley at the instant considered. Vi the quantity by which the velocity varies in an ele- ment of time t. I=. 04551 the moment of inertia of the pulley and of the pieces turning with it. f =0.164: a ratio determined by special experiments, of the friction to the pressure, for the iron axle of the pulley and the ash- wood cushions greased ; B=0.032T the rigidity of the twisted cord, determined also by especial experiments. N the pressure of the axle of the pulley upon the journals. r the radius of the pulley. r' the radius of its journals. If we refer to the principles laid down in No. 186, upon the motion of variable rotation, we shall see that at each instant of the motion of the pulley, the sum of the moments of the exterior forces must be equal to the sum of the moments of the forces of inertia. Now, the sum of the moments of the exterior forces is Pr TrRrN . r f . The sum of the moments of the forces of inertia answering to a velocity v l of angular velocity is easily found ; for, one of these forces, relative to a molecule of the mass m, qj M situated at a distance r^ being m . -i-^, its moment in re- t tf spect to the axis is mr? l , and the sum of the similar mo- u ments is I- 1 , for all parts turning around the axis. t The moment of inertia of the weight P is r, and 9 * must be added to the preceding ; we have then, at each instant of variable motion of the pulley, the relation 266 FRICTION. The pressure "N upon the axle of the pulley being the re- sultant of two perpendicular forces, the one horizontal equal to the tension T, the other vertical and equal to the weight P of the box, increased by the weight of the pul- ley, and diminished by the force of inertia - , which is 9 t developed in the acceleration of the vertical motion of the weight P, and is opposed to its acceleration ; we have then Now, according to an algebraic theorem of Poncelet, the value of a radical of the form yV+5 2 , in which we know beforehand that a > 5 is given to nearly -^j by the formula 0.96 a+QAl. In applying it to the case in hand, where we have always P+# --- ->T, since the weight ^ * P exceeds the resistance T and the friction of the sled, we have to -j nearly $T=0.96 JP +2 -^ ' 9 * The relation of the equality of moments becomes then, in making K=0.032T, .032Tr 0.96// { P+# - ^ ( 0.4//T ' t } P and in deriving from this equation of the first degree the FEICTION. 267 value of T, the tension of the horizontal strip of the cord, we find T j 1+0.032+0.4?^-' 1 =P 1 10.96^ i 0.96/^ g t \ r ) r* t In substituting for the known quantities their values, which are /=0.164, r'= 0.030512"-, r=0.36417 ft -, I=.04551, whence ^=0.34317, if we have for the practical formula which gives the tension T, when we know the weight P of the box, T=0.95 \ P (.34685+-)^ [ 0.1753 lb3 - ( v git > When experiment has demonstrated that the accelera- n\ M tion -i- is constant, and the abstract of the curves, in giv- t ing their equation T 3 =2CE, shall have furnished for the - acceleration the value =^1 in calling 20 the parameter t O of the parabola, we shall have all the elements required to calculate the value of the tension of the cord in the ex- periment. It will be T=0.95 1 P (.34685+?U I .1753 lbs - ( V g'Vy Qy-tf* When the motion is uniform the acceleration =7, t *-> is zero, and the above formula is reduced to T=0.95P 0.1753 lb % 268 FRICTION. or simply T=0.95P, on account of the small value of the second term .lT53 lbs - In extracting directly from the curves of tension of the dynamometer, the values of T relative to more than forty experiments, in which the loads have varied from 26 to 209 \ pounds, we have found that the ratio of the ten- sion to the load, thus furnishing a direct measurement, was at 0.96, which shows that all the data introduced in the above formula leads to a result which accords with this measure, within very satisfactory limits of correctness. 228. Relations between the tension of the cord and the friction of the sled. Knowing the tension of the cord T, by means of the dynamometer, or having calculated it by the preceding formula, it is quite easy to deduce the value sought, of the friction of the sled, in applying directly the principle of action equal and opposite to re- action. In fact, the tension T, and the friction sought F, are two external forces with opposite directions, whose diiference T F produces the motion of acceleration of the sled. On the other hand, the resistance which the inertia of the weight Q of the sled opposes to this accel- eration is (No. 62) %. We have then for the equality of action and reaction, whence P_ T Q 1 When, by direct observation, or by the formula of the preceding number, we shall have determined the tension of the cord, we must for the value of the friction subtract from it the quantity - , easily calculated when we know FRICTION. 269 by the abstract the parameter 2C of the curve of mo- tion. Such is the method which was adopted for the calcu- lation of all the experiments where the motion was accel- erated ; as for those where the motion is uniform, we have simply F=T. We see that the law of the motion being once known by the abstract of the curves, and being that of a uni- formly accelerated motion, we may, after having proven the constancy and the generality of this law, pass to the use of the dynamometer, and rest content with the indi- cations of the chronometric apparatus. 229. Results of experiments. I give here, as exam- ples of the results obtained, some of the tables inserted in my memoirs, successively presented to the Institute and inserted in the " Kecueil des savants etrangers," and as an example of the application of the preceding formula, I select the second experiment of the first of these tables, relative to the friction of oak, sliding upon oak without unguent, with the fibres parallel to the direction of the motion. In this experiment we have Q=295.22 lbs - P=203.38 lbs - The trace of the curve gives for the parameter 2C=0.6339 f % whence and consequently the tension T=0.95 j P (0.34685+?) ^ 1 .1753=173.05 lb8 - 270 FRICTION. The other formula gives for the value of friction The ratio of the friction to the pressure is here then _. Q-295T2- TABLE. Experiments upon the friction of oak upon oak without unguents the fibres of the wood being parallel to the direction of motion. g " g Velocity of motion. 1 'i 1 . 1 g H 1 ll 1 8 1 IH.O g 1 fe 1 i I l!-l 1 fi 3 "1 z *3 & 3 1 If 8 o 1 b 1 F 1*3 Q P T F Q Sq. ft. lb Ibs. Ibs. ft. Ibs. ft. ft. 295.22 148.58 141.15 141.15 0.477 2.264 295.22 203.38 173.02 0.634 3.123 144.1 0.488 ' 7.77 333.52 171.03 162.48 162.48 0.487 TOO 970.63 504.32 479.10 479.44 0.491 1.345 6, iVO 970.63 610.01 536.64 0.850 2.352 466.41 0.480 6.726 1499.13 930.23 819.18 0.862 2.820 709.33 0.472 6.693 2291.56 1273.69 1164.91 1.688 1.184 1080.60 0.471 6.299 2291.56 1114 91 1059 16 1059.16 0462 8.511 102.09 64.95 5417 1.914 1.044 50.86 0.49S 4.495 108.53 56.59 53.77 53.77 0.496 4.20' 120.55 6290 5975 59.75 0.495 4.92 120.55 98.39 76.44 0.384 5.208 56.95 0.472 10.072 1.062 226.81 186.83 152.57 0.472 4.237 110.38 0.486 8.924 227.63 182.64 117.77 1.054 1.897 104.36 0.458 6.102 332.76 162.72 154.58 154.58 0.464 4.ioi 440.03 211.37 200.80 200.84 0.456 2.001 440.24 210.45 199.93 199.93 0.454 2.789 ( 21567 10862 103.19 103.19 0.478 3.478 0.33 4 321.47 175.49 164.74 0.933 2.145 133.34 0.414 6.918 ( 604.06 468.80 389.44 0.506 8.952 293.51 0.484 8.858 When the motion is uniform, as in the sixteenth ex- periment of the same table, we have simply, for P=211.37 lbs -, FKICTION. 2T1 An examination of the different tables relating to these very variable cases, completely establishes the laws of friction, which are to be used in the motion between the greater part of materials employed in the arts. The results of all the other experiments agree with those which we limit ourselves to reporting here. TABLE. Experiments upon the friction of Elm upon Oak, without unguents the fibres of the wood ~being parallel to direc- tion of motion. | i 2 s * b L ij 1 f g J-, S a **- s * 1 .1 ii I 1 1 Ii I' m i S* 9 Sq. ft. Ibs. Ibs. Ibs. ft Ibs. ft. 260.05 161.31 139.19 0.732 2.734 117.18 0.45 7.55 260.05 187.42 153.06 0.469 4.261 118.88 0.45 9.45 921.38 506.69 450.40 0.984 2.031 892.27 0.43 8.60 921.88 480.24 440.46 1.859 1.075 408.78 0.44 4.62 1.838- 921.38 921.38 454.13 664.42 416.77 525.72 1.902 0.377 1.051 5.291 386.62 374.53 0.42 0.41 4.48 12.46 1980.10 1113.83 976.94 0.802 2.494 821.76 0.42 7.41 1980.10 1007.77 927.09 1.993 1.003 865.54 0.44 4.04 1980.10 1118.83 911.42 1.206 1.657 787.42 0.40 5.68 1980.10 1298.70 1104.86 0.600 8.330 899.99 0.45 8.10 244.81 135.42 122.86 1.414 1.414 108.93 0.45 5.25 .063- 389.58 811.19 240.06 0.347 5.756 171.43 0.44 10.50 917.79 479.76 489.82 1.734 1.153 408.60 0.44 4.76 Mean 0.434 272 FRICTION. TABLE. Experiments upon the friction of soft oolitic Limestone of Jaumont, near Metz, upon stone of the same kind with- out unguent. , | j o s s L I If is! i,<; J, j Si 1 J 1! i 1 | 1 I 5 1 1 EH ^ - Sq.ft. Ibs. )ba. Ibs. ft. Ibs. ft. 314.04 254.18 222.40 0.829 2.412 198.89 0.633 6.890 814.04 254.18 218.36 0.682 2.929 187.60 0.597 7.579 0.861 - 1264.18 999.63 853.54 0.621 8.216 727.50 0.575 7.940 1274.94 1034.92 859.41 0.499 4.001 700.86 0.549 8.858 1274.94 1034.92 859.41 0.499 4.001 700.86 0.549 8.858 Mean. .0.580 0.499 309.56 331.62 1257.50 1257.50 1257.50 293.88 293.88 1034.92 1140.78 1140.78 245.40 244.65 925.13 943.99 924.32 0.536 0.524 1.066 0.488 0.426 8.725 3.815 1.874 4.101 4.687 209.56 207.97 851.95 783.89 741.28 0.677 0,627 0.677 0.623 0.589 8.498 8.662 6.070 8.990 9.613 .0.639 298.40 240.95 218.30 1.426 1.402 205.31 0.688 5.249 298.40 240.95 211.02 0.841 2.877 189.01 0.633 6.824 298.40 293.88 239.18 0.451 4.433 198.10 0.664 9.350 597.93 465.91 421.45 1.841 1.491 393.79 0.659 5.413 597.93 465.91 431.15 2.499 0.800 416.28 0.696 3.970 597.93 571.77 485.40 0.597 3.347 423.25 0.709 8.104 .0.675 General mean. When the soft limestone slides upon soft limestone, and especially when the moving body rests upon surfaces of small area, the latter are destroyed rapidly during the experiment. This circumstance, and the presence of the dust powder resulting from it, have not changed the laws observed. FRICTION. 273 TABLE. Experiments upon the friction of strong leather ', tanned, and placed flatwise upon cast-iron. a 1 1 i 2 JS s fi 1 2 y J 1 L | li li 08 i J y 1 I 5 E o2 I" I i P 1 1 Sq. ft. 0.4155 a Iba. 471.02 1106.42 Ib*. 320.35 637.94 Ibs. 291.83 606.04 ft. 1.548 1.292 272.75 606.04 0.579 0.547 5.02 Mean 0.563 0.4155 291.01 291.01 I 291.01 I 1115.03 188.02 161.55 135.08 977.58 154.78 183.85 118.83 689.54 0.497 0.926 0.244 4.024 3.816 2.159 8.196 118.63 96.44 99.49 407.11 Mean. 0.408 0.842 0.842 0.865 .0.364 8.66 6.56 12.70 0.4155 r 1114.10 o J 1114.10 3 1 1114.10 [ 1114.10 0.4155 j 298.49 299.17 1114.10 1114.10 214.48 820.36 426.10 92.19 148.32 214.48 193.80 198.54 279.52 350.41 87.30 76.29 140.91 196.22 2.042 2.584 0.795 0.475 0,548 1.804 0.979 0.776 2.516 4.210 168.21 172.38 192.43 0.146 0.155 0.172 0.164 Mean 0.159 8.649 1.108 87.29 42.52 140.91 157.93 Mean. 0.124 0.142 0.126 0.141 .0.133 4.58 8.87 7.09 8.46 4.59 tin 1114.10 I 320.35 478.92 I 185.08 294.48 123.77 2.011 1.950 0.944 I 260.07 1.025 I 108.66 0.227 Mean... . . .2.30 4.66 4.66 Though leather is a soft and very compressible sub- stance, its friction is none the less proportional to the pressure, and independent of the velocity, throughout the whole range of the experiments. 18 274: FRICTION. TABLE. Experiments upon the friction of brass upon oak, without unguent. Fibres of wood parallel to the direction of motion. fco 1 a i s 1 1 .11 I S 8 | 1 "* |O J~" | 1 'II | 1 I 1 (2 fi 1 1 i H > Sq. ft. Ibs. Ibs. Ibs. ft. Ibs. ft. 257.13 161.46 153.36 .... 153.39 0.60 .... 257.13 161.61 153.61 153.54 0.60 .488 - 1539.90 1539.90 981.99 1114.32 932.69 1068.80 1.548 1.291 932.89 1007.79 0.60 0.65 1539.90 1273.11 1101.97 0.707 2.828 967.05 0.62 748 1989.83 1378.97 1290.72 4.846 0.460 1262.61 0.63 305 248.81 161.72 153.60 153.61 0.61 248.49 188.86 169.56 1.283 i.558 157.58 0.63 5.2l' 0.141 - 763.97 532.07 487.11 1.956 1.022 462.99 0.60 4.92 1531.26 981.89 932.69 932.89 0.61 1531.26 1273.11 1103.69 6.719 2.780 971.73 0.63 7.51 Mean. .0.616 For the experiments where we have not indicated the value of the parameter of the law of motion, and that of the acceleration, the motion was slow and somewhat un- certain. The results contained in this table confirm the three laws before enumerated, but we remark that the mean value of the friction, which is here 616, is more consid- erable than in the case of oak rubbing against oak, or than that of elm upon oak, for which the results are con- signed to the tables of pages 270 and 271. "We shall see, by the following table, that the coeffi- cient diminishes considerably when the friction occurs between two metallic surfaces. FRICTION. 275 Experiments upon the friction of cast iron upon cast iron. 1 1 A I 1 i I 1 J I* *s 1 I J | | J g- s %-. 5 J *d 1 o ^ !r" t s 1 P 1 >a 1 1 of the cord ! tio of fricti locity at 9. 1 1 * H Sq. ft. Ibs. 496.10 Ibg. 108.62 lb. 95.78 0.993 2.012 11,8. 64.49 0.130 ft. 6.37 vj. 496.10 135.09 113.88 0.585 3.417 60.79 0.122 8.20 OO a 1091.14 820.37 283.82 0.938 2.130 211.15 0.193 6.50 0.3874 1 - 1091.14 1104.80 426.08 174.79 336.38 166.05 0.378 5.291 157.18 166.05 0.144 0.150 10.17 Slow. ft 4412.70 796.73 745.58 4.267 0.468 681.74 0.154 8.25 4412.70 929.06 865.85 8.816 0.604 783.54 0.177 8.48 4412.70 1054.77 949.52 1.158 1.726 712.94 0.161 5.81 Mean.... 0.154 0.3874 0.3874 0.3874 .2 0.8874 ^ f 110437 899.74 861.17 1.402 1.426 312.82 0.282 8.90 J 110437 505.61 432.96 0.646 3.095 824.60 0.298 9.25 1 2202,70 770.26 731.36 731.36 0.832 Uniform. ? [2202.70 876.13 806.43 2.036 0.982 739.30 0.386 458 Mean.... 0.811 x ( 1091.14 201.25 191.15 191.15 0.175 Slow. J < 1091.14 820.87 287.77 1.251 1.598 281.00 0.211 5.68 ( 1091.14 373.30 821.78 0.'695 2.878 224.47 0.205 7.09 Mean.... 0.197 f 496.10 52.49 49.87 49.87 0.100 Slow. 496.10 78.96 65.48 1.950 1.024 50.40 0.101 456 1103.43 108.64 103.17 103.17 0.093 Slow. ^ 1103.43 201.25 179.20 l".060 1.885 114.64 0.104 6.17 o 1103.43 240.95 212.87 0.939 2.130 117.81 0.106 6.47 "3 2214.98 293.88 271.14 2.286 0.875 211.30 0.095 420 -1 2214.98 293.88 274.54 4.023 0.497 243.84 0.109 8.08 2214.98 426.10 379.70 0.999 2.000 243.33 0.109 6.30 6185.82 62470 593.47 593.47 0.096 .... Very slow. 1108.14 108.62 103.17 103.17 0.093 Slow. Mean.... 0.101 129.48 118.70 2.011 0.994 84.62 0.076 4.53 129.48 118.13 1.767 1.131 79.44 0.071 4.72 133.89 12119 1.395 1.432 72.16 0.065 5.61 133.89 120.99 1.414 1.414 72.54 0.066 5.58 frj 138.31 126.29 1.767 1131 85.82 0.077 4.59 s 1103.43 138.31 124.41 1.295 1.544 71.55 0.065 5.51 ^ 138.35 123.55 1.341 1.491 72.71 0.066 5.44 138.35 124.41 1.295 1.544 71.55 0.065 5.51 193.44 168.15 0.783 2.553 80.65 0.073 7.12 193.44 167.07 0.731 2.734 72.94 0.066 7.28 , 193.44 168.92 0.823 2.430 85.6S 0.078 6.82 Mean.... 0.070 276 FEICTION. This table, besides verifying tlie laws of the propor- tionality of the friction to the pressure, and its independ- ence of the velocity, shows that water rather increases than diminishes the friction of cast-iron. "We see also that tallow, somewhat hard, does not reduce the friction so much as lard. 230. Consequences of the experiments. The experi- ments made by me upon the friction proper of plane surfaces upon each other, comprise 179 series, answering to different cases, according to the nature or condition of the surfaces in contact ; and they all, without exception, lead to the following results : The friction during the motion is : 1st. Proportional to the pressure. 2d. Independent of the area of the surfaces of con- tact. 3d. Independent of the velocity of motion. 231. Experiments upon the friction at starting, or when the surfaces have been some time in contact. The same apparatus has served for the experiments upon friction at the start, or after a prolonged contact, whose aim was to establish in w r hat cases there is a notable dif- ference between it and that produced during motion. This difference, which, according to the case, arises from very different causes, may in general be attributed to the reciprocal compression of the bodies upon each other, and to a kind of gearing of their elements. The time or duration of the compression probably exerts an influence upon the intensity of the resistance opposed by their sur- faces to sliding. But, generally, this resistance reaches its maximum at the end of a very short period. 232. Results of experiments. We publish here the re- sults of some experiments which we have made. FRICTION. 277 TABLE. Experiments upon the friction of oak upon oak, without unguents, when the surfaces have ~been some time in con- tact. The fibres of the sliding pieces leing perpendicu- lar to those of the sleeper. Extent of the surface of contact. Pressure Q. Motive effort or friction P. Ratio of the friction to the pressure/ Sq.ft. Ibs. Ibs. 120.55 67.15 0.55 282.49 150.23 0.53 0.947 495.01 252.34 0.51 1995.23 1171.10 0.58 2526.65 1287.16 0.51 389.35 203.80 0.52 0.043 402.98 212.44 0.53 1461.08 854.77 0.58 Mean. .0.54 The friction seems to be proportional to the pressure, which varied from 120 lbs - to 2526 lb % and independent of the surfaces of contact, which varied in the ratio of 1 to 22, the smallest being .043 sq - f % and the greatest 0-947 S<1 - ft ' ; this last value exceeds those usually employed for sliding surfaces in mechanical constructions. The ratio of the friction to the pressure is here raised to 0.54, while it was only 0.48 during the motion, as was the result of the table page 270. The friction at the start is raised then about an eighth above that which we first considered. A similar increase occurs in all similar cases. 278 FEIOTION. TABLE. Experiments upon the friction of oak upon oak, without unguents, when the surfaces have been some time in con- tact. The sliding pieces have their fibres vertical, those of the fixed pieces are horizontal and parallel to the direction of motion. Extent of the sur- face of contact Pressure Q. Motive effort or friction F. Ratio of friction to pressure/ Time of contact. Sq. ft. Ibs. 432.12 Ibs. 184.88 0.427 5 to 6" 432.12 184.88 0.427 10' 432.12T 157.43 0.364 1' 696.77 354.59 0.509 6' 696.77 304.31 0.436 30" 6845 696.77 342.03 0.498 8 to 10' 882.01 405.32 0.459 8 to 10' 1106.99 555.73 0.502 10' 1106.99 430.03 0.388 5 to 6" 2205.30 810.24 0.367 15' 2205.30 882.60 0.400 10' Mean 0.434 This table shows that for wood, the friction at the start presents for equal surfaces and pressures great dif- ferences from one experiment to another, and that the resistance attains its maximum in a short time of contact, which seems not to exceed some seconds. We in fact see that the figures answering to 5 and 6 seconds are not inferior to those relating to a contact of 15 minutes, the longest of any recorded in the table. The mean value of the ratio f of friction to the pres- sure is 0.434, but it would be well in application to reckon it at 0.48 or even 0.50. FRICTION. 279 TABLE. Experiments upon the friction of oolitic limestone upon oolitic limestone, when the surfaces have leenfor some time in contact. Surface of contact Pressure Motive effort or friction F. Eatio of friction to the pressure Time of contact. Sq.ft. 0.8611 Ibs. { 314.01 330.85 - 1162.72 1274.93 1274.93 Ibs. 228.88 239.25 949.64 932.87 958.02 0.728 0.723 0.752 0.731 0.751 15' 15' 15' 5 to 6" 5 to 6" Mean 0.737 0.4992 l 309.55 \ 1257.49 ( 1257.49 228.88 983.16 983.16 0.739 0.781 0.781 2' 10' 1' Mean 0.783 Edges rounded 1 298.38 J 602.32 228.88 442.58 0.774 0.740 2' 5 to 6" Mean., General Mean.. .0.757 .0.740 We still see by these experiments that the friction at starting, as well as the friction in motion, is independent of the extent of the surface of contact, and is proportional to the pressures. This conclusion, and even the value deduced from the above experiments, has been since con- firmed by results obtained in similar cases by M. Boistard, Engineer of Roads and Bridges, in 1822. These figures moreover differ so little from each other, that we may place all confidence in the general mean 0.74, and employ it in all similar cases. 280 FRICTION. TABLE. Experiments upon the friction of oolitic limestone upon oolitic limestone, when the surfaces have been some time in contact, with a bed of fresh mortar. Surface of contact. Pressure Q. Motive effort or friction F. Eatio of friction to pressure/ Time of contact. Sq.ft. Ibs. Ibs. 325.66 253.98 0.780 10' 506.08 404.87 0.800 10' 783.98 783.98 580.87 608.22 0.740 0.773 15' 10' 783.98 555.73 0.709 10' 1167.73 983.16 0.841 15' Mean 0.773 ' 309.55 239.21 0.772 10' 489.97 379.74 0.775 10' 781.10 568.30 0.727 10' 0.4992 1164.86 807.15 0.792 15' 1164.86 907.74 0.779 10' 1169.27 807.15 0.690 10' 1548.61 1159.17 0.748 15' Mean 0.745 f 319.82 254.02 0-794 10' niA o 500.25 0.1636 1 79137 304.30 480.26 0.608 0.607 10' 10' 1161.90 731.69 0.629 15' Mean. General mean. .0.659 .0.735 These experiments show that the friction at starting is for these stones, very nearly the same, with the interposi- tion of mortar as without. In recapitulating, recent trials have caused us to see that the friction at the moment of starting, and after a very short time of contact, is : 1st. Proportional to the pressure. 2d. Independent of the area of the surfaces of con- FRICTION. 281 tact ; and that furthermore, for compressible bodies, it is notably much greater than that which takes place during motion. 233. Observation relative to the expulsion of unguents under heavy pressures, and by prolonged contact. "We have observed metallic bodies with unguents of grease or oil, under very great pressure, compared to their surfaces, and find, after a contact of some duration, that the un- guents are expelled, so that the surfaces are simply in an unctuous state, and thus have double the friction of sur- faces well greased. This shows us why the effort required to put certain machines in motion is, disregarding the in- fluence of inertia, often much greater than that required for maintaining a rapid motion, and proves that, for an experimental appreciation of the friction of machines in motion, we need not, as is sometimes done, make use of the same methods as for machines starting from repose. 234. Influence of vibrations upon the friction at start- ing. Another remarkable circumstance noted in the experiments at Metz is, that when a compressible body is solicited to slide by an effort capable of overcoming the friction of motion, but inferior to the friction at starting, a simple vibration, produced by an external and appa- rently a slight cause, may determine the motion. Thus, for oak rubbing on oak, the friction at starting is 0.680 of the pressure, and the friction during motion is 0.480 ; so that, to produce the motion of a weight of 2205 lbs - it is necessary then to exert an effort of 1500 lb % while there is only needed 1059 lbs - to maintain it. Still, under an effort equal, or a little above 1059 lb % and by the effort of a vibration, the body may be started. This important observation applies to constructions always more or less exposed to vibrations, and shows that, if in the calculations for machines for producing motion, we should take into account the greatest value of the 282 FRICTION. friction, we should in those relating to the stability of constructions, on the other hand, introduce its smallest value, that for motion. It seems, finally, to explain how it sometimes happens, that buildings, for the stability of which no uneasiness was felt, have fallen at the passing- of a wagon, and how the firing of salutes from a breach battery may, at certain times, accelerate the fall of a ram- part or a building. 235. Influence of unguents. Tat unguents considera- bly diminish friction, and the consequent wear of surfaces. But from the observations made (No. 230), we see that though the friction is in itself independent of the extent of the surfaces, it is well to proportion them to the pres- sures they are appointed to sustain, so that the unguents may not be expelled. We would also remark that all the experiments in consideration were made under pressures more or less considerable, and their results should only be applied to analogous cases. In fact, we may conceive that if the pressures were so great, in respect to the sur- faces, as to occasion a marked defacement, the state of the surfaces, and consequently the friction, would vary ; or that, on the contrary, if the surfaces were great, and the pressures very slight, the viscosity of the unguents, usu- ally disregarded, might then exert a sensible influence. "We would observe, that in general, and especially for metals, pure water is a bad unguent, and often increases, rather than diminishes the friction. 236. Adhesion of mortar and solidified cements. But, for mortars which have set and acquired a proper degree of dryness, there exists a different condition of tilings. Adhesion and cohesion take the place of friction, and the resistance to separation becomes sensibly proportional to the extent of the surface of contact, and independent of the pressure exerted, either at the moment of rest, or that of separation. FRICTION. 283 For limestones bedded with mortar of hydraulic lime of Metz, the resistance is about 2112 lbs - per square foot of surface. With other limes, undoubtedly common, M. Boistard, Engineer of Koads and Bridges, has found 14:26 lbs - With plaster, the resistance seems to follow the same law, but it varies considerably with the instant of the setting of the plaster, which seems to exert a great influence upon the cohesion. 237. Observation upon the introduction of friction and cohesion in calculations upon the stability of construc- tions. Finally, we would remark that friction cannot, in the case of beddings in mortar, or in plaster, show itself until the cohesion or adhesion is overcome, and that con- sequently these two resistances cannot coexist. In cal- culations upon the stability of structures, we should only reckon upon one of these, and that the weakest. 