.^vy^Jji»5i,r./a-;«ft:,' ;a.. .^.;. ï>»>t'itf,. .'iS THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES GIFT OF lie Library P^ >s^- Jifu.*^^ ft^ ru. > .. %. ^;\ V ^ : A , > ' N »- >3:v /^5>>. y>f\ V 1> -^ -r^ . : '. * AN ELEMENTARY TREATISE ON MECHANICS. TRANSLATED FROM THE FRENCH OF M. BOUCHARLAT. /a)DITIONS AND EMENDATIONS, DESIGNED TO ADAPT IT TO THE USE OF THE CADETS OF THE U. S. MILITARY ACADEMY. BY EDWARD H. COURTENAY, PROFESSOR OF NATURAL AND EXPERIMENTAL PHILOSOPHY IN THE ACADEMY. NEW-YORK; PRINTED AND PUBLISHED BY J. & J. HARPER, NO 82 CLIFF-STREKT, AND SOLD BY THE BOOKSELLERS GENERALLY THROUGHOUT THE UNITED STATES. ^^i ^ Mechanical Ensine^, [Entered according to the Act of Congress, in the year 1833, by J. & J. Harper, in the Office of the Clerk of the Southern District of New-York.] • "^^TkO .00810/. ^«^^'■''' Es^'neeriBg Libiaiy PREFACE. In preparing a translation of Boucharlat's Elements of Mechanics, it has been the principal object of the translator to supply a suitable text-book for the use of the Cadets of the United States' Military Academy. To accomplish this object more effectually, it has been deemed necessary to introduce several subjects which are not noticed in the original, and to extend or modify others, where the methods of investigation adopted by the Author appeared incomplete or obscure. It was also judged proper to omit one or two subjects, the dis- cussion of which is usually reserved for works of a less elementary character. These alterations were adopted with less hesitation as the work was principally designed for a special purpose ; but it is believed that they will render the work more generally useful, by facilitating the comprehension of many of the more difficult investigations, and by affording much valuable in- formation in relation to those subjects which were not dis- cussed in the original, but which are generally admitted to form an essential part of an elementary course of Mechanics. In supplying the deficiencies of the original, reference has been had most frequently to the works of Poisson, Francœur, Navier, Persy, Genieys, and Gregory ; and in some few in- stances, the methods of investigation pursued by those authors have been adopted with but slight alterations. The works of Boucharlat have long enjoyed an unusual share of public favour ; and the hope is therefore entertained A2 4 PREFACE. that the treatise now presented, in our own language, will prove a useful introduction to the study of the higher branches of Mechanics, and that it will be received with in- dulgence by all those who are disposed to cultivate a taste for the most interesting apphcation of Mathematical Science. As the entire work may be found to constitute too exten- sive a course for those students who can devote but a limited time to the study of Mechanics, it was thought expedient to indicate such of the more difficult subjects as might be omitted. These subjects are designated in the table of con- tents by being printed in italics ; and they will be found to be unnecessary in enabling the student to comprehend those which follow. CONTENTS. PART FIRST. STATICS. Introductory Remarks and Definitions 9 Of the Composition and Decomposition of Forces .... II Of Forces situated in the same Plane and applied to a single Point 20 General Remarks on Forces situated in any manner in Space 24 Of Forces situated in Space and applied to a Point .... 27 Of the conditions of Equilibrium of a Point acted upon by several Forces, and subjected to the condition of remaining upon a given Surface 31 Of the conditions of Equilibrium of a Point acted on by several Forces, and subjected to the condition of remaining con- stantly on two curved Surfaces, or on a Curve of double Curvature 36 Of Parallel Forces 39 Of Forces situated in the same Plane and applied to Points con- nected together in an invariable manner 48 Of Forces acting in any manner in Space 60 Theory of the Principal Plane, and Analogy existing between Projections and Moments 71 Centre of Gravity 79 Of the Centrobaryc Method 97 Machines — Cords 99 Of the Catenary 103 Of the Lever 109 Of the Pulley 116 Of the Wheel and Axle 120 Of the Inclined Plane 126 Of the Screw 128 Of the Wedge .131 6 CONTENTS. Page Friction 132 Effects of Friction in certain Machines 136 Of the Stiffness of Cordage 142 On the Resistance of Solids 145 Of the Resistance to Compression or Extension 147 Of the Resistance of a Solid to Flexure and Fracture produced hy a Force acting at right angles to the direction. if the Fibres ,. 150 Of the Figure of the Solid after Flexure 164 Of Solids of equal Resistance , 175 Of the Principle of Virtual Velocities 177 Of the Position of the Centre of Gravity of a System when in ! Equilibrio 184 PART SECOND. DYNAMICS. Of the Law of Inertia 187 Of uaiform rectilinear Motion 188 Of varied Motion , 190 Of uniformly varied Motion 194 Of the Motion of a Body projected vertically upward . . .197 Of the vertical Motion of a Bod/ when acted upon by the Force of Gravity considered as variable 199 Of the vertical Motion of a Body in a resisting Medium , . . 202 Of the Motions of Bodies upon Inclined Planes 205 Of curvihnear Motion 208 Of the Motion of a material Point when compelled to describe a particular Curve . 221 Of the Motion of a rhaterial Point when compelled to move upon a curved Surface . . . 229 Of the Motion of a material Point on the Arc of a Cycloid . . 236 Of oscillatory Motion . . . . . . . 238 Of the Simple Pendulum ..... . 240 Of the Centrifugal Force . . ... . . . . . , . 247 Of the System of the World 253 Of the Motions of Projectiles in Vacuo ........ 272 Of the Motions of Projectiles in a resisting Medium .... 278 Of the different Methods of measuring the Effects of Forces 288 CONTENTS. f Page Of the direct Impact of Bodies 293 Of the direct Impact of unelastic Bodies ....... 293 Of the direct Impact of elastic Bodies 294- Of the Preservation of the Motion of the Centre of Gravity in the Impact of Bodies 297 Of the Preservation of living Forces in the Impact of elastic Bodies — Relative velocity before and after Impact — Loss of living Force in the Collision of unelastic Bodies . . . 299 Principle of D'Alembert 301 Of the Motion of a Body about a fixed Axis 309 Of the Moment of Inertia 316 Of the Motion of a Body about a fixed Axis when acted upon by incessant Forces 320 Of the Compound Pendulum 322 Of the Motions of a Body in Space when acted upon by im- pulsive Forces 328 Of the Motions of a System in Space when acted upon. by in- cessant Forces 333 General Equations of the Motions of a System of Bodies . . 388 General Principle of the Preservation of the Motion of the Centre of Gravity ... ; 345 PART THIRD. HYDROSTATICS. Of the Pressure of Fluids 349 General Equations of the Equilibrium of Fluids 351 Application of the general Equations of Equilibrium to incom- pressible Fluids 354 Application of the general Equations of Equilibrium to elastic Fluids 359 Of the Pressure of heavy Fluids 36 J Of the Equilibrium, Stability, and Oscillations of floating Bodies 366 Specific Gravity — Hydrostatic Balance — Hydrometer . . . 379 Of the Pressure and Elasticity of Atmospheric Air .... 384 Of Pumps for raising Water 386 Of the Air-pump 391 Of the Barometer 398 b CONTENTS. PART FOURTH. HYDRODYNAMICS. Page Of the Discharge of Fhiids through horizontal Orifices . . . 407 Of the Motion of Water in Pipes 429 I i ELEMENTS OF MECHANICS. PART FIRST. STATICS. INTRODUCTORY REMARKS AND DEFINITIONS. 1. Mechanics is the science which treats of the laws of equihbrium and motion. Wlien appUed to solid bodies it is divided into Statics and Dynamics ; the former discussing the conditions of their equihbrium, and the latter those of their motion. In the application of Mechanics to the consid- eration of fluid substances a similar division is likewise made, viz. Hydrostatics, which treats of the equilibrium of fluids, and Hydrodynamics, which investigates the circumstances resulting from their motions. 2. The object proposed in Statics being the determination of the laws of equilibrium, this state of equilibrium may always be regarded as resulting from the mutual destruction of several forces. 3. The term force or power is applied to every cause which impresses on a body or a material point a motion or tendency to motion. 4. A force may act on a material point either by drawing the point towards it, or by pushing the point in advance of it. The first hypothesis will always be adopted, unless the contrary is expressly indicated. 5. A material point being solicited by a single force will naturally move in a right line, since there can be no reason why it should deviate to the right rather than to the left of this line. 10 STATICS. 6. The right line along which a force acts is called the line of direction, 7. The effect of a force depends, 1°. On its intensity ; 2°. On its point of application ; 3°. On its line of direction ; and 4°, On its pushing or pulling along this line. 8. By the intensity of a force, we understand its greater or less capacity to produce motion. 9. If two forces directly opposed to each other sustain in equilibrio a material point or an inflexible right line, the in- tensity of either one of these forces may be assumed arbitra- rily, provided we assign an equal intensity to the second force. A similar remark is equally applicable to a system composed of any number of forces ; and hence it appears that the con- ditions of equilibrium will depend simply on the ratios of the forces, and not on their absolute intensities. 10. Having assumed one force as a unit of measure, we say that a second force is equal to it, when, if directly op- posed to it, an equilibrium would ensue. Two equal forces applied to a material point, acting along the same right line, and in the same direction, constitute a double force : in like manner a triple force may be regarded as resulting from the union of three equal forces, &.c. ; so that the number of these equal forces will constantly be propor- tional to their joint intensity. It may hence be inferred that if several forces solicit the point M {Fig. 1) in the same line of direction from M towards B, we can add into one sum all these forces, since their joint effect will be precisely the same as that of a single force equal to their sum. For the same reason we should subtract from this sum, or we should regard as negative all the forces which tend to solicit the point from M towards A. 11. The unit efforce being arbitrary, it may be repre- sented by any portion of its line of direction. 12. When a force is applied to any point of a body whose several parts are firmly connected together, this point cannot be put in motion without communicating the motion to the other parts of the body ; if, therefore, a force be applied to any point A [Fig. I), it will have the same effect as though it were applied to £m.y other point M, assumed on the line of COMPOSITION OF FORCES. 11 direction AB, Moreover, if we drop the consideration of a body, we may still regard the points in space situated on the line of direction as mathematical points, no one of which can be moved without imparting its motion to all the others. 13. It appears from Art. 12, that by interposing a fixed obstacle on the line of direction of a force, its effect will be entirely overcome. 14. Two equal forces P and Q, applied to the points A and B of an inflexible right line {Pigs. 2 and 3), and acting along this line, but in contrary directions, will sustain each other in equilibrio : for if the force P tends to draw the point A from A towards a, the point B, which is firmly connected with A by means of the intermediate points, will have a tendency to describe the space Bb, equal to Aa ; but by hypothesis, the force Q, tends to move the point B over a space Bb' equal to Aa ; and since B cannot yield to one of these influences rather than to the other, it must remain immoveable, and aji equilibrium will necessarily ensue (Art. 13). In like manner, if the forces P and Q, had been supposed to exert a tendency to push A and B, the same consequences might have been deduced. 15. When the right line AB is reduced to a point, the two equal forces, being directly opposed, are still in equilibrio ; but if the forces are unequal, the point M {Fig. 1) will be moved in the direction of the greater, by a force equal to the differ- ence of the two unequal forces. Of the Composition and Decomposition of Forces applied to a Point. 16. When two forces act upon a moveable point in direc- tions forming with each other an angle whose summit is the point of application, the state of equilibrium cannot subsist. For, if we suppose the two forces P and Q, {Fig. 4) to be in equilibrio, we may introduce a third force P' equal and directly opposed to the force P. The forces P and d being supposed to destroy each other, the force P' must produce its entire effect, and must consequently move the point M in a direction from M towards P'. But P and P', being equal and 12 STATICS. directly opposite, must likewise destroy each other, and the force Q. will therefore act as though it were alone, soliciting the point M in a direction from M towards Q, ; and since it is impossible for the point M to move in two directions at the same time, we cannot suppose that P and Q, are in equi- Ubrio without involving an absurdity. 17. Since an equilibrium cannot subsist between two forces whose lines of direction are not coincident, the point M will tend to move in a certain direction MR, as though it were solicited by a single force R. This force is called the result- ant of the two others, and the original forces are called com- ponents. It may be observed that two forces which have a resultant do not always intersect. For example, if two parallel forces P and Q, be supposed to act on a body, and if a third force R be found which shall produce the same effect, R will be the resultant of the forces P and Q,. 18. Having examined the conditions of equilibrium of two equal forces acting on a point, the most simple case which next presents itself is that of three equal forces applied to the same point. Let P, Q, and R represent these forces ; if they produce an equilibrium, their directions will divide the cir- cumference of a circle whose centre coincides with the point of application, into three equal parts {Fig. 5) : for since the same reasons may be adduced to prove that the point should tend to move in the direction of each of these forces, it fol- lows that it cannot yield to the influence of either in prefer- ence, and must consequently remain at rest. 19. The equal angles PMa, PMR, and QMR {Fig. 5), being measured by one-third of the entire circumference, each of them is equal to f of a right angle, or 120°, Hence, if one of the three lines PM, QM, or RM be prolonged through M, it will bisect the angle formed by the other two. If MS, for ex- ample, be the prolongation of the line RM, the angles PMS, CIMS will be equal, being supplements of the equal angles PMR and GlMR ; whence it appears that MS bisects the angle PMGl. 20. Let us next suppose the two equal forces P and Q, {Fig. 6) to be applied perpendicularly to the extremities A COMPOSITION OP FORCES, 13 and B of a right line AB ; the resultant of these forces will pass through the point O, the middle of the line AB, and will be equal in intensity to the sum of the intensities of the two forces P and Q,. For, draw through the points A and B the four right lines AC, AD, BC, BD, each forming with AB an angle equal to ^ of a right angle : the triangles ACB, ADB will be isosceles, and will have the sides AC, CB, AD, DB equal to each other. The right lines AB, CD will intersect each other at right angles, and the figure ACBD will be a rhombus : the sides of this rhombus and their prolongations determine by their intersections the four obtuse angles ACB, ADB, P'AC, Q'BC, each of which is equal to | of a right angle ; for, the angle CAD being by; construction equal to | of a right angle, its supplement P'AC must be equal to | of a right angle ; and since the opposite sides of the rhombus are parallel, the angle ACB is equal to P'AC, and is consequently equal to f of a right angle. The same may be proved of the angles CBQ,' and ADB. Moreover, since the line CD bisects the angle ACB, which was proved equal to f of a right angle, it follows (Art. 19) that the three angles ACB, ACS, and BCS are equal to each other. In like manner it may be shown that there are three equal angles at each of the points A, B, and D. 21. We will now apply at the points A, B, C, D, which are supposed firmly connected together, twelve equal forces, dis- tributed as follows : At the point A three equal forces P, P', P", At the point B three equal forces Q,, Q,', Q,", At the point C three equal forces S, S', S", At the point D three equal forces V, V, V" ; forming with each other angles equal to | of a right angle : these twelve forces will sustain each other in equilibrio. But the forces P' and V", Q' and V, being equal, and directly opposed, will destroy each other, as also will the forces P" and S', Q," and S". If, therefore, an equilibrium is main- tained in the system, it must subsist between the four forces P, Q, S, and V. The two last, acting in the same direction along the line DC, are equivalent to a single force equal to their sum, which may be applied at O, a point in their line of 2 14 STATICS. direction. Thus, an equilibrium will take place between the forces P and Q, and a force R whose line of direction passes through the middle of the line AB, and whose intensity is equal to the sum of the intensities of P and Q. If we suppress P and Q, the equilibrium will be destroyed, but it may again be established by applying at O a single force R' equal and directly opposed to the force R. The force R' must therefore produce an effect precisely equal to the joint effect of P and Q, and will consequently be their resultant. We hence infer that tlie resultant of ttvo equal and jmrallel forces is equal to their sum, is parallel to them, and divides equally the line AB, which is draivn j^erpendicular to the co?nmou direction of those forces. 22. To determine the resultant of two unequal parallel forces P and Gl applied to the extremities A and B of a right line AB {Fig. 7), we will suppose p to represent the unit of force, and make mp==V, 7ip=Q,. The ratio of m:n will be the same as that of the forces P and d. Let the right line AB be also divided in the same ratio at the point D, and we shall have the proportion P : Gl : : AD : DB («). On the prolongations of AB, take AA'=AD, and BB'=BD ; we shall then have, since A'D and DB' are double AD and DB, P : a : : A'D : DB' : : w : n. If then we divide A'D into m equal parts, DB' Avill contain n such parts, and A'B' will contain one of these parts as many times as p is contained in P+Q,. And since any two of the points of division a', a", a'", 6cc. separate three equal parts, wh ile three points separate four parts, &c.,the number of equal parts in the line A'B' will exceed by unity the number of points of division. A force being applied at each point of division, there will remain one of the number m-{-n, of which one half may be applied at A', and the other at B' ; the several partial forces will thus be distributed throughout the line A'B'. But the points A' and D being equally distant from the point A, the force ^p applied at A' may be combined with one lialf of the force p applied at D, and their resultant, which is equal to their sum, will pass through A. The same remarks will apply to the forces /> and p applied at a' and a^ to the forces COMPOSITION OF FORCES. 15 p and p applied at a" and a,,, &-c. ; thus, the total resultant of the partial forces distributed along A'D,will be equal to their sum P, and will pass through the point A. In like manner it may be shown that the forces applied to DB' may- be replaced by Q, ; and the entire system of partial forces may therefore be replaced by the two forces P and Q, applied at the points A and B. But these parallel forces may be otherwise compounded, by combining them in pairs taken at equal distances from the middle point O of the line A'B' ; and it may thus be easily shown that the resultant of the whole system will pass through the point O, and will be equal to P+Q,. The position of the point O must now be determined. For this purpose, it may be remarked that A'O {Fig. 7), being one- half of A'B', is equal to AB ; and by substituting this value in the equation AO=A'0— A'A, which results immediately from an inspection of the figure, we shall obtain AO=AB— AA', or AO=AB— AD=DB. In a similar manner it may be shown that OB=AD ; and by substituting these values of DB and AD in the proportion (a), there will result Q : P : : AO : OB {b). If P and Gt are incommensurable for the unit p, this pro- portion which results from the division of A'B' into m-{-7i equal parts, might seem to fail : but by diminishing indefi- nitely the value of the unit p, and increasing in the same proportion the number of these divisions, the demonstration becomes applicable to all cases, since the equal parts Aa', a'a", &c. being indefinitely small, the points of division will then become continuous. 23. This proposition is equally true when the two parallel forces P and Gl are applied to the extremities of an oblique line CD {Fig: 8). For, by drawing AB at right angles to the common direction of the two forces, and transferring the points of application to the points A and B in their lines of direction, the proportion {b) will evidently subsist ; but the similar triangles AGO, EDO, give AO : OB : r OC : OD ; whence we obtain 16 STATICS. a : P : : OC : OD : and we therefore infer that when two parallel and unequal forces P a7id Q, are apjAied to the extremities of a right line CD, their residtant will divide this line in the inverse ratio of the intensities of tJie forces. 24. By the aid of this theorem we can readily demonstrate that of the parallelogram of forces, which may be enun- ciated as follows : — If any two forces P and Q, applied to a "point A {F'g- 9) he represeiited in direction and intensity by the lines AB and AC, their I'esultant ivill be represented in direction and intensity by the diagonal of the parallelogram constructed upon the lines AB a7id AC. It is immediately obvious that the resultant will pass through the point of application of the forces ; since the forces conspire to solicit this point, and their resultant, which may replace them, must therefore contain it. 25. The resultant of the two forces P and Q, will likewise be contained in the plane of those forces. For, if it be situ- ated above this plane, a position in all respects similar can be selected below the plane : the same arguments may then be advanced to prove that its direction coincides with either of these lines ; and since the resultant cannot have two direc- tions, we infer that it coincides with neither. 26. It may also be proved that the resultant of two equal forces [Figs, 10, 11, 12) will bisect the angle included between them. For, if we suppose Km to represent the resultant of the two forces P and Q, and draw AD bisecting the angle PAQ., a line An may always be found, whose position with respect to AD, AQ,, and AP shall be precisely similar to that of Am with respect to AD, AP, and AGI ; hence, the same reasons which would prove Am to b§ the resultant, become equally applicable to An, and it might thence be inferred that there are two resultants : this being impossible, we conclude that the resultant coincides with AD. 27. Let the two unequal forces P and Q. be now supposed to act upon the point A [Fig. 13), and let the parallelogram ABDC be constructed, whose sides AB and AC are taken on the lines of direction of those forces, and are proportional to theiE- COMPOSITION OF FORCES. 17 intensities. It has already been shown that the resultant will pass through A, and it remains to be proved that it will also pass through D, the extremity of the diagonal AD. Having taken DE=AB=P,* draw EF parallel to AB, and apply at E and F, in contrary directions, the two forces Q,', Gl", each equal to Q,. Since these forces will destroy each other, we can substitute for P and Q, the four forces P, Q,, CI', and Q,". But by regarding P and Gl' as two parallel forces applied to the extremities of an inflexible line BE, and having obtained by construction the proportion P : a' : : DE : BD, it follows immediately from the preceding theorem, that the resultant R of P and Q,' will pass through the point D; Again, if we transfer the force Q,, and apply it at F, in its line of direction, the two equal forces Q, and Q," will have a re- sultant S, which, bisecting the angle Q^FGl", will pass through D, the opposite angle of the rhombus CDEF. We thus ob- tain two forces R and S which are equivalent to the original forces P and Gl ; aad since the forces R and S pass through the point D, the resultant of P and Gl will likewise pass through the same point. 28. It will now be proved that if the intensities of the forces be represented by AB and AC, the diagonal AD will repre- sent the intensity of the resultant {Pig. 14); If at the point A {Mg. 14), and in the direction AD of the diagonal of the parallelogram constructed on the sides AB=P, AC=Gl, there be applied a force X equal and directly op- posed to the resultant of P and Q,, an equilibrium will take place between the forces P, Q, and X. But we may regard Q, as equal and directly opposed to the resultant of the forces P and X ; hence it follows, that if through the extremity B of the line AB a line be drawn parallel to X, intersecting at * It should be remarked that the expression AB=:P is merely intended as an abridged method of stating that the line AB represents the relative intensity of the force P, when compared with the unit of force whose intensity is likewise represented by a line. In like manner, we speak of the " force AB," denoting thereby that the line AB represents the line of direction and relative intensity of the force. These abbreviations have been sanctioned by usage. B 1 8 STATICS. E the prolongation of the Une AC, which, as has been already shown, coincides in direction with the diagonal of the paral- lelogram constructed on P and X, the line BE, being a side of this parallelogram, will be equal to the opposite side, which must represent X : but BE, being also the side of the paral- lelogram ED, is equal to the opposite side AD, which repre- sents the diagonal of the parallelogram constructed upon P and Q, ; whence X=AD, and the intensity of the resultant is likewise measured by the length of the diagonal. 29. One of the simplest corollaries which may be deduced from the foregoing proposition is the trigonometrical relation existing between the components P and Q, and their resultant R {Fig. 15). To obtain this relation, we will assume on the directions of these forces the parts AB and AC propor- tional to their intensities, and constructing the parallelogram ABDC, we shall have the proportion P : a : R : : AB : AC : AD. And from the equality of the sides DD and AC, we shall have in the triangle ABD, P : a : R : : AB : BD : AD. But the proportionality of the sides of the triangle to the sines of tlie opposite angles gives AB : BD : AD : : sin BDA : sin BAD : sin ABD. Hence we deduce P : a : R : : sin BDA, : sin BAD : sin ABD. The determination of the relations between P, Q, and R is thus reduced to the solution of a case in plane trigonometry. 30. K there be given, for example, the two components AB and AC, and the angle BAC contained between them, and it be required from these to determine the resultant, we shall have, in the triangle ABD, the sides AB, BD, and the angle B equal to the supplement of BAC. With these data we readily obtain the value of the side AD=R, by means of the formula R3 =P2 -j-ds _ 2Pa COS B. If in this formula we wish to introduce the angle included between the two forces, since the angle B is the supplement of the angle BAD, we shall have the relation cos B= — cos Aj COMPOSITION OP FORCES. 19 whence by substitution the following equation is obtained between the resultant, the two components, and the angle included between them, R2 =P- +0,2 +2Pa cos A (1). 31. When the angle A becomes equal to 90°, the parallelo- gram ABDC {Fig-. 16) becomes a rectangle, and cos A=0. The general relation between the resultant and its two com- ponents is then reduced to R2=p2+a^ The solution of the converse problem, or the resolution of a single force R into two components P and d, having given directions, is readily effected by constructing a parallelogram upon the line representing the given force as a diagonal, the sides of the parallelogram having the directions of the re- quired components. 32. When there are several forces lying in different planes, but all meeting in a single point, the resultant of the system can always be determined ; for, by combining these forces in pairs, and substituting each resultant for its two components, the number of forces will be successively reduced, and we shall finally obtain but a single resultant. 33. The method of compounding any number of forces which has just been explained gives rise to a remarkable graphic construction. Thus, let P, P', P", P'", &c. represent any forces whose directions intersect at the point A {Fig: 17), and whose intensities are expressed by the hues Ap, Ap', Ap", Ap"\ &c, assumed on the respective lines of direction ; through the point p draw the line pr parallel and equal to the line Ap', and complete the. parallelogram Aprp' ] the di- agonal Ar=R will be the resultant of the forces P and P': in like manner, by drawing rr' parallel and equal to Ap", and forming the parallelogram Arr'p", the diagonal Ar' will be the resultant of R and P", and therefore the resultant of the three forces P, P', and P". By continuing this process, a polygon Aprr'r" would be formed, having its sides parallel to the directions of the forces, and their lengths representing the intensities of those forces. The distances from the point A to the angles of this polygon will be B2 20 STATICS. Ar=the resultant of P and P', Ar'=the resultant of P, P', and P", A?-"=the resultant of P, P', P", and P'". And by repeating the construction for any number of forces, the distance from the point A to the extremity 7-^''' of the last side of the polygon will be equal to the resultant of the en- tire system. Of Forces situated m the same Plane, and applied to a single Point. 34. Let P, P', P", &c. {Pig. 18) represent several forces situated in the same plane, their directions intersecting at the point A ; through this point let there be drawn the rec- tangular axes Ax and Ky ; then, denoting the respective intensities of these forces by AP, AP', AP", (fcc, let each be decomposed into two components, whose directions shall coincide with the rectangular axes. For this purpose we will represent by «, «', «", (fee. the angles included between the forces and the axis of a;, and by /3, /3', /3", (fee. the angles which they form with the axis of y. In the right-angled triangle ABC {Fig. 19), the^ side AC being expressed by AB cos A, and the side BC by AB sin A, the components of the forces P, P', P", (fee. in the directions of the two axes are readily obtained : for the force P repre- sented by AB, forming an angle ct, with the axis of x, and an angle /3 with the axis of y, will have for its components along these axes, AC=P cos a, BC=P cos /3. In like manner, the forces P', P", P'", (fee. will have for their components in the direction of Aa:, P' cos «', P" cos «", P'" cos «'", (fee, and in the direction of the axis Ay, F cos /3', P" cos /3", P'" cos /3'", (fcc. If the sum of the components acting in the direction of x be taken, as also the sum of those acting in the direction of y, we shall have, denoting these sums by X and Y respectively,. P cos «+P' cos «'-f-P" cos «"-f «fec.=X, P cos |34-F cos |3'+P" cos /3"-f (fec.=Y; FORCES APPLIED TO A POINT. 21 and the entire system will thus be reduced (Art. 10) to two forces, of which one X is directed along the line Ax, the other Y acting along the line Ay. Calling R the resultant of these two forces, its value may be determined from the equation Xa4.Y2=R2. 