238. Experiments upon friction during a shock. From the general opinions which we have published upon the action of forces, and the efforts of compression developed during a shock, and from the verification in Nos. 66 and 67, of the consequences derived from them, we are justi- fied in concluding that the efforts produced during the shock occasion frictions, which follow exactly the usual laws of friction. It is moreover expressly admitted by the illustrious Poisson, who, in the second edition of his " Traite de Me'canique," No. 475, expresses himself in these terms : " Though no observations have been made upon the intensity of friction during a shock, we may suppose, by induction, that it follows the general laws of friction of bodies subjected to pressures, since percussion is only a pressure of very great intensity exerted during a very short time." To verify by direct observation the correctness of this supposition, and at the express invitation of M. Poisson, I undertook many series of experiments, choosing for that purpose the case of strips of cast-iron sliding upon bars 284 FKICTION. FIG. 99. of cast-iron spread with lard, since this had been the sub- ject of careful study in my preceding experiments, and is the case which most frequently occurs in practice. 239. Description of the apparatus employed in the ex- periments. The apparatus which I employed differs from that described in No. 225, only in the following dis- position necessary for sus- pending to the sled, at a desired height, the body de- signed to produce the shock, and allowed to fall at will during the motion. Upon the sides of the box of the sled are raised two frames of firm uprights ab and #'', pierced with holes at intervals of .16 f % through which pass two iron pins ; upon these pins rests a movable crosspiece cd of oak. By raising and lowering the pins, the height of the crosspiece cd above the sled may be varied at will. A screw e and nut passes freely across a hole cut in the mid- dle of the crosspiece, and bears a plyer with ring legs, upon which is suspended a shell to give the shock. The two legs of the plyers are bound with strips of wick with quick match, holding them shut. By means of the screw e the height of the shell above the surface shocked can be exactly regulated. We may easily conceive from this description, the box and uprights being firmly fastened to the sled, that the whole system partakes of a common motion, and that if at any instant of its course, the shell falls upon the sled, it falls there with a vertical velocity due to the height of the fall, and with a horizontal velocity which, as we shall see hereafter, was sensibly the same as that of the sled. By means of the ligature of the legs of the plyers we ac- complish the fall of the shell, without any external con- FRICTION. 285 cussion or disturbance. For this purpose, a man sets fire to the match, and gives the signal for the starting of the sled ; combustion is communicated to the upper part which keeps the plyers closed ; these open suddenly and let loose the shell, without any possibility of disturbing the common motion of the system of the two bodies. 240. General circumstances of the experiments. The experiments were made in impressing the sled, sometimes with a uniform, and sometimes with an accelerated mo- tion. The first of these motions was obtained at will, by giving to the descending box a weight just sufficient to overcome the friction, and in suspending under this box a shell of 110 ib3 - weight, which only descended 1.64"- when its action ceased. As for the accelerated motion, it was produced whenever the motive weight surpassed the fric- tion. The law of these motions was, moreover, deter- mined, in each case, by means of curves traced by the style of the chronometric apparatus. 241. General examination of what occurred in these experiments. We can readily appreciate the mode of ac- tion during the experiments. We take, for example, a case where the system of the sledge and the shell sus- pended above it, is impressed with a uniform motion. At the instant when the combustion of the wick allows the legs of the plyers to separate, the shell is free, and falls ; during its fall, until the moment it reaches the sledge, the latter being freed from the weight of the shell, acquires an amount of motion precisely equal to what would be consumed by the friction due to this weight. The horizontal velocity of the sledge, at the instant of the shock, is then a little greater than that of the shell. After this the forces of compression developed by the shock produce a friction variable as themselves, at each instant, which consumes a certain quantity of motion ; so that the sledge, whose progress was accelerated during the fall of the shell, is afterwards retarded during the action of the shock. 286 FRICTION. 242. FormuloB employed in calculating the results of the experiments. As it is desirable to prove whether the friction remained proportional to the variable pressures produced during the short intervals of the phenomena, we proceed to give some formulae relative to this hypothe- sis, which we will hereafter compare with the results of experiment. We consider first the case of uniform mo- tion, and call Q the weight of the sledge 3 and the suspending appa- ratus of the shell ; q the weight of the shell producing the shock ; f the ratio of friction to the pressure for the surfaces in contact; h the height of fall of the shell above the sledge ; U the velocity due to this height ; T the time of the fall ; Y the horizontal velocity of the sledge and shell at the instant when the latter is let loose by the plyers ; V the velocity of the body after the shock ; At the instant when the shell is freed, the quantity of motion of the system is ff The weight of the shell, when connected with the sledge, produces a friction fq which, in each element of time t, consumes a quantity of motion fqt^ and which, during the time of the fall, would consume the quantity /^T. But since, on the other hand, the shell ceases to press upon the sledge during this time, it follows that the quan- tity of motion gained by the system by reason of this diminution of pressure, is precisely fqT. At the instant when the shell reaches the sledge, the quantity of motion possessed by the system is then FRICTION. 287 From this instant, and during the whole period of the shock, the shell loses, in each element of time, an element of velocity, and consequently a quantity of motion -u, whence results a force of compression - x -, producing a 9 t friction 1 ^ x-. This friction consumes, in an element of 9 t time a quantity of motion -S . ^, and when all relative ' ' . motion in a vertical direction is destroyed, the friction due to the forces of compression has finally consumed a quantity of motion equal to U. y Consequently, when the shock has terminated, we should have between the quantities the relation or Now, the shell falling with a uniformly accelerated motion by virtue of gravity, we have, evidently, TJ=^T, whence it follows that Y V ; that is to say, that in our apparatus the quantity of motion destroyed, by the fric- tion resulting from the forces of compression, must be precisely equal to that which it gains during the fall of the shell. These two effects are successive, but take place in a short interval of time, and therefore occasion in the curve of motion undulations in opposite directions, which do not affect the general law, and are scarcely appreciable, either in the draughted curve or that made from the ab- stract of the table. 243. The acceleration of the motion of the sledge dur- ing the fall of the shell may ~be neglected. It is easy to be 288 FRICTION. assured a priori, that the acceleration of the velocity of the sledge during the fall of the shell, was always very small in our experiments, though the height of the fall has reached 1.97 ft - We observe, then, from what has just been said, that calling V x the horizontal velocity of the sledge, at the moment when the shell reaches it, we shall have whence VTr Making, for example, 2=11 and U=13.80 ft -, Q=590.68 lb % /=0.071, which answers to one of the most intense shocks produced during the experiments, we find ,=0.1829"- Now, the shock of the shell in the horizontal direction taking place only in virtue of this difference in velocity, we see that its effect upon the general motion should be quite insensible, and we may, as we have done in the pre- ceding calculation, neglect its influence upon the general motion of the sledge. 244. Case where the motion of the sledge is acceler- ated. The preceding reasoning applies to the case where the system of the shell and of the sledge is impressed with an accelerated motion, and it follows that if, as we have admitted, the friction during the shock remains pro- portional to the pressure, the general law of motion in our apparatus cannot be affected ; or, in other words, that if, FRICTION. 289 before the fall of the shell, the motion is uniform or accel- erated, according to' a certain law, it will still be so after the shock, according to the same law. The only disturb- ance which will result will be sometimes manifested by undulations, which, in most cases, would hardly be appre- ciable. Moreover, the hardness or compressibility of the body in contact should not have any influence upon the result, and in causing the shell to fall upon the beech- wood joists composing the sledge, or upon a mass of soft loam placed upon it, we should, for circumstances other- wise similar, find the same law of motion, which should be the same as though there had been no shock. 245. Results of experiments. It remains now for us to compare the results of the formulae with those of ex- periments which have been made, some when the sledge was impressed with a uniform motion, and some when the motion was accelerated. In these experiments, we have varied the weight of the shells imparting the shock from 26.43 lbs - to 110 lb % or nearly 1 to 4 ; the ratio of the weight of the body imparting the shock to that of the body shocked, from T V to , and the height of the fall from 0.32S ft - to 2.29 f % or from 1 to 7. The shock was produced upon wood, and upon loam placed upon the sledge. If, then, the laws which we have admitted in the preceding formulas are verified by experiments within such extended limits, we may, I think, conclude that they subsist for pressures developed during the shock, as well as for others without shocks. I shall only publish here the experiments made in the case of uniform motion, where the shock was made upon wood and loam. Experiments made with motive weights which produced an accelerated motion, have led to simi- lar consequences : the acceleration produced having al- ways been sensibly the same in the case of a shock as in those where there was none. 19 290 FRICTION. *S cJ I * I % V ? V o "8 * $ .1 03 O ^ 5* ? 13 s .2 I *o &% 0 >0 00* CO CO CO CO 00 00 00 00 l> 00 00 00 00 00 00 00 00 00 O5 OS O5 .fc'S 5 "" ^ J5 co^ * " o>c * "*' "* **' I dddddr-?r-Jdddi-irH-HTHcy means of lelts. The theory of the transmission of motion by means of cords or endless belts is founded upon two theories. The first, that of M. Prony, relative to the sliding of a cord or belt upon the surface of a drum ; the second, due to M. Pon- celet, refers to the variation of tension in the two parts of the cord or belt employed in these transmissions. I pro- pose to prove, by special experiments, the consequences of these two theorems, and proceed to give a succinct account of the results of these researches. 247. The slipping of cords or lelts upon cylinders. In explaining the first of these theorems, let us consider a cord or belt enveloping a portion of the surface of a cylinder, and acted upon at one end by a power P, and upon the other by a resistance Q. It is clear, that to pro- duce the slipping of the cord, the power P should be equal to the resistance Q, increased by the resistance FEICTION. 293 opposed by the friction of the cord upon the surface of the cylinder. Let us seek to determine this friction. For this purpose, we consider the two consecutive ele- ments ah and be of the cord, and call : T the tension of the cord in the element ab. T r the tension of the cord in the element 5 ; consequently, the weight of each strip, of equal length, was, with its plate, 3.03 lbs> The arc embraced was equal to the semi-circum- ference. At first, equal weights were put in the scales, then gradually was added to one of them the weights necessary to make the belt slide upon the drum. We see from this, that the tension Q of the ascending strip was equal to 3.03 lbs - plus the weight contained in the corresponding scale, and that the tension P of the de- scending strip was equal to Q increased by the weight added, over and above the primitive load. This established, the preceding formula becomes log P= log Q+0.434/. ^ = log Q+0.434/ x 3.1416, whence we deduce fe log P~ log Q^log P-log Q J 0.434 x 3.1416 1.363 By introducing in this formula the values of P and Q furnished by experiments, we are enabled to calculate the different values of the ratio f of the friction to the pressure, and to be assured that they confirm the theoretic consequences which we have unfolded. 249. Results of 'experiments. The two following tables contain the results of the experiments : FRICTION. TABLE. Experiments upon the friction of belts upon wood drums. Width of belt. Condition of the belt. Diameter of drum. Length of arc embraced. Tension of the part. Ratio of 'riction to pres- sure f. rising Q. falling P. ft. ft. ft. Ibs. Ibs. 1 ( { 14.060 66.992 0.497 14.060 64.786 0.486 Dry \ 14.060 64.786 0.492 0.164 - somewhat oily. 2.741 - 4.306 -j 36.114 36.114 167.341 153.336 0.488 0.460 36.114 151.461 0.458 25.087 111.102 0.473 , . I 25.087 95.603 0.426 Mean 0.472 f 14.060 63.683 0.472 14.060 69.197 0.458 2.099 \ 14.060 63.242 0.507 I 36.114 140.875 0.479 I 36.114 140.875 0.433 Mean 0.462 j- 1 14.060 73.608 0.526 14.060 75.813 0.541 Dry 25.087 91.252 0.411 0.164 - somewhat \ 0.328 - 0.514 - 25.087 98.975 0.438 oily. 25.087 94.560 0.422 36.114 161.827 0.477 . J . 36.114 168.576 0,490 Mean 0.472 1' r 11.911 71.458 0.570 11.911 72.560 0.575 0.091 - 4.306 - 22.938 22.938 114.465 104.541 0.512 0.483 33.965 137.622 0.446 . 33.965 136.519 0.443 Mean.... General mean.... ,. .504 ,.0.477 FRICTION. 297 I I 3 - . ^ 1 *i, s |3 mbr S. 1* "o -*J II 21 ^i l-d Jiul af5j 3^2, le II 111 -23 I!! IO1C4CMCO I C 1 -* d o d o I o o d o o < Q$ T H CO O CO O 1O o oo*cj2c>GJ|<5> C5 O T* * OS C'l oS CO CO 00 CO s 298 FRICTION. We see by the results of these experiments, in which the arc of contact varied in the ratio of 8.3 to 1 nearly, and where the tension has reached very nearly the limits assigned to the belts of machinery, that the value of the ratio / of friction to the pressure, remained very nearly constant. The three first series of the first table fully confirm the theoretic considerations. The fourth series relates to a belt quite new, and very stiff, and to this we attribute the small increase presented by it in the mean value. This belt having, moreover, only a width of .091 f % or about the half of the preceding, we see that this last series con- firms, as to belts, the law of the independence of sur- faces. In the experiments of the second table, the extent of arc embraced varied in the ratio of 6 to 1, the breadth of the belt pressed against the pulley in that of 2 to 1, the tension from 1 to 3 and from 1 to 6, and still the value of the ratio f, of friction to the pressure remained sensibly constant, and equal in the mean, for the dry belt and dry pulleys /=0.282. When the pulley was moistened with water we had 250. Conclusions. In considering the results of these two series of experiments upon the friction of belts upon wooden drums and cast-iron pulleys, we see that we are justified in admitting that the ratio of the resistance to the pressure is : 1st. Independent of the width of the belt and of the developed length of the arc embraced, or of the diameters of the drums, or, what amounts to the same, are inde- pendent of the surface of contact. 2d. Proportional to the angle subtended by the belt at the surface of the drum. FRICTION. 299 3d. Proportional to the logarithm of the ratio of the tension of the strips, and expressed by the formula - J ~ 1.363 251. Experiments upon the variation of the tension of endless cords or l>elts used in transmitting motion. We pass now to an experimental proof of the theory given by M. Poncelet, upon the transmission of motion by endless cords or belts, and will first give a description of its nature. When a cord or belt surrounds two pulleys or drums, between which it is de- signed to maintain a conjoint motion, care is taken to give it a sufficient tension, which is usually determined by trial, but which it would be best to calculate, as we shall see hereafter. The primitive tension is, at the commencement, the same for both parts of the belt, and this equality established in repose, is only destroyed by the friction of the axles, which may act in either direction according to that of the FIG. 101. motion of the pulleys. Let us examine how this motion is transmitted in such a system. Let C be the motive drum ; C' the driven drum ; Tj the primitive tension common to the parts AA' and BB' of the belt, from the moment when the drum C be- gins to turn until it commences to turn the drum C'. The point A of primitive contact of the part AA' ad- vances, in separating from the point A', in the direction of the arrow, the strip A A' is stretched, and its tension increased by a quantity proportional to this elongation, 300 FRICTION. according to a general law proved by experiment upon traction.* At the same time, the point B of contact of the part BB 7 , approaches by the same quantity towards the point B 7 , so that the portion BB 7 is diminished by a quantity equal to the increase of that of AA 7 . If, then, we call T the tension of the driving portion AA 7 , at the in- stant of its being put in motion, T' the tension of the driven part BB 7 , t the quantity by which the primitive tension T a is increased in the portion A A 7 , and diminished in the part BB', we shall have T=Tt + t, and T / =T 1 -, and consequently Then, at any instant, the sum of the two tensions T and T 7 is constant and double the primitive tension. Now it is evident, that in respect to the driven, drum C 7 the motive power is the tension T, and that the tension T 7 acts as a resistance with the same lever arm, so that the motion is only produced and maintained by the excess T T 7 of the first over the second of these tensions. If the machine is, for example, designed to raise a weight Q acting at the circumference of an axle with a radius R 7 , it is easy to see, according to the theory of moments, that at any instant of a uniform motion of the machine, we must have the relation !N" being the pressure upon the journals, and r their radius. * See Lessons -upon " Resistance des Materiaux. FEICTION. 301 The pressure is easily determined; for calling a the angle formed by the directions AA' and BB' of the belts with the line of the centres CC'. M the weight of the drum. We see immediately that N= V[M+Q+(T T) sin af+(T+T) cos'a, an expression which, according to the algebraic theorem of M. Poncelet, cited in No. 227, has for its value to ~ nearly, when the first term under the radical is greater than the second, ~J$T=0.96 [M+Q+(T-TO sin a]+0.4 (T+T) cos a. This value" of K being introduced into the formula for equality of moments, we have a relation containing only the values of the resistance Q and of the tensions. But as it may be somewhat complicated for application, ob- serving that in most cases the influence of the tensions T and T' upon the frictions, will be so small that it may be neglected, at least in a first approximation, we proceed as follows : First, neglecting the influence of the tensions upon the friction, we have simply, in the actual case, and consequently (T-T')K=QE'+/(M+Q)r, whence we deduce ^=Q. B which furnishes a first value for the difference of tensions, which is the motive power of the apparatus. But this is not sufficient to make known these ten- 302 FRICTION. sions, and it is necessary to determine the primitive ten- sion T', so that in no case the belt may slip. According to the theory of M. Prony, we have, at the instant of slipping, between the tension T and T' the re- lation The number K being a quantity depending upon the nature and condition of the surfaces of contact, as well as Q upon the angle ^ embraced by the belts upon the drum C'. These quantities are known, and we may in each case calculate the value of K by this formula, or take it from the following table, which answers to nearly all the cases in practice : T?ofi/\ r\f VALUE OP THE RATIO K. JttltlO 01 the arc embraced New belts Belts in usual con- dition. Moistened X/-klfo Cords upon wooden drums or axles. to the DGltS circumfer- ence. wooden drums. upon wooden drums. upon cast-iron pulleys. upon cast-iron pulleys. Bough. Smooth. 0.20 1.87 1.80 1.42 1.61 1.87 1.51 0.30 2.57 2.43 1.69 2.05 2.57 L86 0.40 3.51 3.26 2.02 2.60 3.51 2.29 0.50 4.81 4.38 2.41 3.30 4.81 2.82 0.60 6.59 5.88 2.87 4.19 6.58 3.47 0.70 9.00 7.90 3.43 5.32 9.01 4.27 0.80 12.34 10.62 4.09 6.75 12.34 5.25 0.90 16.90 14.27 4.87 8.57 16.90 6.46 1.00 23.14 19.16 5.81 10.89 23.90 7.95 1.50 111.31 22.42 2.00 535.47 63.23 2.50 2574.80 178.52 By means of this table, we shall have then the value of T=KT', and consequently T-T / =(K 1)T'=Q, Q representing the greatest value which the difference FRICTION. of tensions should attain, to overcome the useful and passive resistances. Froni this relation we may derive the smallest tension to be allowed to the driven portion of the belt, to prevent its slipping : we thus have /_ Q "We should increase this value by T V at least to free it from all hazard of accidental circumstances, and to restore the account of the influence of the tensions upon the fric- tion, which was neglected. This established, we have and consequently rp __ a -~r~ J- - -"-T~ - . All the circumstances of the transmission of motion will then be determined. If these first values of T, T', and T x are not considered as sufficiently correct, we may obtain a nearer approxi- mation by introducing them in the value of the pressure !N, and thus deduce a more exact value of Q, which will serve to calculate anew T', then T and T,. 252. Experiments upon the variations of tensions of endless "belts employed for the transmission of motion. To verify by experiment the exactness of these considera- tions, I placed vertically above the axis of a hydraulic wheel, and of a pulley mounted upon its axle, a cylindri- cal oak drum, 2.74: ft - in diameter, and whose axis was 9.84: ft - from that of the wheel. Around this drum A'B', 304: FRICTION. 102. and the pulley AB, was passed a belt which, instead of being in one piece, was in two parts, joined at each end by a dynamometer wit^i a plate and style, of a force of 441 lbs - Moreover, these dynamometers were easily secured in positions, such that that of the descending portion of the belt was near the upper drum, and that of the ascending near the lower drum. Thus the belt could be moved over a space of 6.56 ft> , without the risk of the in- struments being involved with the drums. A thread wound several times around the circumference of one of the grooves of the plate of each of the dynamometers, and attached by the other end to a fixed point, caused the plate to turn when the appa- ratus was in motion, and the paper with which the plate was covered received thus the trace of the style of the dynamometer. The belt being passed over the two drums, the ten- sions of the parts were varied at will, in either direction, by suspending at the circumference of the upper drum a plate Q charged with weights. As to the primitive ten- sion, it was increased by bringing nearer together the ends of the belt, or in diminishing its length before the experiment. The apparatus being thus prepared for observations, before loading the plate Q, we traced the circles of flexure of each of the dynamometers, so as to have the tensions of the belt at rest, and to obtain by their sum the double of the primitive tension T r We may conceive that these two tensions can never be quite equal, but that is not im- portant, inasmuch as we have to deal only with their sum. This obtained, we load the plate with a weight which, being suspended upon the circumference by a cord of a diameter equal to the thickness of the belt, has the same lever arm as the tensions. That part of the belt opposed to this weight is stretched, and the part on the same side FRICTION. 305 is slackened, and we trace the new curves of the flexure of the dynamometers. For the same primitive tension we may make a series of experiments up to the motive weight under the action of which the belt slides upon either drum. TABLE. Experiments upon the variation of the tensions of end- less belts employed in transmitting motion by pulleys or drums. Number of experi- ment. Weight suspended at the circumfer- ence. Tension of the part. Sum of the tensions T+T^. Remarks. Rising or stretched T. Descending or slackened T'. 1 2 3 Ibs. 0.00 44.61 59.55 Ibs. 38.57 60.07 63.14 Ibs. 32.84 12.84 10.19 Ibs. 71.41 72.91 73.33 The belt slipped. 4 5 6 7 8 0.00 22.56 44.61 66.67 97.54 64.86 75.09 84.51 97.96 109.92 57.41 46.82 36.26 24.17 20.76 122.27 121.90 120.77 122.13 130.67 \ The dynamome- V ters moved *' ) about 3.28 ft. 9 10 11 0.00 55.64 110.78 73.73 99.00 117.77 62.32 41.53 20.38 136.05 140.53 138.15 lid., id. 12 13 0.00 115.19 66.91 103.78 57.94 15.86 124.85 119.64 The belt slipped. 14 15 16 0.00 55.64 110.78 107.53 130.05 157.02 98.91 70.26 47.57 206.44 200.31 204.59 17 18 19 0.00 110.78 174.23 97.24 154.29 170.67 88-75 40.78 43.42 185.99 195.07 214.09 Do. do. 20 21 0.00 88.72 86.72 134.84 71.34 44.17 158.06 179.01* * Besides the load Q there was suspended to the main circum- ference of the floats of the wheel at 6.05 ft. from the axis a weight of 22.56 Ibs., which broke the equilibrium 20 306 FRICTION. In these experiments, facilities were afforded for allow- ing the two drums to turn a certain amount, under the action of the tensions, so that we could realize the three cases in practice, to wit : that of the variation of tensions before motion was produced, that of the variation during motion, and finally, that of the slipping. The belt used in these experiments was very pliable, soft, and little liable to be polished in slipping. In cal- culating the ratio of the friction to the pressure for this belt, by means of experiments 3, 13, and 19, we find re- spectively /=0.578, /=0.596, and /=0.544, the mean being /=0.5Y3. 253. Remarks upon the results contained in the pre- ceding table. We see that the first line of each series corresponds to the case where there was no additional weight, and where each portion of the belt took the primi- tive pressure corresponding to the distance apart of the axes. As the weight suspended from the drum was in- creased, the tension of one of the strips was increased, and that of the other was diminished ; but so that their sum remained constant, as is shown by the fifth column of the table. These results, which completely confirm the theory of M. Poncelet, being relative to tensions whose sum reaches 198 lbs - and more, where the greatest rise as high as 169 lbs - and the smallest fall as low as ll lb % comprise nearly all the cases in practice, and show that this theory may, with safety, be applied to the calculation of transmission of motion by belts. In conclusion, we would add, that belts designed for continuous service may be made to bear a tension of 0.551 lbB - per .0000107 8q - ", or .00155 8q - in - of section, which FRICTION. 307 enables us to determine their breadth according to the thickness. 254. Friction of Journals. Besides the experiments previously reported, upon the friction of plane surfaces, I have made a great number upon that of journals, by means of a rotating dynamometer with a plate and style, the first apparatus of the kind, but which it is not worth while to describe here. The axle of this dynamometric apparatus was hollow and of cast-iron. It could receive, by means of holders exactly adjusted, a change of journals of different mate- rials and diameters. Its load was composed of solid cast- iron discs weighing 331 lbs - each, whose number could be increased so as to attain a load of more than 3042 lbs - A pulley, the friction of whose axle was slight, and which transmitted the motion by the intervention of a spring, received by a belt, the motion of a hydraulic wheel, and the difference of tension of the two parts of the belt was measured by the dynamometer with the style. We used journals from .11 to .22 ft - in diameter. The velocities varied in the ratio of 1 to 4. The pressures reached 4145 lb % and within these extended limits we have proved that the friction of journals is subject to the same laws as that of plane surfaces. But it is proper to observe, that from the form itself of the rubbing body, the pressure is exerted upon a less extent of surface, ac- cording to the smallness of the diameter of the journal, and that unguents are more easily expelled with small than with large journals. This circumstance has a great influence upon the intensity of friction, and upon the value of its ratio to the pressure. The motion of rotation tends, of itself, to expel certain unguents, and to bring the surfaces to a simply unctuous state. The old mode of greasing, still used in many cases, consisted simply in turning on the oil, or 308 FEICTION. spreading the lard or tallow upon the surface of the rub- bing body, and in renewing the operation several times in a day. We may thus, with care, prevent the rapid wear of journals and their boxes ; but, with an imperfect renewal of the unguent, the friction may attain .07, .08, or even .1 erf the pressure. If, on the other hand, we use contrivances which renew the unguents without cessation, in sufficient quan- tities, the rubbing surfaces are maintained in. a perfect and constant state of lubrication, and the friction falls as low as .05 or .03 of the pressure, and probably still lower. The polished surfaces operated in these favorable condi- tions, became more and more perfect, and it is not sur- prising that the friction should fall far below the limits above indicated. These reflections show how useful are oiling fixtures in diminishing the friction, which, in certain machines, as mills with complicated mechanism consume a consid- erable part of the motive work. "We cannot, then, too much recommend the use of appliances to distribute the unguent continuously upon the rubbing surfaces of ma- chines, and it is not surprising that a great number of dispositions have been proposed for this purpose within a few years. We should be careful to select those which only expend the oil during the motion, excluding those which feed by the capillary action of a wick of thready substances. These constantly drain the oil even during the repose of the machine, thus consuming it at a pure loss. 255. Results of experiments. In the following table will be found some of the results of experiments in sup- port of the preceding considerations : FRICTION. TABLE. Experiments upon the friction of cast-iron journals upon cast-iron "bearings. '3 +s J 0.2 sj & o 1 ^> a ll s s - "8 :* i!i Kemarks. 1 ^"d < o* J 1 3 ? 2 D cS 2 ^ I 2 ^ 1 ft. ft. Ibs. 0.1961 r 0.082 0.222 0.082 In these experiments the 0.488 V 0.445 2269.4^ 0.082 0.079 oil was poured only upon the surface of the journals. 0.345 J I 0.079 0.081 r 0.2121 r 0.054 = In these experiments the 0.328 J 0.262 1 0.409 f 2269.4^ 0.052 0.052 oil was poured ceaselessly upon the rubbing surfaces. I 0.488 J I 0.052 Mean 0.053 0.177) s j 0.429 ) 0.409 [ 0.465 ) 2241.8 -j 0.101 ) 0.109 V 0.101 ) In these experiments the oil was expelled hy the pres- sure, and the surfaces were simply very unctuous. Meau 0.104 r 0.190^ - 0.0701 0.268 0.069 In these experiments the 0.177- 1 - 0.328 0.393 - 2240.7- 0.075 0.084 ' surfaces themselves supplied the lard. " 0.445 0.070 0.465 j 0.060 : ; 0.222^ 0.049 s . 0.331 0.050 In these experiments the 0.328- a - 0.380 , 4157.- 0.052 - unguent was renewed. H 0.409 0.040 0.429^ 0.042 ' very, low. ' 0.037 = 0.150 0.039 . 0.238 0.025 In these experiments the 0.328- OS " 0.321 - 2276. - 0.026 unguent was continually re- H 0.321 0.035 newed. 0.380 0.026 . . 0.492 J 0.832 810 FRICTION. The examples contained in this table suffice to show that the friction of journals is in itself subject to the same laws as that of plane surfaces ; but they also show the great influence which the constant renewal of the unguent possesses in diminishing the value of the ratio of the fric- tion to the pressure, which sometimes falls as low as .025. We see also that the diameter of the journals seems to have some influence upon the more or less complete expulsion of the unguent, and consequently upon the friction, so that the dimensions to be given them should not be determined from a consideration solely of their resistance to rupture. Recapitulating, the summary of the experiments which I have made upon the friction of journals, shows that it is nearly the same for woods and metals rubbing upon each other, and that its ratio to the pressure may, accord- ing to the case, take the values given in the following TABLE. STATE OF SUEFACES. With rotten-stone and perfectly greased Continually supplied unguent Greased from time to time / Unctuous 0.025 to 0.030 0.050 0.07 to 0.08 0.150 256. Advantage of granulated metals. It is not true, as is generally supposed, that the friction is always less between substances of different kinds than between those of the same kind. But it is well generally to select for the rubbing parts granulated rather than fibrous bodies, and especially not to expose the latter to friction in the direction of the fibres, because the fibres are sometimes FRICTION. 311 raised and torn away throughout their length. In this respect, fine cast-iron, which crystallizes in round grains, as well as cast-steel, are very suitable bodies for parts subjected to great friction. Thus, for several years past, a cast-iron packing has come into very general use for the pistons of steam-engines. If for the boxes of iron or cast- iron axles, brass continues in use, it is chiefly because it is less hard, and wears out before the axles, and because it is easier to replace a box than an axle. 257. Remarks upon very light mechanisms. In very light mechanisms, and especially with very rapid motion, the viscosity of the unguent may offer a resistance similar to that produced by friction proper ; in such cases, the results of experiments made under considerable pressures in relation to the surfaces of contact, should only be ap- plied with extreme caution. 258. Use of the results of experiments. The results obtained from the experiments at Metz are resumed in the three following tables, which give the ratio of the friction to the pressure, for all the substances employed in constructions. The first of these tables relates to plane surfaces which have been some time in contact. The values which it gives for the ratio/* of friction to the pres- sure, should be employed whenever we are to determine the effort necessary to produce the sliding of two bodies which have been some time in contact. Such is the case with the working of gates and their fixtures, which are used only at intervals more or less distant. 312 FRICTION. TABLE No. I. Friction of plane surfaces which have been some time in contact. Kind of surfaces in contact. Disposition of the fibres. Condition of the surfaces. Ratio of friction to pressure f. parallel . . without unguent . . . 062 do rubbed with dry soap 0.44 054 Oak on elm do wood upright on wood flatwise parallel moisten'd with water without unguent do 0.71 0.43 0.88 Elm on oak < do do do rubbed with dry soap 0.69 0.41 Ash, pine, beech, sorb on oak perpendicular parallel .... without unguent do 0.57 0.53 Tanned leather on oak < Black curried ( on plane oak sur- the leather flatwise the leather on edge parallel . ... do do moisten'd with water without unguent .... 0.61 0.43 0.79 0.74 leather < face, or belt ( on oak drum. perpendicular do 0.47 Hemp matting on oak < parallel without unguent 0.50 do parallel moisten'd with water without unguent. 0.87 080 parallel do 0.62 Oast-iron on oak. do parallel moistn'd with water do 0.65 065 Brass on oak parallel without unguent. . . . 0.62 Oxhide for piston packing on flatwise moisten'd with water 0.62 cast-iron on edge . . with oil lard, tallow 0.12 Black curried leather, or belt flatwise $ without unguent . 0.28 upon cast-iron pulley Cast-iron upon cast-iron. do ( moist'd with water without unguent. . . . 0.38 0.16* Iron upon cast-iron do do 0.19 Oak, elm, yoke-elm, iron, cast- ) iron and. brass sliding two > flatwise spread Avith tallow, O.lOt and two one upon the other. \ Calcareous oolite upon oolite ) limestone . ) do with oil, or lard. . . without unguent. . . . 0.15J 0.74 Hard calcareous stone called J Muschelkalk upon oolite > limestone . . . ) do do 0.75 Brick on calcareous oolite do do do 0.67 063 Iron on do do Hard muschelkalk on muschelk. Calcareous oolite upon do. ... Brick on do Iron upon do Oak on do Calcareous oolite on calcareous 1 oolite . [ do do do do do do do do do do do do do ("with mortar three J parts fine sand and 1 1 part of hydraulic 0.49 0.70 0.75 0.67 0.42 0.64 1 0.74 1 lime. * The surfaces being somewhat unctuous. t When the contact had not been long enough to press out the unguent. $ When the contact had been long enough to press out the unguent and bring the surfaces to an unctuous state. After a contact of from 10 to 15 minutes. 313 TABLE No. II. Friction of plane surfaces in motion upon each other. Surfaces in contact. Position of fibres. State of surfaces. Ratio of friction to pressure f. Oak on oak < parallel 'I without unguent. . . rubb'd with dry soap } without unguent wet with water J without unguent do do do do do wet with water 0.48 0.16 0.34 0.25 0.19 0.43 0.45 0.25 0.36 to 0.40 0.62 0.26 0.21 0.49 0.22 0.19 0.62 0.25 0.20 0.27 0.30 to 0.35 0.29 0.56 0.86 0.23 0.15 0.52 0.33 0.38 0.44 0.18t 0.15t 0.31 0.20 0.22 0.16$ 07to08 0.15 0.64 0.67 0.65 0.38 0.69 0.38 0.65 0.60 0.38 0.24 0.80 do perpendicular do upright on flatwise, parallel ) Elm on oak . < perpendicular > parallel 3 Ash, pine, beach, wild pear and goto on oak ,0 do 1 do j do do do do flatwise on edge.. < I flatwise and on J edge | rubbed with dry soap, without unguent. wet with water rubbed with dry soap, without unguent do do do do wet with water Iron on elm . Black curried leather on oak Tanned leather on oak $ Tanned leather upon cast-ironl and brass i without unguent wet with water unctuous and wet with Hemp strips or cords upon $ oak ; [ spread with oil without unguent. wet with water perpendicular parallel Wild pear on cast-iron do do do do do do do do do do wood upright parallel do do do do wet with water Iron upon iron Iron upon cast-iron and brass Cast-iron on cast-iron and brass. Cast-iron on cast-iron ( OH brass Brass < on cast-iron . . do do lubricated in the usual way with tallow,lard, soft coom, &c r on iron Oak, elm, yoke-elm, wild pear, f cast-iron, iron, steel, steel and j brass, sliding upon each other j Calcareous oolite on calc. oolite < Muschelkalk upon do do.. Common brick upon do do . . Oak on oolitic limestone Forg'd iron upon oolitic limestone Muschelkalk upon muschelkalk. . Oolitic limestone upon do. ... Common brick on do .... Oak on do Iron on do . . \ C slightly unctuous to ' the touch . ... ( without unguent. . . . do do do do do do do do j do (wet with water wood upright parallel do * Surfaces worn when there was no unguent, t The surfaces still being slightly unctuous. % The surfaces slightly unctuous. When the unguent is constantly supplied, and uniformly laid on, this ratio may be lowered to 0.05. 314: FRICTION. TABLE No. III. Friction of journals in motion upon their pillows. Surfaces in contact. State of surfaces. Ratio of friction to the pressure when the unguent is renewed. in the common way. continuously. Cast-iron journals in cast- iron bearings ' unguents of olive oil, of lard, of tallow, or of soft coom 0.07 to 0.08 0.08 0.054 0.14 0.14 o.or to o.os 0.16 0.16 0.19 0.18 0.10 0.14 O.OT to 0.08 O.OT to 0.08 0.09 0.19 0.25 0.11 0.19 0.10 0.09 0.12 0.15 0.030 to 0.054 0.03 to 0.054 * t 0.090 0.030 to 0.054 0.030 to 0.054 * 0.030 to 0.052 0.07 with the same unguents and moistened with water Cast-iron journals on brass unctuous unctuous and wet with water. . "unguents of olive oil, of lard, of tallow, and of soft coom \ unctuous Cast-iron journals on lig- num vitae bearings Wrought-iron journals on cast-iron bearings unctuous and wet with water. . . slightly unctuous C without unguent unguents of oil or lard < unctuous with oil or lard unctuous, with a mixture of lard l_ and black lead j unguents of olive oil, tallow, lard, \ or soft coom Wrougbt-iron journals on brass bearings ["unguents of olive oil, tallow, lard j unguents of soft coom | unctuous, and wet with water. . (^slightly unctuous Iron journals on lignum vitse bearings J unguents of oil or lard 1 unctuous Brass journals on brass bearings J unguents of oil unguent of lard .... Brass journals on cast-iron cushions Lignum vitse journals on cast-iron cushions. Lignum vitae journals on lignum vitse cushions . . . unguents of oil or tallow. . . . unguents of lard unguent of lard * The surfaces began to wear. t The wood being slightly unctuous. $ The surfaces began to wear away. Table No. 2 relates to plane surfaces in motion upon each other, table No. 3 applies to journals in motion upon their bearings. The values given by these tables ought not to be used except to calculate the friction of two sur- faces in motion upon each other, after the period in which the coefficient of friction at the starting has been introduced. FRICTION. 315 259. Application to gates. Let L be the horizontal width of a gate under a certain head of water, and H' the head or height of level above a horizontal section of this gate, of a thickness A' infinitely small. The pressed sur- face of this element will be LA', and the pressure which it will experience will be 62.32LH/A'. The total pres- sure upon the entire surface of the gate being equal to the sum of all the similar pressures upon each of the elements, will have for its value 62.32L(H / A / +H // A // +H /// A /// +&c.). Now, the products LH'A', LH"A", etc., are the mo- ments of the elementary surfaces LA 7 , LA", etc., in rela- tion to the plane of the level, and their sum is equal to the moment of the whole surface equal to LEU. Calling E the height of the gate pressed, and H the distance of the centre of gravity from the surface of the level, or the head upon the centre of the figure. Then the total pres- sure is 62.32 LEH, and the friction which results against the slides of this gate is 62.32/. LEH, /being the ratio of the friction to the pressure for the surfaces in contact, a ratio whose value should be taken from the first table, if we are to calculate the effort required to put the gate in motion. fl EXAMPLE.-]? L=6.56 ft , E=1.148 ft -, H=4.92 ft -, the first table gives for a wood gate of oak sliding with crossed fibres upon oak wet with water y=0.71 ; we have then for the friction 62.32 x 0.71 x 6.56 x 1.148 x 4.92 ft -=1639.4: lb9 - FHJ.IOS. 316 FKICTION. The effort should be transmitted in the direction of the racks fixed upon the gate ; and as it is considerable, it will be proper to arrange a kind of screw-jack, suitably proportioned, for the establishment of which we may take as the effort to be exerted by a man upon the winch, at any instant, from 55 to 66 pounds at most, and during the motion from 22 to 26.5 pounds. When the gate is in motion, the effort to be trans- mitted to the racks is much less, because the ratio of the friction to the pressure diminishes, and is reduced for a gate with moistened slides to 0.25, which gives for the friction during motion 62.32 x 0.25 LEH=62.32 x 0.25 x 6.56 x 1 .148 x 4.92 =577.2 lbs - at the first instant, and a value decreasing with the raising of the gate, or as the head H upon its centre is lessened. We hardly need to say that, in working the gate we must calculate for the maximum effort. 260. Application to saw frames. If we have, for example, the frame of a saw for veneering, subjected to a pressure of 110.274 lb % and provided with iron strips sliding in brass grooves, greased with lard, we have, if the surfaces are well lubricated, for the friction, 0.07xll0.274=7.719 lb % and, if they are unctuous, 0.15xll0.274=16.54 lbs - If the stroke of the frame is 3.936 ft> , and the number of strokes 180 in 1', the space described in V will be 11.81 ft ', and the work consumed by the friction of the frame in V will be, in the first case, 2 x 11.81 x 7.719=182.32 lbs - ft -=4 horse power nearly, o FRICTION. 317 in the second case 2 x 11.81 x 16.54r=390.66 lbs - ft -=| horse power nearly. 3 261. Application to journals. To calculate the work consumed by the friction of the journals of a revolving axle, we begin by seeking the resultant of the forces act- ing around this axle, and decompose this into two, the one horizontal and the other vertical, and we take sepa- rately the resultant of each of these groups. Calling X the sum of the horizontal components, Y the sum of the vertical components, the general resultant will be and the friction produced by it will be The theorem of M. Poncelet, already cited in "No. 227, informs us that when we do not know the order of mag- nitude of X and Y, we may calculate to nearly - of the value of the radical by the formula 0.83 (X+Y), and that if we know beforehand that one of the terms, X for exam- ple, is greater than the other, which is most usually the case, we shall have the value of the radical to nearly, 2o by the expression 0.96 X+0.4Y. Suppose, for example, that we have a hydraulic bucket- wheel weighing 88219 pounds, transmitting a useful effect of 50 horses' power to the exterior circumference, and imparting motion to a pinion, so that the useful resistance may be horizontal and represented by Q. Suppose the radius of the wheel R=:9.84: ft -, the velocity at its circum- ference to be 5.249 ft> , and the radius of the gearing wheel 318 FEICTION. K'=6.56 ft> The effort P transmitted to the circumference of the wheel will be The pressure upon the journals of the hydraulic wheel will be or, since M=88219 lb % and consequently M+P is greater than Q, we may take for an approximate value of the radical to nearly 25 0.96 (M+P)+0.4Q. For uniform motion, the moment of the power P must be equal to the sum of the moments of resistances. We have then, in calling r the radius of the journal =0.393 ft> , /=0.07, 5239 lbs - x 9.84= Q x 6.56+0.96 (0.07) (88219+5239) (0.393) +0.4 (0.07) (Qx 0.393); whence n _5239 x 9.84-0.96 x 0.07 x 93458 x 0.393__ AQ Qlbs *** 6.56x0.4x0.07x0.393 while if we had neglected the friction of the journals, we should have found The velocity of the gearing wheel being 5.249 x?=3.499 ft - o FRICTION. 319 The work transmitted to this circumference in V is 7469.8 lbs - x 3.499=:26137 lbs - ft -=47.5 horses' power. The loss by the friction of the journals is then 50.00 horses' power -47.5=2.5 H. P. If the surfaces of the journals had not been unctuous the loss would have been double. The space described by the rubbing points, being one of the factors of work consumed by the passive resistance, it is important to diminish it the most possible, and con- sequently to give the journals only such dimensions as will ensure a proper strength. To calculate their diameter in the establishment of the wheel, we disregard the friction, which will give us a first value of Q=785S lbs -, a little too much, and conse- quently for the resultant of the efforts to which the jour- nal is subjected of pres- sure, and its diameter, calculated by the formula for journals of hydraulic wheels, will be * d=. 00364 -v/46893; whence This is the value which we have adopted in the pre- ceding calculation. * In original / p ~ r 368156* In English measures ( =/ P x 3.2809' 320 FRICTION. 262. Axles of wagons We should calculate in a similar manner the friction of the axles of wagons against their boxes ; observing that it is the box which slides around the axle, and that the path described by the rubbing points is at the circumference of the box, and its arm of lever the radius of the box. KIGIDITY OF COEDS. 263. Rigidity of cords. When a cord solicited at its extremity by a weight or an effort of tension, is passed over a cylinder or a pulley, movable around its axis, it experiences a diffi- culty in bending owing to its stiffness, and the curve it takes is of a greater radius than that of the cylinder, so that the direction of the part prolonged passes at a greater distance from the axis, than the radius of the wheel in- creased by that of the cord. From this it follows that the moment of ten- sion of this part is increased by a cer- tain quantity arising from the resist- ance of the cord to flexure. This resistance, known by the name of rigidity of cords, has been experimentally investigated by Amon- tons, and more lately by Coulomb, who made use of the apparatus contrived by his predecessor, and of another arrangement similar to that described in No. 223, which served for his experiments upon rolling. We may gather sufficiently exact ideas of these researches, by confining ourselves solely to those of Coulomb, which, though in- complete, are the best we have upon the subject. 264. Experiments of Coulomb, with the apparatus of Amontons. In this apparatus, a free roller LM is en- 21 FIG. 104, 322 RIGIDITY OF COEDS. circled by one turn of each of the two portions of a cord, which passes over two pulleys A and B made fast to a beam. At the ends C and D of this cord are two hooks sustaining a platform loaded with a weight. The roller is Q FIG. 105. a FIG. 106. placed horizontally, and the turn of each of the portions encircling it, is arranged symmetrically in respect to each other. Midway between these turns, a flexible thread passes round the roller, to which it is fixed at one end, while it supports at the other a small plate, in which is placed a weight q, necessary for the slow descent of the roller. In this movement, the lower portion of the cord is enrolled upon the roller, and the upper part is unrolled. The tension of each part is equal to the half -~ of the load 2 of the platform. Moreover, it is readily seen, that the space described by the motive weight q will be double the space described by the enrolled parts of the cord. In fact when the roller has descended from the point of con- tact a to the point 5, (Fig. 107), it is evident that the arc of the enrolled cord, or the space described in the direc- tion of the resistance to rolling will be ab. It is clear RIGIDITY OF COEDS. that in this displacement, the point of the thread of sus- pension of the motive weight <, which shall have come into the vertical or in contact, will be a point d placed at a distance cd equal to the arc c^'=arc db'=db) which the roller itself has described. Then the weight will have descended db by the translation of the roller, and cd'=ab by its rolling, or %db in all. The work developed by this weight will be qx2al>=q. Da,, calling D the mean diameter of the roller, and a l the angle de- scribed at the unit of distance, while the work developed by the rigidity Bj of each portion will be equal to the rigidity itself multiplied by the space described, or to D FIG. 107. We shall have then at the moment of equilibrium, or when the motion is very slow and nearly uniform, by reason of the resistance of the two portions of the cord, qD=^ 1 - ; whence q=E>^ 2i that is to say, the motive weight is equal to the resistance which each of the two parts oppose to the enrolling. 265. Results of the experiments of Coulomb. Before taking other steps, we introduce in the following table the data and results of some experiments made by Coulomb, with the apparatus of Amontons, limiting ourselves to the transformation of the old measures into the new. The cords of 6 threads, of 15 and 30 threads, which he used, were rolled on rollers 1.09 in % 2.18, and 4.37 ins - diam- eter, and the total tensions varied from 55 to 2205 pounds, the motor weights varying also through extended limits. 324 RIGIDITY OF CORDS. TABLE. Results of experiments of Coulomb upon the rigidity of cords, made with the apparatus of Amontons. 