35. For the purpose of rendering the preceding determi- nation of the value of the resultant general, we have attrib- uted the positive sign to all the cosines which enter into the expressions for X and Y ; but it will be necessary in practice to regard the essential signs with which these quantities are severally affected. The following considerations will serve to explain the necessity of this distinction. Let a point M {Pig. 20) be solicited by a force represented in intensity by the line MP. By decomposing this force into two others whose directions shall coincide with the rectangular axes Mx and My, and calling « the angle which the direction of the force makes with the axis Mx, its two components will evidently become MC=MP sin cc, MD=MP cos «. The forces which are directed in the line Mx, being regarded as positive when they act from M towards x, the component MD will obviously be positive. If the force MP should as- sume the position MP', the angle « would be increased, and its cosine diminished ; and if the angle becomes greater than 90°, the direction of the force will fall in the second quadrant. In this case it will assume the position MP", and the cosine of the angle will change its sign. But it is evident that the component MD" of the force MP" becomes also nega- tive, since it solicits the point M in a direction opposite to that in which it was urged by the component MD. Thus it appears that the signs of these two components result from the signs of the cosine of a, and hence the forces MP, MP', &c., which solicit a point, may be always regarded as essen- tially positive, provided we attribute the appropriate signs to the cosines of the angles which they form with the axes. 36. If the force under consideration fall below AB, as in the position MP'", the angle x being measured by the arc ALBP'", will be greater than two right angles. To avoid this inconvenience, it has been agreed to reckon the angles as 22 STATICS. and /J indiscriminately on each side of their respective axes. Thus when the force falls beneath AB, the angle «e will be measured, not by the arc ALBF", but by the arc AP'", which has the same cosine. By this arrangement all the arcs em- ployed are less than 180^. It is true that when the angle « is alone given, the direction of the force would appear inde- terminate, since this angle may be counted either from A to Pj or from A to P'" ; but this ambiguity will immediately dis- appear by considering the value of the angle /3, which is evi- dently acute for the force MP, but obtuse for the force MP'". 37. Whatever may be the direction of the given force, since it must necessarily lie in one of the four right angles formed by the axes around the point M, its position must correspond to some one of those given in Figs. 21, 22, 23, 24. In the first quadrant,» and P being acute give cos« positive, cos P positive, In the second, a, obtuse and/3 acute give cos «negative, cos /J aegBtwe, - In the third, «obtuse and P obtuse give cosa negative, cos (3 negative, In the fourth, «acute and /3 obtuse give cosas positive, cos (} negative. Each of these angles will be less than 180°. 38. It may be observed that the signs of these cosines are similar to those of the co-ordinates x and y of the point B- For example, if the point be situated within the angle .r'Ay {Fig. 22), X will be negative and y positive, while at the same time we shall have cos « negative and cos j3 positive. 39. For the purpose of making an application of the pre- ceding principles, let us determine the resultant of the five forces P, P'j P", P'", P"j which are situated as represented in Fig. 25, and solicit the point A. By attributing to the components of the forces the positive or negative signs cor- responding to the angles which are acute or obtuse, the com- ponents of P ^ C +P cos «, -fP cos/3, F -f P' cos «', -F cos (3', P" i will be I -fP"cos«", -P"cos/3", P'" — P''"C0S«"',-P"'C0S/3'", P- [ — P"'C0S«"',+P"'C0S/3". Having taken the sum of the components which act in one direction, we subtract from it the remaining components which act in an opposite direction, and we thus obtain FORCES APPLIED TO A POINT. 23 P COS «4-P' COS «'4-P" cos «"-F" cos «'"— P^ cos «"=X, P cos /3 + P'' COS /3" — P'COS ^' — P" COS ^"— P'" COS /3"'=Y. 40. If we defer the determination of the signs of the cosines until we wish to make an appHcation of the preceding equa- tions, the several terms may be written with the positive sign, and the general form of the equations will then become P cos «4-P' cos «'+P"cos «"+&c.==X (2), P cos /3+F cos 13' -fP" cos Ii"+ÔÙC.=Y (3). 41. The resultant being represented by the diagonal of a rectangle, the lengths of whose sides are denoted by X and Y, its value will be determined by the equation R=^(X2+Y==) (4). The position of the resultant remains to be determined. If we denote by a and b the angles which the resultant forms with the co-ordinate axes, we shall have X=R cos a, Y=R cos b ; whence cos a=-^, cos 6=^5- (5). K K The positions and intensities of the forces being given, the values of X and Y may be immediately deduced from the equations (2) and (3). These values being substituted in the equation (4), make known the value of the intensity of the resultant, and its position may be determined from the equa- tions (5). 42. Its line of direction passing through the origin A {Pig. 26), will liave for its equation sin a ?/=a; tang: «i or V=a^ ; •^ & > y cos a ' and by substituting cos b for sin a, since a and b are com- plements of each other, the equation becomes cos b y~x , co^« and by substituting in this equation the values of cos a and cos b given in equations (5), we have Y ^=x-^*- 43- When an equilibrium takes place, the intensity of the 24 STATICS. resultant becomes equal to zero ; and the formula (4) then assumes the form V(X2+Y2)=0, or X2 4-Y2=0. But since every square is essentially positive, the preceding equation cannot be true, unless each of its terms is separately equal to zero ; hence X=0, Y=0. Such are the equations which express the conditions of equi- librium of any number of forces situated in the same plane, and acting on a point. 44. If X alone were equal to zero, we should have R=Y, cos rt=0, cos b=±l. These equations prove that the resultant is equal to the com- ponent Y, and is directed along the axis of y. In like manner it might be shown that if Y were equal to zero, the resultant would be equal to the component X, and would be directed along the axis of a;. General Remarks on Forces situated in any marmer in Space. 45. If three forces solicit a point, their directions not being confined to a single plane, a theorem analogous to that of the parallelogram offerees will still serve to determine their re- sultant. Thus, let any three forces P, P', and P" be applied at the point A {Fig. 27), and let their intensities be repre- sented by the lines AB, AC, and AD. If a parallelopiped be constructed upon these three lines, the diagonal AE, of the base of this parallelopiped, will evidently represent the re- sultant of the forces AB and AC ; and by substituting the force AE for its two components, the resultant sought will be that of the forces AE and AD ; it will therefore be termi- nated at the extremity F of the line EF drawn parallel and equal to the line AD ; hence it will be the diagonal of the parallelopiped DE. 46. If the three forces are rectangular, the angle ABE will be a right angle, and hence we obtain AE==AB2-|-BE2; FORCES APPLIED TO A POINT. 25 but the triang^le AEF being also right-angled, we have AF=»=AE2-fEF2. And by substituting for AE* its value given above, we deduce AF='=AB»4-BE='-fEF2. Or by replacing BE and EF by their equals AC and AD, we finally obtain AF=^(AB« 4-AC^ +AD2), or, the resultant of the three forces being denoted by R. 47. It has been shown that any number of forces lying in the same plane may always be referred to two rectangular axes : in like manner, we may refer to three rectangular axes those forces which are situated in different planes. Thus, having assumed three co-ordinate axes passing through any point O {Fig-. 28), we draw through A, the point of applica- tion of a force P, the three rectangular axes Ax, Ay, and Az, parallel respectively to the axes of co-ordinates ; and denoting by X, /3, y the angles formed by AD, the direction of the force P, with the three lines A.v, Ay, Az, the direction of the force will be determined when these angles become known. 48. The values of these angles may also be employed to determine the components of the force P, which act in direc- tions parallel to the three co-ordinate axes. For, DC beins: perpendicular to the plane i/Ax, the angle DCA will be a right angle, and the triangle ADC, having the angle D=y, will give DC=AD cos y (6). In like manner, the components parallel to Ax and Ay will be expressed by AB=AD cos u, BC=AD cos |3 (7). ^nd replacing the line AD by the force P which it repre- sents, we obtain for the three rectangular components of P, P cos «, P cos /3, P cos y. 49. It is important to observe that the values of two of the angles u, jâ, and y will serve to determine that of the third. For, since the square of the diagonal AD is equal to the sum of the squares of the three edges, we have AB='+BC2-1-DC2=AD2 ; and substituting in this equation the values obtained from the o O 26 STATICS. equations (6) and (7), suppressing the common factor AD*, there will remain C0S2<«4-C0S2/3-}-C0S=y = l j whence, cos y=±y/(l — C0S = «— C0S'»/3) (8). And since a similar value may be found for each of the other cosines, it follows that the angle formed by the direction of a force with either of the axes will become known, when the angles formed with the other two axes have been previously determined. 50. The radical in equation (8) being affected with the double sign, the cosine of y may be either positive or nega- tive. The first value will obtain when the angle is acute, and the second when it is obtuse. But the angle y will be acute or obtuse according to the position of the force P ; in the first case, the force falls above the plane xky, and the co-ordinates z of the points in the line representing the force, will therefore be positive ; in the second, it falls below xAy, and the co-ordinates z will then be negative. The same observations may be extended to the angles » and /3 considered with reference to the axes o( x and y ; so that in general the cosines will be affected with the same signs as the co-ordinates x, y, z, reckoned from A. 51. The signs of the cosines may also be determined by a rule which is founded on Art. 10. Thus, if Ax {Fig. 29) represent the line of direction of a component, this compo- nent will be positive when it acts in the direction from A towards x, but negative if it acts from A towards x'. The tendency of the force in the first case will be to remove the point A from the origin O, but in the second to cause its ap- proach. Hence, we derive the following rule : A compoiiem is positive when it tends to increase the co-ordinate of th( point of application, and negative when it tends to diniinisl this co-ordinate. FORCES APPLIED TO A POINT. 27 Of Forces situated in Space, and applied to a Point. 52. Let P, P', P", &c. represent different forces which so- licit a point A, and let there be drawn through this point the three rectangular axes Kx, Ay, Az ; represent by «, /3, y, the angles formed by the force P with the axes of co- ordinates, «', |3', y', the angles formed by P' with the same axes, «*',/3",y", the angles formed by P" with the same axes, &,c. &c. &c. By resolving these forces into components acting along the three axes, we shall obtain (Art. 48) P cos «, P cos /3, P cos y, compoueuts of P, P' cos et'j P' cos Q', P' COS y', Components of P', P"cos«", P" cos ô",P", cosy", Components of P". If we defer, as in Art. 40, the determination of the signs of the cosines of these angles until the formulas are applied to a particular example, and denote by X, Y, and Z the com- ponents of the resultant, directed along the three axes, we shall have P cos «+F cos a'+P" cos «"-F&c.=X (9), P cos /3 + P' cos /3' + P" COS /3" + &c. = Y (10), P COS y+P' cos y' + P" COS y"+&C. = Z (11), 53. But X, Y, and Z being the projections AB, BC, and CD of the right line AD, which represents the resultant R {Fig. 28), we shall obtain (by Art. 46) AB^ -f BC^ +CD2 =AD% and consequently, X2-fY2+Z='=R^ The intensity of the resultant will thus be determined, being expressed by the equation R=^(X2+Y='-f-Z») (12). Again, if we call a, 6, and c the angles formed by the result- ant with the co-ordinate axes, the components of R directed along the axes will be R cos a, R cos b, R cos c ; 28 STATICS. and since these components have been represented by the quantities X, Y, and Z, we shall have X=R cos a, Y=R cos 6, Z=Rcosc; whence, X , Y Z cos «— p"> COS 0=-jîj cosc = :5- (13). If the forces P, P', P", &.C., and the angles a, js, y, «', (3', y', dec are known, the values of X, Y, and Z will result from the equations (9), (10), and (11). These values being substituted in formula (12), the intensity of the resultant will be deter- mined, and its position will become known from the equa- tions (13). 54. If an equilibrium subsists, the resultant becomes equal to zero, and the equation (12) then assumes the form X=+Y3-fZ2=0. And since this equation cannot be true unless the terms are separately equal to zero, we have X=0, Y=0, Z = 0. These values reduce the equations (9), (10), (11) to P cos «-f-P' cos a' + P" cos <«"-}-&.C. = i P cos i3 + P' cos /3'+P' COS /3"-l-&c.=0 > (14). P COS y+P' COS y'-fP" cos y"-|-&c.=0 7 Such are the conditions of equilibrium of a system of forces situated in any manner in space, and applied to a point. 55. If we determine the resultant of all the forces in the system except one, the remaining force will be found equal and directly opposed to this resultant. For, let R' represent the resultant of all the forces except P ; X', Y', and Z its three components, and a, b\ and c the angles which its direc- tion forms with the co-ordinate axes ; we shall have X'=P' cos «'+P" cos «"-f P" cos «'"-f&c, Y'=P' cos /s'-f P" cos i3"4-P" cos /3 "+&C., Z'=P' cos y'-f P" cos y"-f P" cos y"+&C. , and by means of these values the equations (14) may be re- duced to P cos «-f X'=0, Pcos/3-fY'=0, Pcosy-fZ'=0; FORCES APPLIED TO A POINT. 29 and eliminating X', Y', Z', by the equations X'=R' cos a, Y'=R' cos 6', Z =R' cos c', there results P cos «=— R' cos d \ P cos /3= — R' cos 6' > (15). P cos y = — R' cos c j Taking the sum of the squares of these three equations, we obtain P*(cos2«-f cos»/3-f cos'y)=R'2(cos2a'4.cos26'4-cos2c') ; and since the second factor in each member is equal to unity, / ^ this equation reduces to P3=R2, or P=R'. The force P is regarded as essentially positive, its position being determined by the rule explained in Art. 35, &c If the value of P be substituted in equations (15), the factor R' being suppressed, those equations will become cos «= — cos a (16), cos /3=— cos h' (17), cos y= — cos c (18). The relation between the values of cos « and cos a indicates that d and « are supplements of each other. For, if cos a' 1)6 represented by AC [J^ig. 30), cos a. will be represented by AC'=AC ; whence a'=DAC, and «=DAO. But these two angles are supplements of each other ; for, AC being equal to AC, gives the angle DAC=D'AC'; whence, by substituting this value in the equation DAC+DAC'=2 right angles, we get D AC4-DAC=2 right angles, or the angles a and « are supplements of each other. In the same manner may it be proved by the equations (17) and (18), that the angles h' and are supplements of each other, as also are the angles c and y. It results from what precedes that the forces P and R' are directly opposed ; for, if R be supposed situated above the plane of a:, y, having the co-ordinates x and y both positive, 30 STATICS. P will be situated below this plane, and will have the co-ordi- nates X and y both negative. 56. After reducing all the forces to three rectangular com- ponents X, Y, Z, it was shown that the resultant R would be represented by the diagonal of a parallelopiped, whose contiguous edges were respectively equal to X, Y, and Z {Fig. 27). The equation of this resultant, which is repre- sented by AF, will therefore be that of a right line passing through A, the origin of co-ordinates, and through the point F, whose co-ordinates are equal to X, Y, and Z. 57. The case may be rendered yet more general by sup- posing that the point of application of the forces has the three co-ordinates x, y\ and z ; the co-ordinates of the point F will then become {Fig. 31) y-f X, y'-FY, z'-^Z. And the equations of the resultant, being that of a right line in space, will be of the form z=ax-\-h, z=ay-\-b' (19) : substituting in these equations the co-ordinates of the point F; in place of the quantities x, y, and z, we find z'-{-Z=ax'-\-aX+b, z'-\-Z = ay'+àY+b' (20); but the co-ordinates of the point A should also satisfy the equations (19), and therefore we obtain z'=ax'+b, z'=ay-\-b' (21). Subtracting these last from equations (20), we have Z^aX, Z=a'Y; whence, Z , Z a=X> «=Y- Again, by eliminating è and h' between the equations (19) and (21), we find z—z'=a{x—x'y, z — z'=a'{y—y'): and by substituting the values of a and a' previously ob- tained, the equations of the resultant finally become z-z~{x-xl z-z'=^{y-y']. P EQUILIBRIUM OP A POINT UPON A CURVED SURFACE. 3^1 Of the Conditions of Equilibrium of a Point acted upon by several Forces, and subjected to the Condition of re- maining upon a Given Surface. 58. The material point to which the forces P, F, P", &c. were applied, has been supposed hitherto to submit freely to the action which those forces exert ; but if, on the contrary, the point were required to remain constantly on a given sur- face, the equations (14) would no longer be applicable, and the condition of the resultant being equal to zero, which was then necessary, would, under this supposition, be replaced by the condition that the resultant must be normal to the given surface. For, if the direction of the resultant be oblique to the surface, it can be decomposed into two forces, of which one shall coincide with the direction of the tangent, and the other with the normal : the first would cause the material point to slide along the surface, while the second would be overcome by the reaction of the surface. Hence, it follows that the resultant of all the forces must act on the point in the direction of the normal to the surface, and since the re- sultant is destroyed by the resistance of the surface, we may regard this resistance as a force directly opposed to the nor- mal force, and denote its intensity by a quantity N. If the intensity of the force N and the angles 6, 6', 6% which it forms with the co-ordinate axes, were known, it, would be sufficient to add to the equations of equilibrium the compo- nents N cos 5, N cos 6\ N cos 6" of the force N ; we should thus obtain the equations of equilibrium N cos 6-\-V cos «s-f P' cos x-j-V" cos «"-|-&c.— 0, N cos 6' + F cos /3-f-P' cos B' + V COS (3"-f &c. = 0, N COS Ô' + P COS y + P' COS y+P'coS y "-{-&C. = 0. 59. These equations may be simplified by representing, as in Art. 52, by X, Y, and Z, the sums of the components par- allel to the three axes ; the equations will thus become N cos ô-|-X=0, N cos o'-f Y=0, N cos â'-f-Z=0 (22). 60. To determine the values of the unknown quantities cos ê, cos 6', COS ê", and N, we will suppose L=0 to be the equation of the given surface, and x, y\ and z the co-ordinates of the 32 STATICS. material point to which the forces are apphed, and which by hypothesis is required to remain on this surface. The nor- mal being a right hne passing through the point whose co- ordinates are x\ y', z', its equations will be of the form x~x=a{z—z\ y—y = h{z—z') (23). The differences x — x\ y — y\ z — z, which enter into these equations, represent the projections of the right line on the axes of co-ordinates. To determine the relations existing between these projections and the angles =0] and since it must be satisfied by the co-ordinates x\ y\ z', we shall have Ax'+By'-^Cz'+B^O. Eliminating D between these two equations, the equation of the tangent plane to the surface becomes A{x-x')+B(y~y')-{-C{z-z)=0 ; and dividing by C, it may be put under the form ^(x~x'n^{y-y')-{-{z-z')=0 (25). But if the plane be tangent to the surface whose equation is dz' dz' L=0, the values of -j^, and j^ deduced from that equation, will be expressed as follows : dz' A dz' B rf?=~"C' dy'^~C ^^^^• And from the known principles of analytical geometry, when a plane whose equation is Ax-\-By-\-Cz+D=0 is pcrpen- C 34 STATICS. dicular to a right line represented by the equations x=az-\-», y=hz-\-^, the following relations between the constants exist : A B ^ the equations (26) will therefore reduce to £:=_.|;=-......(2T). 62. The values of these coefficients must now be deter- mined from the equation of the surface. We obtain by dif- ferentiating, d\j , , dh , , dlj J ,, dx dy dz whence we infer that dL^ dL dz=z — '^dx — ^dy ; . rfL dL ^ dz dz and by applying this equation to the point of tangency, for which the co-ordinates are x\ y\ z\ we find dL_ rfL_ dz'_ _dx' dz _ dy' dx' ^' ~d^'~~~dO dz' d^ substituting these values in the equations (27), they become dL dL dL ' dL dz dz Replacing a and b in equations (24), by their values found above, we obtain, after reduction, dL cos êz= ± dx' ^/]&-m'^m cos 6'= ± dy' ^liÈr-ar^m'ï EQUILIBRIUM OF A POINT UPON A CURVED SURFACE. 35 dh COS ^ = ± The double sign is here prefixed to the values of cos ^, cos ô', COS 0", for the purpose of indicating that the resistance op- posed by the surface may be exerted either in the direction of the normal or along its prolongation, according as the body is placed on the concave or convex side of the surface. The form of these equations being inconvenient for the purposes of calculation, they may be simplified by making ^ =V (28); s/\m^m-{W\ which reduces them to cosfl=V— ^, costf=V^— , cosfl =¥-—,; ax ay' dz substituting these values of the cosines in equations (22), we obtain NV^+X=0, NV'^.+Y^O, NV^4.Z=0 (29). dx dy dz 63. The value of IN remains to be determined. If we transpose X, Y, and Z in the equations (29), and take the sum of the squares of the three equations, we shall obtain and reducing by means of equation (28) there results N2=X2+Y2+Z2, whence, N=^(X='+Y-+Z=') (30). This value of N is precisely the same as that of the resultant of the entire system ; but its components should be aftected with signs contrary to those of the components of the result- ant, since its action is exerted in an opposite direction. Thus, having determined the resultant of all the forces P, P', P", &c., the reaction of the surface will be equal to this re- sultant, but will be exerted in an opposite direction. C2 36 STATICS. 64. If the direction of the normal force be parallel to the axis of 2r, we shall have N cos o'+M cos V'+Z=0 ) The equations of the surfaces L=0 and K=0 being differ- entiated, make known, as in Art. 62, the values of the quan- tities cos ê, cos s', cos 6", cos a, cos «', cos >)", and by adopting abbreviations similar to those of Art. 62, making ± I _=u, we shall find „c?L ^-,dK cos 6=\-T—, , COS t}=\J--—, , ax ax ,, T7-c?L , -r^dK. cos^=V— ^. cos;î=IJ--;, di/ dy ^^dL „ T,dK cos 6 = V-— , , COS !,'=U— - : dz dz which values, being substituted in the equations(31), give dx dx NV^^+MU^-fY=0 y (32). dy dy' NV^-fMU^+Z=0 dz dz From these three equations the unknown quantities M and N may be readily eliminated ; and since U and Venter into .them in the same manner as M and N, ihey will also disap- 38 STATICS. pear in the elimination : or, to simplify the case, we may regard MU and NV as the unknown quantities, which, being eliminated between the three preceding equations, will give an equation of condition including one or more of the three variables. This resulting equation being combined with those of the surfaces, viz. L=0, K=0, will determine the co- ordinates .X-', y', z\ of the point sought. It may be proper to remark that the radicals, which would serve to complicate the expressions, disappear at the same time as the quantities U and V. 67. When the point is subjected to the condition of re- maining on a curve of double curvature, such curve may be regarded as being formed by the intersection of two curved surfaces. The equations of these surfaces being represented as above by L=0 and K=0, the co-ordinates of the points in which they intersect will necessarily appertain to both sur- faces, and the quantities x\ y\ and z' may therefore be re- garded as having the same values in each of these equations ; but we have also the equation of condition referred to in Art. 66 ; thus by eliminating the values of two of the co- ordinates, the third will be expressed in functions of known quantities : denoting by A, B, and C the values of the func- tions corresponding to each of the co-ordinates x\ y\ and z\ we shall have x'—K, y'=B, z—G. 68. It may occur that the equation resulting from the elim- ination of M and N will not contain either of the variables. This case presents itself when the surfaces become planes ; their equations L=0 and K=0 may then be put under the form Aa:+By+C;::+D=0, and the differential coefficients are then constant. Under such circumstances the values of the intensities M and N determined by the equations (32) become independent of the co-ordinates x', y\ and z ; and since these co-ordinates still apply to any points common to the two planes, it follows that the conditions of equilibrium will be fulfilled, if the forces be applied to any point whatever in the common intersection of the two planes. A similar remark is appli- cable to Art. 65. PARALLEL FORCES. 39 Of Parallel Forces. 69. The forces which have been considered in the pre- ceding paragraphs were supposed to have a common point of apphcation ; but if they were appHed to different points of a body or system of bodies, the points being retained at fixed distances by means of their connexion with the inter- mediate points, we might regard the forces as having their points of apphcation united by means of inflexible right hnes. 70. Let there be two parallel forces P and Q, applied to the extremities of a right line AB {Fig- 34), which intersects at right angles their common direction. It has been proved (Art. 22) that the intensity of the resultant of these forces will be equal to the sum of the intensities of the two components, and that its point of application O will divide the line AB in the inverse ratio of the two forces. This proposition may be demonstrated in another manner, provided we admit that of the parallelogram of forces, which is susceptible of direct proof Let the two parallel forces be represented by the right lines AP and BQ, proportional to their intensities {Fig. 33) ; we can add to the system, without changing the value of the resultant, the two equal and opposite forces AM and BN, and the four forces AP, AM, BQ., and BN may then he re- placed by the two AD and BI, the diagonals of the rectangles MP and NO.. But since these diagonals intersect at the point C, the forces AD and BI may be conceived to be applied at that point, and will be represented by CE=AD and CF=BI. If the forces CE and CF be then decomposed into rectangular components, by constructing the rectangles GL and HK, having their sides respectively equal and parallel to those of the rectangles MP and NQ,, we shall replace CE and CF by the four forces CL, CK, CG, and CH. But the last two are equal, being equivalent to the forces AM and BN; which by hypothesis are equal, and being directly op- posed, they must mutually destroy each other ; there will tlxerefore remain at the point C, the two forces CL and CK 40 STATICS. equal respectively to P and Q,, and having the common direc- tion of the hne CO. The resultant of these two forces must evidently be equal to their sum ; and if ii be denoted by R, we shall have the relation R=P4.Gl: but since the resultant may be applied at any point in its line of direction, we will consider it as acting at 0, the point in which it intersects the line AB ; the position of this point may be determined thus : the two similar triangles CAO, CEL give the proportion CO : AO : : CL : EL, and the triangles COB, CKF give BO : CO : : KF : CK ; whence, by multiplication, suppressing the common factor CO, we have BO: AO:: CLxKF:ELxCK. But KF and EL, being equal to BN and AM, which by hy- pothesiG are equal to each other, the proportion reduces to BO : AO : : CL : CK : and since CL and CK are equivalent to the lines AP and BQ, which represent the intensities of the given forces, the pro- portion may be written BO : AO : : P : a (33). Hence we conclude that the point of application O of the two parallel forces P and Q. divides the line AB into two parts, reciprocally proportional to the intensities of those forces. 71. From the above proportion we obtain {Pig. 34) BO+AO : AO :: P + Q : Q, or, AB : AO : : R : a (34). And from the equations (33) and (34) combined, we find P : a : R : : BO : AO : AB ; from which we derive the following rule : T/ie jyarts AO, BO, avd AB compj'ised between any two of the forces P, Q, and R, will he co'nstanily 'proportional to the tJii)'d force. The term R, for example, in the above proportion; corresponds: to PARALLEL FORCES. 41 the portion AB, which is included between the forces P and Q,. 72. If from the known values of P, Q, and AO, it were required to determine that of BO, the proportion would give a : P : : AO : BO ; whence, BO=P^_^. a 73. Reciprocally, if there were given the force R applied at O, and we wished to resolve it into two parallel components whose points of application should be A and B ; by denoting the unknown components by P and Q, the value of the first would result from the proportion AB : BO : : R : P ; and that of the second would in like manner be obtained by means of the proportion AB : AO : : R : CI. From these two proportions we deduce p_RxBO RxAO ^~~AB~' ^ AB~' In the preceding demonstration, the forces P and Q. have been supposed perpendicular to the line AB ; but if they were oblique to the direction of this line, we might draw through O, the point of application of the resultant {Fig. 35), the right line CD, perpendicular to the direction of the given forces, and the force P applied at A would have the same effect as though it were applied at the point C. In like manner, the point of application of the force Q, may be transferred from B to D : and since we have the proportion P : a : : OD : OC, we shall likewise obtain from the similarity of the triangles OBD, AOC, P : a : : BO : AO. 74. When the forces P and Q. act in opposite directions, the resultant is equal to the difference of these forces. For, let S {Pig. 36) be the resultant of the forces P and R, which are supposed to act in the same direction, we shall then have 42 STATICS. S=P+R (35); and if we replace S by a force Q, equal in intensity, and directly opposed to it, an equilibrium will subsist between the three forces P, R, and Q, : we may therefore regard R as beirig- equal and directly opposite to the resultant of the forces P and Q, and the equation (35) will give for the intensity of this resultant R=S-P; but S and Q, being equal in intensity, we have, by substi- tuting the value of S, R=Q— P. The point O at which the resultant is applied, may be found by the proportion AB : BO : : R : Q, whence we obtain TJ/-V AL XQ, R~' or, replacing R by its equal Q— P, we have QXAB From this value of the distance BO, we infer that the point O will be farther removed from B in proportion to the dimi- nution of the quantity Q,— P ; if therefore Q and P become equal, BO becomes infinite, and R becomes equal to zero : hence, if two parallel and equal forces act in contrary direc- tions, but are not directly opposed, the equilibrium cannot be established except by the application of an infinitely small force at a point whose distance is infinite ; it is therefore im- possible in such cases to find a single finite force which shall sustain in equilibrio the two forces P and Q, ; or, in other words, the two forces P and Q, cannot be replaced by a single resultant. The efifect of these forces will be simply to turn the line AB about its middle point C. 75. These pairs of parallel and equal forces, acting in con- trary directions, but not directly opposed, are called couples. 76. The results obtained in the preceding articles may be applied to any number of forces. Thus, let P, P', P", F", P"', {Fiff. 37) represent parallel forces applied to the points A, B, PARALLEL FORCES. 4S C, D, E, which are connected together by inflexible right lines ; the point of appHcation and the intensity of the result- ant may be readily found. For, the forces P and P' being compounded, their resultant will be applied at a point M, whose position may be determined by the following proportion, AB: AM::P4-P':P'5 whence, J^_ABXF the line MC being then drawn, we can determine the point of application N of the resultant of the forces P-fP' applied at M, and of the force P" applied at C ; for we have MC:MN::P+P'+P":P"; from which the value of MN results, mcxp:^ P+P'+P" By connecting the points N and D, the point of application O, of the four forces P, P', P", P ", may be found in a manner precisely similar, and lastly, by joining the points O and E, we shall determine the point K at which the resultant of the entire system must be applied. 77. When some of the forces of which the system is com- posed exert their efforts in a contrary direction, we reduce the components P, P', P", «fee, which are supposed to act in the same direction, to a single resultant equal to their sum, and likewise the components Q, Q.', Q,", (fee, which are supposed to act in a contrary direction, to a second resultant equivalent to their sum ; then, having determined the points of applica- tion K and L {Fig. 38) of these two resultants, the system will be reduced to two parallel forces, the one applied at K, and equal to P+P'+P" (fee, the other at L, and equal Q,+Q,+(^" «fee. : the resultant of these two forces may then be determined by the method explained in Art. 74. 78. If the forces P, F, P", P'", (fee. {Fig. 39), retaining their points of application, and continuing parallel, assume the positions AQ, BQ,', CQ", DQ,",' «fee, the resultant will be parallel to the new directions of the forces, but its intensity and point of application will remain unchanged ; for, the 44 STATICS. construction employed to determine this resultant, being- dependent only on the intensities of the forces and their points of application, the data of the problem will remain the same. 79. If, for example, the forces P and P' should assume the positions represented by the parallels Ad and BCi,' ; there would be given P, P', and the line AB, to determine the posi- tion of the point M ; and this would be determined from the same proportion as when the forces were directed along the lines AP and BP'. The point through which the resultant of a system of par- allel forces constantly passes, whatever may be the direction of those forces, is called the centre of parallel forces. 80. To determine the co-ordinates of the centre of parallel forces, let P, P', P", &c. represent the intensities of the several forces, and denote by " a;, y, z^ the co-ordinates of the point of application M of the force P, x\ y\ z' those of M', x",y",z" those of M", a:,, yi, Zi, those of the centre of parallel forces. If we represent by N {Fig. 40) the point of application of the resultant of the two forces P and P', we shall have MM' : M'N : : P-f P' : P ; and by drawing the line ML' parallel to HH', the projection of MM' on the plane of x, y, the similar triangles ML'M', NLM' will give MM' : M'N : : ML' : NL ; whence, by combining the two proportions, ML' : NL : : P+F : P ; from which results the equation (P+P')NL=PxML': adding to each member the product (P+P')LK, we have (P+P )(NL+LK^ =P(ML'+LK)+P' X LK ; and since NL-{-LK=NK, PARALLEL FORCES. 45 ML'+LK=MH, LK=M'H', the preceding equation may be reduced to (P+F)NK=PxMH4-FxM'H'. If we denote by Q, the resultant of the two forces P and P', and by Z the co-ordinate of its point of application, this equation may be written under the form QZ=P;r+PV- in Uke manner, representing by Q,' the resultant of the paral- lel forces Q. and P", and by Z' the co-ordinate of the point at which it is applied, we obtain a'Z'=az-hPV; and thence, by substitution, Gi'Z':=Fz+l?'z'+V"z". If the resultant of the entire system be represented by R^ and the co-ordinate of its point of application, parallel to the axis of ;r, by z^, we shall obtain, by continuing the same pro- cess, the general relation Rz,=Pz-fPV+P"z"+&c (36). 81. The 7nom,ent of a force with reference to a 'plane is the jiroduct of the intensity of this force hy the distance of its point of application from the plane. The preceding equation therefore expresses that the moment of the residtant of the parallel forces P, P', P", ^«c, taken with reference to the plane of x, y, is equal to the sum of the moments of the several forces taken with reference to the same plane. The moments being taken with reference to the other two co-ordinate planes, we have Ry.=Py+Py+PV'+(fec (37). R2:,=Px-{-PV-f PV'-|-&c (38). 82. When the co-ordinates x, y, z, x, y\ z\ &c. of the points of application, and the intensities P, P', P', &c. of the forces, are given, the intensity of the resultant will become known, being equal to the algebraic sum of the several intensities ; and the values of the co-ordinates .t,, y,, and sr,, of the centre of parallel forces, will be found from the equations (36), (37)i and (38). 46 STATICS. 83. The forces are affected with the positive or negative sign according to the directions in which they act ; and since the signs of the co-ordinates are Ukewise determined by their positions with respect to the origin of co-ordinates, the mo- ments of the forces must be regarded as positive, when the forces and co-ordinates have the same sign, but negative when the two have contrary signs. 84. If the several points of application M, M', M", &c. were situated in the same plane MM" {Fig- 41), the plane of x, y might then be assumed parallel to that in which the forces are applied, and the co-ordinates z, z', z", &c., being com- prised between two parallel planes, we should have z=z'=z"=" +py +&c.=:0 (45). 48 STATICS. And finally, the values of R, and R„ being substituted in equation (41), give P + F + P" + F" + P"+P'' + + Py; and replacing R by — P", since the forces are equal, and act in contrary directions, the equation becomes Pp+py+py=o. Thus the conditions of equilibrium of three forces situated in the same plane, and applied to different points, will be ex- pressed by the three following equations : — P cos ^+F cos«' + P" cos cc"=Q (49), P cos/3 + Fcos0'+P" cos/3"=0 (50), P/> + Py + P>"=0 (51). 100. If the number of forces be greater than three, we may regard P as being the resultant of the two forces P"' and P"' : we shall then have P cos «=P"' cos *"'+P'^ cos «'", P cos /3 = P"' cos /3"' + P^ cos /3", Pp=P"y"+P'y'; and by substituting these values in equations (49), (50), (51), they become P' cos «' + P" cos «" + P"' cos «'" + P"' cos «"=0, F cos (3'+P" cos ^" + F" cos |3"'+P" cos ^"=0. V'p' + p>" + vy + Yy =0. 101. The same principle may be extended to any number of forces, and we shall therefore obtain for the general equa- tions of equilibrium of forces acting in the same plane, and applied to different points, P cos « + F cos a' + P" cos«" + &c.=9 (52), FORCES APPLIED TO DIFFERENT POINTS. 53 P C0S/3 + F C0S/3'+P" COS /3" + "; <^c., these forces will all tend to turn the perpendiculars in the same direction about the point C ; but if, on the contrary, the centre C be situated within the angle formed by the directions of the extreme forces {Pig. 52), or within the opposite angle, the forces P, P', P", &c., situated on the same side of the point C, will tend to turn the perpen- diculars in one direction, while the forces P'", P'', &c., on the opposite side, will tend to turn the perpendiculars in a con- trary direction. But the expressions —^, — i-, — ^, &c., rep- c c c resented by the lines AD, AD', AD", )=0, expresses the condition that the sums of the moments of the forces wliich tend to produce rotation in the two directions are equal to each other. 106. If, in the system supposed in equilibrio, we suppress one of the components, P for example, the remaining forces will have a resultant R ; and since this resultant should be equal in intensity, but directly opposed to the force P, the equations (52), (53), and (54) will be replaced by the following: R cos a=F' cos a' + P" cos a" + F" cos «"'4-&c., R cos b^V cos /3' + P" cos /3" + P'" cos /3"' + (fcc, Rr = Vy + V"p" + P'>'" + &c. ; or, R cos « = 2(P cos et) =X, R cos b =!:.(? cos /3)=Y, Rr=s(P/>). The double sign is not prefixed to the moment P;?, since we are at liberty to assume arbitrarily the sign of one of the moments. The moment Rr, deduced from this equation, may have either a positive or negative value ; if positive, R and P will tend to turn the system in the same direction ; if negative, in con- trary directions. The forces P and P', being then replaced by their resultant R, this resultant can be combined with a third force P", and we shall obtain, in a similar maruier, R'r'=Rr±PY'; in which equation Rr, whatever may be its essential sign, may be replaced by P^^Py. The sign of the moment P"p" will be similar to that of Rr, if P" and R tend to produce rotation in the same direction, and dissimilar in the con- trary case. But the moments Pj) and Rr will have like or unlike signs, according as the forces P and R tend to turn the system in the same or in contrary direc- tions. Hence the signs of the moments Vp and P"p" in the equation R'r'= T'pizP'p':izP"p'\ will be like or unlike according to the directions in which the forces P and P" tend to produce rotation. The same reasoning may be extended to a greater number efforces. FORCES APPLIED TO DIFFERENT POINTS. 55 107. By means of these equations, the position and mag- nitude of the resultant may be determined. For, the two first equations give R2(cos2a + cos26)=X2 +Y2 ; and since the sum of the squares of the two cosines is equal to unity, we have R2=X2+Y2. The inchnations of the resultant to the co-ordinate axes may also be determined from the same equations ; for we have X Y cos a=— , cos 6 =-5-. K Jti 108. To establish its position in the system, we first deter- mine the position of a right line AB, passing through the origin, and parallel to the resultant. If cos b be affected with the positive sign, the line AB must form with the axis of y an angle less than 90° : it will therefore assume one of the positions indicated in {Fig. 53). But if, on the contrary, this quantity should have the negative sign, the right line AB would then be situated in one of the positions represented by {Pig. 54). Thus, whatever be the sign of cos 6, the line AB may assume two positions, one in which the angle formed with Kx will be obtuse, and another in which this angle will be acute. The sign of the cos a will determine which of these positions the line AB must assume. Having thus established the position of the right line AB, let a perpendicular /■ be drawn to it through the origin A, equal to s(P«) ji • This perpendicular will be represented {Pig. 55) by AO or by AO', according to the sign of the quantity r ; and the line OR or O'R', parallel to AB, will represent the true position of the resultant. 109. To. obtain the equation of this resultant, it may be observed that its line of direction will, in general, intersect the axis of y at a certain point B {Pig. 56), and that the form of its equation will therefore be y=x tang D-f AB (55) ; 56 STATICS. and since the angle which the resultant makes with the axis of a; is denoted by a, we have T>=a, and consequently -r, sin a cos b R cos b Y tanff D= = =c; =-— . cos a cos a K cos a X The value of AB may be obtained from the equation OA=ABxcosOAB. But the angle OAB is equal to the angle D, since they are both complements of OAD. The angle OAB can therefore be replaced in the preceding equation by D or a ; and since the line AO is the perpendicular from the origin on the direc- tion of the resultant, it will represent the quantity denoted by r ; we shall thus obtain ?'=AB cos a ; and consequently, AB=-!1-. cos a Substituting the values of AB and tang D in the general equation (55), it becomes Y r _Y Rr__Y Rr. '^"X'^'^co^ ~ X'^'^Rc^i^^X''^'^ X ' whence, by transposition and reduction, we find yX-a:Y=Rr] or, replacing Rr by its equal ^(Pj)), the equation of the resultant finally becomes yX-xY=i:(Pp). 110. When an equilibrium subsists, X and Y are equal to zero, and the equation reduces to s(P/?)=0, corresponding with the result previously obtained. 111. The data requisite for the determination of the resultant being, 1°. The intensities of the several forces; 2°. Tile angles on which their dii'eciions depend ; and 3". The co-ordinates of their points of application, it will prove con- venient to transform the equation (54) into another, in which the quantities />, p', p", ôcc. shall be replaced by the co-ordi- nates of the points of application. To effect this transforma- tion, let the origin of co-ordinates be assumed at A {Fig. 57), FORCES APPLIED TO DIFFERENT POINTS. 57 and let x and y denote the co-ordinates of the point M to which a force P is applied : the intensity of this force being represented by MP, its components parallel to the axes of x and y will be respectively MN=P cos «, MQ,=P cos /3. From the point A demit the perpendiculars AO, AF, and AE on the prolongations of the force MP and its two components j we shall then have OAxMP=the moment of the force P, AF X MN=the moment of the component P cos «, AE X MQ,=the moment of the component P cos j3. But if we regard the forces as pushing the point M, the resultant MP and the component P cos « will tend to produce rotation in the same direction about the point A. Their moments may therefore be alfected with the positive sign ; while the component P cos /3, tending to turn the system in a contrary direction, must be affected with the negative sign. We shall thus obtain the equation Pp=yP cos » — xY cos /3. For a similar reason, P'jo'=3/'F cos «'-.r'F cos /3', P"p"=y"P" cos a:'—x"Y" COS /3"j )=0 were alone satisfied, an equilibrium could not subsist ; for the quantities X and Y having certain values, a resultant might be found whose intensity would be determined by means of the equation In this case, the equation s(P/»)=0, or its equivalent Rr=0, can only be satisfied by making the factor r equal to zero ; hence, the centre of moments must necessarily be found on the line of direction of the resultant R. 116. If there be a fixed point on the line of direction of the resultant, the equilibrium will be still maintained, and the centre of moments being placed at this point, the condition 2(P/)) =0 will be satisfied ; if, for example, the forces P, P', P'^^ &,c. be supposed applied to the different points of a solid body, and if the point C through which the resultant passes be im- moveable, the effect of this resultant will be entirely destroyed 60 STATICS. by the reaction of the fixed point, and the condition 2(Py>) =0 will be alone sufficient to ensure the equilibrium. It will appear hereafter that the intensity of this resultant is a measure of the pressure sustained by the fixed point. 117. If the system can be reduced to two parallel forces, equal in intensity, but not directly opposed, the addition of an arbitrary force S will render it susceptible of a single result- ant. For the new force S must necessarily be either par- allel or inclined to the direction of the forces ; in the first case {Fig. 60), it may be decomposed into two parallel components P' and Q.' applied at the points A and B (Art. 73), and the sys- tem of three forces P, Q,, and S will be replaced by the two unequal forces P + P' applied at A, and Q, — Q.' applied at B ; these two forces will obviously have a single resultant. If the new force S is not parallel to the other two, its direction may be prolonged {Fig. 61) until it intersects the direction of one of them at A'. This point being then taken as the point of application of the forces P and S, they may be compounded by constructing a parallelogram on their lines of direction, and the direction of their resultant will intersect that of the force Q, with which force this resultant may be combined. Of Forces acting in any tnanner in Space. 118. Let P', P", P'", (fcc. represent different forces situated in space ; x', 2/, z', the co-ordinates of the point of application of P', x'^, y'\ z", those of P", x"', y'", z"\ those of P'", &c. &.C. &c. ; ce'j /3', y', the angles formed by P' with the axes of co-ordinates, tt\ /3", y'', those formed by P" with the axes, «'", 0'", y'", those formed by P'" with the axes, &c. ■ (fcc. (fee. Let us investigate the conditions of equilibrium iw this sj'-s- tem, and endeavour to discover if these conditions cannot be Ïi-ORCES ACTING IN STfACE. 61 rendered dependent on those which have been obtained in the preceding cases. We first attempt to decompose all the forces of the system into two groups, one of which shall con- sist of parallel components, and the second of forces situated in the same plane. Since the axes of co-ordinates may be as- sumed arbitrarily, we will endeavour to decompose the forces in such manner that a certain number of them may be in the plane of a;, y, and the remainder be parallel to the axis of z. 119. If in the given system there be no force parallel to the plane of a;, y, the proposed decomposition may be readily effected ; for, let one of the forces be represented by P', its point of application being at W {Fig. 62); prolong the hne of direction of this force until it intersects at C the plane of .r, y, and transferring the point of application to C, decom- pose the force P' into two others, one C'L parallel to the axis of 2:, the other C'N in the plane of .r, y. 120. But if the force P' is parallel to the plane of a:, y, a similar decomposition cannot be effected, and some other mode of decomposing the forces must therefore be adopted. For this purpose, let there be drawn through the point M' {Fig. 63) a line parallel to the axis of 2;, and to the point M' let there be applied along this line, and in contrary directions, the two forces M'O and M'O', having intensities equal to g' and —g' respectively. The introduction of these forces cannot disturb the condition of the system, since the two mutually destroy each other ; and we shall then have applied at the point M' the three forces P', g\ and — g'. The force P' may then be compounded with — g\ and by calling their resultant R', we can replace in the system the force P', by the two forces R' and g\ each of which must ob- viously intersect the plane of a:, y. 121. Let the force R' be now applied at C, the point in which its line of direction intersects the plane of a:, y, and let it be decomposed into two components, one situated in the plane of .r, y, and the other parallel to the axis of z. The force P' will thus be replaced by a force applied at C, and lying in the plane of a:, y, and by two others parallel to the axis of 2;, one applied at C, and the other at M'. 122. The co-ordinates of the points of application being 6 62 Î3TATICS. necessary to express the conditions of equilibrium, those of the point C must be determined. The equations of the resultant R' which passes through the point .t', y', 2;', have been found (Art. 57) to be of the form Z z — z'=^{x — x') (57); in which X, Y, and Z represent the projections of R' on the co-ordinate axes. These projections being equal to the com- ponents of R' parallel to the axes, the quantities X, Y, and Z may be replaced by the values of the three components. But R' being the resultant of P' and— ^', we may substitute for F its three components F cos «', P' cos /S', P' cos y' ; and R'^ will then be the resultant of the four forces P' cos a, P' cos /3', F cos y', —g^. These forces acting parallel to the axes of co-ordinates, we shall have X=F cos «', Y=F cos /3', Z=F cos y'-g'; and by substituting these values in equations (57), we obtain for the equations of the resultant R', , P' cos y' —ff, ,v 1 F cos «' ^ I Fcosy'-^, ,- ^ (58). F cos /3' 123. To obtain the co-ordinates of the point C (Pig. 63), at which the right line R' intersects the plane of x, y, we make z=0 in the equations (58) ; and denoting by a, and 6, the other two co-ordinates of the point C, we shall have ^,_Fcosy'— ^' —z' Z=:- P' cos «' P'cos y' — £r' F cos /3' (a—x'), (i-yO; from which we deduce a =x h,=t/' Z'V cos a' P' cos y' — g' z'V cos iS' P' cos y'—g" (59): FORCES ACTING IN SPACE. 63 these are the values of the co-ordinates of the point C, at which the resultant R' intersects the plane of x, y. 124. The force R', being represented in intensity by the line M'R' {Fig. 64), may be supposed applied at C, in its line of direction. Then making CD' = M'R', and decomposing CD' into three rectangular forces, applied at C and parallel to the co-ordinate axes, these components will be equal to those of the force M'R' ; and the point C may therefore be considered as solicited by the three forces F cos «', P' cos ^\ and P' cos y'—g\ the two former being situated in the plane of a-, y, and the latter parallel to the axis of z. Thus, instead of the force P' applied at M', we shall have the force g applied at M', parallel to the axis of z, the force P' cos y'—g applied at C, parallel to the axis of Zy the force P' cos » applied at C, and acting in the plane of ar, y, the force P' cos ^ applied at C, and acting in the plane of x, y. 125. By adopting a similar method of decomposition for the forces P", P ", &c., employing the auxiliary forces g'\ g"\ &c., applied at the points M", M", (fee, the system will be reduced to two groups of forces, of which one will have its components parallel to the axis of z, and the other will be situated in the plane of x, y. The forces parallel to the axis of z will be g\ g'\ g"\ &c., applied at the points M', M", M'", «fcc. ; and P' cos y'—g, P" cos y"—g'\ F" cos y"'—g"', &c., applied at the points C, C", C", &c. And the forces lying in the plane of x, y, will be P' cos «', F' cos «", F" cos «'", (fee, applied at the points C, C ", C ", &c. ^ and P' cos (3', P" COS fl'', P'" COS /3"', &C. applied at the same points C, C", C ", (fcc. 126. It will now be demonstrated that when an equilibrium subsists in the system, it will be necessary, 1°. that the forces parallel to the axis of z should be in equilibrio ; 2". that the forces acting in the plane of a:, y, should also destroy each other. 64 STATICS. For since the equilibrium is supposed to subsist, the state of the system will not be changed by supposing a line C'C" assumed arbitrarily in the plane of x, y {Fig: 65) to become immoveable. The forces situated in this plane will then be destroyed by the resistance of the fixed line. For, every force in the plane of .r, y must intersect the fixed line, or be parallel to it. In the first case, let the force be represented by AB, and prolong its line of direction until it intersects the fixed line at a point O : this point being supposed immove- able, the effect of the force AB, which is transmitted to the point, must be destroyed. Again, if the force be parallel to the line C'C ", its point of application E cannot be moved without communicating a motion to the line C'C" which by hypothesis is immoveable. The effect of this force must therefore be destroyed by the fixed line. Thus, the forces lying in the plane of x, y being destroyed, the system will be reduced to the group parallel to the axis of z. These latter forces would obviously tend to turn the system about the fixed line C'C, unless the forces should be in equilibrio, or their resultant should pass through the fixed line. But the position of this line having been assumed arbitrarily, it cannot happen that the resultant of the forces parallel to the axis of z will always pass through this line. These parallel forces must therefore be in equilibrio. The group parallel to the axis of z being in equilibrio, the forces lying in the plane of x, y must mutually destroy each other, since the equilibrium of the entire system could not oth-erwise be preserved. 127. The problem is thus reduced to finding the conditions of equilibrium, 1°. of a system of fr .-_es parallel to the axis of z\ 2°. of the forces acting in the plane of x, y. Conditions of Equilibrmm of the Forces parallel to the Axis of z. 128. These conditions being the same as those enunciated in Art. 87, the following quantities must be equal to zero, — FORCES ACTING IN SPACE. 65 1°. The sum of the forces parallel to the axis of z ; 2°. The sum of the moments taken with reference to the plane of y, ^ ; 3°. The sum of the moments taken with reference to the plane oi x\ z. The first of these conditions gives F cos y—g'+g'J^Y' cos y"—g"+g" + P"' cos y" —g" -]rg"' + «Sec. = ; or, by reduction, F cos y' + P" cos y" + F" COS y"' + &C.=0 (60). The second condition requires the consideration of two dif- ferent sets of moments. 1°. Those of the forces g\ g'\ g"\ &c., applied at the points M', M", M'", &c. 2°. Those of the forces P' cos y —g'^ P" cos y'—g'\ P"' cosy'"— ^"', (fcc, applied at the points C, C", C", (fee. The moment of the force g' applied at M' {Pig. 66), taken with reference to the plane of y, z^ is g X M'N' : but M'N'=B'D'=.t'' ; the moment therefore becomes g'x. The moment of the force F cos y — g applied at C, taken with reference to the same plane, is evidently (F cos y — g'") XE'C, or (P' cosy— ^')a, ; and the sum of the moments of the two forces will therefore be represented by g-'x' + (P' cos y'—g')a,. Substituting in this expression the value of a, (59) deter- mined in Art. 123, we obtain gx^(^ cosy— ^) I X'— ,5; — -1; \ P COSy— ^/ performing the multiplications indicated, and reducing, we get .-pT'cosy'— «T'cos»'. By a similar process, the moments of the parallel forces applied at M", M'", C", C", (fee. may be obtained, and being collected into one sum, the equation expressing the second condition of equilibrium becomes Y{x cos y — 2;' cos «') •\-Y\x" cos y — 2;" cos «") +P'"(a:"' cos y"'—z" cos «"') +(kc.=0 (61). To obtain the third condition of equilibrium of parallel! E 66 STATICS. forces, we find tlie moment of the force g' applied at M', talccn with reference to the plane of a;, z, and that of the force P' cos y — g' applied at C, taken with reference to the same plane : the first of these will be equal to ^'xM'L'=^' xB'G' =g y^y ; the second will be (P' cosy' — g')h^ ; and their sum will be expressed by ^y+(Fcosy'-^')5,. Substituting for b, its value (59) found in Art. 123, and re- ducing, we obtain yV cos y' — z'P' cos /3'. And by finding the moments of the other parallel forces, taken with reference to the plane of x, z^ we shall have for the third condition of equilibrium, F(2/' cos y'—Z cos /S) +P "(2/'' cos y"—z" COS |3") +F"(y" cosy'"— ^"' cos/î"')+&c.=0 (62). Conditions of Equilibrium of the Forces situated in the Plane of x, y. 129. These conditions being such as arise when the forces act in the same plane, it is necessary, 1°. That the sum of the components parallel to the axis of X should be equal to zero. 2°. That the sum of the components parallel to the axis of y should be equal to zero. 3°. That the sum of the moments of the forces taken with reference to the origin should be equal to zero. The first two conditions are expressed by the equations, F cosa'+P" cos «" + P'" cos«'"+(fcc.=0 (63), P'cos /3' + P" C0S/3" + F" cos/3"' + &c.=0 (64). "With regard to the third,, it may be observed, that the two forces F cos a! and P' cos /3' are applied at the point C {Fig- 67) ; the moment of the fir«tj being taken with reference to the origin A, will be F cos «' X AE'=F cos «' X C'F'=F cos «' . 6, ; in like manner, the moment of the force P' cos /3', taken with reference to the origin A, will be F cos ^'XAF'=:F cos p'xE'C'^rF cos/3', a,. FORCES- ACTING II» SPACE. 67 These moments should be taken with contrary signs, since the two components P' cos «' and P' cos /s' tend to turn the system in contrary directions about the point A. Thus, by regarding that momeni as positive in which the component P' cos «' enters, the sum of the moments may be written F cos u X b^—V cos /3' X a, ; substituting in this expression the vahies of a, and Z>, (59), we get •n, > t ' ^'P' cos /3' \ ^, ,/ , z'Y cos <*' \ F cos a! ( y — _ —'— ) — P cos /3 ( x'—^, -, ) ; \ PcoSy— o-/ V Pcosy— ^/ and by performing the multiphcations, and reducing, we- obtain y'Y cos x—x'Y cos /3'. The moments of the forces apphed at C", C", &c., being found in a similar manner, the third condition of equilibrium of the forces which lie in the plane of x, y becomes F(y' cos «' — x cos /3') + P"(y" cos x' — x" cos /3") + P"'(2/"' COSa:"—x" COS|3"') + (fec.=0 (65). 130. The six equations of equilibrium (60), (61), (62), (63), (64), (65), may be written under the following form : 2(P cos«)=0 ^ x(P cos /3) =0 V . (66). 2(Pcosy)=0> 5:[F(y cos at, — X cos (S)] =0 ^ s[P(x cosy — z cos«)]=0 > (67). 2[P(y cos y — z cos /3)] =0 3 131. If there be a fixed point in the system, the six equa- tions will not be requisite to express the conditions of equi- librium. For, if the origin be placed at the fixed point, the equilibrium will subsist between the forces acting in the plane of X, y. when the system has no tendency to turn about this point. This condition will be fulfilled when we have 2[P(2/ cos « — X cos /3)J=.0. It remains to discover the conditions of equilibrium of the forces parallel to the axis of z. Let .t„ y,, and be the co- ordinates of the point at which the resultant of the parallel forces intersects the plane of a-, y ; the moment of this result- E2 68 STATICS. ant taken with reference to the planes of x, z, and y, z, will be equal to the sum of the moments of the several forces taken with reference to the same planes ; whence we have 'Rx=-l\P{x cosy— 2; COS a)], Ry^ = 2.[P(y COS y — Z COS /s)]. If an equilibrium subsists between the parallel forces, their resultant must pass through the fixed point, which, by hy- pothesis, coincides with the origin of co-ordinates, and we therefore have :r,=0, y=0. The preceding equations will thus be reduced to Y.\P{X cos y — Z cos «)J=0, 5;[P(y cos y—z cos/3)]=0. We therefore conclude that when the system contains a fixed point, the equilibrium will subsist, if the equations (67) are alone satisfied, the origin being taken at the fixed point. 132. Wlien the system contains two fixed points, one of the co-ordinate axes may be drawn through them ; this axis will thus become fixed, and the system can only be subject to a motion around it. A similar case will be examined in the succeeding paragraph. 133. When there exists a fixed axis about which the system may turn, this axis may be assumed as the axis of z, and the forces parallel to it will produce no effect. The remaining forces are situated in the plane of x, y. But the condition of equilibrium of these forces requires that their resultant should pass through the point A {Fig. 67), which point is immove- able, being on the axis of z ; and the condition of the result- ant'spassing through A is expressed, as above, by the equation 2:[P(y cos a—x cos /3)]=0. This equation expresses that the system is in equilibrio, when the axis of z is supposed fixed. 134. If we suppose, successively, the axes of y and x to become fixed, it may in like manner be demonstrated that the system will be in equilibrio, in the first case, when 5;[P(.'r cos y—z cos «)]=0, and in the second, when 5;[P(y cos y—z cos /3)]=:0. 135. When the body is capable of shding along the fixed FORCES ACTING IN SP^ACE. 69 axis, supposed to be that of z, an additional condition of equilibrium becomes necessary ; this condition is expressed by the equation 2:(Pcos y)=0. 136. By comparing the conditions of equilibrium of a sys- tem moveable about a fixed axis, with those which obtain when the system turns about a fixed point, we iiifer. That an eqniUhriiirii icill take place about tJie fixed point tvlien, by vegarding the axes passing through this point as fixed in succession, the equilibrium is tnaintained with reference to each of them. 137. If the forces be supposed to act against a fixed plane,, which may be assumed as the plane of x, y, the components perpendicular to it will be destroyed by the reaction of the plane, and the conditions of equilibrium will thus be reduced to those of forces acting in a plane ; we consequently have ^(P cos «)— 0, 5:(P cos iS)=0, 2[P(y cos »—x cos /3)]=0. 138. If a body be supposed placed on a fixed plane, being at the same time liable to be overturned by the action of thei forces exerted upon it, we must add to these three equations the condition, that the résultant of the perpendicular forces shall pass through a point in which the body touches the plane, or that it shall intersect the plane within the polygon formed by connecting the points of contact. 139. The discussion of this subject will be terminated by the solution of the following problem : To find the analyti- cal condition expressive of the existence of a single result^ ant of any number of forces situated in space. The system will admit of a single resultant, when the resultant of the components parallel to the axis of z intersects the plane of X, y, in a point situated on the resultant of the forces lying in that plane. To express this condition, we remark, that in case of an equilibrium, the following relations must subsist between the forces parallel to the axis of z (Art. 128) : P cos y-}-P' cos v'+P" cos y"-]-&:c,=Q. 70 STATICS. P(:r cos y— Z COS «)-f-P'(a;' cos y' — z' COS «') +P"(a;" COS y"-z" COS «")+ may he determined. If we draw through the point C {Pig. 69) the lines CA and CB perpendicular to the given planes, these lines will contain between them the angle ^, and its value will result from the formula cos p=cos£ cos «-f cos t cos /î 4- COS î" cos y (71). 144. When the angle ^ is a right angle, its cosine will be equal to zero, and the equation becomes THEORY OF THE PRINCIPAL PLANE. 73 COS t COS «-fcOS s COS jS-fCOS t" COS y = 0. 145, From the formula (71) we deduce a very remarkable property of projections. For, let there be two planes, the first of which forms with the co-ordinate planes the angles a, 6, and c, and the second the angles «, /3, y ; the angle