03 O 9"te * a 2 VALUES OF THE MOTOR WEIGHTS FOUND FOR CORDS OF THE DIAMETERS d=0.02SSft. or 6 threads rolled round rollers with diameters D equal to inclination will furnish the values of A and for each 2 cord and each roller. M. Navier, in a discussion of the experiments of Cou- lomb, which is published in the second edition of the " Architecture hydraulique " of Belidor, has attributed to the constants A and B particular values for the different diameters of the cords, which are respectively as follows : DIAMETER of cords d. VALUES OF COEFFICIENTS* A. B. 0.065ft. 0.047 0.0288 1.6097 0.4596 0.0767 0.031949 0.018104 0.007808 But for comparing M. Javier's formula with the re- sults of experiments, we must introduce in the formula R=A+BT the values of T= , to derive the weight #, lound with the apparatus of Amontons. It is thus, we have made this comparison in the figures of 108, taking for a graphic representation of the results of experiments, the abscissa equal to the total load, and the ordinates *equal to the values of qD, or to the resistance of rolling upon a cylinder 1 foot in diameter. Then, to compare these results with the values of the coefficients deduced by M. JSTavier, we have taken for the same abscissa ordi- nates equal to the values of A+ - Q, deduced from the 2t values of A and B as given by him in the preceding table. An examination of Fig. Ill shows that the values of A and B adopted by M. Kavier accord very well with the results of observations for the cord of 0.065 f % the * The coefficients of M. Navier have been changed to suit the English units of Ibs. ft. ; and the following tables are worked for a drum of 1 foot diameter, instead of 1 metre. KIGIDITY OF COEDS. 327 straight line traced according to the values of his coeffi- cients, diverging but slightly from all the points, obtained FIG. 108. Fig. I. Cord of 0.0288ft. Fig. II. Cord or 0.0472ft. Fig. III. Cord of 0.065Gft. 100 200 300 400 500 600 700 800 Total weight in Pounds. Straight lines indicate M. Navier's formula. 1000 1100 graphically by figures directly derived from experiments; but for the cords of 0.(M7 ft - and 0.029 f % the values adopted by this engineer are too small, especially for the number B, the points corresponding to the figures of the table in Figs. I. and II., relative to the cords of .029 ft - and .047 ft< be- ing all situated above the straight line which represents the formula. The figures of M. Javier seem to have been determined solely by means of the last series of experiments made upon each cord, and with the value q obtained for the greatest load. 328 KIGIDITY OF COEDS. 267. Other experiments of Coulomb. Coulomb also made use of another method of experimenting upon the FIG. 109 See page 852. Fig. IV. Battery wagon with gun upon the route from Montigny to Metz, dry. Fig. V. Battery wagon with gun upon the pavement of Metz. Fig. VI. Stage coaches without springs upon the pavement of Paris. Fig. VII. Stage coaches with springs upon the pavement of Paris. 0.020; 0.01811 0.016 0.014 1 0.012 1 0.01 Ol 0.008 3.38ft 4.92ft. 6.56ft. 8.20ft. Velocities. rigidity of cords, and their resistance to rolling ; having placed rollers upon the horizontal bench previously used in his experiments on friction, he loaded them with equal weights, suspended upon the two strips of the cord. A gentle motion was produced by weights upon a flexible cord, whose rigidity was not regarded. Previous experi- ments having enabled him to appreciate the resistance due to the rolling of the rollers, by subtracting it, he ob- tained that arising from the rigidity of the cords. The results of these experiments are given in the following table, which is reduced to measures in feet. KIGIDITY OF COEDS. 329 TABLE. Experiments upon the rigidity of cords made ~by Coulomb with movable rollers upon a horizontal plane. Loads or tensions Q. Values of the resistance to rolling for cords with diameters of ' i5 * r'+r In the application to wagons heavily loaded, which are the most important for our consideration, the weight of the wheels is but a very small fraction of the load, and the weight proper of the body of the wagon, and it may be neglected alongside of the total load, which reduces the ratio to w= / ,, f r vehicles with four wheels, or to jTj T -\-T ~= J l for vehicles with two wheels. *i T These experiments will hereafter serve to determine by experiment the ratio of the draught to the load for the most usual cases. 278. Influence of the pressure. To ascertain the influ- ence of the pressure upon the resistance to rolling, we set in motion the-same vehicle with different loads, upon the same road, in the same condition. We give the results of some of these experiments made at a walking pace. 346 DRAUGHT OF VEHICLES. TABLE. Experiments upon the influence of pressure upon the draught of vehicles. Vehicles used. The routes run over. Pressure. Draught Ratio of draught to load. Ibs. Ibs. 1 13215 398.4 QQ 1 Artillery Ammu- nition wagon. Road from Courbevoie to Colombes, dry in good order and dusty. 13541 352.6 dO. J. 1 "381 1 10101 250.7 40^2 1 15716 306.3 5T.3 Wagon without , springs. Road from Courbevoie to Bezons, solid, hard gravel, very dry. 12037 9814 245.9 205.5 1 1&9 1 47.7 1 7565 150.8 50.1 1 1 From Colombes to Cour- 1 3528 86.6 40.8 Wagon with springs. bevoie, paved, in good ( order, with wet mud. 7260 196.7 1 36.9 1 11018 299.9 "3678 " 1 6616 306.3 21.6 Wagon with six 1 equal wheels. Two wagons con- - nected with six ?rom Courbevoie to Co- lombes; deep ruts; wet detribus. 10348 494.0 ~2l7 1 equal wheels. 13232 630.3 TiT 1 13232 632.3 ~2l7 It results from an examination of this table that upon solid metalled roads, and upon pavements, the resistance to the draught of wagons is sensibly proportional to the pressure. DRAUGHT OF VEHICLES. 347 We would observe that the experiments made with one only or two wagons with six wheels have given the same total draught for the same load of 16232 pounds, vehicles included. It follows from this, that the draught is, all else being equal, and moreover within certain limits, independent of the number of wheels. We may also draw the same consequence from results given in the following table, relative to the same wagon employed successively with six and four wheels. The resistance was the same in the two cases for the same load. 279. Influence of the diameter of the wheels. To study the influence of the diameter of wheels upon the draught, we have respectively set running over the same part of the road, in the same condition, wagons having the same weight, and having equal widths of tires, but with diameters differing in very extended limits. We publish in the following table some of the results ob- tained. We have also compared with the artillery ammunition wagon, whose wheels have a diameter of 6.654: ft -, different kinds of vehicles, including drays whose wheels had a diameter not above from 1.94:2 ft - to 1.378 ft - The ratio of the draught to the pressure varied from ^ for the largest wheels to % for the smallest, upon a road paved with sandstone from Fontainbleau. Upon the metalled road from Courbevoie to Colombes, the experiments were made upon diameters, comprised between 6.654 ft - and 2.86 ft - These examples show that upon solid roads we may admit, as a law for practice, that the draught varies in the inverse ratio of the diameter of the wheels. 348 DRAUGHT OF VEHICLES. 00 IrH ^ ^ IO - Hi* [co |t~ rH IrH OS CO I N I CO IrH t~ |os r ~ l OS|OS* r " l |O r ~'H> r "' co jco |co |os |os p |o - 3 1 5 CO |CO ' COCO W3H OSlM O Oi ^H CO ^^ "^ ^ "^ ' W5 01 H IH d d d d 111 Ifj III IsJ HI | -o ^ a o CO "* bO tiO I ^ I i ft p DRAUGHT OF VEHICLES. 349 General Piobert, who gave his attention to theoretical and experimental researches upon the resistance to rolling, concluded that this resistance varied in the inverse ratio 2 of a power of the diameter included between - and unity, o approaching more nearly to the last limit as the ground is harder ; and that upon pavement this resistance varies in the inverse ratio of the radius of the wheel, increased by the roughness of the pavement. If we would apply this law to experiments made upon roads of solid metalling, dry or moist, admitting, for ex- ample, that the power of the radius to be employed is 4. -, we find the results inserted in the last column of the 5 preceding table. An examination of these results, which are expressed in common fractions, shows that the law of Coulomb represents the results of an experiment made upon the road from Courbevoie to Colombes with an exactitude of - ; while by varying the resistance in the 12.5 4. inverse ratio of the power - of radius we obtain an ap- 5 proximation of . 15 Tow, in such researches, we seldom obtain direct results of experiment, which do not differ more than from to - , corresponding to the limits calculated by these 12 15 two laws. It follows, therefore, that in practice, we may for solid roads adopt the simple law of Coulomb, without fear of committing a grave error. 280. Influence of the width of the rims. This influ- ence was first investigated with an apparatus composed of a cast-iron axle, upon which were placed cast-iron discs turned at the periphery, and forming at once the load 350 DRAUGHT OF VEHICLES. and the wheels, whose total width was thus proportional to their number ; subsequently they used common wagon wheels, having the same diameter, but unequal breadths. Some of the results of these experiments are recorded in the following table : Experiments upon the influence of the widths of felloes upon the resistance to rolling. 1 s - 2 J* *. Vehicles employed. Ground passed over. Diameter of 3 I S pT .2 "g PS s wheels. P* ft 1 Polygonal enclosure at / ft 2.5827 ft. ft. 0.1476 .2953 Ibs. 2298.1 2944.3 Ibs. 853.3 461.4 0.1982 .2023 Apparatus with cast-iron axle. Metz. f Manoeuvring shed at the Metz school, sand from 0.39ft. to 0.49ft 2.5827 f .4429 .1476 .2958 .4429 .6069 3191,3 2306. 2988.4 8178.3 3043.5 861.2 556.2 589.5 597. 448.1 .1603 0.3114 .2547 .2425 .2070 i .7382 8671. 571.8 .2011 Artillery ammuni- Eoad from Courbevoie 4.7179 4.7179 .5741 7639.7 166.5 0.0514 tion wagon. to Colombes, wet 4.6458 4.6458 .1968 7957.3 166.3 .0496 Artillery ammuni- tion wagon. "Wagon 6 wheels- Paved with sandstone of } Fontainbleau. 4.7179 4.7671 4.7671 2.8215 4.7179 4.7671 4.7671 2.8215 .5741 .8773 .8773 .1968 12165.4 12169.8 10131.9 7211.9 183.0 160.1 128.4 157.9 .0355 .0314 .0302 .0309 These examples show : 1st. That upon soft foundations, the resistance in- creases as the width of the tire decreases, and for farming purposes we should use tires of alout 0.33 ft - width. 2d. That upon solid roads, metalled or paved, the re- sistance is nearly independent of the width of the tire. 281. Influence of the velocity. To appreciate the influence of the velocity upon the draught of vehicles we have put in motion, upon different roads in different con- ditions, the same vehicles, changing in each series of experiments only the velocity, which was successively, that of a walk, a fast walk, a trot, and a smart trot. Some * These values of A are for the unit of foot of radius. DRAUGHT OF VEHICLES. 351 of the results of these experiments are reported in the following table : Experiments upon the influence of velocity upon the resistance to the draught of vehicles. Vehicles used. Ground run over. Load. Gait 1 Draught. Eatioof draught to load. Ibs. ft. Ibs. 1 " *< walk 4.59 363.9 6.3 1 trot 9.19 370.5 Apparatus Polygonal en- 6.2 with cast- iron axle. closure at Metz, wet and soft. 2944.3 \ walk 4.19 474.18 1 6.21 1 ( trot 11.09 434.48 (Trr 41 O i - ' walk .13 202.90 40.76 Road from Metz 1 Carriage No. 16, with its" piece. to Montigny, metalled, very smooth, and " dry. 8270.5 - fast walk trot 4.99 8.04 202.90 224.96 40.76 1 36.76 1 - quick trot 12.40 396.99 30.99 1 } r walk 4.07 317.59 22.83 " 1 Stage wagon Paved with fast walk 5.57 337.44 21.49 hung on six sandstone from }- 7251.6 - 1 springs. Fontainbleau. trot 7.74 355.08 20.42 1 ; J - quick trot 11.81 382.65 iaw We see, by these examples, that the draught does not increase sensibly with the velocity^ upon soft bottom, but that upon solid and uneven roads, it increases with the inequalities of the ground, the stiffness of the wagon, and the rapidity of the motion. 282. Approximate expression for the increase of resist- ance with the velocity. To ascertain the relation existing 352 DRAUGHT OF VEHICLES. between the resistance and the velocity upon hard and uneven ground, we have taken the velocities for the ab- scissa, and for the ordinates, the values of the number A, furnished by experiment, and this graphic representation of the results has shown that all the points so determined were for each series of experiments situated very nearly upon the same straight line. Thus the experiment rela- tive to a battery carriage, loaded with its piece, a very rigid carriage, moved at different velocities upon a very well metalled road, and upon the pavement of the city of Metz, are represented in Fig. 109, IY. and Y., and we see that the value of the ordinates, or of the number A, in- creased with the abscissa or velocities according to a law which, in the limits of experiment, may be expressed by a straight line cutting the axis of ordinates at a certain height, which indicates that for a velocity zero, the resist- ance has still a certain value, or generally is composed of one part independent of the velocity, and of one part pro- portional to it, this resistance, or rather the value of the number A may then be generally represented by an ex- pression of the form in which a is a constant number for each kind of road-bed, in a determined condition, and which expresses the value of tiie number A for a velocity Y=l metre =3.2809 fti , which is that of a gentle walk ; d is a constant factor for each kind of road-bed and of carriage. In the particular case of the two series of experiments above cited, we have for a battery carriage with its gun Upon the road to Montigny jj^ j^ f t> very well metalled ........................... A=0.010 + 0.002 (0.3047 V 1). Upon the Pavement of Metz l^ s j^ s ft> of sandstone from Sierck ................... A=0.0066 + 0.0057 (0.3047 V 1). DRAUGHT OF VEHICLES* 353 These examples suffice to show : 1st. That at a walk, the resistance upon good pave- ment is less than upon a very good metalled road, quite dry; 2d. That with good speed, the resistance upon pave- ment increases rapidly with the velocity Y. Thus at a walk, upon the pavement of Metz, with wheels 3.28 ftj radius, the resistance would be 6.6 lbs - for every 1000 lbs - of the total load, vehicle included, while at a smart trot, V=4 m -=13.12 ft -, it would be Ibs. 6.6 lbs --h5.7 lbs -x 3=23.7 that is to say, nearly fourfold. Upon the pavement of Fontainbleau, with wide joints, and rounded edges, which offers many inequalities whose elements may be displaced under the action of the load, the resistance at a walk is much greater than upon the pavement of Metz, and the increase of the resistance with that of the velocity is still more rapid. Fig. VI., which represents the results of the experiment obtained with a. wagon of the general coach establishment, whose springs were wedged up, shows that the inclination of the straight line or the increase of the velocity is much greater than upon the pavement of Metz, and we deduce from it, for the representation of the value of A, the formula A=0.0092 lb9 -+0.0089 lb9 - (0.3047 Y ft l), which shows that for wheels of 3.28 ft - radius the resistance at a walk of 3.28 ft - velocity will be 9.2 lbs - for every thou- sand pounds of load ; that is to say, nearly one-half above that upon the pavement of Metz, and for a smart trot at a velocity 4 m =13.12 ft - in V it will be equal to, for every 1000 pounds, 9.2 lbs -+8.9 lbs -x3=35.9 lb % while that upon the pavement of Metz. was only 23.7 lb8> 23 354 DRAUGHT OF VEHICLES. As for wagons on springs, experiment shows that the resistance also increases, but much slower with the ve- locity upon roads with an uneven surface. Thus upon the pavement of Paris (Fig. VII.), the same stage wagon, whose springs were restored to free action, has given only for the value of A the expression Ar=:0.0098 lbs -+0.0025 lbs - (.3047 V-l), so that for a trot, at the velocity of 4. metres, and wheels l m - radius, the resistance for a load of 1000 pounds would be but 9.8 lbs -+2.5 lbs - x3=17.30 lb % that is to say, one-half of that experienced by the same wagon, unhung, upon the same pavement and with the same velocity. 283. Practical consequences of these experiments. These experiments show, on the one hand, the great advantagewith respect to traction and economy of motive power possessed by wagons with springs over those with- out them, and on the other hand, the superiority of pave- ments with narrow tight joints and smooth surface, over those with wide joints and uneven surface, generally used in Paris. These results, obtained in 1837 and published in 1838, attracted the attention of engineers, and we may suppose have stimulated trials, which have since been suc- cessful, for the use of cut stone blocks of regular forms, whose advantages the public can easily appreciate. 284. Comparison of paved and metalled roads. The same experiments show us that, if, for rolling at a walk, paved roads offer an advantage over the metalled, it is not so for great velocities upon good metalled roads, dry, and in perfect order; but that when these roads are wet, the pavement resumes its advantage. In fact, we find for DRAUGHT OF VEHICLES. 355 this last case, that upon the road from Metz to Nantz, wet, with some mud and pebbles upon a level, the value of the number A, which represents the resistance for 1000 pounds of load, with wheels 3.28 ft - radius, for the dili- gences of the general stage department, with springs, is given by the formula A=0.014: lbs - + 0.0022 lbs - (0.3047 V 1). Comparing this with that obtained for the pavements of Paris, we find that the draught per 1000 pounds with wheels 3.28 ft> radius would be : ft. ft. ft. ft. For velocities of. 3.28 8.20 9.84 13.12 Ibs. Ibs. Ibs. Ibs. Upon the wet metalled road from Nantz H. 17.30 18.40 20.60 Ibs. Ibs. Ibs. Ibs. Upon the pavement of Paris 9.8 13.55 14.80 17.30 The excess of draught experienced upon wet metalled roads, arises chiefly from their compressibility ; and natu- rally increases in proportion to the softness of the mate- rials, the moisture of the road, and to a want of proper maintenance. This last circumstance has upon the resistance to trac- tion a great influence, whose consequences injurious to the economy of transportation have not met with a proper attention. Experiments made in September and October, 1841, with the same wagon, running successively over various parts of the same road, show, the mate- rials and season being the same, the draught of this wagon, upon portions in good condition, to have been from to do of the load, while on the parts badly managed it rose OD from A to k- 356 DEATJGHT OF VEHICLES. 285. Influence of the inclination of the traces. To study the influence of this element of the question, we made use of a battery carriage with equal wheels, the pole being inclined 135', 335', 630', 830', 11, and 1330', and set this carriage in motion upon ground covered with wet grass, preserving otherwise the same weight and the same velocity in all cases. The effort of traction F, measured by the dynamome- ter, and exerted in the direction of the traces, is evidently resolved into two forces, the one F' horizontal and par- allel to the ground, which produces the motion, and sur- mounts all the resistances, the other vertical, F", which diminishes the pressure of the fore wheels upon the ground. From this it follows, in preserving the notations of No. 276, that the pressure upon the ground may be ex- pressed by 0.96 [P'+P"] +0.4F. So that calling f the ratio of the friction to the pressure, T\ the mean radius of the boxes, T that of the wheels, L the total space run over, The equation of motion of this carriage, upon level ground, was approximately Moreover, we have ^=0.124"-, r=2.565 ft -, /=0.065. Now, before going further, we will observe that on DBAFGHT OF VEHICLES. 357 account of the smallness of the term we may evidently neglect the value of this expression, and reduce that of R'-f-B", derived from the preceding equation, to F'-- L (p'+p"_ r =F' .00303 (P'-fP" F"), and, on the other hand, we know that we have then to compare the results of the above formula with experiments, the relation . _ F/ _ 0.00303 (P / +P // --F // ) P.-F" Now, this comparison has given the following results, which are the means of many experiments repeated for each case : Inclination of draught ....135' I 335' I 630' I 830' I 11 I 1330' Value of uumber A* ....0.1145 j 0.1145 [ 0.11778 | 0.1145 | 0.1145 | 0.1017 The agreement of all these values shows that the me- chanical effects take place exactly as indicated by the formula, when account is taken of the resolution of the efforts; consequently, to ascertain the inclination an- swering to the maximum of effect, we find, by known methods of calculation, that calling h the height of the forward point of attachment of the traces, above * The value of A is for a wheel one foot radius. 358 DRAUGHT OF VEHICLES. that of the hind attachment, Id, the horizontal projection of the distance of these two points, the ratio of the quanti- ties answering to the maximum effect of the motive power will be ~b~ rOAfr l ' This expression shows that for a given wagon, the in- clination of the traces, or the value of j increases with that of A or the resistance of the road bed, and in this respect the site chosen for the experiment was very suita- ble for the purpose. Moreover, for the same ground, the inclination increases as the radii of the wheels are dimin- ished. Applying the above relation to the battery carriage and to the ground of the experiment, for which we have -, A=0.1145, we find 20.9' If we had with the same data ^=0.820 ft , as for drays, we should have 6.71' a quantity much smaller than that generally in use. Upon metalled roads, for which A =.0492, with the usual state of moisture and maintenance, we should find for artillery wagons ^=0.022=i which is nearly the inclination adopted for battery car- riages designed for long marches. DEACJGHT OF VEHICLES. 359 It is not worth while to extend this discussion, to which constructors generally attach more importance than it deserves, and we limit ourselves to saying that, within the limits where it is necessarily constrained, the inclina- tion of the traces has but little influence upon the draught, and that, in common cases, it must be very slight. 286. Recapitulation and application of the general results of experiments. The following table gives the values of the ratio of draught to the load, for a great num- ber of different circumstances, the formula of No. 277, combined with the direct results of experiments, enabled us to calculate approximately the value of this ratio, for the usual proportions of wagons employed in trade. 360 DRAUGHT OF VEHICLES. STATEMENT OF THE DEAUGHT AND THE LOAD OF CAEEIAGE8, Artillery carriages Artillery wagons. Comtois 1 ' wagons. and carts. W=0.229ft to 0.246ft W=0.196ft. to .229ft. )esignation of the route passed over by the carriage. W=0.32ft. to 0.89ft r l =0.124ft. r'=2.565ft. r'+r"=5.130ft r 1 =0.124ft. r'=1.885ft. r"=2.558ft r'+r"=4.4426ft r 1 =0.087ft r'=2.049ft. r"=2.377ft r'+r"=4426ft. /r 1= .00806 //i=.00806 /r 1= .00574 Earth driftway in good order, nearly dry 1 ] 84:8 1 soil 1 81 Solid driftway, with bed of gravel .09ft to .13ft. 1 1 1 thick 11.9 Solid driftway, with bed .of gravel .16 ft. to 0.19ft. 1 1 1 tt Fin el lick 11.6 loii iol n ground, with grav- bed, 0.33 to 0.49 ft new road i Driftway with untrod snow i 1 fi A. i i a i Firm earth bed of fine JUUI lo i sand, with gravel 0.33 to n J.Q ft. thir-t m2 8iT 8i9 - in good order, very dry and smooth.. I walk g2J jtrot JL 1 54.3 1 [ 50.5 some wet covered 1 1 with dust^and some pebbles on surface. 448 8i7 40i8 led Eoads. very solid,with large pebbles on level of W6t SUrfilCG * 1 541 1 46.8 1 49.1 1 D 3 solid, slightly trav- elled, and soft mud. 1 848 1 30.1 1 Tl solid, with ruts and mud 1 28.5 1 24.6 1 25.2 W is width of tire. The other notations are given in Art. 276. DRAUGHT OF VEHICLES. 361 FOE DIFFERENT SOILS AND VEHICLES. " Charrettes de roulage." W=0.82ft to 0.89ft. Carts. Diligence of the imperial and gen- eral coach estab- Wagon on springs with seats. r 1 =0104ft. W=0.33ft. to 0.39ft. lishment. W=.229 to .262ft. r 087ft. r'=1.476ft r'=1.80ft. r \__L 1 ^ W=0.33ft. to 0.39ft. r"=2.480ft. r"'=2.79ft. r'=2 62 r-'=8.28 r,=0.104ft. r'=2.29ft. r'+r"=3.936ft r'+r"=459ft. /r^.00682 // 1 =O06S2 r'+r"=3.77ft. /' +r" ?.77fti /r 1 =.00682 yj- 1 =.00682 ^=0,00682 yrj=.00574 1 27.2 1 8L7 1 86.8 1 45.4 walk & trot ^ walk & trot 1 26.4 105 12^8 sr m walk & trot ^ walk & trot ioi 1 &f 1 10.4 i 11.9 i 149 1 walk & trot "g-g walk & trot 1 8.6 1 1 1 1 1 1 "as ill walk & trot ~g" walk & trot ~T i i i I 1 143 f6.7 IlT 23.8 iar 1 7.9 1 1 105 1 ial walk & trot y walk & trot i 6.9 i 'walk JJ-Q walk 1 49 1 1 1 i 1 1 49^9 "58 6O2 8^8 . trot 40.9 trot 4T8 1 1 fast trot 89j fast trot 406 1 1 walk 387 walk 848 1 1 1 1 1 1 35^2 IT 47.0 58i6 trot 2678 1 trot 2L2 1 fast trot 248 fast trot 24.6 1 walk 1 40.8 4T8 1 i 1 1 i trot 26~5 trot 1 4278 49^8 5^9 7T fast trot fast trot 27 1 22.6 22.8 : walk walk 1 26.1 204 1 1 1 1 1 trot 1 21.7 22 2L2 81J 86^2 4^2 1 1 fast trot 2^- fast trot 20.3 1 1 "walk -gj- walk 2T5 1 1 1 1 1 1 . trot ias trot 185' 22^2 25.8 29^5 86.9 i 1 fast trot iTl fast trot 17.2 362 DRAUGHT OF VEHICLES. Artillery carriages and carts. Artillery wagons. " Comtois " wagons. Designation of the route passed over by the carriage. I W=0.82ft. to 0.89ft. r,=0.124ft. r=2.565ft. r +r"=5.130ft. W=0.229ft. to 0.246ft. r 1 =0.124ft. *-'=1.885ft. r"'=2.558ft. r'+r"=4.4426ft. W=0.196ft. to .229ft. r,=0.087ft. f=2.049ft. r"=:2.377ft, r'+r"=4.426ft. 1 /r^.00806 /r^.00806 /r 1= =.00574 ( with detritus and 1 1 1 thick mud 241 20T8 2T3 FH 13 <3> ' much worn,deep ruts from .196ft.to. 262ft. 1 1 1 3 2 thick mud. 18.4 15.9 16^2 ring of bridge with 1 1 1 jo ists 54.1 46TS 49.1 287. General conclusions. From a general inspection of all experiments, we derive the following practical laws: 1st. The resistance opposed to the rolling of wagons, by solid metalled roads or by pavements, and referred to the axis of the axle, in a direction parallel to the ground, DKAUGHT OF VEHICLES. 363 " Charrettes de roulage." W=0.32ft. to 0.39ft. Carts. Diligence of the imperial and gen- eral coach estab- "Wagon on springs with seats. /!=:0.104ft. W=0.33ft. to 0.39ft. lishment. W=.229 to .262ft. r 1 =0.104ft. W=0.83ft. to 0.39ft. rj=0.087ft. r'=1.4Sft. r'=1.476ft r'=1.80ft. p"=2.480ft. r"=2.79ft. ' r'=2.62 r ' 3 gg r=.893 ft + f" 3 936ft r'+r"=4.59ft. //,=a.0682 1 />' 1 =0.06S2 r > _j_^,"_lg 77ft /r^.00682 /r^.00682 /r 1 =0.00682 y^jszO.00688 fwalk walk 1 17.9 8 1 1 1 1 1 1 _i^ 18J 2T8 2lL9 31.1 isTs trot j smart trot - smart trot i I 14.9 15 f 1 i walk jg- 7 walk las 1 1 1 1 trot JL trot i 14.8 16.7 IsT 28.8 12.4 laB quick trot _L quick trot i 11.8 .1.8 1 1 i 1 (j_ Wftlk 12~.2 walk 1 12JJ 12.7 14.9 17 2T2 1 1 trot ^ trot 9i9 walk -1. walk 1 64.2 1 1 1 1 . trot JL ;rot 1 64.7 75.5 86.3 107.9 42 43" 1 1 fast trot gg-g fast trot "37 1 I walk ^Q walk 59 1 1 1 1 trot trot 1 59.6 69.5 79.9 99.9 88.1 "89" fast trot fast trot 1 32. i sag i Walk 5771 walk i 59 trot 1 40.9 trot 4L8 fast trot fast trot 1 85.8 86T5 walk walk 1 44 45.1 1 1 1 1 1 trot 1 00 K 46 53.5 74.4 76.5 1 oo.O 1 fast trot 2^2 fast trot 2^8 1 1 1 1 1 1 4^8 w n walk & trot ^ walk & trot a3| is sensibly proportional to the pressure or total weight of the vehicle, and inversely proportional to the diameter of the wheels. 