is equal (Art. 142) to the projection of the area x on the second plane, and the products x cos a, x cos 6, and X cos c are, in like manner, the projections of the same area on the co-ordinate planes. 146. The equation (72) therefore gives rise to the following theorem : The j)rojection of a plane surface on any plane is equal to the simi of the jiroducts of its projections on each of the co-ordinate planes, nudtiplied respectively by the cosines of the angles o, (S, and y, which measure the inclinations of the plane of projection to the co-ordinate planes. This theorem becomes much more general, if, instead of the area x lying in a single plane, we consider several areas X, a', a", «fee. situated in different planes, and projected on a plane whose inclinations to the co-ordinate planes are de- noted by u, j3, and y : to avoid repetition, let us call the plane of projection x, /3, y, and denote by ç and ) the inclinations of the ( to ttie plane «, /j, y, and a, b, c, ) area a ( to the co-ordinate planes, the inclinations of the \ to the plane «, ,3, y, and area a' ( to the co-ordinate planes. ç" and } the inclinations of the C to the plane u, /s, y, and a", b", c", ) area a" I to the co-ordinate planes, ^' COS a" COS <* + /' cos 6" cos /S + a" cos c" cos y, (fcc. (fcc. &.C. , >,', x", &c. be now projected on two other planes which form with the co-ordinate planes the angles «', /3', y', a.", li", y" ; and denote by P' and P" the sums of the projections of x, x', x", (75). F =A cos a"+B cos 0"-\-C cos y" } 150. If the planes upon which the projections P, P', and P" are made be supposed rectangular, their intersections will be perpendicular to each other, and may therefore be regarded as three rectangular axes, which intersect at a point O ; consequently, by representing these new axes by Ox' Op', and Oz', they will be respectively perpendicular to the new planes of co-ordinates ; but the axes of x, y. and z were likewise perpendicular to the primitive co-ordinate planes ; hence, the angles formed by the primitive axes with the new, will be measured by the inclinations of the primitive co- ordinate planes to the new. These angles of inclination are, by hypothesis *, /?, y ; *', ^\ y ; «", /2", y" 5 and since each of the primitive axes corresponds to the same letter although differently accented, we find that The axis of x forms with the new axes the angles «, «', <»", The axis of y forms with the new axes the angles /;, /3', /3", The axis of z forms with the new axes the angles y, y', y". The following relations will therefore subsist between the cosines of these angles, COS^ « -{-cos'' *'-|-C0S2 a"=l ^ C0S2 /8 -fcos* yfi'-f cos2 ^"=1 > (76). COS^ y-4-COS' y'-fC0S2 y" = l ^ Again, since the angle formed by any two of the primitive axes is a right angle, we shall obtain (Art. 144) cos * cos yg-fCOS *' COS i^'-j-cos a' COS (S"=0 ^ cos tf COS y-l-COS «' COS y'-f COS a' COS y"=0 > ^'l^)' COS ^ COS y-f-COS /j' COS y'-f COS ^" COS y"=0 J 151. If we take the sum of the squares of the equations (75), reducing by means of (76) and i^l)^ we shall obtain the relation P» 4.F3 -j-prrs ^p^i -}-B^ -f C^ (78) ; 76 STATICS. which expresses that the sum of the squares of the projec- tions of the areas a, a', a", -^^^2_|_Bs_j_C3) These angles express the inclinations of the plane of maxi- mum projections^ which is called the principal plane. The determination of this plane being dependent only on the angles «, 0, y, the same property will be enjoyed by every parallel plane. (81). THEORY OP THE PRINCIPAL PLANE. 77 153. It may also be demonstrated that the sum of the pro- jections of the areas a, a', V, (fee, on every plane which is equally inclined to the principal plane, will be equal to a con- stant quantity. For, let Q, be the sum of the projections on any plane whose inclinations to the co-ordinate planes are denoted by a, 6, and c : if we represent, as heretofore, by A, B, C the projections of these areas on the co-ordinate planes, we shall have Cl=A cos a-f B cos b + C cos c ; but if d, /3, y denote the inclinations of the principal plane, the equations (80), which are A=Pcos'', B=Pcos/3, C=Pcosy, will reduce the preceding equation to Q=P(cos a cos *-fcos b cos /3-1-cos c cos y). The quantity within the brackets being equal to the cosine of the angle included between the principal plane «, (3, y and the assumed plane a, b, c, we shall have, by calling this in- clination 6, Q,— P cos 6; and since P represents the sum of the projections on the prin- cipal plane, which, by Art. 152, is equal to .y/iA' -f B" +0^), the substitution of this value gives a=v/(A2-fB='-fC2)Xcos<» (82). But the projections A, B, and C remaining the same, it follows from the equation (82) that the value of Q,, the sum of the projections on any plane, will be constantly the same for all planes having the same inclination to the principal plane. It also appears that this sum will increase or diminish in the same ratio as cos $. 154. Lastly, it may be remarked that the sum of the pro- jections on every plane perpendicular to the principal plane is equal to zero ; for 6=90° gives cos fl=0, and Q,=0. 155. The several theorems relative to projections which have just been demonstrated are likewise applicable to the case of moments. For, let the centre of moments be sup- posed to coincide with the origin of co-ordinates, and con- ceive the plane «, 0, y to pass through the origin : if from the 78 STATICS. points of application of the several forces we take upon their respective hnes of direction, portions wliich shall be propor- tional to the intensities of these forces, these lines may be represented by the letters P, P', P", &.c. The centre of mo- ments may then be regarded as the common vertex of several triangles, of which P, P', P", (fee. represent the bases : the projections of these triangles upon the plane «, /3, y, and on the co-ordinate planes will likewise be triangles, their bases 2^, ])', p", (fee, being the projections of the lines P, P', P", (fee, and their altitudes h, A', h", (fee, being the perpendiculars de- mitted on the lines p, p', p", (fee. from the centre of moments. These values behig substituted in eq^nation (73), which may be written under the following form : 2(the projections on the plane «, /3, y) = { The projections on the co-ordinate planes multipHed ) ( respectively by the cosines of the angles of inclination y convert it into Iph + \p'h' + \p"h!' -1- (fee. = ( The projections on the co-ordinate ^ s ^ planes multiplied respectively by the j> (83).. 1^ cosines of the angles of inclination J The second member of t'his equation will contain similar pro- ducts, and the factor \ will therefore be common to the two members ; this being suppressed, the first member will re- duce to ph-{-p'h'^p"k!'^6i.c. But p, p', p", (fee, being the projections of the right lines P, P', P", (fee, the products j^h, p'h', p"h", tfee will be the mo- ments of the lines />, p', jy", (fee, taken with reference to the origin of co-ordinates. The same remarks being applicable to the second member of equation (83), it follows that the sum of the moments of the projections of the forces on the plane «, /3, y, which passes through the origin of co-ordinates, is equal to the sum of the moments of the projections of the same forces on tlie three co-ordinate planes, multiplied re- spectively by the cosines of the angles of inclination. 156. By making similar substitutions in equations (78), it may likewise be proved that the sum of the squares of the CENTRE OF GRAVITY. 7*^ moments of the different forces, when projected on three rectangular planes, is a constant quantity. The equations (80) make known the position of the plane in which the sum of the moments will be the greatest pos- sible. And the equatian (79) determines the sum of the moments on the principal plane. Centre of Gravity. 157. The particles of matter are constantly subjected to the action of a force which tends to draw them towards the earthy in directions perpendicular to its surface. This force is called the force of gravity. The earth being nearly spherical, the lines of direction in which material points tend to move, will converge towards its centre ; and since the distance of this centre from the sur- face is exceedingly great when compared with the dimen- sions of those objects which we usually consider, the direc^ tions of the forces which act on the different particles of the same body^ may, without sensible error, be regarded as parallel. 158. It is known from observation that, as we recede from the centre of the earth, the intensity of gravity diminishes in the inverse ratio of the square of the distance included be- tween the centre and the place of observation. For example, if a body be placed at a certain distance from the centre of the earth, assumed as unity, and be subsequently transported to distances represented by 2, 3, 4, &c., the intensity of the force of gravity will become _,_,_, &c., or -^ ^, --, &c., of what it was at the distance of unity. 159. The earth being flattened towards the poles, and pro- tuberant at the equator, it follows, that in going from the equator towards the poles, we must necessarily approach the centre of the earth, and the intensity of gravity will therefore increase. It will appear hereafter in discussing the subject of centrifugal forces, that from another cause, the intensity of the force of gravity is greater at the poles than at all other places on the earth's surface. 80 STATICS. 160. The action of gravity being- exerted on all the particles which compose a body, these particles may be regarded as solicited by forces whose directions are parallel ; the resultant of these forces is equal to their sum, and constitutes what is called the weight of a body. Hence, if the bodies considered are homogeneous with each other, their weights will be pro- portional to their volumes. 161. The term density is used to express the gi*eater or less number of particles contained in a body of a given volume, when compared with the number of particles con- tained in some other body assumed as a standard. If we assume as the unit, the quantity of matter contained in a cubic foot of a given substance, distilled water for example, and compare this quantity with that contained in a cubic foot of any other substance, their ratio will express the density of the second substance. Let this ratio be denoted by D. If the second substance considered were gold, by calling D the density of gold, we should have The quantity of matter in ) _ ( D X T/ie quantity of matter a cubic foot of gold ) ( in a cubic foot of water ; whence y^_ (Quantity of 'matter in a cubic foot of gold Quantity of matter in a cubic foot of water 162. In the preceding article we have considered bodies of the same volume ; but if we wish to estimate the quantity of matter contained in a homogeneous body whose volume is V, the quantity D must be taken as many times as there are units of volume in the volume V ; we shall thus have M=DV (84). The quantity M is called the mass, and evidently expresses the relation between the quantity of matter contained in the body, and that contained in the unit of volume of the sub- stance assumed as the standard. 163. If the intensity of gravity were the same at all places, the weight of a body would be proportional to its mass, and might be represented by the same quantity. For, if ^^ denote the effect exerted by gravity on the unit of mass, or the weight of the unit of mass, and W the weight of the body, we CENTRE OP GRAVITY. 81 shall have, from the definition of the weight, W=M^ ; in which expression the quantity g will be constant, and may be assumed as the unit ; we shall thus obtain the relation W=M (85). This equation merely expresses that the number of units of weight is equal to the number of units of mass. But, if by transporting the mass to different distances from the earth's centre, the intensity of gravity be subject to varia- tion, the quantity g will be variable, and the equation ex- pressing the relation between the mass, weight, and intensity of gravity, must then be written under the general form W=M^ (86). 164. From the equations (84) and (86), we deduce which indicates that the weight varies proportiorially to the gravity- g, the volmne V, and the density D. 165. If, for example, two bodies of the same volume be subjected to the action of the same force of gravity, their weights will be in the direct ratio of their densities. The intensity of gravity varying only with change of place, it follows that^ will be constant for all bodies at the same place. 166. If there be any number of points firmly connected together, and solicited by the weights P, P', P", &c., we may regard these weights as parallel forces ; and denoting the co-ordinates of the respective points by x, y, z, a/, y', z\ x",y", z"j 6cc.,we shall obtain, from Art. (80) and (81)," the expressions for the co-ordinates of the centre of parallel forces ; these co-ordinates being represented by X/, yi, Zj, we find 'Px + F'x' + V"x" + ôcc. x,= P-fP' + P"+, ^h, y = ¥^^- 182. To find the centre of gravity of a plane surface,, bounded by the arc of a cvjrve^ and the axis of abscisses. Let Xj and yj be the co-ordinates of the centre of gravity of the entire surface, and let G be the centre of gravity of an element MP' {Pig. 81); the area of this element being equal to ydx, its moment with reference to the axis of x will be GNxy^dx, and that with respect to the axis of y will be ANxydx. But since the element MP' may be regarded as a rectangle whose side PP' is mdefinitely small, we shall have PM AP=AN=a:, and GN=— — =^y : hence the moments with reference to the two axes become \y^dx^ and xydx. If we represent by x the surface DBMP, its area and the co-ordi- CENTRE OP GRAVITY. 91 nates of its centre of gravity will be determined by means of the equations x=fydx, ^ xxt=fxydx, V (95'), 183. To apply these formulas, let it be required to find the centre of gravity of a circular segment CDE {Pig. 82). The origin being assumed at the centre of the circle, and the axis of abscisses AD a line bisecting the arc CE, the centre of gravity of the segment will evidently be situated upon this line ; it will therefore be only necessary to calculate the value of the absciss AG=X;. If g and g' represent the centres of gravity of the semi-segments, they will be found at equal dis- tances from the axis AD, on a line gg' perpendicular to this axis, since the entire segment is divided into two symmetri- cal portions ; the line gg' will therefore intersect the axis of abscisses at a point G, the centre of gravity of the entire segment. The question is thus reduced to determining the absciss of the centre of gravity of the semi-segment CDB, and, its value may be foimd by integrating the equation (95). For the purpose of eliminating one of the variables in this expression, we assume the differential equation of the circle, ydy— — xdx ; from which, by substitution in equation (95), we obtain xx^^f-y^dy (96); and by integrating, and introducing a constant A, we have f-y^dy=-\y'+K (97). To determine the value of this constant,, the integral must be taken from the point C to the point D ; or, if we denote by c the value of the chord CE, the limits of the integral will be 2/=ic and 2/=(X Thus, if we suppose the integral to become zero, when y=\c, the constant A will result from the equation and the equation (97) will therefore become Oj2 STATICS. Putting y=0, to obtain the value of the entire integral from C to D, we have This value substituted in equation (96), gives ""'-24^ = but since x represents in this expression the area CDB, we have A=i area CDEB, whence, -___£!___ • ^'""12 area CDEB' and we therefore conclude, that the distance from the centre of gravity of a circular segment to the centre of the circle is equal to the cube of the chord divided by twelve times the area of the segment. 184. To find the centre of gravity of a circular sector CAE {Fig. 83). The centre of gravity is evidently situated on the radius AB which divides the sector into two equal parts ; it will therefore be only necessary to determine the value of the absciss AG. If we regard the sector CAE as composed of an infinite number of elementary sectors, the centre of gravity of each will be situated at a distance from the point A equal to two-thirds of the radius AC, since these sectors may be considered triangular. Hence, if from the centre A, with a radius equal to two-thirds of AC, we describe the arc HK, the centres of gravity of all the elementary sectors will be distributed uniformly along this arc ; and consequently, the centre of gravity of this arc will coincide with that of, the circular sector. But if X) denote the absciss AG, we have, by Art. 179, _ AH X chord HK, *' arc HK ' and from the similarity of the sectors AHK and ACE, we find CENTRE OP GRAVITY. 93 AH=|AC, chord HK=| chord CE, arcHK=|arc CE; which values substituted in the preceding equation give by reduction, _ |ACx chord CE *' arc CE 185. To find the centre of gravity of an area OBO' {Pig. 84) comprised between tioo branches of a curve. Let y and y' represent the two ordinates PM and PM' cor- responding to the same absciss AP=x : the element MN' of the surface, being the difference of the areas PN and PN', will be expressed by ydx—y^dx—{y — y'^dx ; and if we represent by a a portion of the area included between the chords MM' and 00', we shall have The element MN' being regarded as a rectangle having one of its sides indefinitely small, its centre of gravity will be situated in the middle of the line MM' ; and the ordinate of this point will therefore be PM' + iMM'=y' + i(y-2<')=i(2/+y'); hence, the moment of this element with reference to the axis of X will be \{y^y'){y—y')dx=\{y ^ —y'^)dx ; and the moment with reference to the axis of y will be ^{y—y')d^- Thus, if Xj and y, denote the co-ordinates of the centre of gravity of the entire surface, their values will become known from the equations xx,=fx{y—y')dx, >^yi^My'-y")dx. 186. To find the centre of gravity of a surface of revolution. Let the surface be supposed generated by the revolution of the curve AM {Fig. 85) about the axis of x. The element of the surface, or the zone generated by the elementary arc Mwz, 94 STATICS. will be expressed by 2iryds : hence, by calling x the entire surface, we shall obtain x-=f2Tryds. But since the centre of gravity is evidently situated on the axis of revolution, the co-ordinate x, will be alone necessary. To determine its value, we take the sum of the moments with reference to the plane yz, which sum being equal to the moment of the whole surface supposed concentrated at its centre of gravity, we find xx=fxy,2fryds] whence, f2fryxds x= - ; substituting for x and ds their respective values, and suppress- ing the factor 27f common to both terms of the fraction, we obtain for the absciss of the centre of gravity, fxy^{dx^ +dy^) X.— (98). ~fyV{dx^-^dy^) 187. For the purpose of applying this formula, let it be required to determine the centre of gravity of the surface of a spheric segment. This surface being generated by the revolution of a circular arc BC {Fig. 86) about the axis of a:, we may eliminate one of the variables in the preceding formula by means of the equation of the circle ^ which gives, by differentiation, , , x'^dx'' ^ y^ hence, This value being substituted in the integrals of equation (98), we find fxy^{dx'^ + dy' ) =frxdx — \ rx' + C, fy^{dx'-\-dy^ )=frdx =rx-\-C'. Taking the integrals between the limits a;=AD=a, and x==AB=r, we obtain CENTRE OF GRAVITY. 96 Sy^idx^ -\-dy^)=r{r—a). These values transform the equation (98) into x,^\{r-{-a)=a-{-\{r—a) ; thus, the centre of gravity is situated at the middle of the line DB. 188. To find the centre of gravity of a solid of revolution Mjhounded hy tivo planes perpendictdar to the axis, (Fig:87). The centre of gravity being necessarily situated upon the axis of revolution, which is supposed to coincide with the axis of X, it will be sufficient to determine its absciss x,. The element of the solid is expressed by Try dx, and we therefore have M=fry*dx (99). The moments being taken with reference to the plane ofy,z, we shall obtain Mx=f7ry''xdx (100) ; and by dividing this equation by the preceding, we find We must eliminate one of the variables in this formula, by means of the equation of the curve, and then integrate be- tween the limits a;=AP and a.-=AQ,. 189. This formula being applied to the determination of the centre of gravity of a cone, it will be necessary to obtain the two integrals fy' dx and fxy^ dx. Eliminating y^ by the equation of the generatrix y=ax, we obtain, after integration, fy^ dx—fa^x^ dx=^^, -, - , «^a* fy^xdx=fa^x^dx=—j—. There are no constants introduced by integration, since tlie volume is equal to zero at the origin A (Pig: 88). These values, being substituted in the formula (101), give 96 STATICS. a-x* 3 from which we conclude that the centre of gravity of a cone is at a distance from the vertex equal to three four tits of the altitude Ax. 190. As a second example, let the required centre of gravity be that of the volume of a paraboloid generated by the revo- lution of the parabolic arc AM {Pig. 85) about the axis Ax. The equation of the curve being y^ =px, we have fy'^ dx =fpxdx = \px^ , fy- xdx=fpx^ dx=-\px^ : these values substituted in formula (101), give x,='^^^ = ^x. \px^ ' The constants introduced by integration are equal to zero in the present instance, for the reasons assigned in the preceding paragraph. 191. Let the solid of revolution be an ellipsoid, the equa- tion of whose generatrix is this value ofy^ being substituted in the integrals of equation (101), we obtain, since the constants are equal to zero, /y^dx=—/ (a^dx — x'^dx) = — la^x——-j, r, J ^" /? , J ,j N h"" (a'^x^ x^\ ly^xdx=—:; l{a^xdx—x^dx)=-^ I — ^ -r- I . These values reduce equation (101) to _\a'X — \x^ _&a'^x — 3x2 '~ a^ — ix2 12a2— 4x- ' and by taking the integral between a;=0 and x=a, we find, for the absciss of the centre of gravity of the semi-ellipsoid, x, = ^a. 192. To find tlie ceîitre of gravity of a volume generated CENTROBARYC METHOD. Ô7 by the revolution of an area embraced by a curve BMCM' {Fig> 89) about the axis of x, this axis being situated entirely without the curve. Represent by y and / the ordinates MP and M'P : the volume generated by the revolution of the element Mot', will be equal to the difference of the volumes generated by the elementary rectangles Mp and M'p ; thj expressions for these volumes being ^y'^dx and vy'^dx, that of the element of the solid will be 7r(y2 — y'^)dx] hence, if we denote by M the entire volume of the solid generated, we shall have M=-!rf{y'' —y"')dx. By taking the moments with reference to the plane of y, z, we obtain Ma:, = vrfiy '^ —y'^) xdx. The value of x^ will be alone necessary, si ice the centre of gravity must be situated on the axis of abscisses. Of the Centrobaryc Method. 193. Let X, and y, represent the co-ordinates of the centre of gravity of a plane surface MPP'M' {Fig. 90), the area of which is represented by x. The moment of the element of this surface, taken with reference to the axis of re, is, by Art. 182, {yxydx ; and by making the sum of the moments of all the elements equal to the moment of the whole body supposed concentrated at its centre of gravity, we have f\y^dx=y^x. The two members of this equation being multiplied by the quantity 25r, it becomes f'>Fy^dx=2-7ryi\: The expression f^ry^ dx represents the volume generated by the revolution of the given surface about the axis of x, and the second member 2xy/ is the product of the generating surface by the circumference described by the centre of gravity; hence, we deduce this general theorem : The volume of every solid of revolution is equal to the product of the generating wea by the circuTnference described by its ceidre of gravity. 194. Let it be required, for example, to determine the G 9 98 STATICS. volume of the solid generated by the revolution of an isosceles triangle ABC {Fig. 91) about the axis of x. Denote CD by A, and AB by a ; the generating area will then be ex- pressed by ia/i. But the centre of gravity of the generating triangle being at a distance from C equal to fCD, the circum- ference described by this point will be |/iX2îr. Hence, the volume will be expressed by the product |/iX2jrX^a/i = fw-a/i-. As a second example, let us determine the volume of a right cone generated by the revolution of the right-angled triangle ABC [Fig. 92) about the line AB. The area of the generatrix will be iAJBxBC. The line CE being drawn to the middle of the side AB, the centre of gravity G of the generating area will be situated upon this line at a distance from the point E equal to iEC (Art. 172) ; its ordinate GD will therefore be determined by the proportion 3 : 1 : : EC : EG : : CB : GD ; whence, GD=iCB. The path described by the centre of gravity will therefore be expressed by f^rxCB; which, multiplied by the area of the generating triangle gives the volume of the cone equal to |«-xCB=^ XiAB=:iABx^xCB2. 195. Again, let the volume be that of a right cylinder : the ordinate GE of the centre of gravity of the generating rectangle {Mg. 93) being equal to ^AC, the path described by this point will be ;rAC. This expression being multiplied by the generating area which is equal to AB X AC, we have srxAC^ xAB for the volume of the cylinder. 196. The area of any surface of revolution may be found by a rule analogous to the preceding. For, if we consider the surface generated by the revolution of any curve MN [F\g. 94) about the axis of abscisses, and denote by y, the ordinate of its centre of gravity G, we shall have, by Art. 178, fy^{dx"- +c?y=*)=y,Xarc MN (102) ; and by multiplying each member by 2v, this equation be- comes MACHINES — CORDS. 99 f27ry^{dx^ + dy')=2vy, X arc MN. The expression f2Try^{dx^-\-dy'') representing the area of the surface generated, we conclude, that the area of a surface of revolution is equal to the product of the generating arc by the circumference described by its centre of gravity. 197. Thus, to determine the surface of a conic frustrum generated by the revolution of the right line CD {Mg. 95) about the axis of x, we have the ordinate EG of the centre of , , AC+DB , „ AC+DB , , ,, gravity equal to ^ ; and 25rX ^ equal to the circumference described by this point : hence, the product of this expression by the length of the generatrix CD gives 271- X — X CD=25r . GE . CD for the convex surface of the conic frustrum. 198. The two preceding theorems may be included in a single enunciation, viz. : Every solid or surface of revolution is equal to the product of its generatrix by the circumference described by the centre of gravity of the generatrix. Machines. 199. Machines serve to transmit the action of forces in directions different from those in which the forces are applied, and to modify the effects of those forces. The force applied to a machine is called the power, and that which tends to oppose the effect of the power is called the resistance. The most simple machines are the cord, the lever, and the inclined plane. To these are sometimes added the pulley, the wheel and axle, the screw, and the wedge, which may be formed by very simple combinations of the first three. These machines are usually called the Mechariical Powers. Cords. 200. We shall adopt the hypothesis that cords are perfectly flexible, that they are inextensible, without weight, and re- duced to their axes. If the extremities of a cord be sohcited G2 100 STATICS. by two equal forces P and Q, {Pig. 96), Avhich tend to stretch it, the tension of the cord will be measured by one of these forces ; for, since the equilibrium subsists, we may regard A, the middle of the line PQ,, as a fixed point, and drop the con- sideration of that portion of the cord included between A and Q, ; thus, the force P, acting alone against the fixed point A, will measure the tension of the cord PQ,. 201. When the force Q, exceeds P, a portion of Q. equal to P is employed to stretch the cord, while the remaining part of the force tends only to move the cord in the direction from P towards Q, : thus the tension will be measured by the least of these forces. 202. If three cords be united by a knot, the conditions of equilibrium are similar to those which obtain when any three forces act on a point. The force acting in the direction of each cord must be equal and directly opposed to the resultant of the other two ; hence, the conditions of equilibrium require that the three forces be situated in the same plane, and bear to each other the following relations {Fig> 97), P : Q, : R : : sin 7> : sin g- : sin r. 203. This proportion will be insufficient to establish the equilibrium, if the cords are united by a sliding knot. For, by regarding P and R as fixed points {Fig. 98), to which the cord PCR is attached, if the force Q, be supposed to act upon this cord by means of a ring or sliding knot, the point C will describe an ellipse, the plane of which will pass through the points P and R. But the revolution of this ellipse around the axis PR will generate an ellipsoid, having its transverse axis equal to PC |-CR, and the point C will necessarily be found upon the surface of the ellipsoid, or, in other words, at some point of the moveable ellipse ; but the point C being only subject to motion when the force Q. has a component in the direction of the elliptical arc, the equilibrium will be maintained when the direction of the force Q, is normal to the ellipse. If the line T^ be drawn tangent to the curve, wo shall have, from the well known property of the ellipse, Z.TCP=ZRC^; and by subtracting these angles from the right angles TON, if ON, there will remain CORDS. • 101 z:pcn=zncr; thus the angle PCR must be bisected by the direction of the force Ct, and the proportion P : R : : sin NCR : sin PCN becomes, in the present case, P : R : : sin NCR : sin NCR; whence, P and Q, are equal to each other. 204. The funicular machine consists of a number of cords united to each other at several knots, and maintaining an equilibrium between the forces applied to these cords. 