2d. Upon paved ov metalled causeways, the resistance is very nearly independent of the width of the tires. 3d. Upon compressible bottoms, such as earths, sands, 364 DRAUGHT OF VEHICLES. gravel, etc., the resistance decreases with the increase of width of tire. 4th. Upon soft earths, such as loam, sand, earth-drift- ways, etc., the resistance is independent of the velocity. oth. Upon metalled roads and upon pavements, the resistance increases with the velocity. The increase is sp much less, as the wagon is better hung, and the road more smooth. 6th. The inclination of the draught should approach the horizontal for all roads and for common wagons, as far as the construction will admit. Let us bear in mind that these simple laws are not strictly mathematical, but merely approximate, which, for the most common cases of practice, and for the usual dimensions of carriages, will represent the results of experiment, with sufficient exactness, and very nearly equal to what may be deduced from the direct experiment. It is in this sense only that I have proposed and applied them. 288. Consequences relative to the construction of vehi- cles. From what precedes, it follows that the transporta- tion business has for its interest to use for vehicles, wheels of the greatest diameter comporting with their construc- tion and destination. Carts being more readily adapted than other two- wheeled vehicles to the use ot great diam- eters, afford in this regard a marked advantage. But, on the other hand, if the roads are in bad condition,, pro- ducing jolts, the shaft horse being knocked about by the thills, becomes fatigued, is soon ruined if he is spirited, or if lazy will leave the draughting to be done by the other horses. Now, by bringing the axle of the hind wheels nearer to the forward wheels, and thus placing them more under the load, the proportion of the load borne by the hind wheels will be more considerable, and so the draught will be diminished. "We may then considerably reduce the DRAUGHT OF VEHICLES. 365 draught of the small wheels, which will thus be relieved, and the chariot is nearly transformed to the cart. Still, we must place in the front a sufficient preponderance of the load, so that in ascending, the box may run no risk of rising and turning around the hind axle. This obser- vation shows that the weighing of vehicles in the lump would be fallacious, if it is pretended that the weight is distributed in equal parts upon each wheel. It is well known that wagoners have for a long time appreciated the necessity of loading the hind truck in a much greater proportion than the fore truck. But we see that for a given and nearly constant distribution of the load, as with diligences, omnibuses, etc., there is an advantage in bringing the hind axle as far as possible under the vehicle, and this explains why, all else being equal, short vehicles require so much less draught than long ones.* 289. Destructive effects produced lyy vehicles upon roads. The destructive influence of vehicles upon roads has for a long time called for the attention of governments and of engineers ; but whatever may have been the im- portance of this question, for the interest of the state and trade, there has been, to the present, but little time de- voted to a profound study of facts, in place of which we have theoretical considerations more or less plausible, but often quite contradictory to nature. Without entering into a discussion which would take us beyond our pro- posed limits, we propose to examine successively the con- sequences which we may deduce from direct experiments upon the draught, as far as concerns the preservation or destruction of roads, and we will afterwards publish the principal facts which we have directly observed. 290. Preserving influence of great diameters of wheels. * For further details see "les Experiences sur le tirage des voitures et sur les effets destructeurs qu'elles exercent sur les routes." 366 DRAUGHT OB' VEHICLES. The resistance experienced by a wheel from the ground, being evidently a more or less immediate measure of the efforts of compression or of disintegration which it exerts upon the ground, we see at once that, since wheels of a great diameter occasion less draught than those of small diameter, they must also produce less disintegration upon roads. A very simple observation confirms this fact. If we take stones from 0.22 ft - to 0.26 ft - diameter, and upon a road somewhat wet and soft, place some before the small wheels of a diligence, and others before the great wheels, we see the first are pushed forward by the small wheels penetrating the ground, which it ploughs and disin- tegrates, while the second, simply pressed and borne down by the great wheels, seldom experience a displacement. This results evidently from the fact that if we resolve the effort exerted by the wheel, upon the stone at the point of contact, into two others, the one vertical, tending to bury the body in the ground, the other horizontal, tending to push it forward, the second effort, which pro- duces the tearing away of the road, is evidently much greater proportionally for the small wheels than for the great. From this simple observation we infer, as I had already done in 1838, that the effects of disintegration produced ~by the wheels of vehicles, are so much the greater, as the wheels are smaller. Experiment having also proved that the draught on solid bottoms increases but slightly with the decrease in breadth of tire, we may also infer that the loads capable of producing equal destruction should not increase pro- portionably with the width of the felloes, as all the rules of the carriage police have admitted, and that the loads permitted, according to these rules, for the broadest wheels, must produce more wear than those with narrow wheels. Finally, the resistance increasing with the velocity, it DRAUGHT OF VEHICLES. 367 was natural to suppose that wagons going at a trot do more mischief to roads than those going at a walk. But suspension, by diminishing the intensity of the shocks, may compensate for the effects of velocity in certain pro- portions. 291. Direct experiments upon the destructive effects produced by wagons upon roads. However rational these deductions from experiments upon the draught of vehicles may seem, it was necessary to verify them by other special experiments, executed upon a great scale, and having in view a direct study of the destructive effects exerted upon routes by vehicles, according to their differ- ent proportions and the circumstances of their action. These experiments commenced at Metz in 1837, by order of the minister of war, were continued in the envi- rons of Paris in 1839, 1840, and 1841, by order of the minister of public works. To distinguish the separate influences of the width of felloes, of the diameter of wheels, and of the velocity, upon the wearing of roads, I have studied their respective effects, and to establish them, I made use of a direct ab- stract of the route by means of cross sections, and a measurement of the draught before, during, and after the experiments, and in a great number of instances by the measure of the quantity of materials used in repairs. The general method of experimenting consisted in causing the vehicles to circulate upon a particular track, always the same, and kept, by sprinkling, in a nearly equal state of moisture for all, until the same total weight had been transported upon each track, and this total weight nearly always was from 11 to 13,000,000 pounds, and often beyond that. 292. Experiments upon the influence of the width of tires. All the rules of the administration and the laws 368 DRAUGHT OF VEHICLES. proposed by the carriage police having admitted that, to obtain for all vehicles an equal action upon the roads, it was necessary to charge them with loads proportioned to their widths of tire, it became necessary to see if this basis of tariff was exact. For this purpose, three artil- lery train wagons, having each wheels about 4.75 ft. diam- eter, for the fore and hind trucks, with tires of 0.196 ft., 0.362ft., and 0.573 ft., were loaded, proportionably to these widths, with the following weights respectively : Carriage No. 1, with tires of 0.196 ft -, 5309 pounds. Carriage No. 2, " 0.362 f % 10129 " Carriage No. 3, 0.573 ft -, 15417 " These carriages, thus loaded, were made to traverse three tracks, each 984 ft. in length. On account of farms bordering the road, it happened that the track of No. 1, with narrow tires, was generally more moist than that of the other two, and that consequently this carriage was in less favorable circumstances. Observation has shown that the draught upon the track of carriage No. 3, with broad tires, was increased with the number of trips, much more rapidly than upon the other two tracks, that it was also increased, but in a much less ratio, upon the track of carriage No. 2, with tires 0.362 ft., and that finally, upon the track of carriage No. 1 it remained stationary, and only varied by reason of the state of moisture of the road. Moreover, an examination of the state of the road, of the abstract of the cross sections, and of the measurement of the draught, all -agree in showing that after the trans- portation of the same weight of materials the carriage No. 3, with tires of 0.573 ft, loaded with 15417 pounds, car- riage included, produced much more injury than the two other carriages ; that the carriage No. 2, with tires of 0.362ft,, loaded with 10129 pounds, had produced more than carriage No. 1, with tires of 0.196ft., loaded with DRAUGHT OF VEHICLES. 369 5309 pounds, and that the last produced no ruts and no apparent wear. 293. Consequences of these experiments. It seems then that we may conclude from these experiments, made upon carriages exactly alike in all respects, saving in the width of tires and magnitudes of loads, which were proportional to this width, that the proportionality of loads to the widtJis of tires, admitted as the basis of tariffs, was more unfavorable than useful to roads. 294. Experiments made with the same carriages under equal loads. Two similar experiments were made upon the same carriages, loaded with an equal weight of 12228 pounds, carriage included, and upon three tracks also identical as possible, and always kept very moist by abundant sprinkling, they were made to circulate until they had transported each, 18356625 pounds. The abstract of profiles, and above all the result of the experiments in traction, have shown that, with equal weights, upon metalled or gravelled roads, the wheels, with tires of 0.196 ft., produced disintegration far more considerable than those of 0.362ft., but that beyond this last width there is very little advantage in the interest of maintenance of roads to increase the dimensions of the rim of the wheel. 295. Experiments upon the influence of the diameter of wheels, in their destructive effects upon roads. Similar experiments have been made with the same carriages with wheels of a common width of 0.362 ft., but with diameters varying from 2.859 ft. to 4.766 ft. and 6.655 ft., and which were loaded with the same weight, equal to 10870 pounds. The tracks passed over by these carriages were 656 ft. in length, and they were sprinkled during the last part of the period of the experiment. An examina- 24 370 DRAUGHT OF VEHICLES. tion of the road, of the abstract of the profiles, and the measure of the intensity of draught, proved by the self- same carriages upon three tracks after a transportation of 22040562 pounds upon each of them, has shown that the track passed over by the carriage with the small wheels, 2.859 ft. in diameter, was much the most worn, and that of the carriage with great wheels of 6.655 ft. diameter, was nearly intact ; which evidently proves the considera- ble advantage afforded by great wheels for the preserva- tion of roads. 296. Influence of velocity upon the destructive effects. It was proposed to compare the injury produced upon roads by carriages on springs, going at a trot, with those occasioned by wagons unhung, going at a walk. For this purpose, we used two chariots exactly similar, one of which was suspended, and the other, by the wedging of the springs, was transformed into a chariot unhung. The load of these two vehicles was at first fixed at 13232 pounds, vehicle included, then at 11027 pounds, when the road was in bad condition. The chariot suspended was carried at a trot of 10.5 ft. to 11.81 ft. in 1", or from 7.15 to 8.05 miles per hour, and the chariot not suspended at a gait of 3.28 ft. to 3.937 ft. per 1", or from 2.23 to 2.68 miles per hour. An examination of the road, the abstract of profiles, and the measurement of the draught, has shown that the dis- integration, as well as the increase of draught upon the two tracks, was sensibly the same after the transportation of about 10253000 pounds upon each of them. These results have completely confirmed the experiments which had previously been made at Metz, and prove that, in considering only the preservation of roads, the law should not impose upon spring carriages going at a trot more restricted limits than upon wagons without springs going at a walk. DRAUGHT OF VEHICLES. 371 297. Comparative experiments upon the wear pro- duced J>y carriages (comtois), carts, and wagons, without springs. It was admitted, for a long time, that one horse wagons, and voitures " comtois" with narrow tires, were more destructive to roads, than the large wagons and carts with broad tires, drawn by several horses, and in 1837 these vehicles, so light, so useful, and by means of which the power of horses is so well utilized, came near being abolished. The results of our first experiments upon the influence of the widths of tires, have doubtless sufficed to show how erroneous these impressions were, but it seemed none the less useful to institute a series of direct experiments, upon business wagons, loaded in their usual proportions. For this purpose, I procured four wagons, with tires 0.196ft. broad, having fore wheels of 3. 64ft., and hind wheels 4.46ft., while the true Franche-Comtois wagons have wheels with diameters respectively 4.26ft. arid 4.75 ft., and are in the most favorable conditions. Each of these wagons, when empty, weighed 1378 Ibs., and when loaded 3971 Ibs. A cart with wheels 6ft. diameter and 0.54ft. breadth of tire, weighing, empty, 2260 pounds, and when loaded 11044 pounds, and a wagon with wheels 3.313 ft, and 5.675 ft. in diameter, with 0.54 ft. breadth of tire, weigh- ing, when empty, 7000 pounds, and when loaded 17496 pounds, were, along with the chariots (comtois), put on trial, upon three tracks 492 ft. in length, all in the same condition, and kept constantly wet. By means of observations made in the previous experi- ments we have connected with them a measurement of the quantity of material required for repairs. From all these means united it resulted : 1st. That the (comtois) wagons, after a transportation of about 15438360 pounds, upon a tivsk always wet, pro- duced less wear than the wagon and f cart. 372 DRAUGHT OF VEHICLES. 2d. That in the same circumstances, the wagon with four wheels produced less wear than the cart. These results prove, incontestably, the advantage of a division of the load upon wagons with narrow tires, and they demonstrated that the transportation of heavy goods should be made in four-wheeled wagons. All the consequences just published do, then, justify and establish those derived solely from the experiments upon traction. 298. Experiments to determine the loads of equal wear and tear. The researches, whose results we have so mi- nutely recorded , having proved that the bases, until then admitted in the laws and regulations of the carriage police, were incorrect and incomplete, it becomes us to investigate the new ratios to be established between the dimensions of wheels, as to their diameter, width, and loads, so that all the vehicles employed in trade may produce nearly the same wear upon the roads. New experiments were made at Courbevoie in 1841, and their results served as bases for the draft of a law presented to the Chambers in 1842, and especially for the report of the commission of the chamber of deputies, who had studied the question conscientiously. This is no place to enter into a detailed study of these researches, which belong more properly to legislation and public administration than to industrial mechanics, and I shall limit myself with a reference to a publication made by me in 1842, under the title of " Experiences sur le tirage des voitures et sur le& effets destructeurs qu'elles exercent sur les routes." KESISTA1STCE OF FLUIDS. 299. The resistance of fluids. When a body moves in a fluid mass, it necessarily displaces the molecules of the latter, impressing upon them velocities in a certain ratio to its own, and we easily conceive that the inertia of the molecules, thus set in action, develops a resistance which increases with the velocity of the body. Similar effects are produced when a body at rest or in motion is shocked by a fluid. The manner in which the fluid particles are divided at their meeting with the body, depends -much upon the form and proportions of the latter, and we see that the resistance in question must vary notably with these cir- cumstances. This important question of physical mechanics has for a long time engaged the attention of philosophers, and we find in the introduction to the industrial mechanics of M. Poncelet, a complete analysis of all the ancient and modern researches upon this matter. I only propose, in these lectures, to call attention to the most important cases in practice. 300. Theoretic considerations. When a body of any form mnpq (Fig. 87) moves in a stream in a direction xy, if we project it upon a plane perpendicular to the direction of motion, it is easy to see 374 RESISTANCE OF FLUIDS. that ill moving an elementary space s=inm', the body will displace a volume of liquid which will be represented by As, obtained from multiplying the projection of the body upon the plane perpendicular to the direction of its motion, by the path described. In fact, in its two suc- cessive positions the body occupies in the fluid, the same volume, and in each of them there is a part of this vol- ume which has not changed its position, which corres- ponds to mop'r, so that the anterior volume on'm'q'rm is necessarily equal to the volume onpqrp ' . It is no less evident that each of these volumes is also equal to the volume mn'q'q, made by the greatest cross section of the body, or by the area of its projection A ; for these three elementary volumes may be regarded as composed of an infinity of small prisms of the same base, height, and number, whose edges are parallel to the direc- tion of motion, and which only differ in their respective positions. Thus, when the body describes in relation to the fluid, or when the fluid describes in relation to the body, an elementary space s, the volume of the deviating fluid, which passes from the front to the rear of the body, is ex- pressed by q=As, and its mass is calling d the density or weight ot a cubic foot of the fluid ; this deviating mass effects its relative displacement with a velocity depending essentially upon that of the body in relation to the fluid, in the case when it is the body that moves, and which it is natural to suppose is proportional to the velocity of the body. It will be the same for the vis viva imparted to the deviating fluid ; so that in the case of a fluid at rest, in which moves a body impressed with a velocity Y, the vis viva imparted to the RESISTANCE OF FLUIDS. 375 displaced fluid, for an elementary motion of the body will be proportional to and if we call Ic the unknown ratio, (to be determined by experiment) of the vis viva F really impressed upon the fluid, to the above expression, we shall have On the other hand, if we call R the total resistance which the inertia of the fluid molecules opposes to their displacement, the work of this resistance for the elemen- tary displacements will be Rs, and must, according to the general principle of vis viva, be equal to one-half of the vis viva imparted to the displaced fluid. We should then have the relation whence MAY 3 In the case where the fluid displaced by the body is moving with a velocity of its own, if the body moves in an opposite direction to the motion of the fluid, the rela- tive velocity with which the molecules are met and dis- placed is Y+v ; and when the two velocities Y and v are in the same direction, the relative velocity is Y v ; simi- lar reasoning to that already used will give us then, for the case in which the body moves in an opposite direction to the fluid and for that in which they move in the same direction . 376 RESISTANCE OF FLUIDS. 301. Work developed per second ~by the resistance of a medium. When all the circumstances of motion remain the same, and the phenomena occur constantly in the same manner, the work developed in each second by the resistance of the medium opposed to the motion of the body is, in the case of a fluid at rest, and in the case of a fluid in motion , which shows that, in the first case, the work of the resist- ance increases as the cube of the velocity. 302. Equivalent expressions of the resistance. In the preceding expression of the resistance, applied to a liquid whose density d is constant, and in places where the value of 2^=64,3634, we may place 0-=^, and it then takes the form K=K.AV 2 , or in which it is frequently used. Some authors, and in particular Dubuat, calling H the height due to the relative velocity V or V v, and put- ya (V+vf ting, consequently, H= , or H= v - , and K'= 2<7 2<^ write this formula under the form It is evident that the three formulas are equivalent, and I have pointed out the two last, which less convey RESISTANCE OF FLUIDS. 377 the idea of the law of resistance, only to facilitate the un- derstanding of other authors. 303. Case of a ~body at rest in a fluid in motion. The body being at rest in the fluid, if we suppose them to be impressed with a common motion of translation, whose velocity is precisely equal and opposite to the actual velocity of the fluid ; this would give us the fluid at rest, and the case would revert to the preceding ; therefore, the expression for the resistance should be the same. The consideration of the physical phenomena presented by the displacement of the fluid molecules situated in front of the body, and the return of those flowing to the rear to fill up the void formed by its passage, induced Dubuat to infer that the resistance experienced by a body moving in a fluid at rest, was not the same as the effort exerted upon a body at rest, by a fluid in motion, all else being equal. This would be contrary to the preceding results, but there is further need of its confirmation by experiments ; it seems, however, to accord with those of M. Thibault upon the resistance of air, which we shall notice hereafter. However, the difference, if it exists, must, in most cases, be so small, that it may be neglected. 304. Experiments upon the resistance of water to the motimi of variously formed bodies. Though these experi- ments may be of small importance in an industrial point of view, which is the main object of this work, I will report those which were made at Metz, in 1836 and 1837, principally on account of the methods of observation em- ployed. The bodies subjected to these experiments were : 1st. Thin iron plates of different sizes, which were made to ascend from the bottom to the surface of the water by the action of a counter weight. 2d. Solid or hollow brass spheres, with diameters 378 RESISTANCE OF FLUIDS. ranging from 0.341ft, 0.387ft., 0.422ft,, 0.485ft, to 0.530 ft 3d. Tin plate cylinders, with altitudes equal to their diameters, which were 0.324ft., 0.656ft, and 0.984ft 4th. Cones terminating upon cylinders, with the same diameter and height as the preceding, and whose angles at the summit varied as follows : Half angle at the summit, 64.48 / ; 46.50 / ; 26 P .01 / . 18.49'; 14.19'.48". 5th. Cylinders of the same dimensions with the pre- ceding, and terminated in front by hemispheres. 305. Mode of observation. The experiments were made upon the Moselle, in front of the dam at Pucelles, in a place where the water was, at least at its surface, nearly without a current, and had a depth of 16.4 ft It was the most suitable place that could be found in the neighborhood. The vertical motion of the body was made in the descent, by its own weight, with an occasional ballast to increase its velocity, and in the ascent by means of a counter weight. In all cases, the law of the motion was observed and determined by means of a chronometric apparatus with a style, similar to those used in the exper- iments on friction. In the first experiments it was at once apparent that the resistance of the water increased so rapidly with the velocity, that the motion very readily became uniform. Then, knowing, in each case, the velocity and the motive weight, and keeping account of the passive resistances, it was easy to calculate the value of the corresponding re- sistance of the fluid, and to investigate its law. A graphic representation of the results, taking the resistances for abscissae, and the squares of the velocities for ordinates, has shown in each case, as in the preceding, that the resistance is composed of two terms, the one RESISTANCE OF FLUIDS. 