205. When several forces P, R, S, T, &c. (Fig: 99), act conjointly at a single knot, their number will be reduced by unity, if we substitute for any two forces P and R their resultant R'; and by a repetition of the same process the entire system may always be reduced to three forces united at a single knot. 206. Let there be several forces P, P', P", F", P'^, (fee. {Fig. 100), acting at the knots A, B, C, &c. of the cord ABC. The conditions of equilibrium of these forces may be reduced to those of a system acting on a single point : for, let R represent the resultant of the forces P and P' ; since its effect must be destroyed by the third force acting in the line AB, the direction of this resultant must coincide with the pro- longation of AB : but the point of application of a force may be assumed any where on its line of direction, and hence we may transfer the force R to the point B. If it be there de- composed into two components parallel and equal to P and P', the effect will be the same as if the two forces P and P' had been transported parallel to their original directions, and applied at the point B. In hke manner, by transporting the forces P, P', P", (fcc, which are supposed to be applied at B, to the point C, the entire system may be considered as acting on this point. Thus the conditions of equilibrium are, (Art. 54), 2(P cos «) =0, 2(P cos /3) =0, s(P cos y)=0. To determine the ratio of the extreme tensions P and P,^, we will denote by t and t' the tensions of the portions AB and BC, and by 102 • STATICS. a the angle PAP', a' the angle ABP", a" the angle BCP'", b the angle P'AB, b' the angle P"BC, b" the angle P"'CP" ; we shall then obtain, Art. 202, P : ^ : : sin 6 : sin a, t : i' : : sin b' : sin a', i : P" : : sin b" : sin a" ; whence, by multiplication, suppressing the factors which are common to the two first terms, we have P : P" : : sin 6 xsin 6'xsin b" : sin a Xsin a' Xsin a". We may, in like manner, determine the relations between aiiy other two forces. 207. If the forces P', P", P'", (fcc. be supposed parallel, we shall have b+a'=lSO% 6' + «"=180°; and since the sine of an angle is eaual to the sine of its sup- plement, we must have sin 6= sin «', sin b'—sin a" ; and the preceding proportion will then reduce to P : P" : : sin b" : sin a. If the forces P', P", and F" represent weights {Mg. 101), the entire system will be situated in the same vertical plane ; for, the right line AF being vertical, the plane of the forces P, P', and t will be vertical. For a similar reason, the plane of the forces t, P", and t', will be vertical ; but the line AB not being vertical, it is impossible to pass more than one ver- tical plane through it : hence, the forces P, P', t, P", and i' will be situated in the same vertical plane. The same rea- soning may be extended to a greater number of forces. 208. The extreme forces P and F" being required to sus- tain the resultant of all the others, this resultant must be directly opposed to that of the forces P and P", and must consequently f>ass through the point G, at which the direc- tions of those forces intersect. Moreover, its direction must be vertical, being parallel to the components P', P", and P"', and it will therefore be represented by the vertical line GH drawn through the point G. 209. If we regard a heavy cord as a funicular polygon, CATENARY. • 103 loaded with an infinite number of small weights, it results from what precedes that the effect produced on the fixed points by the weight of the cord may be estimated by drawing the tangents PG and QG {Fig. 102), and applying at G a weight equal to that of the cord ; since if we denote this weight by G, we shall then have P : Gl : G : : sin LGa : sin LGP : sin PGQ. Of the Catenary. 210. The catenary is the curve which a perfectly flexible cord assumes when it is suspended from two fixed points A and B {Fig. 103), and subjected to the action of the force of gravity. We will suppose that the cord is uniformly heavy, and that the force of gravity is exerted on every particle : it will readily appear, as in Art. 207, that the curve will be situated in a vertical plane. Let the origin of co- ordinates be assumed at A, the horizontal line AC being the axis of abscisses ; the co-ordinates of a point M will then be AP=a:, and PM=y. Through the point M, and through the origin A, let tangents AH and MH be respectively drawn, intersecting at the point H, and through this point draw the vertical line HL. If we consider the portion of the cord MA, we shall have, by Art. 209, tension at A : weight of the portion AM : : sin LHM : sin AHM (103). Let s denote the length of the arc AM ; A the tension of the cord at the point A, which is exerted in the direction of the tangent AH ; and « the angle included between this tangent and the horizontal line AC. The quantities A and « will remain constant. The tension at A, being a quantity of the same kind as that contained in the second term of the preceding proportion, will necessarily be expressed by a weight ; and if we repre- sent by p the weight of a portion of the cord whose length is equal to unity, sp will express the weight of the part AM, and the tension at A will be of the form ap. Thus the two first terms in the above proportion will be replaced by the ratio ap : sp, or by its equal a : s] hence, a : 5 : : sin LHM : sin AHM (104). 104 • STATICS. 211. To determine the analytical expressions for the sines which enter into this proportion, we remark, that in the elementary triangle 7nMn, we have Mm X sin mMn =mn, Mm X cos m'M.7i=M.n ; or, __ mji _^ M;t sm mMfi= ^r-r— 5 cos mM?i=r — — ; Mm' Mm ' and replacing these elementary lines by their analytical values, these equations become dx di/ sin ?n'M.fi——r-, cos nïM.n=—r (105). as as Bat the angle mMn included between the vertical and the arc of the curve, is equal to the angle LHK formed by the vertical with the tangent at M ; hence, sin LHK=4^ , cos LHK=4^ (106). as as The first of these equations may be reduced to sinLHM— (107); as for the angles LHK and LHM being supplements of each other, we have sin LHK=sin LHM. Again, the angles AHK and AHM being supplements of each other, we obtain sin AHM^sin AHK=sin (LHK-LHA) ; and from the well known trigonometrical formula for the sine of the difference of two angles, we have sin AHM=sin LHK cos LHA— sin LHA cos LHK ; eliminating sin LHK and cos LHK by means of the equations (106), we find sin AHM^'^-^cos LHA-^'sin LHA (108). as as The triangle LAH being right-angled at L, the angles LHA and HAL are complements of each other, and the latter hav- ing been denoted by a, we obtain cos LHA=sin «, sin LHA=cos «. CATENARY. 105 212. These values substituted in equation (108) give » sin AHM=^sin «-"^cos « (109) ; as ds and the equations (107) and (109) convert the proportion (104) into dx dx . dy a: s '.: -y : -y-sin « — fcos *, as ds ds From this proportion we deduce the equation 5=asin« — a-^cos» (110). dx This equation contains three variables, one of which may be eliminated by means of the relation ds=^y/{dx'i-\-dy^). For, by differentiating equation (110), regarding dx as con- stant, we find ds= — a cosa— -^ ; dx and by equating these values of ds, and dividing each mem- ber of the equation by dx, we obtain \/(l + -r^ )=— a cos«— ^, ^ \ dx^/ dx""' or, by division, d'^y — a cos«~ dx^ v/O+g) This equation will become integrable, if we multiply its two members by 2dy ; we shall thus obtain 2dy^^ 2dy— — a cos «__ ^ ; ^ 4. L whence, by integration, jV^'^ y=-acos«^(l-F^)+c. This equation being multiplied by dx gives {c—y)dx=a cos*y/{dx^-\-dy'^) ; 106 STATICS. and by reduction dy^ ^/[{c-yy—a^ cos»c«] .^^^^ dx a cos « 213. The constant c may be determined by the consider- ation that at the point A, x=0, y=0, and ^=tang«. dx These values reduce the equation (111) to tang« — ^ ^ i.; a cos« from which we deduce but a tangae cos »=^{c^—a^ cos* «) : whence, tang* cos «= sin a; a^ sin^ *=c2 — a^ cos^ *, and consequently c^=a2 (sin' a+cos^ «)=a'. Thus the constant c is equal to a, and by substituting its value in equation (111), we find for the differential equation of the catenary, dy_ ^[{a—yY-a^ cos^ a\ ,^^^y dx a cos <* 214. It appears from a comparison of this equation with (110), that the catenary curve is rectifiable ; for, if the pre- ceding value of Y ^^ substituted in equation (110), we shall obtain s=a sin a— ^[{a—yY —a^ cos^»] (113) : from this expression the value of s may be readily found in terms of y, when the constants a and » have been determined. . 215. To integrate the differential equation of the catenary, we make a—y=z, a cos et=b (114) ; and we then obtain dy = — dz ] these values substituted in equation (112) give MACHINES. 107 dx = --4^—— (115); this expression becomes integrable by making ^{z^-h^)=z—t (116): which by squaring and reducing, gives By the differentiation of this equation, we obtain xdt-\-tdz=tdtj or, dz _ dt z—t~~T' This relation, in connexion with that assumed above (116), converts the equation (115) into , bdt dx — — J t which gives, by integration, x=-h log^+e; and by substituting for t its value expressed in terms of sr, we obtain x=h\o%\z—^{z''—h^-)\^e\ or finally, by replacing the quantities h and z, by their values given in equations (114), we find ar=acos<«log|a— y— ^[(a— y) = — a^ cos2<«]| +e (H^). 216. To determine the value of the constant e, we observe that at the point A, a;=0, and y=0 ; which conditions reduce the equation (117) to e=— «coscelog |a[l— y/(l— C0S2 «)]|. This value substituted in equation (117) gives a;=acos«log[a— y— -v/(a— y)"— a^cos* « ] —a cosee log [a(l — v/l— cos= «)] ; or by reduction. Such is the equation of the catenary. 217. The values of the constants c and e have been deter- mined in functions of a and «f ; but these two quantities are 108 STATICS. Still unknown. To determine their values, we will suppose that jc' and y' represent the known co-ordinates of the second point of suspension B, and I the length of the curve AMB ; these values being substituted in the equations (113) and (118), we obtain l=a sin »— ^[{a—y'Y —a^ cos* «], X —a cos«» ^\ â[l— /(l-cos^«)] ) 218. These equations, in connexion with the relation C0S2 «+sin' «=1, determine the values of a, cos «, and sin ec,in functions of^.y', and /. But another difficulty still presents itself; this consists in the proper choice of the signs with which to affect cos «, and the radicals which in the preceding expressions have not received the double sign. To resolve this difficulty, we will determine the co-ordinates of that point to which the max- imum ordinate appertains. The characteristic property of this point is that -^=0, which reduces equation (112) to dx ^/[{a-yy—a'' cos'' ^] ^Q . acos« ' and consequently, a — 2/=acos« (H^)- To establish the condition that this equation belongs to a maximum, rather than to a minimum value, we attribute the proper sign to the second differential co-efficient -j-^. But by squaring the equation (112), we obtain dy'^ _{a—yY —a^ cos» « dx'^ a^ cos 2 a. ' and by differentiating, and dividing each member by 2c?y, we find d^y__ a— y . dx^ a'cos'^tt'^ substituting in this equation the value oia—y determined in equation (119), we obtain dry^^ l__ dx' a cos <*' LEVER. 109 219. This equation indicates that the condition of a maxi- mum will be fulfilled by attributing the same sign to a and cos « ; but these signs must be positive ; for, if they were negative, the value of y determined hj the equation (119) would be also negative, which is evidently inadmissible in the hypothesis adopted, that the positive ordinates are reckoned from the line AC downwards. From the equation (119) we likewise infer, that the quantity a exceeds the maximum value of y, and therefore that it exceeds all other values. Let EF represent the maximum ordinate {Fi^. 103); it is evident that between the limits â;=0 and a-=AE, as y in- creases, the arc of the catenary will likewise increase. But it appears from equation (113) that the increase of y will not necessarily involve that of the arc 5, unless the radical in that formula be affected with the negative sign. For, as y in- creases, the quantity a — y will decrease, and the value of the radical will therefore decrease ; but the smaller the value of this radical, the less it will diminish the positive part of the expression a sin «, and the greater will.be the value of the arc. The equation (113) is therefore in perfect accordance with the hypothesis that the co-orJinate has not attained its maximum value. But from :2;=AE to a;=AD, the arc 5 should increase while y diminishes, and since this decrease in the value of y augments the value of the radical expres- sion, the required condition can only be fulfilled by affecting the radical with the positive sign : thus, between the limits ar=AE and ar=AD, the sign of the radical must be changed in the formula (113). Of the Lever. 220. The lever is a bar of wood or metal moveable around a fixed point, which is called the fulcrum. To simplify the considerations which relate to this machine, we shall regard the lever as destitute of thickness, and will therefore represent it by a simple line, either straight or curved. Let a lever AB {Fig. 104) be sohcited by the two forces P and P' ; the effect of these forces cannot be destroyed by the resistance of a fixed 10 110 STATICS. point C, unless they are situated in a plane passing through this point. If this condition be fulfilled, the equilibrium will be maintained, when the sum of the moments taken with reference to the point C is equal to zero. 221. If the lever is capable of sliding along its point of support, it will also be necessary that the resultant of the forces acting on the lever should be perpendicular to the lever at the point of support. 222. When the lever is straight and the two forces parallel to each other, ifp and p' represent the lengths of the portions AC and BC {Pig. 106), we shall have from the theory of parallel forces (Art. 73), V :V' ::p' :p] from which we infer, that when the forces are in equilibrio, their intensities will be inversely proportional to the arms of the lever. 223. If the lever be curved, and a right line ED {Fig. 105) be drawn through the fulcrum C, the forces may be conceived to be applied at the points E and D taken on their respective directions ; we shall thus obtain P : F : : CD : CE. 224. Levers are divided into three kinds. In the first kind, the fulcrum C {Fig. 106) is situated between the power and the resistance : in the second kind, the resistance R {Fig. 107) is situated between the power and the fulcrum ; and in the third kind {Fig. 108), the power is between the fulcrum and the resistance. The balance and steelyard are examples of the first kind of lever ; a bar of iron used in raising weights and having its fulcrum at one extremity, forms a lever of the second kind ; the treddle of a turning lathe is a lever of the third kind. 225. The effect produced by the weight of a lever may be readily estimated by regarding it as a force S applied at the centre of gravity of the lever. For example, let P and P' {Fig. 109) be two weights suspended from the extremities of the lever AB, whose centre of gravity is situated at G ; we LEVER. Ill shall have, by virtue of the principle of the moments, P'xCB+SxCG=PxAa This equation will determine either P or F; and the weight sustained by the fixed point will be P + P' + S. If the power and resistance act in opposite directions, regard must be had to the directions in which they tend to turn the lever; thus, in Fig. 110, the equation of the moments be- comes PXCA + SXCG=P'XCB (120); and the weight sustained by the fulcrum is P+S-F. 226. Let the lever CB {Fig. 110) be supposed homogeneous, and of uniform weight throughout its length : represent by tn the weight of a portion of the lever whose length is one foot. If X represent the length of the lever expressed in feet, its weight S will be expressed by mx^ and should be regarded as a force acting at its centre of gravity, which corresponds to the middle point G : thus, if we make CA=a, the equation (120) will then become Va + \xXmx=.V'y.x] from which we deduce F==— -fima: (121). X \i, therefore, x be assumed arbitrarily, this formula will make known the value of P' ; but it may be required to assign the value of X which shall render P' the least possible ; we must then regard P' as a function of x, and make the differential 0^ co-efficient -r- equal to zero ; we shall thus obtain ,J^ \ dx whence, — — 4-im=0; ^^ 2Pa , //2Pa x'=- and X m -^m By substituting this value in equation (121), we obtain 112 STATICS. P = Va or, by reduction, 2Pa ^ \ in / 227. The cornmon balance is an important application of the lever. It consists essentially of a lever having equal arms, from the extremities of which are suspended scales of equal weight. The lever of the balance, which is called the beam, is sustained by a horizontal axis perpendicular to its length, which rests upon a firm support, and the substance to be weighed, being introduced into ne of the scales, is counter- poised by the addition of known weights in the opposite scale. The figure of the beam is so chosen that its centre of gravity will be found immediately beneath the axis, or centre of motion, when the beam has assumed a horizontal position ; and the weights suspended from its two extremities are known to be equal when they will retain the beam in this situation. If the centre of gravity were found upon the axis, the beam would obviously rest in any position, and there would be nothing to indicate the equality of the weights io the two scales ; and if this centre were situated above the axis, the beam would have a tendency to overturn if deranged in the slightest degree from the horizontal position. 228. When the balance has been constructed with such accuracy that the lengths of the arms are exactly equal, the beam will assume the horizontal position if equal weights be introduced into the two scales ; but in the false balance, where the lengths of the arms are unequal, the weights necessary to maintain the beam in this position are likewise unequal. In this case, the weight of the body may be obtained by counter- poising it successively in the tv/o scales : the true iceight will be a geometrical mean between the two ajiparent weights. For let 2^ and p represent the lengths of the two arms, and W the true weight of the body. Then, if a weight P, sus- pended from the extremity of fhe arm y>, be supposed to sustain the weight W when suspended from the extremity LEVER. 113 of the arm jo', the conditions of equilibrium in the lever (Art. 220) will give But if the weight W be transferred to the extremity of the arm j», it will be necessary to apply a dijEFerent weig-ht P' to the extremity of the arm p', in order that the equilibrium may be preserved. Thus we shall have and by multiplying the corresponding members of these two equations, we obtain or, by reduction, W=y(PP'); hence, the truth of the proposition enunciated becomes appa- rent. 229. It is frequently necessary that the balance employed should possess great sensibility, or should be capable of indi- cating very minute differences in the weights of the substances placed in the two scales. The sensibility of the balance is measured by the smallness of the weight necessary to produce a given inclination of the beam, when the scales are charged with a given load. The sensibility depends upon the following particulars. 1°. The beam should be as light as is consistent with a proper degree of strength, in order that the friction at the axis, which is proportional to the pressure, may oppose the least possible resistance to the motion of the beam. For the same reason the axis is constructed of hardened steel, and has the form of a knife-edge, or triangular prism, the lower edge of which rests upon polished steel or agate planes. 2°. The lengths of the arms should be as great as possible, other things remaining the same, since the moments of the weights introduced into the scales, taken with reference to the centre of motion, will be directly proportional to these lengths. Thus, the same weight, placed at twice the distance from the centre of motion, will exert a double effort to turn the beam. H 114 STATICS. 3". The sensibility will be increased by diminishing the distance between the centre of gravity of the beam and the centre of motion. For, when the beam has been deranged from the horizontal position through a given angle {Fig: 111), the weight of the beam W, which acts at its centre of gravity G, will exert an effort to restore it to its former position, which effort will be directly proportional to the moment of the weight W, taken v ith reference to the centre of motion D; this moment will be expressed by Wxdg. But the derangement of the beam having been made through a given angle, the distance dg will evidently be proportional to DG, the distance between the centre of gravity of the beam and the centre of motion. Thus, in proportion as the distance DG is diminished, the tendency of the weight of the beam to counteract the derangement which would be produced by ah inequality of the weights in the two scales will likewise be diminished, or the sensibility will be increased. 4°. The line joining the points of suspension of the two scales should pass through the centre of motion. For, if the centre of motion be found at C above the line AB, and the beam be supposed to have assumed the inclined position represented in Fig. Ill, the effective arm of lever CE' of the scale P' will evidently be greater than the arm CE of the scale P. Thus the beam may have a tendency to return to the horizontal position, although the weight P' be less than P. And if, on the contrary, the centre of motion be placed at a point C below the line AB, the lever-arm C'F of the scale P will exceed that of the scale P', and the beam would therefore have a tendency to overturn, although the weights in the scales were equal to each other. When the centre of motion is situated at the point D, the equality of the two arms will be preserved, whether the beam be in a horizontal or inclined position. 5°. The sensibility of the balance will be increased by diminishing the load with which the scales are charged, since the friction at the axis will be diminished in the same pro- portion. 230. A very accurate balance will be sensibly affected by the addition of jg^ part of the load with which the scales are charged. LEVER. 115 231. The steelyard, represented in Pig. 112, is a balance having unequal arms, and is so constructed that a moveable weight P, applied successively at different points of the longer arm, shall sustain in equilibrio different weights suspended from the extremity of the shorter arm. The longer arm GB is so graduated as to indicate the weight which will be supported by the moveable weight P^ when placed at each of these divisions. 232. To discover the law according to which this arm should be graduated, we will denote by W, W, W", &c., the weights suspended successively from the extremity of the shorter arm , j9, p', p", &c., the corresponding distances at which the weight P must be placed to maintain the equilibrium, r, the length of the shorter arm , IV, the weight of the beam, r', the distance of its centre of gravity from the fulcrum^ Then, if the centre of gravity of the beam be supposed to lie on the side of the longer arm, as usually happens, the con- ditions of equilibrium will give Wr=wr' + P/?, Wr—tvr' + Fp\ W"r=ur' + Fp", &c. &c. &c. ; and by subtracting each of these equations from that which follows, we obtain (W'-W)r=(p'-p)P, (W"-W')/-=(p"— _p')P, ( W" — W")r = (p'" —//')?• If the weights W, W, W", &c. be supposed to increase in arithmetical progression, we shall have W— W=W"— W'=W"'— W"=&c. ; and therefore 2)' —2i=2}" — p'=p"' — ;y' = (fcc. ; thus the distances ^?, />', p", -]-h'cos6±V[{psk2+h'^cos6y—{k^p^—h'>){k''s^ — h^)] ^ and by developing and reducing the terms contained under the radical signs, we obtain ps k^ -\-h^cos6±h^[k^{2r- -\-2pscos6-\-s'')—h''{\—cos''6)] ^ ^'^~~ k'p^-h' ' and finally, by substituting for z and 1 — cos^ e their respect- ive values, we shall have P psk'' +h- cos 6 ±hy[k''(p'' +2ps cos 6 + s'')—h' sin'' O ] S l^2p2_h,2 142 STATICS. 281. If the radius of the cyUnder be very small, its square h^ maybe neglected, and the preceding ratio will then become P _5 A^(jJ^+2jJgCOS0 + 5') If the perpendiculars p and 5, demitted from the point C on the respective directions of the power and resistance, become equal to each other, the results will apply to the case of the pulley ; and by still neglecting the quantity h^, we shall find P^ AvW+cos^] . S kp ^ ^* 282. Finally, when the power and resistance act in par- allel directions, the angle 6 becomes equal to zero ; whence, sin ^=0, cos^=l ; and the equation (143) then reduces to ?=1±?^. S kp 283. The same principles will serve to determine the con- ditions of equilibrium in the other mechanical powers, when regard is had to the effects of friction ; but the results obtained would in general prove much more complicated. Of the !Stiffness of Cordage. 284. In employing the cord as a means of transmitting the effect of a force to a machine, we have hitherto supposed the cord to be perfectly flexible. But as this hypothesis is inad- missible in practice, it becomes necessary to estimate the ad- ditional force that will be necessary to overcome the rigidity of the cord. Let P and d {Fig. 138) represent two weights which are applied to the extremities of a cord passing over a fixed pul- ley : if the weight P be supposed to prevail, and the cord be regarded as perfectly rigid, the extremity Q. will evidently be brought into a position Q,', such that the vertical line Q'O will intersect the horizontal line CO drawn throusfh C, at a distance CO from the centre, greater than the radius CG. The extremity P will at the same time assume the position P', STIFFNESS OF CORDAGE. 143 such that the vertical Hne drawn through P' will intersect the radius CF. Hence the arm of the lever to which the force Q, is applied will now be longer than that of the force P, and the condition of equilibrium will therefore require that the force P shall exceed Q., 285. If the cord be supposed imperfectly rigid, similar effects will be produced, though in a less degree ; and in practice, it is found that the decrease in the arm of lever, to which the preponderating weight is applied, is wholly insen- sible. Hence, in estimating the effects produced by the rigid- ity of a cord employed in a machine, it will simply be neces- sary to increase the arm of the lever to which the resistance Q is applied, by a proper quantity q. 286. To determine the value of q^ we remark that the re- sistance to flexure opposed by a given cord arises from two distinct causes, — viz. 1°. The tension of the cord, or the force Q, which is employed to stretch it ; and, 2°. The materials used in the construction of the cord, and the degree of twist which has been given to it. The resistance arising from the tension of the cord is found to be proportional to this tension, and may therefore be represented by an expression of the form 6Q, in which h represents an indeterminate constant. The resistance produced by the second cause may be repre- sented by a quantity a. Thus, for the same cord bent over the same pulley, the expression (« + 6Q,) may be supposed to represent the effort necessary to bend it. But if we suppose the diameter of a second cord to be greater, the force necessary to bend it will become greater, and we can assume that this force will increase according to some power n of the diameter D. The force will also increase as the curvature increases, or as the radius of the pulley is decreased, and hence (a + 60,) may be taken as an expression for the force necessary to overcome the rigidity of the cord. This expression represents the increment that must be given to the power P, in order that it may be on the point of overcoming the resistance Q, : but we also have Vr=Gi{r-\-q) ; 144 STATICS. and since the forces P and Q, become equal when the cord is supposed destitute of rigidity, P— Q, or Q i will also express r the value of this increment. By making these values equal to each other, we obtain D"(a+6a)=%; whence, q^^l.{a-\-hGi) (143 a). 287. This equation should only be regarded as furnishing an approximate value of the quantity q, since the above rela- tion has been obtained by considerations of a very general character. It moreover contains certain unknown quantities a, b, and ?i, which vary with different cords. For the purpose of verifying the truth of the preceding formula, and at the same time determining the values of the unknown constants, we proceed as follows. Having selected a cord, we pass it over a fixed pulley, and attach to its extremities two equal weights : we then increase one of these weights until it is about to prevail over the other, and the difference k will give one value of the quantity r By repeating the experiment several times, changing the weights, the cord, or the pulley, we can obtain a number of similar equations, in which the quantities a, b, and n will be the same, and the quantities D, r, and Q., although different, will be known by observation. Three such equations will serve to determine a, b, and 7i, and their values being sub- stituted in the general relation expressed by formula (143 a), the accuracy of the formula can be tested by comparing it with the results furnished by other experiments. The quantity n was found by Coulomb to be usually about 1.7 or 1.8 ; and the resistance to flexure must therefore vary nearly as the square of the diameter of the cord : but the quantity n is itself subject to some variation, becoming nearly 1.4 when the cord has been long used. The following results, expressed in French poinids, were obtained in the experiments of Coulomb. RESISTANCE OP SOLIDS. 145 ibs. Tin lbs. J)rt iOS. jy„ 30 threads in a yarn . . — Xa=4.2 White rope < 15 threads f 6 threads Tarred rope 30 threads in a yarn . . — Xa=6.6 15 threads 6 threads i4.2 . . _ixl00=: 9 r 1.2 . . . . . " 5.1 0.2 . « 2.2 lbs. ;6.6 . D» lbs. . —6X100=11.6 r 2.0 . « 5.6 0.4 . " 2.4 On the Resistance of /Solids. 2S8. The particles of every solid body are found to oppose a certain resistance to any force which tends to separate them. This resistance arises from the mutual actions exerted by the particles upon each other ; and if the nature of these actions, as well as the arrangement of the particles which compose the body, were accurately known, it might be possible to estimate the force necessary to separate the particles, or to produce a given change in the figure of the body. Bat as we are entirely ignorant of these particulars, it becomes neces- sary to adopt some hypothesis relative to the manner in which bodies are constituted, and the nature of the actions exerted by the particles upon each other. Then, by reasoning upon such hypothesis, we can obtain results which, compared with those derived from experiment, will serve to test the accuracy of the supposition. 289. The hypotheses most generally adopted are — 1'^, That of Galileo, which supposes all solid bodies to be niade up of fibres, disposed parallel to the length of the body, and sus- ceptible of being ruptured without undergoing flexure, ex- tension, or compression ; or, 2°. That of Leibnitz, modified by Bernoulli and others, which regards the fibres of all bodies as elastic ; being susceptible of extension and compression, and capable of opposing a resistance directly proportional to their extensions or compressions. The force required to produce a given extension is, moreover, supposed to be equal to that which is capable of producing an equal compression. 290. It is very certain that neither of these hypotheses is strictly correct ; but as the results given by the latter difier but K 13 146 STATICS. little from the truth, when the extensions or compressions are inconsiderable, we shall adopt it, and apply it to the investiga- tion of the resistance which a solid will oppose under different circumstances. 291. The kind of resistance which the body offers will de- pend in a great measure upon the manner in which the force is applied. Thus, the force may exert an effort to extend or compress the solid in the direction of its length, or it may tend to produce a flexure of the solid, or it may operate as a force of torsion ; and in each of these cases it may be required to determine the force necessary to produce a rupture or separation of the particles, or simply that necessary to effect a given change in the figure of the solid. The cases which more generally occur are, 1°. That in which the solid sustains an extension or compression in the direction of its length, without undergoing sensible flexure ; and, 2°. That in which flexure is produced by the applica- tion of a force perpendicular to the length of the solid. As it is the object of the present article merely to exhibit the general methods in which the hypothesis assumed may be applied to the determination of the strength of bodies, or the resistance which they are capable of opposing, we shall confine our investigations to the consideration of these two cases. 292. The resistance of a body to a change of figure de- pends upon its force of elasticity, which is measured by the effort necessary to compress or extend the body by a given quantity. Its resistance to rupture depends upon its force of tenacity, or upon the effort necessary to rupture or crush the body. The values of these forces having been determined experi- mentally for a body composed of a given substance, and having a simple form, we can calculate the compression, ex- tension, or flexure produced in another body, of the same substance, by the application of a given force. The methods of effecting this calculation will now be explained. RESISTANCE OF SOLIDS. 147 Of the Resistance to Compression or Extension. 293. When a solid is stretched or compressed in the direc- tion of its length, being at the same time prevented from experiencing flexure, the lengths of its fibres are found to undergo very slight variations, and we can therefore assume, in conformity with the hypothesis adopted, 1°. That the extensions or compressions of all the fibres will be equal to each other, and uniform throughout the extent of each fibre ; and that the force necessary to produce a given extension will be capable of producing an equal compression. 2°. That the variations in the lengths, and the resistances opposed by the fibres, are constantly proportional to the forces which produce them ; and that this proportion obtains even for those forces which rupture or crush the body. 294. Let a cubical mass of any substance be placed upon a horizontal plane, and subjected to the action of a weight which rests upon its upper surface, compressing the substance in the vertical direction. Denote by a, the length of one of the edges of the cube ; a', the quantity by which its vertical dimension is com- pressed, and which is always extremely small in comparison with a ; P, the force which produces the compression. Then, since the compression of each fibre is supposed uni- form throughout, or since the particles which compose any one fibre are supposed to approach each other equally at every point of such fibre ; it is obvious that the entire compression a', sustained by any fibre, will be directly proportional to its length a. For example, if the length of another solid be supposed equal to 2a, its transverse section remaining the same, and if the same force P be applied to its upper surface, the number of particles in the length 2a will be twice as great as the number contained in a ; and each pair of consecutive particles being caused to approach each other to within the same distance, in order that the resistance of the fibre may be uniform throughout, the whole variation in the length 2a will evidently be twice as great as that which was produced K2 148 STATICS. in the length a, and will therefore be expressed by 2a'. And, generally, the compression of the solid, whose length is 7ia, and whose transverse section remains the same, will be ex- pressed by im', when the same force P is applied to its upper surface. Let the quantity a be supposed equal to the linear unit, — one foot, for example ; then 71 will express the number of feet contained in the length of the second solid, and 7ia' will express the variation produced in the length of a solid whose transverse section is equal to one square foot, and whose length is equal to 71 feet. 295. The preceding remarks have been confined to the case in which the solid suffers compression, but from the nature of the hypothesis, they must apply with equal force to the case in which the eflbrt is exerted to extend the body. 296. If the transverse section of a second solid, whose length is likewise equal to 7i, be supposed greater than that of the first, the number of its fibres will be increased in the same proportion, and the total effort exerted by these fibres when compressed to the same degree will evidently be pro- portional to their number : thus, if P' represent the force necessary to compress a prism whose length is 71, and whose transverse section contains ??i square feet, by a quantity equal to na, we shall have the proportion section 1 : section w^ : : P : P' ; whence, P'=wP. 297. If the force P' be increased, the solid will undergo a greater compression, and the quantity by which the length /* of the fibre is compressed will no longer be represented by 'iia\ but by an unknown quantity 7ia". To determine this quantity, we recur to the hypothesis which assumes that the compressions are proportional to the forces which produce them ; hence, by calling P" the value of the force which pro- duces the compression 71a", we shall have 7ia' : 7ta" : : P' : P", and therefore, P"=P'Z^- na' ' RESISTANCE OP SOLIDS. 149 or, replacing P' by its value mP, we have p„^P^m^" ^^3jj d n p 298. The quantity — is called the coefficient of the elas- a ticity : its value will depend only on the elastic force of the substance of which the prism is composed, and will therefore be independent of the dimensions of the particular prism under consideration. If we denote this coefficient by A, we shall obtain, for the entire compression of the prism, „ nV" na = — r-. mA This expression will determine the quantity by which a given prism will be compressed under the influence of a given force, when the coefficient of the elasticity has been previously ascertained. It should be remembered, however, that this formula is only applicable when the compressions are exceed- ingly small ; and that the solid is ruptured or crushed before its length undergoes a very sensible change. 299. The preceding expression is equally applicable when the force P" tends to stretch the solid. 300. To determine the force necessary to rupture a given prism, when exerted in the direction of the length of the prism, we shall denote by B the force necessary to rupture a prism of the given substance whose transverse section is a square foot. Then, if the transverse section of the given prism be supposed to contain m square feet, the number of its fibres will be m times greater than the number contained in the prism whose section is equal to one square foot ; and since each fibre in the two prisms must oppose the same resistance at the instant of rupture, we shall determine the force P" necessary to rupture the given prism, by the propor- tion section 1 : section w : : B : P" ; whence, P"=mB. 301. The quantity Bis called the coefflciejit of the tenacity, and depends only on the nature of the substance under consid- 150 STATICS. eration. Having determined ihis quantity by experiment, we can readily calculate the force necessary to rupture a given prism of the same substance. This investigation is equally applicable whether the force be exerted to compress or extend the solid. The methods of determining experimentally the coefficients of the elasticity and tei.acity will be explained hereafter. Of the Resistance of a Solid to Flexure and Fracture produced by a Force acting at right angles to the direction of the Fibres. 302. When the length of a solid body bears a certain pro- portion to its thickness, the body is found to undergo a cer- tain degree of flexure before breaking. Th^s flexure becomes more perceptible as the length of the solid is increased : thus a bar of wrought iron whose length does not exceed twelve or fifteen times its thickness gives very slight indications of flexibility ; but when its length is increased to forty or fifty times its thickness, it yields readily to an effort exerted to bend it, and becomes susceptible of taking a very consider- able flexure before breakingf. 303. If a force P be applied in a direction perpendicular to the length of the solid AB {Fig. 139), which is supported at its two extremities, and if this force be supposed to produce a certain degree of flexure in the solid, causing it to assume the form represented in Fig. 139 a, the fibres aa, &c. situated on the convex side will be extended, their lengths being in- creased, and those situated on the concave side will suffer a compression, and will undergo a diminution in length. This effect is readily observed : for, if the force P be gradually increased until it become capable of breaking the solid, the rupture will be found to commence at a point D on the con- vex side, thereby indicating that the fibres aa on that side have been most extended ; and if some of the fibres situated on the convex side be previously separated by cutting them through transversely, it will be found that a smaller force than P will be required to fracture the solid. But if, on the contrary, the fibres bb situated near the opposite side of the RESISTANCE OP SOLIDS. 161 solid be cut transversely to a certain depth EF (Fig. 139), and if a thin plate of some unyielding substance be intro- duced into the cut EF, so as to fill it entirely, it will be found, upon subjecting the solid to the action of the force P, that the thin plate will be -retained by a strong pressure tending to compress it, and that the strength of the solid will not be diminished, the rupture commencing at the convex side, when the force P has been increased in the same degree as was necessary to rupture the solid before severing any of its fibres. As we proceed from the convex towards the concave side of the solid, the extensions of the fibres will gradually dimin- ish, and at a certain distance from the surface, their lengths will undergo no variation ; beyond this distance the exten- sions will be changed into compressions, and these will again increase until we arrive at the concave side. 304. The flexure of the fibres being supposed to take place entirely in planes parallel to the axis of the solid and the direction of the force applied, it is evident that the change of figure experienced by the solid will require that those fibres whose lengths undergo no variation should be contained, previous to the flexure, in a plane perpendicular to the direc- tion of the force which produces the flexure ; and that, after the flexure, these fibres will form a cylindrical surface, whose elements will be parallel to the same plane. More- over, the fibres situated at equal distances from this plane will undergo equal extensions or compressions. 305. Let us now conceive a right prism AB to be firmly fixed at its extremity A, in such manner that its axis shall be horizontal, and that a vertical plane passing through the axis shall divide the solid into two symmetrical parts. Let a weight P be applied at the other extremity of the solid, causing it to undergo a certain degree of flexure, and to assume the form represented in Fig. 140. If two planes, ariv, a'u'v', be drawn infinitely near to each other, and normal to the curve Auu'B assumed by the fibres whose lengths re- main invariable, such planes will include between them an elementary portion of the solid, and if the system be sup- posed in equilibrio, the state of equilibrium will not be dis- 162 STATICS. turbed by regarding the portion of the solid included between the sections ACD and uav as absolutely immoveable, and the portion of the solid included between the sections BEF and 2iav as constituting a distinct system. The conditions of equilibrium in this system will evidently require that the force Pj together with the force necessary to retain the part DCAïiav in its position, shall be just capable of sustaining the efforts arising from the compressions and extensions of the fibres, or, in other words, that all these forces should reduce to two that are equal to each other and directly opposite. 306. If we assume any two rectangular axes Ax and Ay situated in the vertical plane passing through the axis of the solid, we can resolve each of the several forces into two com- ponents respectively parallel to these axes ; since these forces are all situated in planes parallel to the plane of the axes. Moreover, since the solid has been supposed to be symmetri- cally divided by the vertical plane passing through the axis, the forces of elasticity arising from the extensions or com- pressions of the different fibres will be symmetrically disposed with respect to this plane, and the conditions of equilibrium will therefore be the same as though the forces were all situ- ated in this plane. These conditions are, 1°. That the sum of the components parallel to each axis shall be equal to zero ; and, 2°. That the sum of the moments of all the forces taken with respect to any line perpendicular to the plane of the forces shall be equal to zero. 307. We shall assume the origin of co-ordinates at the fixed extremity A of the solid, and refer the points in the curve Ann' Bto the axes of x and y, which are respectively hori- zontal and vertical. 308. The normal plane auv intersects the cylindrical sur- face which contains the fibres of an invariable length, and the vertical plane passing through the axis of the solid, in two lines au and nv, at right angles to each other ; and the points in the section anv will be referred to two rectangular axes, one of which au will be called the axis of ii, and the other, parallel to irv, and passing through the origin a, will be desig- nated as the axis of v. Thus the two co-ordinates of the point 7)1 will be ao=u, and om=v. The moments of the sev- RESISTANCE OF SOLIDS. 153 eral forces will be referred to the line au, which is frequently called the axis of equilibrium. 309. This being premised, we shall denote by A and B, the coefficients of elasticity and tenacity (Arts. 298 and 301), R, the radius of curvature wr, of the curve of flexure, at the point u, s, the length of the arc Au of the curve of flexure, X and y, the co-ordinates Ap and jm of the point u re- ferred to the origin A, x' and y', the co-ordinates of the point ^ referred to the same origin, U and U', functions of the absciss ao=t(., expressing the values of the corresponding ordinates ol and ol' of the curve of intersection, reckoned from the axis of equilibrium au, towards the convex and concave sides of the solid, a, the dimension of the solid estimated along the axis of equilibrium, V, the greatest value of U or U', or the distance from the axis of equilibrium to that fibre which is most stretched or compressed at the instant of rupture. Then, if we consider an elementary portion of the solid, in- cluded between the consecutive normal planes, whose base is represented by the element inm" = du. dv, of the normal section auv, its original length will be equal to uu'=ds ; and after the flexure, this length will be increased or diminished, according to its position with reference to the axis of equili- brium, and will be represented by mm' or nn' {Fig. 141). But from similarity of the figures rmm\ rim', run', we have the proportion rii : rm : m : : uu' : mm' : nn' \ or, R : R + v : R — v : : uu' : inmf : w/i' ; and therefore, R : V : r : : uu' : mm' — uu' : uu! — nn' \ whence, , v.uu' vds mm —uiv=tiu—nn= -5— = -5-. R R 154 STATICS. This expression will represent the variation in the length of the element whose base is equal to du.dv {Pig. 140) and whose original length was equal to ds. To determine the resistance opposed by this element when thus extended or compressed, we employ the expression (143 b) in which we replace 7ia", the variation in length, by mm'—uu', or nu'—nn' ; the transverse section ni, by dii . dv ; and the length ii, by ds : we shall thus obtain an expression for the resistance P" opposed by the element, jy„_vds dudvK _Avdvdu ,-...y v "^ ~R^ ds KT" ^ ''^'' and the moment of this resistance taken with reference to the axis of equilibrium au, will be A A -vdvdu'><.v = -v^dvdu (143 d). SX K 310. The other elementary portions of the solid included between the consecutive normal planes will give similar expressions for the resistances and their moments ; and by taking the sums of these expressions, we shall obtain the value of the entire resistance, and that of its moment with reference to the axis au. To determine the value of these sums, we must integrate the expressions (143 c) and (143 d) throuo^hout the limits of the section a7iv. This intégration is effected, first with reference to one of the variables, v for example ; and its value being then substituted in terms of u, we integrate a second time with reference to the other varia- ble. The limits of the first integration will evidently be v=0, and ^=11, for those fibres which sufier extension ; and v=Q, v—V, for those which suffer compression. The limits of the second integration will be 7i=0, and u=a. 311. This being premised, the sum of the resistances arising from the extensions of the several fibres will be ex- pressed by * An expression of the form y du is intended to indicate that the integral of da is to be taken between the limits u=0, and u=a. In like manner, f vdv sig- nifies that the integral of vdv shonld be taken between the limits v=0, and v=U. RESISTANCE OP SOLIDS. 155 and the sumofthe resistances arising from thecompressions of the fibres will be The sum of the moments of these resistances, taken with reference to the axis au, will be £(/"''"/''"■*+/"''"/"'"'''') (1*3^)- 312. For the purpose of resolving the resistances (143 e) and (143/) into components parallel to the axes of x and y, we must multiply them respectively by -— and--^, the cosines CiS (tS of the angles which their directions form with the axes : but as the curvature assumed by the solid is always found to be exceedingly small even at the instant when the rupture takes place, the expression -— will be very nearly equal to unity, as and the components in the direction of the axis of x may there- fore be assumed equal to the entire resistances. These being the only forces in the system which have components par- allel to the axis of x, the condition of equilibrium which requires that the sum of the components parallel to this axis shall be equal to zero, will be expressed by the equation /a /»U pa /»U' duj vdv—J duj vdv^O (143 /t). The negative sign is given to the resistances offered by those fibres which suffer compression, because they are exerted in a direction contrary to the resistances of the extended fibres. This equation will determine the position of the axis of equilibrium au when the figure of the transverse section is known. 313. A similar condition may be obtained for the com- ponents parallel to the axis of y] but as it will not be required in the succeeding steps of this investigation, it will be unne- cessary to express it analytically. 314. The moment of the force P taken with reference to the axis au will be expressed by V{x'—x), and since this force tends to turn the system about the axis au, in a direô- 156 STATICS. tion contrary to that in which the resistances of the fibres would cause it to turn, the condition that tfie algebraic sum of the moments of all the forces taken with reference to the axis of equilibrium shall be equal to zero, will be expressed by the equation ^{jduj v^dv^jdaj v^'dvS —V{x' —x)^{) .{\^Zi). 315. When the radius of curvature becomes equal to unity the expression (143 g) becomes KyJ duj v^dv^-J dvj v-dv) (143 A-). This quantity is called the moment of elasticity of the solid, and will depend upon the elasticity of the substance, and the figure of the transverse section. Its value will evidently determine that of the force P, which, acting at the extremity of a given arm of lever, will be necessary to produce a given curvature in the solid ; thus, the moment of elasticity becomes a proper measure of the resistance to flexure opposed by the solid. 316. If the flexure of the solid be supposed such that the extreme fibre, or that which undergoes the greatest extension or compression, is about to be ruptured or crushed, the resist- ance opposed by this fibre will be that due to the tenacity of the substance : hence, if dudv denote, as in Art. 309, the base of an elementary portion of the solid included between the consecutive normal sections, and if the distance of this ele- ment of the solid from axis of equilibrium au be equal to V, that of the fibre which is most extended or compressed, the resistance opposed by such element will be expressed by Bdiidv ; B denoting the coefficient of the tenacity. This element being at the distance V from the axis of equilibrium, its original length ds will undergo a variation represented (Art. 309) by Yds R ' tind the corresponding variation in the length ds of the ele- RESISTANCE OF SOLIDS. 157 ment, whose distance from the same axis is denoted by v, will be vds ~K' But the resistances opposed by the two elements being by hypothesis (Art. 293), proportional to their extensions or com- pressions, we shall have the proportion Yds vds r»j J r»// —5- : -5- : : Bdudv : P", SX JK P" denoting the resistance opposed by the element at the distance v from the axis of equilibrium. From this propor- tion we deduce T?"=l.Bdudv. 317. Similar expressions may be obtained for the resistances offered by the other elements ; and by taking their moments with reference to the axis of equilibrium, and adding them mto one sum, we shall obtain for the moment of the entire resistance, at the instant when a fracture commences. This expression is called the moment of rupture, and will depend upon the tenacity of the substance, and the figure of the transverse section. This moment must evidently be equal to the moment V[x' — x) of the force P, which is just capable of causing rupture. Thus, we shall have Y{/,"di/\^^dv+fJduf\^dv'^=V{x'-x)...{U3m). The value of the moment of rupture will serve to determine that of the force P, which, acting at the extremity of a given arm of lever, will be just capable of producing fracture. Thus, the moment of rupture becomes a proper measure of the resistance to fracture opposed by the solid. 318. By comparing the expression (143 k), for the moment of elasticity, with (143 I), which represents the moment of rupture, we shall perceive that the latter may be deduced from the former by merely substituting — for A. 41 ^ 158 STATICS. 319. When the transverse section can be divided sym- metrically by a horizontal line, that line will be the axis of equilibrium, since the equation (143 A) will evidently be satis- fied by regarding that line as the axis of ti. The moment of elasticity will then be expressed by 2Af dut v-clv] t/ t/ and the moment of rupture by — - / dut v^dv, 320. In other cases, it will be necessary to determine the position of the axis of equilibrium by the condition (143 /i), and then to calculate separately the two integrals which enter into the expressions for the moments of elasticity and rup- ture. 321. To apply these principles, we shall determine the moments of elasticity and rupture for those solids whose transverse sections are such as are more commonly adopted in practice. 322. Let the transverse section be a rectangle {Fig. 142), whose breadth and height are denoted respectively by a and b. The value of the moment of elasticity will then become 2 At dut V'dv. pa pib At dut V' •/ t/ and by integrating with reference to v=om, between the limits v=0, and v=ot=^b, we shall obtain double the sum of the moments of all the elements, whose bases constitute the ele- mentary rectangle oq. Performing the integration, we have 2Ar duy."^ =2Ar duy.^^l=i-^Ab^r du. Integrating a second time, with reference to w, between the limits M=0 and u=a, we shall obtain for the moment of elasticity «, «=y_Aa&' (143 w). Hence it follows that the resistance to flexure opposed by a solid whose transverse section is rectangular, will be propor- tional to the breadth and the cube of the depth. T> 323 If we replace A in this expression by —, we shall RESISTANCE OF SOLIDS. 159 obtain the moment of rupture $ of the rectangle ; and since V is in the present case equal to 16, we shall have i3=JBa6= (143 o). Thus the resistance to fracture is proportional to the breadth and the square of the depth. 324. If the solid be disposed in such manner that the dimension a shall become vertical, and the dimension b hori- zontal, the expressions for the moments of elasticity and rup- ture will become respectively and by comparing these expressions with those obtained, when the dimension a was supposed horizontal, we shall deduce the proportions «:»':: ab^ : ba^ : : b* : a*, /3 : /3' : : ai^ : ba' : : b : a. It thus appears that the resistance to flexure when the broader face b is placed vertically, will be to that exerted when the narrower face a is vertical, as the square of the broader face to the square of the narrower. But that the resistances to frac- ture in similar cases are proportional simply to the first powers of the same quantities. 325. If in the expressions (143 n) and (143 0), we make az=b, we shall obtain for the moments of elasticity and rup- ture of a prism with a square base, «=^TVAa% 0=lBa^ (143 jo). 326. Let the transverse section of the solid be a rhombus, {Fig. 143), whose diagonals are represented by 2p and 2q, and let the diagonal 2q be placed vertically. If we first determine the moment of elasticity of the triangle oBG, that of the rhombus can be immediately deduced by simply multi- plying by the number 2. The limits between which the first integration with reference to the variable v=ow, should be effected, are -«=0, and v=ot. But from the similarity of tri- angles, we have the proportion ao : ot : : oD : DC, or, u: ot ::p : q] 100 STATICS. whence, ot=— ; P and the limits of the first integration will therefore be v=o, and v=—. Making these substitutions in the general for- mula for the moment of elasticity, we shall obtain 2Af''duf"^v'>dv=2Af"duX^(^-^) '=tA^' f\=du. t/ot/o «/o \p / ]J^»/ Integrating a second time, with reference to the variable u, between the limits 71 = 0, and u=p, the moment of elasticity of the triangle aBC becomes and by doubling this expression, we find for the moment of elasticity « of the rhombus, T> 327. If in this expression we replace A by —, we shall ob- tain the value of the moment of rupture &, which, since Y=q, will become ^ = L-Xpq= = ^Bpq='. 328. If we make 2y=q, the rhombus will become a square, and the values of » and p will reduce to or if the side of the square be denoted by a, we shall have the relation a^=2p'', and therefore «=iAx^=-i-Aa*, '5=iBx 2^ =-g-^ Bad- aud by comparing these expressions with those obtained (Art. 32o) for the moments of elasticity and rupture of a prism with a square base, when the sides of the base are respectively vertical and horizontal, we shall find that the resistance to flexure will be the same whether the diago- nal or side of the square be disposed vertically; but that the resistance to fracture when the side is vertical, will be RESISTANCE OF SOLIDS. ^61 greater than when the diagonal is vertical, in the ratio of v^(2) to 1. 329. When the section is a circle whose radius is equal to r, the integration with reference to the variable v must be effected between the limits v=0, and v=^{2ru — u"); and the second integration, with reference to u, between the limits u=0, and u=2r. Thus, the expression for the moment of elasticity will be «=2A/ duj v^dv = \kj ^ {2ru—u''ydu..{U3q). For the purpose of effecting the second integration, we make r—u—z, which gives du= — dz, 2ru — w^ =r^ — z^. Substituting these values in the expression for «, and observ- ing that the limits u=0, and ?«=2r, correspond to the values z=-{-r, and z=—r, we shall obtain {2ru—u^ydu=—J ^ {r'—z^ydz = -f[^{r^ —z^ )(r= -z^ fdz \ or, r i^ru—u'' y du =J z^ {r' —z' fdz -J^j'ir^ -z^Ydz (143 r). The first term of the second member, being integrated by parts, gives f'z^ (r« —z^fdz^-r^r^ —z^Y2zdz= -{r^-z^f .'^^■^\fl{r^-z^fdz (143 s). 3Z The quantity {r^—z")"'^ will reduce to zero, when z=-f-r, or z=—rt this term will therefore disappear; and the last term being resolved into factors will reduce equation (143 s) to [[z^r^ —z^fdz=lj*jr* -z^fr^ —\J_J^'' -z^Vz'dz'r whence, by transposition and reduction, we obtain 16^ STATICS. This value being substituted in (143 r) gives J \2ru—u''ydu— — lr^j\r^ —z'^fdz. But the integral / (7'^ —z^ydz represents the area of a «/ semicircle whose radius is equal to r. This area being ex- pressed by ^?rr", we shall have •2r {2ru—u^ydu=—^^^\ and by substituting this value in the expression (143 q) for the moment of elasticity «, it will become 330. To determine the moment of rupture j8, we replace A by — or —, and thus obtain V r 331. By comparing these values with the expressions (143 jo),we shall find that the moments of elasticity and rup- ture of a square are to those of the inscribed circle as 1 "> Î6- 332. The moment of elasticity of a tube or hollow cylin- der whose exterior and interior diameters are represented by r' and r", will be determined by taking the difference of the moments of the exterior and interior sections. Thus we shall have «=iA5r(r"'— r"*), and the moment of rupture /3 will be found by replacing A by B B , ^or~; hence, (8-iBT ~ — r' 333. If the section of the hollow cylinder be supposed equal tothat of a solid cylinder, the radius of the latter being denoted by r, we shall have the relation f2 ^=,f'2 y>"3. RESISTANCE OP SOLIDS. 163 and the resistances to fracture opposed by the two will be to each other as J— : {r'^-— r"^y, or as ~^~ : {r'^—r'"')' ; replacing r'^ _ r"^ by its value r^, this ratio will be reduced to r The first term of this ratio must always exceed the second : thus the resistance to fracture opposed by the hollow cylinder will always be greater than that offered by the solid cylinder ; and since the value of the first term may be increased indefi- nitely without affecting that of the second, it follows that the resistance of the hollow cylinder may likewise be in- creased indefinitely without changing the area of its section. 334. Let a and b represent the breadth and height of a rectangle inscribed in a circle whose diameter is denoted by D : we shall have the relation a=^ -|-6=^ =0^ ; and therefore, «62=«^D2— «3). But the moment of rupture of a rectangle being proportional to the breadth and the square of the depth (Art. 323), if we wish the resistance to fracture to be a maximum, we must differentiate the preceding expression with reference to a, and place the first differential coefficient equal to zero : we shall thus obtain da and therefore, :I)'-—3a'=0: Hence, the strongest rectangular solid which can be cut from a given cylinder will be that in which the diameter of the cylinder, the depth of the rectangular section, and its breadth, shall be to each other as the square roots of the numbers 3, 2, and 1. L2 16^ STATICS. Of the Figure of the /Solid after Flexure. 335. We will now consider the form of the curve Aiiu'B {Fig. 140) assumed by the fibres whose lengths remain inva- riable. For this purpose, let AM {Fig. 144) represent the solid which is firmly fixed at its extremity A, and subjected to the action of the weight P, applied at the other extremity, in a direction perpendicular to the original direction of the axis of the solid. Then denoting by « the moment of elas- ticity, the equation (143 i), which expresses a condition of equilibrium, when the solid merely undergoes flexure, with- out being ruptured, will become or by substituting for the radius of curvature R its general value J '^ , this equation will reduce to _d^ d^ (^-'1^) 336. In Uke manner, when the solid is about to be rup- tured, if we substitute /? for the moment of rupture, in equa- tion (143 m), we shall obtain li=V{x'—x) (143 w). 337. Let c denote the horizontal distance AB between the extremities of the solid, /, the ordinate BM, s, the length of the arc AmM^ », the angle included between the tangent to the curve at the point M and the horizontal line. Then, since the curvature is supposed to be extremely small, even at the instant when fracture takes place, the expression -^, which represents the tangent of the angle formed by the RESISTANCE OF SOLIDS. 165 element of the curve with the axis of x, will also be extremely- small, and its square may therefore be neglected in compari- son with unity. Thus the equation (143 1) will be reduced to Multiplying by dx, we obtain »--^dx=Vlc — x)dx ; dx^ ^ and by integration, we have -l=K— Ï) (^*^''>- The arbitrary constant introduced by integration is equal to zero; since, when ar=0, —^, which represents the tangent of the dx angle included between the element of the curve and the axis of abscisses, is likewise equal to zero. Multiplying again by dx we have »-~dx~Vl ex — ^ \dx: dx V 27 ' and performing a second integration, there results the constant will be equal to zero, since x—0 gives y=0. 338. If in this expression we make x—c, the ordinate y will become equal to/; hence we shall have ^-(2-6)=.-^ 3 ("^">- In like manner, by making x—c in equation (143 v), we shall have '^= tang <», and therefore dx P/ , c2\ P ça P 3/* or, replacing — by its value — deduced from the preceding equation, we have 3/- tang «= ;j- (143 x). i6ê STATICS. 