379 independent of the velocity and simply proportional to the wetted surface, the other proportional to the square of the velocity ; but here the first terra is so small, that it may be neglected .in relation to the second, as soon as the velocity has become 3.28 ft. per second. According to this, the resistance opposed by the water to bodies, as proved by these experiments, would be rep- resented by the formula A being the projection of a body upon a plane perpen- dicular to the direction of motion. The values of K derived from experiment, are entered in the following table : Values of the coefficient K of the foi^mula K=KAV a . Bodies used. Values of I. Thin plates (rising vertically upwards) 2.724 Spheres 0.41969 Ric'ht cylinders with height equal to diameter 17715 Cylinders with same proper- f 0.94 to 1 ") n orrest)0n( i ] 64 48' c^nes^vlSse^eightsar^to J 405 to 1 1 in S an S les I 26 1' the radii of their bases in 5.92 to 1 18 49' the ratio of [7.66 to IJ * j 14 19' 48" 1.8944 1.0276 0.90868 0.84802 077449 Cylinders of the same proportion terminated by spheres 077487 306. Observations upon these results. The value of the coefficient K found in the experiments for thin plates is considerable, and nearly double that found by Dubuat in causing a vertical plane to move in a horizontal direc- tion, thus producing a displacement of water entirely different from that in our experiments, and occasioning fcthe difference of results. It is remarkable that of all the bodies used, the spheres offered the least resistance, and that cylinders, terminated by half spheres, have experienced less than those with acute cones. 380 RESISTANCE OF FLUIDS. This result shows, that in regard to the resistance of a medium, the spherical form for projectiles, and the semi- circular for piers of bridges, are the most favorable. 307. Influence of the acuteness of the angles of cones upon the resistance. In comparing the values of the half angles at the summit of the cones, expressed in fractions of the semi-circumference, with the values of the resist- ance, we see that the coefficient K of this resistance in- creases proportionally with these angles ; starting from a certain value answering to the angle zero. It may be given by the formula K=0.59005 +2.29980, a being the half of the angle at the summit in terms of a fraction of the semi-circumference. The comparison of the values of K given by this formula, with those deduced directly from the experiment, is established in the follow- ing table : Comparison between the values of the coefficient K as de- duced by formulas and by experiments. Half angles at the summit Values of the coefficient K deduced. * semi-circumference. From the formula. From experiment. 0.500 1.739 1.771 0.362 1.422 1.394 0.262 1.192 1.027 0.145 0.923 0.908 0.105 0.831 0.843 0.080 0.774 0.774 "We see that, with the exception of the case relative to the cone whose half-angle at the summit was measured by an arc equal to 0.262 of the semi-circumference", the RESISTANCE OF FLUIDS. 381 results, including even that pertaining, to the plane base of the cylinder, are quite correctly represented by the formula, and that we may use it for intermediate cases, which have not been experimented upon. 308. Experiments upon the resistance of water to the motion of projectiles. Without entering into details, for which this is not the proper place, I must say something about the remarkable results of experiments made by me in common with MM. Piobert and Didion, at Metz, in 1836, upon the penetration of projectiles in water. These experiments were made at the basin which had served for the beautiful hydraulic researches of MM. Pon- celot and Lesbros, in firing horizontally beneath the sur- face of the water projectiles which penetrate the water after having traversed an orifice formed by spruce scant- ling. A horizontal flooring, placed at the bottom of the basin, and marked with strips, received the projectiles, which always reached it with a very small velocity. "We found, with this arrangement, the resistance offered to solid balls with diameters of 0.354 ft., 0.328 ft., 0.530 ft, and 0.72ft., and to shells of the same diameters, having different thicknesses and weights, the initial velocities of the projectiles varying from 229ft. to 1640ft. in V '. From the general view of all the experiments made, the results of which are published in ]STo. VII. of the " Memorial de 1'artillerie," we conclude that the resistance of water to the motion of these projectiles may be repre- sented by the formula K=0.453 AV 21b % while the experiments above cited (No. 305) gave us K=0.4197AV albs - On the other hand, the ancient experiments due to 382 RESISTANCE OF FLUIDS. Newton, and made by observing the time of the fall of spheres in water lead to the value R=0.46498AV 21bs , and those which Dubuat made in causing spheres placed at the end of the arm of a horse-gin to pass in a circle through the water, furnish the formula K=0.419TAV 21bs - An inspection of all these researches, made by pro- cesses so different, and within limits so extended, enables us to conclude that, in liquids, the law of the proportion- ality of the resistance to the square of the velocity, is ap- plicable to spheres, even when moving with the highest velocities. 309. The resistance of water to the motion of floating bodies. The preceding theoretic considerations apply to boats which navigate the sea, rivers, and canals ; but their results are influenced by different circumstances, of which it is important to take an account ; some are per- manent, others accidental. 310. Influence of the form of floating bodies. We readily conceive that, when a floating body penetrates and displaces a liquid, in throwing right and left the fluid molecules, the form of the prow must exert a great influ- ence upon the facility with which it cleaves its way. So also the form of the stern, in facilitating, more or less, the return of the liquid to fill the void formed by its passage, affects the difference of level existing between the bow and stern, and consequently the resistance. It is clear, from a mere inspection of figures 111 and 112, that a boat whose front forms, in horizontal planes, are such that the fluid filets are at first separated by a RESISTANCE OF FLUIDS. 383 nearly vertical edge a (Fig. Ill) of the form of a blade, and then divided laterally by curves grad- ually approaching the sides, would ||| a, experience a much less resistance, than one with a bow formed simply of vertical planes more or less inclined to the sides. The first form is that to be given to fast boats navigating FIG. in. FIG. 112. rivers or canals, with steam or horse-power. 311. Flat-bottomed boats with raised fronts. Boats are used upon rivers, whose fore part is formed by the prolongation of the bottom, which rises at an inclination of from 25 to 30 with the horizon, and narrowing very much in a horizontal direction. This form is very un- favorable for speed, for which these boats are not constructed, but they ad- , mit of an easier approach to the banks, and diminish FIG. 113. the violence of the shocks against obstacles concealed in the river. But one could scarcely believe that they are yet retained for common wherries with oars. In fact, we readily see (Fig. 113) that the resistance of the water acting horizontally, being decomposed into two forces, the one tangential, the other normal to the prow, the last tends to raise the front and incline the boat. This effect was quite apparent in the numerous experi- ments which I made at Metz, in 1838, upon boats of this kind, among which there was one whose length could be changed by the addition of other pieces. These experi- ments were made in the trench of the curtain of Fort Saint-Vincent, about 984 ft. long by 98.4 ft. broad, and having a depth of water varying from 2.62 ft. to 3.93 ft. To facilitate various observations which I shall speak of 384: RESISTANCE OF FLUIDS. hereafter, the escarpment wall of this curtain had been divided into portions of 32.8 ft. in length, marked by very plain vertical lines. The motion of the boat was produced by the descent of a box loaded with weights, hung from the top of a crane stationed upon the parapet of a neighboring bastion, by a rope passing round the smallest diameter of a windlass with two drums. Upon the largest of these drums, which was 8.45ft. in diameter, the smallest being only 1.64 ft., was wound a towing line 984 ft. long, its end being fast- ened to the boat by means of a dynamometer with a style. An observer placed in front, provided with a watch indicating tenths of seconds, observed the instant of pass- ing the equidistant divisions of the. escarpment, and thus determined the velocity, while he watched the dyna- mometer. Finally, to observe the inclination of the boat, and other circumstances of its progress, there was placed at the ends of the stem and stern two vertical uprights, ter- minated by small laths aa! , W (Fig. 114,) movable around U a horizontal screw. *' J Perpendicular to the l\ Ji direction of the course of the boat, and in a horizontal position, FIG - 114 about 5.25 ft. above the surface, there was firmly fixed a plank, having at its lower edge a triangular bracket 0, whose level edge was whitened with chalk, while the laths aa! and W were blackened. Before commencing the experiments, the boat was gently brought under the bracket, and the height of the bracket above the two points at the bottom of the boat, corresponding with the laths, was marked. The two heights thus marked were generally equal, or nearly so, when the boat was at rest. RESISTANCE OF FLUIDS. 385 This done, the boat was taken to its starting-point, and the experiment of its trip began. It is readily seen that the small movable laths coming in contact with the fixed racket and falling, as they pass, will preserve the print of the shock against the bracket, and so give the height by which each end was raised or lowered during the motion. By this arrangement, the inclinations of the boat for its different velocities, could easily be compared, and the amount of its rise or fall from a position of rest, ascer- tained. The experiments in question were made with a wherry, with a deck boat, and with a boat with lengthen- ing pieces, having a breadth of only 1.968 ft. at the bottom, and 2.296ft. at the gunwale, a depth of 2.62ft, and suc- cessive lengths of 22.14ft., 32.8ft., and 33.63ft. The draft of water varied for this last boat from 0.92ft. to 1.38 ft., and the velocities from 6.07 ft. to 16.4ft. in 1". The first fact presented by these experiments is, that the area of the greatest submerged section during the trip, is generally superior, or at least equal, to that of the submerged section in repose. The second is, that the inclination of the boat to the horizon increases, at first, rapidly with the velocity, and then increases less promptly with the velocity, varying with the length of the boat and the draft of water ; in general, it increases up to ve- locities of 16.4 ft., even for a boat whose width at the bot- tom is only T 1 T of its length ; which proves how ill adapted this form is to fast sailing, and that even for wherries it should be abandoned. 312. Velocity of waves. Floating bodies in their dis- placement form a principal wave, to which J. Scott Rus- sell (who has given much time to these researches, and to whom we owe some important improvements in the con- struction of fast sailing boats,) has given the name of the great wave, or the solitary wave. This wave spreads more or less upon the sides, according as its highest point is more or less near the middle of the length of the body, 25 386 EESISTANCE OF FLUIDS. and according as the ratio of the width of the body to that of the canal is greater or less. Upon ordinary canals it forms a swell whose highest point, when it is near the middle of the length of the boat, rises from 0.65ft. to 0.98 ft. above the general level of the canal ; but as it is found farther forward, the wave shortens and rises some- times 2.95 ft. above the level of the canal, forming there a true prow wave, in which the bow of the boat seems to be. We may readily conceive, that the form, development, and position of this wave must exert a great influence upon the intensity and upon the laws of the resistance, and thus become important subjects for investigation. J. Scott Russell inferred from observations that the velocity of propagation of the solitary wave was always equal to that corresponding with half the depth of the water in the canal, increased by the height of the wave itself. Now, in order that the wave shall maintain the same position relative to the length of the floating body, and that the resistance shall follow a regular and normal law, the boat must navigate with a velocity equal to that of the propagation of the wave; in other words, the velocity of the boats must be regulated by the depths of water in the canal which virtually amounts to forbidding navigation in very deep water. In fact, the greatest speed to be obtained with horses exerting an inconsidera- ble effort hardly reaches from 14.76 to 16.40 ft. per second, which answers to heights of 3.51 ft. and 4.2 ft, and con- sequently, according to the law of J. S. Russell, to depths of 7.02 ft. and 8.49 ft., from whence it follows that beyond these depths, the navigation of these boats would not be possible. Inversely, upon canals with small depths, the velocity of the boat must be limited so as to lose for this kind of carriage the advantage of speed. It appeared necessary that I should make various ex- RESISTANCE OF FLUIDS. 387 periments upon this preliminary question, which I did first at Metz, taking advantage of the favorable disposi- tion afforded by the long trench of Fort Saint Vincent, and afterwards upon the Canal de 1'Ourcq. When the boat had acquired a uniform velocity, the draught suddenly stopped, the motion of the boat slack- ened, the wave spread, and passed on in virtue of its own velocity of propagation, which was observed from the bank, by means of marks and a time-piece indicating tenths of seconds. We may conceive that these observa- tions, in making which it was difficult to seize the true time of the passing of the culminating point of the wave, are not very precise. We shall see, by the results given in the following table, notwithstanding the difficulty of precise observa- tions, that there existed between the velocities of the boat and of the wave, at different depths and drafts of water, a nicety of agreement sufficient to admit that the velocity of the propagation of the solitary wave, is sensibly the same as that of the translation of the boat which pro- duces it. 388 BESISTANCE OF FLUIDS. dd III &* It 4 it II? S od -J IH O 00 N O Oi O ?O T< r-t t^ t- * 05 rH CO O i-! U5 (M N -* W5 t> CO (N . 00 O O rH b- OS OS RESISTANCE OF FLUIDS. 313. Results of experiments upon the resistance of boats to towing. After this preliminary examination of the circumstances of the phenomena, we give the results of direct experiments upon the intensity of the resistance in its ratio to the velocity of motion. All the experiments made with flat-bottomed boats, of five different forms or proportions, have shown that, on account of the gradual increase of the longitudinal incli- nations of the boat, the resistance increases much faster than the square of the velocity. Furthermore, if, keeping account of the observed inclination, we determine for eacli case, the projection of that part of the boat which lies under the line of floatation, upon a plane perpendicu- lar to the direction of motion, or the area of the immerged section, and introduce it into the formula, we shall still" find that the ratio of the resistance to the square of the velocity does not remain constant, so that it does not seem possible, in this case, to assign any simple law of resist- ance. 314. fast boats. But when the boat presents a sharp prow, nearly vertical, and forms that cleave the water easily, in proportion to the advance of the boat, the resist- ance follows these laws with the greater closeness. "When- ever the speed is well regulated, and the principal wave is spread upon the sides of the boat, so that the latter re- mains nearly horizontal, the numerous experiments which I have made with many boats, upon the Canal de 1'Ourcq the boats being constructed after the model of those of the Paisley canal, in Scotland prove that, from velocities of towing by men at a walk, up to those of a gallop of 14.76 ft. and upward, the resistances follow the law of the square of the velocities. A graphic representation of the results of experiments, made by taking the squares of the velocities as abscisses, and the efforts exerted for ordinates, gives all the points 390 RESISTANCE OF FLUIDS. thus determined upon a straight line, which cuts the line of ordinates above its origin. This circumstance shows that, in this case, as in that of wheels with plane paddles, the resistance is composed of two terms, the one constant and independent of the velocity, and simply proportional to the area of the wetted surface, and the other propor- tional to the square of the velocity and the area of the irnmerged section. But the first term is always so small that it may be neglected in practice, especially in all cases of high velocity. Fig. 115, which represents the general results of the FIG. 115. 9 12 15 18 21 Squares of Velocities in sq. yds. 24 27 30 1st, 2d, 5th, 6th, 9th, 10th, 13th, 14th, 15th, and 16th experiments made in 1838, upon the Canal de 1'Ourcq, in the district of Meaux, affords an example of this law. We RESISTANCE OF FLUIDS. 391 see from a general inspection of all the resistances meas- ured by the dynamometer, when the boat remained hori- zontal, that they are represented by a straight line cutting the axis of ordinates or of resistances in a point which indicates that the constant resistance was about 16.75 Ibs. There are seen upon this figure a certain number of points marked O widely separated from the straight line, show- ing that they correspond to anomalous cases. In fact, all these points, which answer to velocities of from 7.34 ft. to 10.8 ft., or to a moderate trot, express the observed resistances at a time when, by the displacement of the wave, the latter was found in front of the boat, which was inclined and deeply submerged towards that part. The following table contains the results of all the ex- periments which I made upon the Canals de 1'Ourcq, and of Saint Denis, with three models of boats. The most numerous refer to the model of fast boats, which have been a long while in service between Paris and Meaux. The draft of water of these boats varied from 0.9ft. to 1.4ft., and their displacement from 202 c. ft. to 344.6 c. ft. The experiments were made upon the ascent and descent, and the observed resistances were compared with the results of the formula 392 RESISTANCE OF FLUIDS. iad pn ' o o o r- e !.* l I l-S-3 .1 * o o o o o o o o o to to to o- o o o' o o o o o o' o o 00 Bdiqspi =>' o o o o' o" d jo inaraao[dB;a -^ GO C t-i *-J 00 cd5 tl I'l , 5 I ^1 gjf ^1 i 5 1 ^f ^ b l ^ D I ^9 ^2 ^ ropj'^o'Oo'Oa'^n'Oc'oa'cia'^fl'Ofl'a S|ifl9(49flO04>af>fl90Ofl9a0 ojoooci illlil 1 i i i i i 1 1 t. : t.2 : tjtjt.^tjt.d OaQO3OcoOc/aC)OeflOc M^pq^pqjpq^pqjpq^pq^ 25?^^|5|^|2|3| lllllilis a a-a a-a 1 cs^=i c3X3 1, .^ L- L^ , aaaaaaaasasaeaaaaaaaaa oooooooooooooocooooooo IfilllfillifiiSllllif 8 ---**- 3J RESISTANCE OF FLUIDS. 393 315. Consequences of the experiments. From a gen- eral view of these results we may infer that the resistance to towing of the boat upon trial, is represented with all the exactness requisite for practice by the formula K=0.38A'+16.04A(V'y) 2 , units of yds. ; or K=0.04301 + 0.1985A(V0)', units of feet; and if we neglect the term 0.04301A', independent of the velocity, and which seldom exceeds 15.4 or 17.6 Ibs., the formula will be R=0.19S5(V?;) 3 units of feet. This value of the resistance to towing, in the most favorably constructed boat, in narrow canals with small depths, is much greater than that experienced by sea vessels in deep water, which, according to the usual esti- mate of naval engineers, seems to correspond with a value of the coefficient K', equal to 4.6 Ibs. or 6.2 Ibs. per sq. yd. of area of midships for a velocity of 1 yard per second. 316. Accidental variations of the resistance. The pre- ceding results refer to cases where the navigation was in a normal condition, without any marked disturbance in the position of the wave along the sides of the boat, which thus preserved nearly a horizontal position. But when, by accident, the horses slackened their efforts and speed, the motion of the boat was momentarily reduced ; the wave, which had a velocity of propagation equal to the previous velocity of the boat, advanced towards the prow with a dif- ferential motion, at the same time shortening more and more, and raised the bow, which was found to be deeply submerged, inclined the boat, and arrived at the prow, forming a kind of watery hill about 6.56 ft. at the base by 2.62ft. and 2.95ft. in height. We may conceive that in circumstances so anomalous, 394: KESISTANCE OF FLUIDS. the resistance must increase, though the velocity might be lessened, and then we might truly say, that the resist- ance at a slow trot was greater than at a smart trot or gallop. The following results show how effectually the position of the wave exerts an influence upon the in- tensity of the resistance at an equal velocity : Comparison of the resistance to towing of mail-boats, when the wave is spread along the sides and when it is towards the bow. Kesistance when the Displacement wave is of the boat. Portion of the canal passed over. Velocity. A oward the toward middle of the the boat. bow. *tons. ft. Ibs. Ibs. 5.716 ) 7.147 f 8.20 14.10 150 211.7 220 to 264 397 to 441 ^rom the circular basin to the 6.23 110.2 265 basin of Bondy 6.56 132.3 253 (ascending.) 13.44 99.2 390 9.758 do 6.23 176.4 297 87KQ j From the basin of Bondy to the 14.43 251.3 421 . 4 OO circular basin (descending) 13.89 264.6 406 r 13.79 240.3 615 8.758 J do 14.43 14.79 266.8 280. 609 600 I 15.14 295.5 565 * The ton here =1000^.= 2205.48 Ibs. From these results we see how important it is, in this kind of navigation, to maintain a regular speed, and it is because the slow trot is less stable and more liable to va- riations than a very brisk gait, and from the perturbations in the position of the wave being more frequently pro- duced in a slow pace, that we find for it a greater resist- ance than for a gallop. But this is not true where there is no disturbance of the wave. "We have asserted that in the exceptionable cases, RESISTANCE OF FLUIDS. 395 where the wave is wholly in front of the boat, that the resistance goes on increasing so that the wave cannot be so traversed as to replace the boat in its normal position ; this is not exactly so. It has frequently happened that a boat loaded with 9.67 tons, including its own weight, has raised a wave in front 2.95ft. in height, and, after having, in this extraordinary situation, run a distance of 656 ft. to 984ft., has surmounted the wave and brought itself back to a horizontal position, and the wave to the middle of its length. Then it was true to say, that the resistance was less for a velocity of 16.4ft. than for one of 13.84ft. ; for in the first case, the boat moving horizontally, experi- enced, in descending from Bondy to the circular basin, a resistance of 331 Ibs. only, while in the second, when the prow was plunged in the wave, it met with a resistance of 617 Ibs. But this difference is due entirely to that of the circumstances of the phenomena. 317. Recapitulation. We see by these experiments, for which it seemed to me proper to enter somewhat into the details, that when the form of boats is suitably de- termined, so that their position in relation to the surface shall experience no sensible change, the resistance follows the law of the square of the velocity, and that conse- quently the fatigue of horses employed in rapid towing must be very considerable. We are thus obliged to shorten greatly the length of the relays, and, notwith- standing this precaution, we still lose a great number of horses. 318. Work developed ly horses in hauling fast boats. It follows from the experiments, or from the formula ex- pressing their results, that supposing the boat has only 60 passengers, and goes, for example, in the district of Meaux, where the velocity of the water is v 0.984ft. at a velocity of 13.779 ft. in 1", or 9.32 miles the hour on 396 RESISTANCE OF FLUIDS. the ascent, and at that of 14.107ft. per second, or 9.62 miles per hour on the descent, the total resistance sur- mounted by the 3 horses would be, since A 6. 5124 sq. ft. On the ascent, K=.1985 x 6.5124 (13.779 ft - + 0.984 ft -) 2 =281.74 lb3 - On the descent, K=.1985 x 6.5124 (14.107-0.984) 2 =222.62 lb3 - or, per horse, On the ascent, 93.91 lbs - On the descent, 74.2 lbs - Consequently, the work developed by each horse in V is as a mean, in this case, On the ascent, 93.91 lbs - xl3.779 ft -=1293.98 lbs - ft - On the descent, 74.2 lbs - x 14.107 ft 1046.74 lbs - ft - Now, from the results of direct experiments upon the work developed by horses employed in other modes of transportation, some of which are inserted in the follow- ing table, we see that horses employed in hauling fast boats develop per second, during their service, a quantity of work more than triple as a mean of that of the carriage horse, and equal to one and a half times that of the dili- gence horse, which occasions excessive fatigue, producing diseases of the lungs, of which they nearly all die. In exceptional cases, where the wave is in front, we have said that at a velocity of 13.84 ft. the resistance has some- times equalled 617 pounds, which exacts from each horse an effort of 205.66 pounds, and the excessive work of 205.66 lbs - x 13.84 ft -=2846 lbs - ft - in V, during a time of more than one or two minutes, whence results straining of the hams and other accidents. RESISTANCE OF FLUIDS. 397 It* | P | i OS W D * O CO O CO CO O W5 O t- rH OS OS (N CO 398 BESISTANCE OF FLUIDS. 319. Observation upon the daily work of horses. These examples show how the work developed by animal motors may vary, but at the same time they enable us to see, that when we exact but for a short period an unusual work, it is at the sacrifice of the daily work which may be obtained from animals, without fatiguing them beyond measure, or speedily ruining them. Thus in the service of the mail-boats from Paris to Meaux, the distance run for each relay was at a mean 12375.5 ft., which was accom- plished twice a day by horses, on the ascent, and twice a day at the descent, and consequently, according to the values previously found for the resistance, the day's work of a horse, in the district of Meaux, was : At the ascent, 93.91 x 2 x 12375.5=2324366 lb3 - ft At the descent, 74.2 x 2 x 12375.5=1836524 lb9 - ft - Total . . . 