339. To determine the length s of the arc A^nM, we take the general expression for the element ds of this arc, which, being developed, rejecting all but the two first terms as inconsiderable, gives and by replacing — ^ by its value (143 v), this equation ax becomes pa Integrating, we obtain F- {c^x^ cx'^ , x'\ and by making x=c, the value of the entire arc AwiM becomes pa /^6 f.s c5v pa c« ^='=+^i6-8+ro)='^+^^r5' pa or, replacing — by its value deduced from equation (143 w\ this expression reduces to 5=c + |: (143 2/). 5c 340. When the weight P is just sufficient to fracture the solid, the rupture will take place at the supported end ; since the moment V{c—x) of the force P will be the greatest when a:=0: the equation (143 ii) will then become /3=Pc (143 z) ; diJ^ or, if the curvature be still supposed so small that — - may be neglected in comparison with unity, the equation of the curve will be the same as when the flexure was extremely slight, and we shall therefore have /3 P= 3/^ ^ 5c 341. Let it now be supposed that the solid is loaded with RESISTANCE OF SOLIDS. 167 weights distributed uniformly throughout its length. Denote by z the absciss of any point between M and wi, and by p the weight supported by a portion of the solid which corresponds to a unit of length of the absciss : then since the distribution of the weights is supposed uniform, we shall have the proportion \: J) w dz '. pdzj the weight supported by the element of the solid whose pro- jection on the axis of x is represented by dz. The moment of this weight, with reference to the point w, will be pdz{z — x), and the sum of the moments of all the weights supported between M and m, taken with reference to the same point m, will be .fp{z — x)dz. This integral should be taken between the limits z=c and z=x, the quantity x being regarded as invariable : thus we shall have •/ X q2 ^2 p{z—x)dz=p — ~ px{c—x) (143 a'). 2 But the condition of equilibrium requires that the sum of d^V these moments shall be equal to « -^, the sum of the mo- dx'' ments of the resistances offered by the several fibres. Hence, we obtain «-^ =p (t-^\ —px{c—x) = i/?c3 —pcx+lpx^ . d integrating, we obtaii Multiplying by dx, and integrating, we obtain dy_ dx and multiplying a second time by dx^ and integrating, there results »y=p{\c^x^ —\cx^-[-yc'). Making x=c, y=f-, and -^=tang -c.r+i^-^) (143 <^'); or, if the solid be supposed on the point of being ruptured, the fracture taking place at the point A, for which a;=0, the condition of equilibrium will be /3=Pc-f ipc^». RESISTANCE OF SOLIDS. 169 345. The expression (143 d') gives, by two successive integrations, and by making x=Cj y=/, and ~^= tang «, we obtain tano-a;=— (iP + i«c')= ^P+P' 4/ ,5=c(P+i;^c) = c(P+iF) 346. When the soHd is supported in a horizontal position at its two extremities M and M' (Fig: 145), and loaded with weights at its middle point A, the results obtained Arts. 337- 340 will apply to each half of the curve assumed by the solid ; for we may regard either half as perfectly immoveable, and suppose the other portion to be solicited by a force acting at its extremity and equal to the resistance offered by one of the points of support. Hence, if we denote by 2P5 the weight suspended at the middle point, 2c, the distance between the points of support, 25, the length of the curve, /, the sagitta CA, a, the angle included between the line MM' and the tangent to the curve at M or M' ; the resistance exerted by each fixed point in the vertical direction will be equal to P, one-half the weight applied at A, and the formulas (143 w), (143 x), (143 y), and (143 z) will become immediately applicable to the present case. Hence, P c2j2c^ 2P •^ «'3 « -48 (^"^^^h tang .=g 25=2c+^^ oc /î=cP (143/). 15 170 STATICS. The value of / indicates that the depression of the soHd at tlie middle point, or the sagitta AC, will be proportional to the weight 2P, and the cube of the distance between the points of support. 347. The expressions deduced in the preceding article have been obtained upon the supposition that the resistances op- posed by the fixed points were exerted in a vertical direction ; whereas, the resistance is actually exerted in the direction of the normal to the curve at the point M or M' ; and in some instances the inclination of this normal to the vertical line is too great to be neglected. This circumstance will seldom occur except in the case of fracture, the curvature of the solid being then greater than in the case of a mere flexure. If we represent the resistance exerted at M' by the hne M'F, and resolve this force into two components which shall be respectively vertical and horizontal, the latter com- ponent ME will be equal and opposite to the similar compo- nent of the resistance at the point M, and the vertical component M'D will be equal to P, or to one-half the weight supported at the ?Tiiddle point of the solid. The value of the horizontal component M'E may be readily found ; for we have M'E =DF=M'DX tang DM'F=P .tang«. When the equilibrium subsists, and the solid is on the point of being ruptured, the moment of rupture must be equal to the sum of the moments of the vertical and horizontal com- ponents. The moment of the former, with reference to the point A, has been found equal to cP ; that of the latter will obviously be P tang a xAC=P tango- ,/; thus, the con- ditions of equilibrium will become j8=Pc + P tang*./; or, if we suppose the curve to be represented by the same 3f equation as in Art. 337, in which case tang*'=^, this rela- tion may be written 348. If the weight be uniformly distributed throughout the RESISTANCE OF SOLIDS. 171 length of the solid, we may regard each half as firmly fixed at the point A, and solicited at the same time by a system of parallel forces applied at every point of the solid, and acting downwards ; and by a single force equal to their sum, or to the resistance offered by the point of support, applied at the extremity of the solid, and acting upwards. Thus, the case will be the same as that considered in Art. 344, with the ex- ception that the forces arising from the weights uniformly distributed along the solid are exerted in contrary directions. The equations obtained in that case will therefore become applicable to the present one by simply changing the signs of the moments of these forces, and replacing P by pc ; we shall thus obtain a,^=,Cp{cX — \X')—p{\c^X — \cX^-{-\x'^) {^^'^ g')'\ ay—cp{\cx''—\x^)—p{\c''x''—\cx''-\-^^x^) ..... (143 /i'); ç>—cp . c — cp .\c (143 Ï) ; making x=c^ y=-fi -f— tang a», we obtain ^■x C»=- 24' tang. = ^(l-i-i+^-i)c3=i?Ç=|', ^=cp.\c (143/:'). By comparing this value of/ with that obtained in equa- tion (143 e'), it will appear that the depression of the solid at its middle point produced by a weight 2pc uniformly dis- tributed throughout the solid, will be less than that produced by the same weight suspended at the middle point, in the ratio of 5 to 8. And by comparing the values of — given by equations (143 k') and (143/') we shall perceive that the solid will be equally liable to fracture by the action of the weight 2pc distributed uniformly, or by half that weight applied at its middle point. 349. The preceding expressions, like those in Art. 346, have been obtained upon the supposition that the resistances offered by the fixed points are exerted in vertical directions. 172 STATICS. In the case of riiplure, the hne of direction of the resist- ance may deviate so far from the vertical as to render the above supposition inadmissible. We then resolve this resist- ance, as in Art. 347, into two components respectively vertical and horizontal ; the former will be represented by pCy and the latter by jjc • tang a. In case of equilibrium, it will simply be necessary to add to the second member of equation (143 i') the moment jjc . tang « X/, of the horizontal component; thus, we shall have ^ = cp .c—cp. Ic-jrcp . tang « ./=rc/v(|e+/tangc=P+P': thus, when we suppose the resist- ances exerted by the fixed points to act vertically, we shall obtain, by substituting P-f-i>c for pc in the first terms of the second members of equations (143 h') and (143 i'), «y=(P+pc)(ic.r2 —}z-^)—p{lc^x- —}ca;^+^\x*), ^ = (P+pc)c—cp . ic (143 r) ; which give, by making a:=c, y=f, and pc=P', 351. But, if regard be had to the oblique direction of the resistance, as may be necessary in the case of rupture, we must add the moment of the horizontal component RESISTANCE OP SOLIDS. 173 to the second member of equation (143 1% which thus becomes ^=(P+2?c)c— cp . ic+(P+pc) tang a, ./; and therefore, gp_ 2^-F(c+2/tang^) ^ c-\-f tang a The equation (143 g') likewise gives, by replacing pc in the first term of the second member by P+^c, and making |=tang », 3P + 2P' 4/ and this value of tang a may be regarded as sensibly equal to that employed in the preceding expression for the value of2P. 352, To apply the several results which have been ob- tained to particular cases, it will be necessary to substitute the values of the moments of rupture and elasticity apper- taining to the figure of the transverse section. We must likewise assign to A and B the coefficients of elasticity and tenacity, their particular values which depend upon the nature of the substance, and which are supposed to have been pre- viously determined by experiment, 353. The best method of determining the values of A and B consists in supporting a prismatic solid at its two ex- tremities in a horizontal position, loading it with weights at its middle point, and observing the sagittas which correspond to different weights ; or simply, the weight and sagitta at the instant when the fracture is about to take place. If the transverse section of the solid be a rectangle, whose breadth and height are denoted respectively by a and 6, we shall have (Arts. 322 and 323), et=j\kab'', ^=iBa62 ; and if we neglect the weight of the solid (Arts. 346 and 347), and by eliminating « and /3, we obtain, for the case of simple flexure, 174 STATICS. /=2P-^?^, orA=2pi?$4. (143^0; and for that of fracture «=^Pa^O+i?) ("=>"')' 2c being the interval between the supports, and 2P the weight with which the solid is loaded. The values of A and B are thus expressed in functions of quantities which are readily determined by observation. 354. If the weight of the solid 2P' be likewise taken into consideration, it will simply be necessary (Art. 350) to add 1 . 2P' to 2P in equation (143 m% and to replace equation (143 n') by the formulas of Art. 351 : we shall thus have, i» the case of flexure, /=(2P + |.2P')-i?^, A=(2P + f.2F)-^^')' and for that of fracture, 6g _ (2P+2FXc+/.tang<^)— P^c 3P + 2P' 4/ *""^^=8P+5P'-T 355. If the solid be loaded with a weight 2Q,, and if the corresponding sagitta be denoted by f, we shall obtain a value for/' similar to that of /in the preceding article : thus "we shall have, and by taking the difference between / and /', the weight of the solid 2P' will disappear, and we shall obtain /'_/=(2a-2P)_^, A=(2a-2P) — (^^^' 4Aa63' ^ -^ ' 4.abHf'—f) Thus, it will only be necessary to observe the increase /'— / in the sagitta, which corresponds to a given increase 2Q,— 2P in the weights suspended at the middle point. RESISTANCE OF SOLIDS. 175 Of Solids of equal Resistance. 356. When a solid having the prismatic form is subjected to an eflbrt which tends to break it, there will always be a particular point at which the fracture will be most likely to take place. For, the moment of rupture will be the same at every point, whilst the moment of the force applied will de- pend upon its distance from the point with reference to which the moments are taken. Hence, if the strength of the solid be sufficient at that point where a rupture is most likely to occur, it will be unnecessarily great at other points. 357. It becomes an object, therefore, to determine the figure of the solid which shall be uniformly strong through- out, since the adoption of such a figure may frequently effect a material reduction in the quantity of materials employed. Solids having such figures are called solids of equal resist- ance. 358. As an example, let a body ABM {Fig. 146), whose upper surface AB is horizontal, and. whose two lateral faces are vertical, be firmly fixed at its extremity A, and subjected to the action of a weight P suspended from its other extrem- ity. It is required to determine the form of the under surface BmM such that the solid may be equally strong throughout, or that the moment of the weight P taken with reference to any point in the length of the solid, shall be equal to the mo- ment of rupture of the transverse section at the same point. Denote by a the breadth of the solid, h the height AM, c the length AB, x the variabl-e absciss Bjo, and v the corre- sponding ordinate jmi : the moment of rupture of the section ah^ AM will be (Art. 323) B— - ; and since this must be equal to the moment of the force P, we shall have In like manner, the moment of rupture of the section pm av^ will be B-^ , and the moment of the force P with reference 6 to a point in this section will be Vx. These moments being 176 STATICS. equal by the conditions of the problem, the general relation between the quantities v and a: will become P:r=B-— , v^ = — . D C This equation evidently appertains to a parabola, the axis of which will be the line AB. 359. To determine the figure of the curve assumed by the solid when bent, we observe that the moment of elasticity of the section p?n will be (Art. 322) A— — =A —, Hence, if ^^ 12c^ y denote the ordinate of the curve of flexure corresponding to the absciss Ap=c—x, the conditions of equilibrium in case of flexure will be (Art. 337) A X — 4=P^. Performing two successive integrations, and remarking that when x—c^ -/-=0, and y=0, we obtain ax and by making a;=0, and y=/, we find, for the depression of the extreme point B, P 8c^ ''~'A.'ab^' By comparing this expression with that obtained in equa- tion (143 w), it will appear that the depression / is twice as great in the present instance as when the solid had the pris- matic form. 360. If the weight supported by the solid be distributed uniformly along its length, each unit of length being sup- posed to support a weight jh '^he sum of the moments of these weights, taken with reference to the point A, will be (Art. 342) pc.ic; and the condition of equilibrium will therefore be B-—=pc.\c. PRINCIPLE OF VIRTUAL VELOCITIES. 177 In like manner, the sum of the moments of the weights sup- ported between the points p and B, taken with reference to the point /?, will be fx . \x. Hence, we shall have „«v^ , hx b ^ ^ c the equation of a right line. 361. The preceding examples will be sufficient to illus- trate the manner in which the form of the solid of equal re- sistance may be determined when the distribution of the load is previously known. Of ike Principle of -Virtual Velocities. 362. The principle of virtual velocities, which was dis- covered by Galileo, and very fully developed by John Bernouilli and Lagrange, may frequently prove of great utility in stating the analytical conditions of statical problems. Indeed, it is regarded by Lagrange, who has adopted it as the basis of his "Mécanique Analytique," as so essential, that he considers all the general methods which can be employed in the solution of questions relating to equilibrium, as being nothing more than applications more or less direct of this general principle. 363. A virtual velocity is the path described by the point of application of a force, when the equilibrium is disturbed in an infinitely small degree. Thus, by supposing that the point of application vi of a force P {Fig. 147) is, by an instantaneous derangement of the system, transferred to ??, the small line mn which it describes is called the virtual velocity of the point m. 364. If this virtual velocity be projected upon the direction of the force, it will occupy thereon the small space ma, and the product of the force P by this projection ma is called the moment of this virtual velocity, or, sometimes, the moment of the force ; it should however be observed, that the term moment is here employed with a very different signification from that usually implied. The principle of virtual velocities, as will be demonstrated, M 178 STATICS. consists in this, that when the system is in equihbrio, the sum of these moments is equal to zero ; thus, if P, P', P", &.C., represent different forces appUed to a system, and p, p', ;j", &c., the projections of the virtual velocities on the directions of these forces, we must have in case of equilibrium, Fp + P'p'+F'Y +Ôcc.=0 (144). It is necessary to remark that when any one of these pro- jections p, p\ p'\ (fcc, falls upon the prolongation mh {Fig. 148) of the force P, applied at m, this projection must be regarded as negative ; and since the forces P, P', P", '+P>"+(fcc.=0 ; from which we conclude that the principle of virtual velocities is true when the forces are applied to a single point. PRINCIPLE OP VIRTUAL VELOCITIES, 179 366. The most general case of this principle which usually presents itself, is that in which the several forces P, P', P", (fcc, are applied to different points of a body or system of bodies : these points preserving their distances invariable, may be regarded as connected with each other by inflexible right lines. Before examining the general state of the system when the equilibrium has been slightly disturbed, we will consider singly one of these inflexible right lines mw', at the instant when the point m has been brought into the position denoted by ?i. The other extremity m! of this right line will at the same time change its position, and may be situated either above mm' {Fig. 150), or beneath it {Fig. 151) : let it be first supposed above mm', and the line 7nm' will then assume the position mi' {Fig. 152) : the lines mn and on'n' may be regarded as infinitely small when compared with the lines min' and nn', since the derangement of the system is supposed infinitely small. If the points m and n' be con- nected by a right line we shall form a triangle m/m'n', in which the side m'n' being infinitely small, the angle n'mm' will like- wise be infinitely small, and the arc w'a, which measures this angle, may therefore be regarded as a right line. But this arc beinof described with a radius ma, if we assume mb=zma {Fig. 153), the angle bn'a being an angle in a semicircle, will be equal to a right angle, and may be considered equal to the angle mn'a. For, since the angle ii'nia is infinitely small, the angle m,7i'b must be so likewise, and the angles bn'a, mn'a, will therefore differ by an infinitely small quantity. Thus, the triangles mn'a and ii'la {Fig. 152) being right-angled and having a common angle a, will be similar, and we shall there- fore have the proportion m,a : n'a : : n'a : la. But n'a being infinitely small with respect to ma, la must be infinitely small with respect to n'a ; and since n'a is an in- finitely small quantity of the first order, la will be one of the second order. Hence, the quantity la may be neglected, and mrî may be regarded as equal to ml ; thus we shall have mn'=mm'-±m'l. In a similar manner may it be proved that if with the point nf M2 180 STATICS. as a centre, and radius n'm, we describe the arc ma', we shall obtain tmi'=7in'-^7ih, and by placing these values of m?t' equal to each other, we find vnn' -\- ni'l=7in' + nh ; but the right line min' being supposed inextensible, it must preserve its length invariable in its new position ; hence, mm'—nn' ; and by suppressing these equal terms in the pre- ceding equation, we obtain '}n'l=nh. Again, the lines Qnm' and nn' form with each other an infi- nitely small angle ; for, if they intersect at a point o {Fig. 154), we shall have a triangle m'on', two of whose sides are of finite extent, the third side m'n' being infinitely small ; thus, the angle o will likewise be infinitely small. It results from the preceding remarks, that if the perpendicular iik be demitted on the side oTim' (Fig. 152) we shall have nh=9nk] and by substituting this value of nh in the preceding equation, we find 'm'l=mk, which proves that the projections mk and m'l of the virtual velocities mn and m'n' of the points m and m' are equal to each other. 367. Let us now suppose that the point w {Fig. 155) is transported to n, and that the extremity m' falls at n' below Tnm'. It may be proved as in the former case, that the angle is infinitely small, and consequently that the projections ol and oh may be regarded as equal to o?i' and 07i ; whence, on'=om'-{-m'l, on =07n — mA ; by the addition of these equations, we obtain 07i'-{-o7i—om'-\-om-\-m'l— mh ; or, 7in' =mm'-\-m'l — mh ; but 7in' and mtn' are equal to each other, and therefore 7n'l—7nh, PRINCIPLE OP VIRTUAL VELOCITIES. 181 which proves that the projections of the virtual velocities are still equal. 368. In this demonstration it has been supposed that the derangement of the system is such as to preserve the lines mm' and nn' in the same plane. This restriction is however entirely unnecessary. For,if we suppose that mm' and nn'^re not contained in the same plane, we can draw through the points n and 7t' {Fig. 152) planes perpendicular to the line mm', intersecting this line at the points k and I, the projections of n and n'. Then, if a line be drawn through any point of m>m' parallel to nn', and terminated by the perpendicular planes, such line will evidently be equal to nn', and its ex- tremities will likewise be projected on the line mm', at the same points k and I. Hence, if the property be true for the parallel line which intersects mm', it will likewise be true for the line nn'. 369. It should be observed, that in each of these cases, the projections will be affected with contrary signs, one falling upon the line mm', the other upon its prolongation. This appears from an inspection of the figures 152 and 155, and it likewise results from the consideration that if the two projections fell upon the line or upon its prolongations, the length of nn' would necessarily be greater or less than that of mm', which by hypothesis, is impossible. 370. It follows from the preceding remarks, that if we sup- pose two equal and opposite forces to act in the direction of the line mm' on the points m and m', and denote by v and v' the projections of the virtual velocities mn, and m'n' on the line of direction of the forces, we shall have V— — v' ; and consequently, that if we represent by {mm!) each of these equal forces, we shall obtain {mm')v + {mm')v'=0 ; which proves that the forces represented by {mm') being applied at the extremities of the right line, and being regarded as sustaining those points in equilibrio, the sum of the moments of the virtual velocities of these points will be equal to zero. 371. By the aid of this proposition it will be easy to 16 182 STATICS. establish the principle of virtual velocities in the case of any number of forces applied to different points. For, let P, P', P", «fee. {Fig. 156), be several forces applied to the points m, in\ m", " + P"'7/"=o. PRINCIPLE OF VIRTUAL VELOCITIES. 183 The same demonstration is evidently applicable to a greater number of forces. 373. As an example of the manner in which the conditions of equilibrium in any machine may be inferred from the principle of virtual velocities, we will suppose the relation between the power and resistance in the lever to be unknown. The forces exerted upon the lever are the power P, the resistance P', and the reaction of the point of support. If a slight motion be communicated to the lever, causing it to turn about its fulcrum, this fulcrum will remain immoveable, and the moment of the reaction exerted by this point will there- fore be equal to zero. Hence, the principle of virtual velo- cities will give or, Pp = —Fy (146). This being premised, let the values of the quantities jo andp' be now determined. Let C represent the fulcrum of a lever mm' {Pig. 157), which being slightly removed from its po- sition of equilibrium has assumed the position mi' ; the angles at C being equal to each other, the arcs mn, m'n' will be pro- portional to the radii with which they are described, and we shall therefore have mn : m'n' : : Cm : Cm' (147). But if through the points n and n' perpendiculars be drawn to the directions of the forces P and P', we shall have mr = — p , m'?-' —p ', the negative sign being prefixed to p, because it falls on the prolongation of the force P. The arcs being regarded as in- definitely small right lines, the right-angled triangles mm, m'r'n' will be similar ; for the isosceles triangles 7nCn, 7n'Cn' give angle 7imC= angle n'm'C : and by subtracting these equal angles from the right angles rmO, r'm'C, there will remain anffle 7-mn=an2fle r'm'n'. Thus, the triangles rmn, r'7n'n' will be similar, and will give the proportion 184 STATICS. mn : m'n' :: mr : mY ; or, mn : m'n' : : — p : p' ; and therefore the proportion (147) may be converted into Cm : Cm' : : — p : p'. But the equation (146) which expresses the principle of vir- tual velocities gives rise to the proportion F :P:: -p :p'; whence, by the equality of ratios, Cm : Cm' : ; P' : P, or the forces are in the inverse ratio of the arms of the lever. Of the Position of the Centre of Gravity of a tSystem when in Equilibrio. 374. Let m, m', m" (fee, be the centres of gravity of different bodies which are connected together in an invariable manner ; let perpendiculars z, z', z", &c., be demitted from these points on the plane of xy, supposed to be horizontal ; the weights P, P', P", &.C., of the several bodies, which may be regarded as suspended from the points m, m', m"y ace, will act along the directions of these perpendiculars. If z/ denote the co- ordinate of the centre of gravity of the whole system, we shall have (Art. 166) P^-fP^^^-fP^s'^+&c. ^/-- p+P'4-P"+&c. • When the system of bodies changes its position, the ordi- nate z becoming z + hyOr z — h, the increment of 2; will affect the values of z', z'\ z", (fee, since the points m, m\ ?n", (fcc^ being connected in an invariable manner, the value of z. cannot change without the values of z', z", (fee, undergoing a corresponding alteration. Although we are generally unac- quainted with the law of dependence which exists between the positions of the different bodies composing the system, the preceding equation may nevertheless be written under the form _ Pz-^V' 388. Before investigating the expression for the value of the incessant force, it will be necessary to discover the rela- tion which exists between the force and the velocity. If a force P be supposed to communicate a velocity v to any body, a force 7i times as great will communicate to the body a velocity equal to nv. The truth of this proposition might well be questioned, since the nature of forces being entirely unknown, we cannot affirm that a double force will necessarily produce a double velocity ; or, in general, that a single force equal to the sum of two others, will necessarily produce a velocity equal to the sum of the velocities which the two forces would separately produce. But the fact being confirmed by universal experience, we adopt it as a principle. Thus, by supposing different forces applied to the same body or material point, their relative intensities can be estimated by comparing the velocities which they would severally com- municate. The proper measure of an incessant force will be the velocity which it can generate in a given time ; but the in- tensity of the force being constantly variable, we must sup- pose the force to become constant at the instant when we wish to estimate its value, and the measure of the force will then be the velocity generated in the unit of time succeeding this instant. The velocity communicated by this incessant force during the unit of time, when it is supposed to retain a constant value, will obviously be unequal to that which would have been communicated b]f the variable incessant force, in the same time. 389. The preceding remarks indicate the method of meas- VARIED MOTION. 198 uring the incessa,nt force ; since they determine the ratio in which the intensity of the force varies in different times. If, for example, at the expiration of the times t and t\ the incessant force, having become constant, can generate in a second of time velocities represented by the numbers 60 and 20, we infer that the intensity of the force at the end of the time t is triple its intensity at the end of the time t' . 390. To deduce from the above definition the analytical expression for the incessant force, let v represent the velocity acquired by the body at the end of the time t ; then, at the expiration of the time t-\-dt^ the velocity will become v^dv ; consequently, dv will be the velocity communicated during the time dt ; but if at the end of the time t the intensity of the force be supposed to become constant, there will be com- municated to the body in the instant dt which succeeds the time /, a velocity represented by dv ; and the same effect will be repeated during any number of succeeding instants ; so that the velocities communicated after the expiration of the time t^ in the instants dt,2dt, 3dt, &c., will be expressed by dv, 2dv, 3dv, &.C. : and consequently, the velocity communicated in the unit of time which succeeds the time t, will be equal to dv repeated as many times as dt is contained in unity. This number being expressed by -— , it follows that — x dv, or — , dt dt dt will express the effect of the force or the velocity generated in a unit of time. If, therefore, we denote this force by ^,we shall obtain for the second equation of varied motion, ^4 ("«)• The character ç will hereafter be used to designate the inten- sity of the force ; the force being represented by the effect which it produces. 391. From the preceding equation we obtain çdt—dv] thus, if the incessant force be given, the increment to the velocity in the time dt can be readily calculated. 392. By eliminating dt between the equations (148) and (1 49), we obtain a third equation of varied motion, çds=vdv. N 17 194 DYNAMICS. Of Uniformly Varied Motion. 393. The incessant force imparting at each instant a new impulse to the body, if these impulses are equal in intensity, the body will acquire the same velocity in a unit of time after the expiration of the time t, as it would after a time t'. Let this velocity which is constantly generated in a unit of time, be denoted by ^ ; we shall then have Substituting this value in the equation ^ dv '^=dt^ we shall obtain dv=gdt] and by integrating and denoting by a the constant which will thus be introduced, we find v=a+gt (150).* We have likewise obtained for'the value of the velocity ds hence, if we eliminate v between these two equations, we shall have ds={a-i-gt)dt, from which, by integration, we find s=b + at+^gt' (151), the quantity b being an arbitrary constant. * This equation might also have been obtained from the following considera- tions : Let it be supposed that a body in motion has acquired a velocity a: if it then be solicited by a constant force which communicates to it a velocity g in each second of time, the velocity of the body will become a-\-g, at the end of one second, a-\-2g, at the end of two seconds, a-{-3g, at the end of three seconds, a-\-tg, at the end of t seconda : thus, if we represent by v the velocity of the body at the expiration of the time /, we shall have UNIFORMLY VARIED MOTION. 195 If ^ be supposed positive in this equation, the motion will be uniformly accelerated, but if negative, the motion will be uniformly retarded. 394. If we make ^=0, we find h=s\ thus, h will represent the initial space, or the distance of the body from the origin, at the instant from which the time is reckoned. The constant a is equal to the initial velocity of the body, as appears by making ^=0 in equation (150). 395. When the initial space and initial velocity are each equal to zero, the equations (150) and (151) become •^^gt (152), s^\gt^ (153), and the body then moves from rest, under the action of the incessant force. 396. Let s and s' represent the spaces described in the times t and t', under the action of a force g ; the equation (153) gives s=^\gf, and s'==\gV* (154) ; whence we obtain the proportion s:s' \:t^ '.t'^ (155). Consequently, the spaces described by a body in different times, when it moves from rest, being solicited by a constant accelerating force, are proportional to the squares of those times. 397. The equation (152) gives v=gt, and v'=gt', whence, V :v' ::t: t', and by comparing this proportion with (155), we have v:v'::^s: ^s'. Hence it appears that the tim,es elapsed are constantly pro- portional to the velocities, or to the square roots of the spaces described in those times. 398. If we make ^=1, the equation (153) becomes sz=ig. In this case, s represents the space described by the body in the first unit of time, and it appears that this space is N2 1^ DYNAMICS. equal to one-half the quantity g, which represents the mea- sure of the accelerating force. It has been found, for example) that a body subjected to the action of gravity, would describe in the first second of time, in the latitude of New-York a distance equal to 16.0799 feet, or nearly 16j^ feet ; this value being substituted in tht? place of s in the preceding equation, we find ^=32.1598 feet, or nearly = 32i feet. 399. The equation (153) will determine the space described in a given time ; for example, if t=6", we shall have 5 = 1^^2^1(321") x36=579 feet; thus a body being elevated to the height of 579 feet, would require six seconds to fall to the surface of the earth. 400. The velocity acquired by this body, when it has reached the surface, may be determined from equation (150), in which we make a=0, g=32i feet, ^=6". We thus find v=32i"x 6=193"-. 401. If it be required to determine the height from which a body must fall to acquire a given velocity, we eliminate ( between the equations and we thus obtain ^=v/(2^*) (156). Let it be supposed, for example, that we wish to détermine the space through which a body would fall in acquiring a velocity of 386 feet per second ; we shall have 386"=.,/(2 X 321"- X5)= its intensity at the point B ; by r the radius of the earth CM, and by x the distance from B to C : for the purpose of simplifying the calculation, let the known distance AC be assumed as the linear unit. The force being supposed to vary in the inverse ratio of the square of the distance from the earth's centre, we shall have g i