4160890 lbs - ft - But as each relay was accomplished by four horses, one of which rested for four days, the mean daily work was but 0.75 of the preceding result, or equal to 3120667 lbs - ft -, while the table of E~o. 318 shows us that by the other modes of transportation, and without an excessive fatigue, which quickly ruins the horses, we may obtain as a mean for a day's work of a draught horse 12758606 pounds-feet, that is to say, a work four times as great as that obtained, with considerable loss, from horses used for hauling the mail-boats of the Canal de 1'Ourcq. 320. The resistance of water to the ^notion of wheels with plane paddles. We use, to transmit motion to steam- boats, wheels with plane paddles, which, impinging upon and pressing the water, experience a resistance which is precisely the motive power by means of which the boat RESISTANCE OF FLUIDS. 399 is impelled. Direct experiments for ascertaining the laws and determining the intensity of the resistance, seemed to me necessary, and I made, in 1837, many experiments, of which I give a succinct analysis. For these experiments we employed two models of wheels, the one 3.31 ft. diameter at the crown, received at will the paddles, variable in number, up to twenty at most. The paddles used upon this wheel had successively for their dimensions : In width parallel to the axis, 0.32S ft -, 0.656"-, 0.984: ft -, 1.968 ft - In the direction of the radius, 0.328 f % 0.659 ft -, 1.148 ft -, 0.659 ft - The shaft of the wheel formed a windlass around which rolled a cord, which passed to the summit of a crane, 55.77ft. in height, which supported a box in which was placed the motive weight. The wheel was established upon a fixed frame, and the depths of immersion were varied at will, in raising or lowering the level of the reservoir in which we operated, and which had dimen- sions indefinite in relation to those of the wheel. The velocities of rotation of the wheel varied from the smallest in which it was possible to observe a regular mo- tion up to 19.68ft. per second. They were observed, when the motion had become uniform, by means of a Breguet timepiece, indicating tenths of seconds. The whole apparatus was so arranged as to reduce, as much as possible, the passive resistances arising from the friction of the axles, the rigidity of the cord and the displacement of the air, and a reckoning made in the cal- culation of the results, by simple formulae, whose details it would be superfluous to publish. The second wheel employed was 8.567ft. in exterior diameter, with paddks 2.29 ft. wide in the direction of the 400 RESISTANCE OF FLUIDS. axis, by 1.659 ft. in the direction of the radius in which they were placed. The depth of immersion of these pad- dles was successively 1.659 ft., 1.325 ft, and 0.937ft. For each number of paddles, and each depth of im- mersion, the motive weights, and consequently the veloci- ties, were gradually changed, so as to have a series of experiments in which one element only was variable. Having thus, for each case, the values of the resist- ance corresponding with the different velocities, we made a graphic representation of all the results in taking for' abscissae the motive weights, and for ordinates the squares of the velocities of the middle point of the immersed sec- tion. In all the series so represented we observed that up to a certain velocity, which we shall indicate hereafter, all the points were always (Fig. 116) upon a straight line, which cut the line of abscissae in front of the origin at a point O, vari- able for each curve, which shows that the abscissas or the resistance was in each case represented, as for boats, by an expression of the form O FIG. 116. calling always A the immersed surface of the paddles ; Y the velocity of the middle submerged section of the paddle ; K, and K/ constant coefficients. The immersed surface A of the paddles was determined from the number of paddles simultaneously submerged, in whole or in part, by calculating the sum of the im- mersed portions of the floats for many successive positions RESISTANCE OF FLUIDS 401 of the wheel, and taking the mean of the sums of surfaces thus obtained. It thus really represents the mean value of the total surface of the paddles acting upon the water. The mode of representation of iigure 116 has given us the value of the constant coefficient K/ ; since the abscissae AO of the point O of the straight line expressing the law of resistance was that of the term K/A. It is thus we have obtained the following values : Dimensions of paddles Total surface Constant resistance 20 in number. submerged. derived from the trace KVA. per square feet K!*. ft. 0.65 by 0.65 0.98 by 1.15 0.97 by 0.65 sq. ft. 1.4693 4.677 4.4241 Ibs. 0.2867 0.88219 0.86013 Ibs. 0.19513 0.18862 0.19442 Menu... 19272 The trace enabled us to supply the value of the coeffi- cient Kj of the term proportional to the square of the velocity, since the inclination of the straight line express- ing the law of resistance is given by the expression E-K/A R K/A being the value of the abscissae of this straight line diminished by AO, and Y 2 being that of the ordi- nates. Dividing, in each case, the values of K X A, given by the experiment by the known surface A, we obtain the values of the coefficient K r 321. Causes which alter the law of resistance. But, before giving the values of the coefficient K^ of the resist- ance, furnished by a summary of the experiments, we should point out a circumstance, which, in altering the 26 4:02 RESISTANCE OF FLUIDS. conditions of the phenomena, exerts a considerable influ- ence upon the results. In order that with different veloci- ties and depths of immersion, the wheel and its floats may be in comparable conditions, it is necessary, as we have hitherto implicitly admitted, that the void formed by the paddles, which have driven before them the water upon which they have acted, should be constantly replaced, so that the next paddle submerged may meet the same resistance. Now, in observing the motion of the return of the water into the void, we readily understand that the refilling must be accomplished by the flowing of the sur- face, as it were over a dam at the sides, and that a certain time is required for its operation. If, then, the wheel turns so rapidly that the void has no time to fill, the pad- dles no longer finding the same quantity of water to drive Fio. 117. 251bs. 10 20 30 40 50 60 70 80 90 100 110 120 Squares of Velocities. as in less velocities, the circumstances of the phenomena are changed, and accordingly the law of resistance must RESISTANCE OF FLUIDS. 403 be modified. This change increasing more and more with the velocity, it happens that the paddles meet a less amount of water, which may, so to speak, be naught, so that finally the wheel turns in air instead of water. All these effects are perfectly manifested by the trace repre- senting the results of experiments, as we may see by an inspection of figure 117, relating to a series of experiments made with 20 'paddles submerged 0.344 ft. The line representing the law of resistance is at first nearly a straight line prolonged up to a certain velocity depending upon the depth of the immersion and the dis- tance apart of the paddles ; but beyond this velocity it departs, more and more, showing that the resistance no longer maintains its proportion with the square of the velocity. All these facts are highly important to steam navigation, for they show that it is necessary to establish between the depths of immersion of the paddles, their distance apart, and the velocity with which they are im- pelled, such relations that the water may always have time to fill up the voids, and that for each wheel con- structed, there is a limit of speed, adapted to the best effect. 322. Proper distance, of the paddles apart. Without going into further details upon these remarkable effects, I content myself with saying that the law of proportion- ality of the second term of the resistance to the square of the velocity has been verified up to velocities ot 5.444 ft. and 6.232 ft, when the spaces of the paddles on the outer circumference of the wheel, were comprised within from two to three times their depth of immersion. This pro- portion, moreover, is conformable to the ordinary practice. 323. Value of the coefficient K t of the second term of the resistance. Having regard to the circumstances which we have pointed out, and consequently restricting the law of. the resistance within the limits of our ability to verify 404: RESISTANCE OF FLUIDS. them, we will make known the results relating to experi- ments in which the velocity of the wheel, and the spaces of the paddles allowed a complete return of the water. The values of the coefficient E^ furnished by the ex- periments were as follows : Values of the coefficient K^ of the formula K=A(K 1 / +K ) V !I ). Number and dimensions of the paddles. Values of Kj. " 10 paddles 33ft by 033 ft * 1 990 5 " " 2.0748 Wheel 3 31ft 10 " 66 ft 66 ft 1 9319 5 " " 2.0794 5 " 098ft 1.15ft 2.2365 10 " 197ft 066ft 1.9928 5 " 2 2460 Great wheel 8.567ft. in diameter. fl.659 8 paddles of 2.29ft. submerged... -j 1.325 (0.937 General mean... 2.2269 2.1698 2.4078 2.1355 The general mean does not differ over T \ from the partial results, and we see that when the spaces of the paddles were within the indicated limits, that the effect exerted by wheels with plane floats upon the axle may be repre- sented by the formula K=A [.19272+2.13559V 2 ], A being the mean of the surfaces of paddles simulta- neously submerged at rest. V the absolute velocity of the wheel. 324. Case where the wheel turns in running water. The wheel which had served for the above experiments having been placed in a small wooden canal, 3.7 ft. wide by 2 ft. deep, the same course was taken to ascertain the RESISTANCE OF FLUIDS. 405 law of resistance. Without going into further details, I will simply state that the results of these new experiments are also represented, with sufficient exactness for practice by the same formula, by adding or subtracting the velocity v of the current to or from that of Y, so that the general expression of effort exerted upon still or running water, by the paddles of wheels with plane floats will be R=A [0.1927+2.0785 (Y^) 9 ], in taking for the coefficient of the second term a number which conforms best to all the experiments. 325. Influence of the presence of a boat near the wheels. The experiments in question were made upon isolated wheels, and it was proper to ascertain whether the presence of a boat near the wheel would exert any influence upon the intensity and law of the resistance. For this purpose, we placed near the wheel, at a dis- tance of 0.13ft.. parallel to the exterior vertical plane of the floats of the wheel, 8.567 ft. in diameter, a boat sub- merged an equal depth with the floats, and made two sets of experiments, with depths of immersion of 1.325 ft. and 0.84 ft., to compare the results with those of the series made in the case when the wheel was isolated, and its paddles immersed 1.325 ft. and 0.937 ft. The results of these experiments seem to show that by reason of the obstacle which the presence of the boat op- posed to the return of water into the void formed by the float, the resistance diminished somewhat, but so small a quantity that it may come within the limits of the errors of observation. In fact, we found At the depth of immersion of 1.325 ft. without boat Ki=2.1698 " " " " " " ' with boat K,=2.1413 At the depth of immersion of 0.937ft. without boat KI =2.4078 " " " " "0.84ft. with boat K! =-2.1580 406 RESISTANCE OF FLUIDS. "We see, then, that the preceding formula derived from a summary of the experiments may be still applied to the case where the wheel is placed at the side of a steamboat. 326. Application to the wheels of steamboats. The formula of the resistance experienced, and the effect trans- mitted by the paddles of a wheel turning in water being when the axis of this wheel has no motion of translation, it is clear that if this axis is borne upon a steamboat going with a velocity Y, the paddles will only impinge upon the water with a velocity U Y, and that in this case the formula expressing the resistance experienced by the floats will be Y) 2 in still water, and finally, that if the boat bearing the wheel ascends or descends a stream running with a veloc- ity v, the expression of resistance will be -Y v? on the ascent, K^K.A (TJ Y+ v)* on the descent, If we examine particularly the case of navigation in still water, the work of this resistance, or that of the ma- chine moving the wheel in \" will be and if we express in horse powers of 550 lbs - ft - the effect- ive force of the motor will be 550 550 An observation of existing constructions will allow us to judge whether the value of the coefficient K, derived RESISTANCE OF FLUIDS. 407 from the experiments above reported, agrees with the observed facts of navigation. Indeed, we have for each boat the dimensions of the wheels and floats, and the number of the latter, from which we can deduce the sub- merged surface of the paddles. Observation gives us the velocity U of the paddles, which, by reason of their small height compared to the radius of the wheels, may be regarded as the point of ap- plication of the resistance, as well as the velocity Y of the boat, and if we introduce the value of 1^=2.13559, derived from our experiments, the above formula should give the effective force of the machine, such as observa- tion has furnished. Direct experiments, made by hauling upon a fixed point, in giving the effort exerted and trans- mitted by the paddles to set the boat in motion, enable us to verify directly the formula R=2.13559AU 2 , by introducing the particular data of each case. In making this comparison upon the steamers the Sphinx, the Mentor of 160 nominal horse-power, the Medee, and the Veloce of 220 horse-power, for which the dimensions and different velocities are given by M. Cam- paignac, in his work upon steam navigation, we have the following data and results :* 1*4 l| o 2 Names of steamers. Horse power of each of the two engines. N. s&g ci^ 11 *?| tl1 P S-o < w ** 11 1! ^ %** ga IS > B nominal, effective. sq. ft. ft. ft. 80 80 851 41 227 19.993 15.190 2.3074 80 80.854 37.136 20.862 15.528 1.9897 Medee 110 110.380 56.371 20.823 16.201 2.4318 V61oce 110 111.350 42.928 20.948 15.948 2.6879 2.3542 * This table was calculated for the French H. P. of 75 kilometres, or 543 Ibs. ft. 408 RESISTANCE OF FLUIDS. AVe would observe that the value of the whole simul- taneously submerged area of the paddles was determined by tracings, and on the supposition of the vertical floats being entirely submerged a little below the surface, but probably less than it was in reality, so that the values of K x are undoubtedly greater than they should be. It is not, then, surprising that the mean of these values sur- passes those derived from our direct experiments. 327. The resistance of the air. The phenomena pro- duced by bodies moving in air, are similar to those pre- sented by liquids, and the resistance which it opposes to the motion of these bodies is of the same kind. Still, it is proper to distinguish between what occurs in uniform motion from that which takes place in variable motion. In the first case, the velocity remaining the same, the fluid molecules, successively driven aside by the body, experience the same displacements, receive the same velocities, and in different instants of its motion, the body meets the same resistance. But in variable motion, accel- erated, for example, the fluid molecules receive greater and greater degrees of velocity, and as they belong to an elastic fluid, the fluid prow formed in front of the body acquires a density and mass continually increasing, whence it follows that the mass displaced increases in the same time as the velocity imparted to it. We conceive then, a priori, that the greater the acceleration of motion , so will be t the resistance, and so we may foresee that, in accelerated motion, the expression of the resistance of the air must comprise, besides other terms, one peculiarly due to the acceleration of motion itself. It was reserved, however, for the experiments at Metz for the first proving of this matter, as we shall see anon. 328. Results of experiments. The celebrated Borda made, in 1763, experiments upon the laws of the resist- RESISTANCE OF FLUIDS. 409 ance of air, by means of a kind of fan- wheel, with a ver- tical axle and horizontal arms, a little over 7.15ft. in length. He placed at the end of this arm the surfaces and different formed bodies on which he wished to oper- ate, and he observed the uniform velocities of the fly- wheel under the action of different weights. lie thought the influence of the friction of this apparatus might be overlooked, which has occasioned some uncertainty in his results ; for it is difficult to admit that in dealing with so small a resistance, the portion of the motive weight engaged in overcoming the friction, should not be com- parable to that surmounting the resistance of the air. Borda placed in succession at the ends of the arms of his apparatus, square surfaces of *9.56, 6.38, and 4.25 inches at the sides, and set them in motion with weights of f8.8, 4.4, 2.2, 1.1, and 0.5 pounds, and consequently with different velocities. From the dimensions and data rela- tive to this apparatus, the author has calculated the resist- ances of the air corresponding with the different veloci- ties, and the results expressed in yards are given in the following table : Results of JB or da's experiments upon the resistance of air. Surface of 9.591 in. each side, or of .07099 sq. yds. Surface of 6.894 in. each side, or of .08155 sq. yds. Surface of 4.268 in. each side, or of .01402 sq. yds. Eesistance of air. i ** "3 ^3 iil all* g p. II If 11 fS cooocT5-<*<-ia5cowiO-*'-<-*<'-ioo-*iait- 000 o o* o* ^v t- t^ > > 00 Ci O < O_ i-H CO 1C CO CO i-J r-J SO O5 s^ comprises implicitly this third term is a little too great, and should be diminished. The existence of a constant term in the expression of the resistance was manifested in the experiments made upon a wheel with wings 1.09 yd. internal diameter, bear- ing square wings 0.2187 yd. by 0.2187yd., 20 in number, presenting thus a total surface of 0.9568 sq. yds. The results of these experiments were very exactly represented in the case of uniform motion by the formula and K=0 lbs .008892 A+0.001907AY 8 , in units of feet, K=0 lbs .08002 A+0.1548 AY 2 , in units of yards, as may be seen in the following table, in which the values found, at different uniform velocities, for the coefficient of the term proportional to the square of the velocity, are very nearly constant. Experiments upon the resistance of air to the motion of a wheel with plane plates. Uniform velocity of the centre of resistance of wings in yards per second, yds. 2.89 yds. 4.11 yds. 5.17 yds. 5.89 yds. 6.69 yds. 7.20 yds. 7.83 yds. 8.28 Resistance of wings reduced to Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. the mean density of the air, 1.338 2.602 3.941 5.183 6.502 7.867 9.166 10.458 Coefficient ~K 1 of the square of .15818 .15618 .15077 .15355 .14986 .15711 .15494 .15818 the velocity, Mean Kj =.1548* Velocity answering to the for-] yds. I yds. I yds. yds. I yds. yds. i yds. i yds. mula, I 2.918 1 4.129 i 5.108 5.87 ! 6.58 7.25 | 7.83 | 8.37 * A review of the coefficient gives slight variations from those recorded by Morin, the mean of which would be K^.1004 instead of .1002. RESISTANCE OF FLUIDS. 4:19 This comparison of the results of experiments with those of the above formula, show within what limits of exactness the latter represents the real eifects. 337. Method of reckoning the effects of acceleration. We have already shown, in No. 327, that in elastic fluids, the resistance must depend upon the acceleration of mo- tion, and if these considerations are admitted, it follows that the resistance of the air in variable motion, must be represented by a formula of the form of The experiments upon uniform motion having already furnished the approximate values of K/ and K,, it re- mains to find that of K a , or rather the term 2 t Without going into the details of the calculations, we limit ourselves to pointing out the method followed, since it shows a remarkable example of the advantages to be derived from a graphic representation of the law of motion. In the actual case, this law being represented by a continuous curve, whose abscissae indicate the number of turns, or the spaces described, and whose ordinates express the times, it is clear that for one of these tangents, MP, for example, the ratio of NP to MN, in the trian- gle MNP, will be the same as that of e to , representing by e j> E N the infinitely small increase of the abscissa in passing from the point M to the infinitely near point M', and by t the corresponding increase of time or of the ordinate : this ratio - of the elementary 420 RESISTANCE OF FLUIDS. path to the element of time in which it was described, is precisely what is termed the velocity, which we express by the relation Y=-, and we see that we may, by means t of the graphic trace of Fig. 121, form a table of the simul- taneous values of the times and velocities, and so construct a new curve, whose abscissae shall be the times T, and whose corresponding ordinates shall be the velocities Y. This new curve (Fig. 122) yields to analogous considera- tions ; the tangents, at the differ- AB ent points, give us the ratio -^^, which is equal to the acceleration c T IB v , . ,, , FIG. 122. -> v being the elementary increase t of the ordinate or of the velocity Y, and t being always the elementary increase of the time. Consequently, knowing at each instant the total re- sistance R, or the portion of the motive effort employed in overcoming the resistance of the air, as well as the coefficients K/ and K 15 we may calculate the term K a A- t and so deduce the value K 2 . This process may be abridged, by operating upon that part of the curve relating to the end of the fall, since the variations of inclination of the tangents of the first curve are so small, that instead of tracing them, we may deter- 17 TJV mine them by the value of the quotient = ~, of the dif- ference of two consecutive spaces divided by that of the corresponding times. I dwell no longer upon this matter, and close with stating that this delicate and ingenious mode of discussion has led M. Didion to assign to the coefficients of the for- mula, which represents the law of the resistance of air to RESISTANCE OF FLUIDS. 4:21 the accelerated motion of descent of a plate 1.196 sq. yds. of surface, the following values : K=O lbs .06633+0.1295Y 2 +0.27652-, t which is reduced in case of uniform motion to B=0 lbs .06633-}-0.1295Y a , for one square yard of surface, Y being in yards. 338. Proof of the exactness of this formula. To show a posteriori that this formula, composed of three terms, represents the law of the resistance in accelerated motion more exactly, than those which only contain a term pro- portional to the square of the velocity, or two terms, the one constant, and the other proportional to the square of the velocity, M. Didion has first sought for the values of the constant coefficients which it was proper to admit for each of these formulae, so as to render them as exact as pos- sible, and, after having found them, he calculated, by a very simple analytical method, the values of the times corresponding to the regularly increasing spaces described by the bodies, such as would be furnished by these for- mulae, and he has compared them with the real times furnished by the curve of the law of motion. From the results of this comparison, which for one particular case are entered in the following table, we see that the formula with three terms of resistance, represents, quite truly, the law of accelerated motion of the descent of a body in air, while the suppression of the -term depending upon the acceleration - does not admit of so exact a representation TJ of this law, even in determining the coefficients so as to reproduce the calculated duration for one of the spaces, and that is also the case when we suppress the constant term. 422 RESISTANCE OF FLUIDS. The only results inserted in the table are those of ex- periment No. 6, during which the temperature was at 62.24' (Fah.) and the barometric pressure at 2.465 ft, of mercury. Comparison of times and velocities of the fall of a plate one metre square =1.196 sq - yds - observed and calculated. Spaces described. Observed dura- tions. i 1 1 Durations calculated by the formula Velocities calcula- ted by formula 0) S 9 fi g yds. seconds. yds. seconds. seconds. seconds. yds. 0.0999 0.176 0.178 0.160 0.160 0.1993 0.254 0.253 0.227 0.226 0.2998 0.306 0.310 0.278 0.277 0.3994 0.359 0.358 0.322 0.321 0.4809 0.400 0.400 0.360 0.358 0.5997 0.428 0.428 0.394 0.393 0.6996 0.474 0.473 0.419 0.417 0.7996 0.508 0.506 0.460 0.457 0.8995 0.537 0.536 0.488 0.487 0.9995 0.566 0.566 0.518 0.515 1.2018 0.619 0.622 0.570 0.567 1.3998 0.679 0.679 0.619 0.617 1.5988 0.725 0.723 0.665 0.663 1.7990 0.771 0.771 0.710 0.707 1.9991 0.815 0.820 0.748 0.746 3.46 2.9987 1.013 1.013 0.947 0.943 5.46 3.9919 1.187 6.07 1.186 1.123 1.120 6.05 4.8098 1.346 6.50 1.346 1.289 1.287 6.49 5.9974 1.493 6.91 1.497 1.452 1.451 6.86 6.9970 1.636 7.25 1.639 1.607 1.606 7.14 7.9966 1.771 7.50 1.776 1.760 1.760 7.37 8.9962 1.910 7.60 1.912 1.912 1.912 7.56 9.8133 2.034 7.62 2.042 2.062 2.064 7.69 339. Influence of the extent of surfaces. To establish this influence, M. Didion used a square plate 0.5468 yds. per side, and so having an area of O sq - yds -299, or equal to a quarter of that of the first plate. In calculating the time of the fall by the same method as for the plate of 1.196 sq. yds., and by means of the same formula = lbs .06638+0.1295V 3 + 0. RESISTANCE OF FLUIDS. 423 he found between the results of observation and those of calculation a coincidence quite sufficient to permit him to conclude, that between, the extended limits in which he had operated, the resistance of the air is proportional to the extent of the surfaces. The temperature and barom- etric pressure were sensibly the same as in the experi- ments reported in ~No. 338. Comparison of the times and spaces described in the fall of a plate of O sq - yds> 299 surf ace, from observation and calculation. Spaces described. DIJKATION. Observed. Calculated. yds. seconds. seconds. 0.0995 0.174 0.173 0.2001 0.246 0.242 0.2996 0.301 0.297 0.4002 0.356 0.343 0.4811 0.387 0.384 0.5993 0.425 0.420 0.6999 0.460 0.454 0.8322 0.490 0.485 0.9000 0.519 0.515 0.9995 0.547 0.543 1.993 0.775 0.767 2.998 0.951 0.939 3.998 1.102 1.085 4.809 1.240 1.215 5.997 .361 1.330 6.997 .476 1.412 7.996 .586 1.527 8.996 .693 1.646 9.546 .799 1.738 340. Consequence of these results. "We see by this table, that the calculated times of the falls are sensibly the same, though a trifle less than the observed times, which shows that if the coefficient of resistance varies with the extent of surface, it tends to diminish with the diminution of surface, rather than to increase, as some 424 RESISTANCE OF FLUIDS. authors have concluded from experiments made by obser- vation of the motion of rotation. In recapitulating, we may, without fear of notable error, admit in practice, that the resistance of the air is proportional to the extent of the surfaces. 341. Experiments upon parachutes. One of the most, useful questions among our researches upon the resistance 1 of air, which our means of observation enabled us to re- solve, was an exact determination of the resistance expe- rienced by parachutes. Their concave form causing, with the same surface, a marked increase of resistance, it was easy, in this case, to obtain a uniform motion of descent, which was indicated by the curve representing the law of motion, which, in this case, degenerated into a straight line, whose inclination furnished the value of the uniform velocity. The parachute employed was composed of a frame of whalebones, disposed into four equidistant meridian planes, fastened upon a common rod, and strengthened by stays. This frame was covered with taffeta, strongly stretched, and it was suspended upon a rod, at the lower part of which was attached the additional weights. The exterior diameter of the parachute was 1.461 yds. measured perpendicularly from the sides of the polygon, and 1.312 yds. measured between the nearest points of the arcs formed by the rim. Its perpendicular projection to the direction of motion varied from 1.433 sq. yds. to 1.444 sq. yds. of surface. The versed sine of curvature of this parachute was 1.41 ft. to the plane of the ends of the whalebones. A discussion of the experiments in which the velocity was uniform has shown that the resistance of the air to the motion of this parachute could also be represented by an expression composed of two terms, and that it was equal to 1.936 times that of a plane of the same surface, that is to say, nearly double. RESISTANCE OF FLUIDS. 425 It follows, from this, that it may be expressed by the formula -y d8 - [O lbs .06638+ 0. S . (0 lbs .1285+0.2507Y 2 ), for units of yards of surface and velocity, at the ordinary density and temperature of the air. 342. Case where the parachute presents its convexity to the air. In reversing the parachute, and causing it to descend with its convex surface downwards, we found a much less resistance, and equal O.T68 of that of the plane surface with the same area. So that, in this case, the re- sistance is represented by the formula K=0.768A 8q - ^' (0 lbs .06638+0.1295V 2 )= A(0 Ibs .0509+0.0994V 2 ). We see by this that the resistance of the same body varies in the ratio of 1.936 to 0.768, or from 2.5 to 1, ac- cording as it presents to the air its concavity or convexity. 343. Case where the motion of the parachute was ac- celerated. In this expression of resistance we also admit the necessity of introducing a term dependent upon the *M acceleration of motion-, and this expression for the para- t chute employed is K= A (o lbs .1290+0.2513V a +0.2394-), in units of yards for area and velocity. A comparison of the observed times of the fall with those deduced from this formula has shown that it repre- sents the circumstances of motion with all desirable accuracy. 426 RESISTANCE OF FLUIDS. 344. Resistance to the motion of inclined planes in air. These experiments were made by means analogous to those above described, by causing to descend two jointed planes, 1.0963yds. long by 0.5486yds. wide, whose angles were varied at intervals of 5 from 5 up to 180, where they form a single plane. The results regu- larly observed from 180 to 130 have shown that the resistance decreases proportionally with the angles, so that, calling a the angle of one of the planes with the direction of motion, the resistance was expressed for uni- form motion by the formula K=-^A(0 lb9 .06638-{-.1295V 2 ), in units of yards. 90 A comparison of the observed resistance with those calculated by this formula show a satisfactory agreement. Comparison between the observed and calculated resist- ances, for differently inclined planes. Angles formed by each of the planes with the di- rection of motion. KESISTANCES in the ratio to those of a plane perpendicular to the direction of motion. -^. Observed. Calculated. 90 87.5 82.5 80. 77.5 70. 67.5 65. 1.0000 0.996 0.865 0.856 0.846 0.773 0.737 0.728 1.000 0.972 0.917 0.889 0.861 0.778 0.750 0.722 We remark that these results relate to the case of two equal and jointed planes, moved in the air, with the edge of intersection in front, and are by no means applicable to the case of isolated planes. The law of the variation of the resistance proportion- RESISTANCE OF FLUIDS. 427 ally to the angles, is also that which we found for water, in operating upon cones of different acuteness (No. 305). 345. General conclusions from the experiments at Metz. In conclusion, the reported experiments which have been made with chronometric mechanism, giving the times, to nearly some thousandths of seconds, and the velocities acquired at any instant nearly to a hundredth, in observing the law of descent in air of different sized plates, of two plates inclined towards each other, and that of a wheel with wings, for which the velocities have not exceeded from 29 to 33 ft. per second, have con- ducted us to the following conclusions : 1st. In the uniform motion of a body in air, the resist- ance experienced is proportional to the extent of its sur- face, and to another factor composed of two terms, the one constant and the other proportional to the square of the velocity. As it was easily foreseen, that the number of molecules of the air shocked by the displacement of the body must increase in the same ratio with its density, the general expression of the resistance should contain a factor relative to this density ; so that calling d the density of the air at the temperature and pressure observed, and d^ its density at 50 (Fah.) and at 76 centigrades (or 29.92 in.) of barom- etric pressure, and preserving t^e preceding notations, this resistance is represented by the following formulae : Thin plates perpendicular to the 1 O lbs .066 + 0.129V 2 I I - 129 + - 251V2 ( ) -051+0.994V 2 > Two jointed plates, inclined towards each other ............................................. R=A|^ j .066 + 0.129V 2 I The wings of a fan wheel ....................... R=A J 1 } 0.08002-fO. 1545V 2 i direction of motion ........................... R=A 7 Parachutes ......................................... E=A d l Parachutes reversed ........... . .................. ' R=A d^ l 428 RESISTANCE OF FLUIDS. It may be observed that this last formula accords in a satisfactory manner with the results of M. Thibault's ex- periments. 2d. In accelerated motion we must add to the pre- ceding expression a term proportional to the acceleration of motion, and the resistance is then represented by the following formulae : Thin plates perpendicular to the direction of motion E= A^ $ O lbs .066 + 0.129V 2 + 0.276- V Parachutes U=A d l \ > 12 8+ 0.251V 2 + 0.2394y I 34:6. The effort exerted ly the wind upon immovable surfaces opposed to its direction. We have but few re- sults of experiments upon the law and intensity of the efforts exerted by the wind upon the surfaces exposed to its action. Smeaton, in his researches upon wind and water, mentions a table which was communicated to him by Rouse, an English philosopher. It is reported in many works, and especially in Jamieson's Dictionary of Me- chanical Science. Smeaton says that it was constructed with great care by Rouse, after a considerable number of facts and experiments. He observes that for velocities over 50 miles per hour, or 73.33 ft. per second, these experiments do not merit the same amount of confidence as for inferior velocities. The comparative numbers given in this table for the efforts seem to have been calculated, in admitting that the effort exerted is proportional to the square of the velocity of the wind, and would, in general, be represented by the formula F=0.00228AV 2 in units of feet, A being the surface perpendicular to the action of the wind ; Y the velocity in feet per second ; F the effort exerted. RESISTANCE OF FLUIDS. ^Efforts exerted ly the wind upon a surface one foot square, placed perpendicularly to its action. Common designation of the wind. Feet per second. Force exerted upon a square foot. ' ft. 1 47 Ibs. 005 < 2.93 .020 ) | 4.40 5.87 .044 f .079 J .J 7.33 14.66 .123 J .492 i Very brisk ( \ 22.00 29.34 1.107 f 1.968 i 36.67 44.01 3.075 J 4.429 i 51.34 58.68 6.027 f 7.873 ; 66.01 73.33 9.963 j" 12300 88.02 17.715 11736 31490 Hurricane that tears up trees and over- ( 146.66 49.200 347. Observation upon the velocity of the wind. The velocity of the wind attains, and even sometimes exceeds the values above indicated, as is proved in aeronautic ascensions. "We cite, among others, that of Lunardi, who, in an ascent made at Edinburgh, when the air was cairn at the surface of the earth, was, at a certain height, borne by a current of air with a velocity of TO miles per hour, or 102 ft. per second ; that of Garnerin, from London to Colchester, in 1802, where the velocity rose to 80 miles per hour, or 117 ft. per second ; and, finally, that of Green in 1823, who was carried at the rate of 210 ft. per second * I have copied this directly from the English table. The data in this case lead to the formula instead of F=0.120lAV 2 in killogrammes and metres, F=0.1163AV 2 , as Morin has it. 430 RESISTANCE OF FLUIDS. without accident. These velocities suffice to show what difficulties attend the directing of balloons. We will shortly return to this matter. 348. Means employed to measure the velocity of air. The difficulty of measuring the velocity of the air with precision has been, and still is, the chief obstacle against the attainment of conclusive experiments which shall indicate the laws of the effort exerted. The most general mode adopted by experimenters consists in throwing to the wind light bodies, such as feathers, thistledown, the smoke of powders, or the essence of turpentine, and in observing the distances described with the corresponding times, in the movement of trans- lation. But this simple method affords but little precision on account of the small distances in which they can be observed. Anemometers, composed of a small light fan-wheel, whose motion is transmitted to a counter which registers the number of turns, are most certain, and convenient for use, though they must previously be tested, or the rela- tion existing between the^ velocity of the wind and the number of turns of the wings must be accurately deter- mined; this determination presents great difficulties. Most generally, we accomplish this test by placing the instrument upon the horizontal arm of a species of horse-gin with a vertical axis, which is made to turn as uniformly as possible. We then observe simultaneously the number of turns of the wings and the velocity of translation of the instrument, and then suppose that the effect produced by this movement of the apparatus in the air, the same as that which would be due to the action of the wind impressed with the velocity of transport of the anemometer, upon the wings of the instrument at rest. I shall shortly point out another process which I have suc- cessfully used in great velocities, but first will describe a RESISTANCE OF FLUIDS. 431 very light anemometer which M. Combes, Inspector Gen- eral of Mines, constructed to measure the small velocities of air, principally in the ventilation of mining works. 349. Anemometer of M. Combes. We copy from this learned engineer the description which 'he has given in the " Annales des mines" third series, of the instrument which he used for the experiments in question. "This instrument is similar to "Woltiman's mill for gauging streams of a considerable section. It is composed of a very delicate axle, (turning in agate caps,) upon which are mounted four plane wings, equally inclined as to a plane perpendicular to the axis. In the middle of the axle (figure 123) is cut an endless screw, which drives a small wheel B, with a hundred teeth, so that the latter advances one tooth for each revolu- tion of the axle bearing the wings. The axle of the first wheel carries a small cam, which acts upon the teeth of a second wheel E/. The last held fast by a claw is or FIG. 123. very flexible steel spring, which is attached to the hori- zontal plate upon which the instrument is mounted. At each revolution of the first wheel with a hundred teeth, driven by the endless screw, the cam starts one tooth of the second wheel with fifty teeth ; the two wheels are marked at intervals of 10 teeth. The first from 1 up to 10, the second from 1 to 5. The index pointers fixed upon light uprights, which bear the axle of the wings, serve to mark the number of teeth which each wheel has advanced, and thus to indicate the number of revolutions of the axle of the wings. By means of a detent and two cords, which 432 KESISTANCE OF FLUIDS. move it, we may, at a distance, arrest the rotation of the wings, or allow them to tarn, under the impulse of the current of air which strikes them." The manner of using this instrument is easily under- stood after this description. We place the limbs at zero, and the instrument in the axis of the air tubes, keeping the limbs immovable, by means of a catch, which is loosened at the moment of commencing the observation, and made fast at the end of the same. It is well to prolong the observation as long a time as possible, and for two or three minutes at least, if it can be done. The division of the limbs does not admit of counting over 5000 turns, which, for a velocity of air 9.84ft. per second would only correspond with a duration of about 2.8 minutes. The test or error of these instruments may differ very much from each other, though their dimensions may seem identical in all points. It should then be made for each one in particular, and repeated, as far as possible, when- ever we wish to use it after an interruption. Thus the anemometer No. 3, whose trial was reported by M. Combes, gave , since the coefficient a is independent of the velocity or the number of turns. Tt follows from these experiments : 1st. The observations made with the ventilator have extended the test of the anemometer with small wings up to velocities of about 131ft., which exceeds the usual wants of experiments. 2d. That there exists a constant ratio between the velocity of rotation of ventilators and that of the air which they drive in, or exhaust from the tube. This ratio, moreover, depends upon the dimensions of the pipes, and also upon those of the central openings of admission into the ventilator. 3d. That in future, and when the ratio of the velocity of the air expelled by a ventilator, given by the number of turns of its wings, is known, we may easily adjust ane- RESISTANCE OF FLUIDS. 4:39 mometers of different kinds, whether they are those giv- ing the velocity by the number of turns of their wings, or whether they are pressure anemometers, which will be far more convenient than the first means which we used, and admit of an extension of the test to great velocities. 355. Experiments of M. Thibault upon the effort ex- erted by the wind upon immovable surfaces exposed to its action, perpendicular to its direction. We are indebted also to M. Thibault for some experiments which he under- took as initiatory to some researches upon the action of the wind upon sails, in which he used ingenious devices to measure the efforts of the wind upon surfaces of a given extent ; he used an anemometer provided with a dyna- mometer, and determined the velocity of the wind in giving to it light feathers or thistle-tufts, and observing the time of their passage over a given space. This mode is not exact, and may occasion some errors of a nature tending to influence the final results of experiments. Admitting, conformably to the experiments at Metz, of which an account has been given in Nos. 336 and 339, that the resistance may be expressed by the formula in which K/=O lbs .068 will be the coefficient of the con- stant term, we find that the experiments of M. Thibault conduct to the following results, which give us a general mean of the values of the coefficient K 1} 1^=0.16601 for units of yards. EESISTANCE OF FLUIDS. Experiments of M. Thibault upon the action of wind upon plane surf aces perpendicular to its direction. 2 1 13 . 1 d a i I i * FT* 1 | | i 1 fj B Ci 5o^ M a B s 33 . *S- 2'o _| o A f > H 1 'i O W 2 M M sq. yds. ft yds. Ibs. Ibs. Ibs. 2.512 2.463 66.2 64.4 4.568 5.308 0.4907 0.55048 0.4821 0.5419 0.019316 0.016077 0.17886 0.14886 0.1291 2.446 59 5.419 0.56835 0.008601 0.5597 0.015937 0.14756 2.430 59 6.124 0.99004 0.9814 0.021877 0.20257 2.466 58.1 8.988 0.20308 0.1945 0.020926 0.19376 Mean 0.17432 0.1701 2.449 2.413 58.1 4892 4.651 2.000 0.98827 0.22511 0.011564 0.9767 0.2136 0.03757 0.03949 0.18476 0.19418 Mean 0.18947 It follows that the mean value of the action of the wind upon plane surfaces, perpendicular to its direction would be expressed by the formula E= A (0.068+0.18189V 2 ), for units of yards, if no account is made of the variation of density corresponding to the barometric pressure and to the temperature, which, in common applications, is but of little importance though easily made. 356. Agreement of these results with those of Profes- sor Rouse, cited ly Smeaton. We would remark that, with the exception of the constant term 0.068A, which, for mean velocities of the wind, has a very small influ- ence, the preceding formula gives for the resistance nearly the same value as that of No. 346, representing the ex- RESISTANCE OF FLUIDS. 441 periments of Rouse, which seem to have been made with velocities far superior to those observed by M. Thibault. Either formula may therefore be confidently employed for great velocities. 357. Observation. The experiments of Metz having given for the coefficient the value 1^= 0.1295, when the body moves in the air at rest, it would follow, according to the ideas of Dubuat, that the effort exerted by the air in motion upon a body at rest, would be to the resistance experienced by the same body in motion in the air, with equal velocities, nearly in the ratio of 0.1819 to 0.1295 or of 1.40 : 1. 358. Influence of the curvature of surfaces. M. Thi- bault has made a comparison of efforts exerted by the wind upon a plane surface, and upon a canvass of double curvature, 0.1302 sq. yds. of total surface, and, in the last case, capable of taking a curvature whose last elements make, with the direction of the wind, an angle of from 50 to 55 degrees. He has found that, at the same day, and under the action of the same wind, the effort exerted upon the plane surface was to that exerted upon the curved, in the ratio of 0.1079 to 0.1135 or of 0.951 to 1, which shows the difference to be very slight. 359. Influence of the inclination of surf aces towards the wind. In presenting successively to the action of the wind, surfaces perpendicular or oblique to its direction, the author has established the fact, that the effort exerted upon a given surface was not influenced by their inclina- tion, except wlien the latter attained the angle of from 45 to 50 with the direction of the wind. "We may re- member that a similar result was obtained in the case of surfaces moving in the air at rest. 442 RESISTANCE OF FLUIDS. Other experiments of M. Thibault were made relative to a comparison of the velocities of wind and of a vessel under sail, which were but initiatory to those which he proposed undertaking upon this important subject, when a fatal accident deprived the navy of this young and ac- complished officer. 360. Difficulties in the directing of balloons. The fre- quent balloon ascensions made within the past few years, have stimulated a great number of attempts to direct them in calm air, and even against the wind, and it may not be useless to say a few words, serving to show the difficulties of this problem, and even the impossibility of its solution with the mechanical means at our disposal. We would first remark that observation proves that a calm at the surface of the earth is by no means a guarantee that the same repose exists in the upper strata, even at small heights, and that consequently an apparatus suffi- cient for a calm is by no means so for all heights. General Meusnier, of the military engineers, who has devoted much time to the subject of balloons, has left a memoir upon the subject, a succinct analysis of which may be found in the journal " le Conservatoire," No. 1 of the second year. We see in this memoir that this learned officer has already presented the difficulties of the prob- lem in these terms : " We have examined the possible effects of many of the machines proposed for the direction of balloons : these machines must be moved by men whose weight is great compared to their force ; it follows that they will have but little effect in overcoming the resistances of the air, on account of the great surface of the balloons. u Calculation applied to the means of direction, of whatever character they may be, shows in general, that they can never afford for balloons a velocity over a league (3.64 ft. per V) an hour independently of the wind" RESISTANCE OF FLUIDS. 443 Colonel Didion, in a paper read before the Scientific Congress at Metz, in 1838, has shown, by very simple calculations, that the velocity impressed in calm weather, cannot exceed this limit, even in the most favorable hy- potheses, as to the weight of men borne, of balloons and their rigging. A cubic foot of air at the temperature of 32 (Fah.) and at a pressure of 2.492ft. of mercury, weighs 0.08118 pounds, while the same volume of hydrogen gas, impure and moist, such as is made for general use, weighs 0.000624 Ibs. The difference, 0.0805 Ibs., is the weight which a cubic foot of this gas can sustain in air. But as the air and gas are in elevated regions subject to a less pressure, they are then dilated, and the volume which the same weight of the gas occupies will be greater, and it must be so also with that of the balloon. If we admit, that to pass above ordinary moun- tains we must rise 2624 ft. above the level of the sea, and that then the pressure will not be over T %- of that at the surface of the earth, and if the temperature is at 50, it would follow that the weight of 0.000624 Ibs. of hydrogen, instead of occupying 1 cubic foot, would have a volume of 1.15 cu. ft., and one cubic foot of this gas will only weigh 0.005433 Ibs. On the other hand, the cubic foot of air, whose pressure is T 9 , with a temperature =50, will weigh Consequently, 1 cubic foot of the gas of the balloon can only be in equilibrium with a weight of O.OT0486 lb8 - 0.005433 = 0.065053 1 bs - If we admit that the weight of a man is not over 143.36 Ibs., and that of his skiff is but 11.02 Ibs., without supplies, the total weight to be raised per man will be 444 RESISTANCE OF FLUIDS. 154.38 Ibs., and the balloon should have a volume of 154.38 0.065053 =2373- ft - per man to be raised, which corresponds to a sphere of 16.53 ft. for each man. In taking account of the weight of the covering, which cannot be less than 0.05122 Ibs. per square foot, we find that the diameter should be 18.34 ft. Calculating from this basis the minimum diameters to be given to balloons designed to carry different num- bers of men, M. Didion found as follows : Number of men ......... 1 2 34567 89 10 Weight to be raised... 154.41b. 308.7 463 617.5 772 926 1080 1235 1389 1544 ft. ft. ft ft. ft. ft. ft. ft. ft. ft Diameter of balloons... 18.3 22.5 25.4 27.8 29.9 31.8 33.3 34.7 36.1 37.1 From known experiments the resistance of the air to the motion of spherical bodies, for velocities within 3 and 32 ft. is approximately represented by the formula If, for example, there is a balloon designed for one man, we have TV 1 8 34. 2 D=18.34 ft. and == 264.22 * ft, E=0.1886V 2 , which gives the following resistances and quantities of work for different velocities : Velocities in feet per second ...... 3.28 6.56 9.84 13.12 16.4 Resistances in pounds .............. 2.030 8.125 18.267 32.476 50.746 Ibs. ft. Work expended in 1 second ...... 6.659 53.227 179.81 426.19 832.49 Now, a man in a day's work of 8 hours cannot, under the most favorable circumstances, and with the most ap- RESISTANCE OF FLUIDS. 445 proved mechanism, develop a work over from 43 Ibs. ft. to 58 Ibs. ft. in 1". "We see, then, even admitting that there is no loss of work arising from the passive resistances of the machinery, which can never be, the greatest velocity that a man can impart to his balloon is 6.56 ft. per second, or 4.4 miles per hour in a calm. As for other motors, such as steam-engines, their own weight, that of the fuel, and of the water, would tend to give such dimensions to the balloon, that the work of the resist- ance of the air in small velocities, would prevail over that which could be developed by the motive apparatus. Finally, in the present state of our knowledge, and progress in the mechanical arts, the solution of the ques- tion of aerial navigation is shut in within a circle beyond which we cannot pass, without discovering some new motor at once light and powerful, in relation to the quan- tity of work to be developed. THE END. FIG. 16'. Longitudinal Elevation of Figure 16. See page 37, Article 40. FIG. 20'. FIG. 20". The counter. Art. 46, page 46. Side Elevation Rotating dynamometer with counter Scale Vao (Article 53). FIG. 22'. FIG. 22". Rotating dynamometer with counter Article on CD Scale Van. FlG. 86'. Apparatus for tabulating curves Scale Ye See Article 79, page 88. or THE UNIVERSITY RETURN TO the circulation desk of any University of California Library or to the NORTHERN REGIONAL LIBRARY FACILITY Bldg. 400, Richmond Field Station University of California Richmond, CA 94804-4698 ALL BOOKS MAY BE RECALLED AFTER 7 DAYS 2-month loans may be renewed by calling (510)642-6753 1-year loans may be recharged by bringing books to NRLF Renewals and recharges may be made 4 days prior to due date. DUE AS STAMPED BELOW MAR 9 W98 12,000(11/95)