g', or-<--; ff g and from these inequalities we deduce 2h 2h' 7 F' which proves that the value of t' exceeds that of t. 421, In general, if t' and t" represent the times of describ- ing two inclined planes A' and h", having a common altitude A ; and if g' and g-" represent the components of gravity respectively parallel to these planes, we shall have whence, or by replacing g' and g" by their values (Art. 418), we obtain Thus, the times of describing different inclined planes hav- ing a common altitude will be proportional to the lengths of those planes. 422. The motions of bodies upon inclined planes give rise to a remarkable mechanical property of the circle : it consists in this,— that if the plane of the circle be supposed vertical, the body will require the same time to describe a chord AC (Fig. 163), as is necessary to fall through the vertical diam- 208 DYNAMICS. eter AB. For, the equation (176) gives, for the time of descent through AC, ^ g and by substituting for g-' its vaUie ^ , this equation will be- come t'=x/^ (177). gh But if the diameter of the circle be denoted by d, we shall have, by the property of the circle, AB : AC : : AC : AD ; or, d:h':: h' : h ; and consequently, This value substituted in equation (177), gives, after reduction, ^ g but this value is precisely the same as that which has been found for the time t, in which the body would fall through the diameter AB : for, the height AB being expressed by d, we shall have whence, /2d Of Curvilinear Motion. 423. We have hitherto supposed the motion under consid- eration to be rectilinear ; but if it be curvilinear, the space described, and the velocity acquired in a given time, will be insufficient to determine all the circumstances of the motion: it will likewise be necessary to know the nature of the curve described by the body, and tlie point of this curve at which the body is found at the end of a given time. CURVILINEAR MOTION. 209 424. In the resolution of this problem, we employ the prin- ciple of the parallelogram of velocities, which is similar to that of the parallelogram of forces. It may be enimciated as follows : If two forces P and Q, {Fig. 164) communicate^ in a imit of time, to a m,aterial point m, velocities represented by wB a?id mC respectively, the resultant R q/" P and Q, will communicate to the point, in the same time, a velocity 7wD, which will he represented hy the diagonal of the parallelo- gram constructed on the lines mB and mC. The truth of this proposition may be thus established : — Let the force P be represented by the line ?nB ; then, since forces are proportional to the velocities which they communicate in a given time, the force Gi will be represented by the line mC. But, by regarding mBDC as the parallelogram of forces, the diagonal mD will represent the resultant of the forces P and Q, ; and it is required to prove that the velocity resulting from the composition of the two velocities wiB and tnC is the same as that which is due to the force R. Let a; represent the velocity which the force R can communicate to the point m in a unit of time ; then, since forces are proportional to the velocities which they generate, we shall have V :R::mB:x. But from the parallelogram of forces, we deduce P : R : : »iB : mD ; hence, mB : niD : : mB : x ; and therefore, x=mD. 425. In the preceding remarks the forces P, Q., and R have been supposed to act incessantly, communicating new im- pulses at each successive instant of time. The results obtained will however be equally true if we regard P, Q,, and R as impulsive forces which communicate their effects in- stantaneously, since the velocities imparted by such forces are proportional to the intensities of the forces. 426. The composition of three velocities by the construc- tion of a parallelepiped, results immediately from the pre- ceding principle ; for, let P, Q, and R {Pig. 165) represent O 210 DYNAMICS, -three forces which communicate the velocities mp^ mq, and mr to the material point m ; let the velocities mp and mq be compounded into a sino;le velocity mp\ which, by the pre- ceding demonstration, will be the same as that communicated by the force P', the resultant of the two forces P and Q, : in like manner, the resultant ms of the two velocities mp' and mr, will represent the velocity communicated by the force S, the resultant of the two forces P' and R, or of the three forces P, Q, and R ; hence, the diagonal of the parallelepiped con- structed on the lines representing the three velocities will represent the velocity communicated by the resultant of the three forces P, Q,, and R. 427. We will now examine the circumstances in which a material point will describe a curvilinear path. For this purpose, let the material point m {Pig. 166), at rest, be sup- posed to yield to the effect of an impulsion which causes it to describe the right line mK in the time 6, and at the end of this time let it receive a iiecond impulsion capable of making it describe the line AB in the same time 6 ; the material point will not entirely yield to the action of this second force, which tends to draw it in the direction of the line AB ; since, by the law of inertia it would have described the line AC=mA in the time 6, if the second impulsion had not been communi- cated to it ; but it will describe the diagonal AD of the paral- lelogram ABDC. If it should receive at D a third impulse capable of moving it over the line DG in a third time 6, it will, for a similar reason, describe the diagonal DP of a parallelogram constructed upon DG, and DE the prolonga- tion of AD, (fee. ; thus, at the end of a time equal to 116, the material point will have described a polygon having n sides. The velocity being constant so long as the material point remains on the same side of the polygon, it follows, that if at its arrival at the extremity of either side, it be not sub- jected to a new impulse, it will continue to move in the direction of this side, with a constant velocity. 42S. If the time 6 be supposed indefinitely small, the im- pulsions will be communicated in consecutive instants, and the polygon will then be transformed into a curve. The time é being supposed indefinitely small, it may be CURVILINEAR MOTION. 211 represented by dt, and the side of the polygon which is passed over in this time, will become the element of the curve : consequently, to determine the velocity, which will be measured by the space which the body would pass over in the direction of the tangent, in a unit of time, if the in- cessant force should cease to communicate new impulses, we must multiply ds, the element of the curve, by the num- ber of times that dt is contained in unity ; that is, we mul- tiply ds by—-, and we thus obtain ^ ^ ^ dt _ds ~di' 429. Let the body be supposed to describe the polygon m, 711', m", m'", &c. {Fig. 167), receiving increments to its velocity at the points 7n, m', J7i", m'", &c. Let v, v', v", v'", &c. represent the velocities which the body has acquired at the points m, m', m", m'", «fcc, and ê, ô', ê", ô'", (fcc. the times employed in describing the sides mm', m'm", m"m"', &c. Since each of these sides is supposed to be described with a constant velocity, we shall have, by the principles of uniform motion, m.m'=v0, m'm"=v'6', m"m"'=v"6", &c. ; and the perimeter of the polygon will therefore be expressed by vê+v'6' + v"6"-^6cc. If we project the sides of this polygon on the co-ordinate axes, denoting by «, /S, y, «', /3', y', (fcc. the angles formed by the sides mm', m'm", m"m"', &c. with these axes, the projections of the sides will be expressed by v6 cos «, v'ô' cos «', v"ô" cos «", , parallel to the three axes, we can regard these components as forces which impress on the projections of the material point motions which are entirely independent of each other. 430. To determine the analytical expressions for these incessant forces, we remark, that while the material point describes the space ds, its projections describe the spaces dx, dy, and dz respectively : the velocities of the projections will therefore be represented by — -, -^, and-- : and since the dt dt dt incessant force is equal to the differential coefficient of the velocity considered as a function of the time, we shall have, by regarding dt as constant, ^^— X 'dt^~ dt^ d'z „ (180) CURVILINEAR MOTION. 213 Such are the equations which serve to determine the circum- stances of the motion of a material point describing a curve. 431. When the functions X, Y, and Z are given by the lature of the problem, and if the integrals of the equations ^^180) can be obtained, these integrals will give three relations between the four variables x, y, z, and t : the quantity t being eliminated, there will remain two relations between x, y, and z, which will represent the equations of the trajectory, or curve described by the 7naterial point under the influence of the incessant forces. When the forces are situated in a single plane, which may be taken as that of x, y, the trajectory will he contained in the same plane, and it will then only be necessary to use the two equations dt^~~ ' dt^~ ' When, by the nature of the problem, the quantities X and Y are known, and if the integrals of these equations can be obtained, they will contain no other variables than x, y, and t ; thus, by eliminating t, we shall find a relation between x and y, which may be written under the following form, this relation will be the equation of the plane curve described by tha material point. 432. The velocity of the material point at any instant is expressed by ds dt hut the element ds of the arc of a curve situated in space, being considered as an indefinitely small right line, whose projections on the co-ordinate axes are represented by dx, dy, and dz. the value of this element will be ^{dx^+dy'^+dz''). Substituting this value in the preceding equation, we have v=\ ^{dx^ ^dy^ +dz.^), dt or, since the difierentials are taken with reference to ; as a variable, 214 DYNAMICS. ,;=V^(^^ + ^>^\ (181). The angles formed by the direction of the motion with the co-ordinate axes will result from the equations dx V cos « = — -, dt' dy V cos & = —-, ^ dt' dz V cos y=-^-. dt 433. The velocity may likewise be determined in the following manner. Let the equations (180) be multiplied respectively by 2dx, 2c/y, and 2dz ; the sum of these pro- ducts will give 2dx.d^x-\-2dy.d''y + 2dz.d'z ^,^, , ^ , , v-j ^ ^ = 2(Xc?x + Ydy + Zrfz) : and si^ce the first member is the differential of dx'-^-dy" •{■dz^t^ divided by df^, we shall have or, rejjîacing dx' +di/' +dz- by its value ds-, and integrat- ing, wi^Éibtain '■Jl iÇ--m^d.v-hYdy + Zdz) + C; ^- ds and by substituting î? for — , we find v'=2f{Xdx-{-Ydy-{-Zdz)+C (182). 434. It thus appears that the determination of the velocity will depend on the integration of the expression J\Xdx + Ydy-hZdz) (183). When this integration is possible, the integral will be a function of the variables x, y, and z, and the equation (182) may be written under the form v"-^2F{x,y,z)-\-C (184). To determine the value of the constant, we must know the velocity of the moveable point, at a given point of the trajee- CURVILINEAR MOTION. 215 tory. Thus, if V be the velocity at that point which cor- responds to the co-ordinates ar=a, y = 6, z=c, we shall have The value of C being deduced from this equation, and sub- stituted in equation (184), we shall obtain v--N-=2F{x, y, ^)-2F(a, 6, c). 435. The expression (183) is integrable when the move- able point is subjected to the action of a force which is constantly directed towards a fixed centre. To demonstrate this proposition, we will represent the resultant R of the several forces acting on the material point by CD, a portion of the line CM drawn from the point to the fixed centre {Fig. 168) ; let this centre be assumed as the origin of co-ordi- nates, and denote by x the distance of the point M from the origin, and by a, /3, y the angles formed by CM with the axes of co-ordinates : the direction of the resultant forming the same angles, we shall have X = RC0S«, Y=RC0S|3, Z=Rcosy, and consequently X COS» Y_C0S/3 Z_C0S7 (\QK\ COS /3 Z COS y X COS a. But if X. y, and z denote the co-ordinates of the point M, we shall have a;=ACOS«, y=Acos/3, 2;=Acosy; whence, by division, a:_cos« y cos (S r cos "/ y cos /s' z cos y X cos « ' these values substituted in equations (185), give yX— .rY=0, 5;Y-yZ=0, .rZ— ;^X=0. If in these equations we replace X, Y, and Z by their vn1nf>«! deduced from equations (180), we shall find d^x d^u _ y- :r— ^=0, ^dr- dt^ ' d^y d-z „ d' z d'X ^ x^ - — z-, — =0. dt^ dr~ 216 DYNAMICS. Multiplying the first of these equations by dt, integrating and reducing, we obtain ydx-xdy^^ dt ^ ' The other two equations being treated in a similar manner, we find ydx — xdy = Cdt, zdy — ydz = Cdt, xdz — zdx= Cdt. If we multiply each of these equations by the variable which it does not contain, and take the sum of the products, there will result d^Cz-\-C'x+C"y)=0, or, C^ + C'.r + C"y=0. This equation being that of a plane passing through the ori- gin of co-ordinates, or centre of attraction, it follows that the point will describe a plane curve. In the resolution of this problem it will therefore be unne- d'^ z cessary to employ the equation Z=-—, and it will simply be necessary to integrate the equation (186), which may be written thus : ydx — xdy=Cdt\ and from this we deduce f{ydx-xdy)=--Ct+C' (187). To determine the value of this integral, we remark that ydx being the element of a surface bounded by a curve, we can suppose this surface to be included within the limits x=0 and x=-CV {Pig. 169); thus, the expression Tydr will be repre- sented by the area LCPM. If from this area we subtract the triangle CPM, there will remain sector LCM=area LCPM-triangle CPM, or, sector LCM=yyo?ar—'^; differentiating and reducing, we find CURVILINEAR MOTION. 217 the equation (187) can be reduced to the following: 2 . sector LCM=Ci (188) ; the constant C is here suppressed, since we may always re» gard the times as reckoned from the instant when the move- able point is situated at the point L, in which case the sector will become equal to zero. If we make C=2A, the equation (188) will become sector LCM=A^ ; from which we conclude, that when a material point solicited by a force which is constantly directed towards a fixed centre^ describes a curve LM about this centre^ the area of the sector LCM described by the radins vector drawn to the material point is co7istantly jiroportional to the time which the point employs in describing the curve. This property is called the principle of areas proportional to the times. 436. The formula (183) i. always integrable when the forces are directed towards fixed centres, their intensities being at the same time functions of the distances of the material point from these centres. Let M represent the place of the material point {Fig. 170), which is attracted by the forces P, P', P", (fcc. towards the fixed centres C, C, C", &c. : denote by A-, y, z^ the co-ordinates of the point M, a, 6, c, the co-ordinates of the centre C, a', 6', c', the co-ordinates of the centre C, a", 6", c", the co-ordinates of the centre C", &c. (fee. (fee. p, p\ p'\ (fee, the distances CM, CM, C"M, (fee. ; «, 0, y, the angles formed by p with the axes of co-ordinates, a', /3' y', the angles formed by p' with the same axes, «", |8", y", the angles formed by p" with the same axes, &c. (fee. (fee. (fee. The total resultant of the attractive forces will have the fol- lowing components parallel to the three axes, 19 218 DYNAMICS. X=Pcos«4-P'cos«'+P"cos«" + (fcc. ^ Y=P cos /3+P' COS /s' + P" COS 0" + (fee. V (189). Z = P cos y + F COS y' + P" COS y " + (fcc. ^ The projection of the right Hue CM on the axis of .r being rep- resented by BD {Fig. 170), we have BD=AB-AD; and by observing that AB and AD are the co-ordinales x and a of the points M and C, and that BD, being the projection of MC on the axis of x, is expressed by j) cos «, we shall find, by substituting these values in the preceding equation, p cos ct=x — a ; the same remarks being applicable to the projections on the other two axes, we shall have 2^ cos u^=x — a, J) cos/3=y — h, j) cosy=2r — c And in like manner, p' cos oc'=x — a', p' cos i3'=2/ — b', p' cos y'=.z — c', J)" cos ct"=x — a", jj" cos (i"=y — b'\ p" cos y"=-z — c", &c. &c. (fee. By eliminating the cosines of these angles, the equations (189) become X=P^^^+P'^^+P"^^'+&c., p p' p" Y=pfc-VP'^+F'^ + &c., J) J) p Z^-p^^+F'^^+V^^ + acc. p p' p' These values substituted in the formula (183) give f{Xdx+Ydy + Zdz)=^fp{^^^dx-\-'^^dy+''-^dz\ +(fec. (fee. (fee (190). But the distances of the point M from the centres C, C, C", (fee. being given by the equations {x-aY->r{y—hy-{-{z-cY=p^, &c. (fee. (fee, CURVILINEAR MOTION. 219 we shall obtain, by differentiating, dx-\-- ay-\ az=ap, P P P ax •\-- dy-\ dz = dp . p' P P' &c. &c. &c. ; and substituting these values in equation (190), we find J{Xdx-\-Ydy->rZdz)^f{Vdp-^V'dp' + V"dp"^&cc.)....{l<èl). But the forces P, P', P", (fee. are, by hypothesis, functions of the distances jo, p', p", (fee; the expression Vdp-\-Vdp'-\- V'dp" will therefore contain but a single variable in each term, and its integral may be effected by the method of quadratures. It should be observed that the factors dp, dp', dp", (fee. may become negative, if the expressions x — a, y — h, z — c, X — o', (fee. should be transformed into a—x, b—y, c—z, a'—x, (fee. 437. For the purpose of making an application of the pre- ceding theorem, let it be required to determine the velocity of a material point which moves from rest, under the influ- ence of a force of attraction which is constantly directed towards a fixed centre, and which varies in intensity in the inverse ratio of the square of the distance from the position of the point to the fixed centre. Let the direction of the force be supposed to coincide with the axis of z : the co- ordinate axes being disposed as in Pig. 171, the intensity of the force and the co-ordinate z will increase together, and we shall have p=AG—AM=c—z, dp——dz. If g represent the intensity of the force at the distance r from the centre C, and P its intensity at the distance p, we shall have the proportion P . 1 1. ^■^''■7^''p^^ whence, but dp being negative, the quantity Vdp should be replaced 28d DYNAMICS. by —- — df ; integrating, we reduce the equation (191) to r This vahie being substituted in formula (182) gives v2-?ll!4-C (192). V To determine the value of the constant C, we suppose the body to commence its motion at a point whose distance from the centre of attraction is represented by a ; the velocity at this point being equal to zero, we have a or, 2g^r2 0= the equation (192) will therefore become ..=2^r»(i_i). If a be regarded as the unit of distance, the value of v^ will become identical with that determined in Art. 409. 438. To apply the formulas (180) we will first investigate the trajectory described by a material point which moves under the influence of a single impulse. In this case, the incessant forces being equal to zero, we shall have X=0, Y=0, Z=0; and the equations (180) reduce to ^=0 ^-0 '-^=0^ multiplying by dt^ they become ^=0 ^=0 — =0 dt ' dt ' dt The integrals of these equations are dx dy dz -T-=a, -~=h, -i-=c (193). dt ' dt ' dt ^ ^ Substituting these values in equation (181), we find MOTION UPON A GIVEN CURVE. 221 v=^{a^ +b' -\-c')=Si constant; and denoting this constant by A, we have ds . dt ' consequently, s=At-{-B] and the motion of the material point will be uniform. The motion is likewise rectilinear ; for the equations (193) give, by integration, x=at-{-a\ y=ht-\-b\ z=ct-\-c', whence, by eliminating t, az , a'c — ac' bz , b'c — be' c c ^ ^ c c These equations evidently appertain to the projections of a right line on the planes of x, z and y, z. Of the Motion of a Material Point when compelled to describe a partictdar Curve. 439. When a material point m, without weight, has received an impulse K (Fig. 172), and is subjected to the condition of moving upon a particular curve, we can resolve this impulse into two components, one mN=K' normal to the curve, the other mT=K" in the direction of the tangent : the normal force will be destroyed by the resistance of the curve, and the tangential component will produce its entire effect in com- municating motion to the material point. If we regard the curve as a polygon mm'm"m'", &c. {Fig. 173), having an infinite number of sides, the angle tm'm" formed by the prolongation of the side mm' with the consecu- tive side m'm" is called the angle of contact ; it will be denoted by « ; the plane tm'm" is the osculatory plane at the point m', and in plane curves coincides with the plane of the curve. The material point m, being solicited by a force K, receives a primitive velocity v, causing it to describe the side mm' ; but having arrived at the point m', it is deflected from its course, and describes the side ?n'm". By this deflection it ^^ DYNAMICS. necessarily undergoes a loss of velocity which will now be estimated. For this purpose, let the velocity v be represented by the line m'q. This velocity being resolved into two components 9?i'n and m'l, respectively parallel and perpendicular to the side 7ii'j}i", we shall have m' 1=971' q . sin tm'm", 'm'n=m,'q . cos Vnilml'^ or, m'l=v . sin », tn'n^v . cos t». The component v . sin » being destroyed by the resistance of the polygon, the velocity v will be reduced to v . cos a ; and consequently, the velocity lost, being equal to the primitive velocity diminished by the velocity actually remaining, will be expressed by ^(l— cos i"). When the polygon is supposed to become a curve, the angle tm'm" becomes infinitely small, and the quantity v(l — cos u) is at the same time an infinitely small quantity of the second order. To prove that this is the case, we observe that 1 — cos» represents the versed sine DB of an angle » {F^g- 17^4), measured by the arc BC ; and we have the proportion AD : CD : : CD : DB. But when the arc CB becomes infinitely small, CD will be so likewise ; and since CD is then infinitely small with respect to AD, it follows from the above proportion, that DB must be infinitely small with respect to CD, or that it is an infi- nitely small quantity of the second order. Thus, the velocity lost at each side ot the polygon being an infinitely small quantity of the second order, it may be neglected, since the sum of these velocities, although infinite in number, will con- stitute but an infinitely small quantity of the first order, which may be neglected in comparison with the original velocity v. Hence, we conclude, that a material point which is compelled to describe a curve, preserves undiminished the velocity which was originally communicated to it. 440. The component of the velocity v . sin a with which the material point is pressed against the curve, and which is destroyed by the curve's resistance, varies constantly as the MOTION UPON A GIVEN CURVE. 223 point changes its position, since sin * is constantly variable : we may regard this resistance exerted by the curve as an incessant force constantly acting upon the point and deflecting it from the tangent along which it would otherwise tend to move. When there are several forces acting on the material point, we resolve each in a similar manner, and the sum of the nor- mal components must then be added to the pressure arising from the component of the velocity. 441. Let it be supposed that a force N equal and directly opposed to the resultant of all the normal forces is applied to the material point : this force will produce precisely the same effect as the resistance offered by the curve, and the latter will therefore be represented by N. Let «, /s, y be the angles formed by the direction of the force N with the co-ordinate axes ; the components of N in the direction of the axes will- be respectively N cos «, N cos /3, N cos y, and should be added to the components of the incessant forces in the general equations of motion (180) : we shall thus obtain - — =X4-N cos « ^=Y+N cos ,3 ^ (194). - — =Z+N cos y dp To these equations may be added two others which result from the relations existing between the angles u, /3, and y ; the first of these equations is cos- «+C0S2 /3-{-COS2 y=l (195). The second is cos « . cos a'-f-COS /3 . COS /3'-fC0S y . COSy' = 0, in which », /3', y' represent the angles formed by the tano-ent to the curve with the co-ordinate axes. The cosines of these last angles may be expressed as follows : , d.v , dy , dz cos «'=--, C0S/3'=-/, COSy'=--: as as ds 224 DYNAMICS. these values substituted in the preceding equation convert it into -^ cos u-{-^ cos /3+-^ cos y=0 (196). as as its 442. To determine the velocity of the material point, let the equations (194) be multiplied respectively by 2dx, 2dy, and 2dz : their sum being taken, we shall obtain 2dx'^+2dy^ + 2dz^^2(Ldx^Ydy+Zdz) dt'' dt^ dP 4-2N(c?a; . cos a+f/y . cos ^-l-dz . cos y) : the last term of this equation being equal to zero, by formula (196), there remains 2dx'^-V2dy'^+2dz'^^=2{X.dx-^Ydy+Zdz) ; or, d{dx^+dr 'rdz^)^2{Xdx^Ydy-^Zdz) : whence, by substitution and integration, we find 1^ =2f{Xdx+Ydy^Zdz)+C ; or, v'=2f{Xdx-^Ydy+Zdz)+C (197). 443. When the material point merely receives an impulse, without being acted upon by incessant forces, we have X=0, Y=0, Z=0; and consequently, v' =a constant. Thus, when the material point is compelled to describe a curve, being acted upon only by an impulse, its velocity will remain invariable. This result accords with that which has been already obtained (Art. 438), on the supposition that the motion is perfectly free. 444. Let the material point which is supposed to describe the curve, be acted on by the forceof gravity ; we shall then have X-0, Y=0, Z=^; and the equation (1 97) will be reduced to v''=2fgdz-{-C. MOTION UPON A GIVEN CURVE. 229 If the velocity v be supposed equal to V, when «=0, we shall find This value substituted in the preceding equation gives whence, v=^{2gz+N^) (198). This expression for the velocity being independent of the relations which may exist between the co-ordinates x, y, and z^ it follows that the velocity will be the same for the same value of z, whatever may be the form of the curve. To determine the expression for the time employed by the material point in describing a given portion of the curve, we ds replace v by its value -r^, and thus obtain whence, <"=7(^JW,---<^«^>^ or, by substitution, ^'^ VC^gz + Y^) (''^^- To integrate this equation, it will be necessary to reduce it, by means of the equations of the curve, to one which shall con- tain but two variables ; thus, if the equations of the curve are f{x,z)=0, At/,z)=^ (201), we may, by the aid of these equations, in connexion with equation (200), eliminate two of the three variables x, y, and z ; and it will then only be necessary to integrate an equation expressing the relation between dt and one of the co-ordi- nates of the moveable point. 445. If, for example, the curve be supposed to become a right line, the equations (201) will be of the form x=az+u, y=bz+^ (202): from which we deduce dx^adz, dy=bdz\ P 226 DYNAMICS. and by substituting these values in the formula (200), it is transformed into dz y{l + a'- \-b') If the point be supposed to move from rest, its initial velocity V will be equal to zero, and we shall have, by division, dt dz whence, by integration, ^ ]■^/{2gz) (203). The constant introduced by integration becomes equal to zero, since, by hypothesis, when ^=0, -y^O, and z--0 (Art. 444). 446. To determine the space passed over in the time t, we suppress V in equation (199), which then becomes and eliminating z by means of equation (203), there results , g-t .dt and by integration, 10-/3 9— "- 4-C • which proves that the motion is similar to that of a body on an inclined plane, as might have been anticipated. 447. The co-ordinates .r, y, and z are readily determined in functions of the time ; for the latter is given by formula (203), and this, taken in connexion with equations (202), will determine a; and y in functions of t. 448. If, as in the present instance, the point be supposed to describe a plane curve, and if the incessant forces act en- tirely in this plane, we may, by placing the axes of t and y in this plane, dispense with the consideration of the third of equations (194) ; the formulas (195) and (196) will then be reduced to cos'«-l-cos2j3=l, ^-cos«4— /cos/3=0; as as MOTION UPON A GIVEN CURVE. 227 and the two equations of the curve will be replaced by the single relation 449. The velocity being given by formula (198), without the aid of equations (201), we conclude that the velocity ac- quired by the moveable point is independent of the form of the curve, being determined by the value of the vertical ordi- nate. Consequently, if from the point O {Fig. 175), where ^=0, and v— V, we draw the arcs of different curves OM, OM', OM", (fcc, terminated by the horizontal plane KL, the ordinates z of the first and last points of all these arcs being equal, it follows that different bodies departing from the point O with the common velocity V, and describing the several curves, will all have acquired the same velocity when they shall have arrived at the points M, M,' M", (fee, situated in the same horizontal plane. 450. In general, whatever may be the number of forces acting on the moveable point, if the equation (197) be inte- grable, we can determine the velocity v without knowing the nature of the curve described. For, the values of the in- cessant forces X, Y, and Z, expressed in functions of the co- ordinates X, y, and z. being substituted in equation (197), if the expression f{X.dx + Ydy + Zdz) becomes integrable, we may represent it by /(^j y, z) ; and the equation (197) will then reduce to 'v'=2f{x,y,z)+C. If we denote by a, h, and c the values of x, y, and z which correspond to the velocity V, the value of C will become known ; thus, C=V2-2/(«, 6,c); and replacing C by this value in the general expression for the velocity, we find v^ =V* -\-2f{x, y, z)-2f{a, b, c) ; an expression which depends only on the initial velocity, and the co-ordinates of the first and last points of the curve described. P2 228 DYNAMICS. 451. It has been explained (Art. 440) that the normal pressure exerted against the curve arises in part from the nor- mal components of the incessant forces, and partly from the normal force due to the velocity. To determine the value of the latter, let perpendiculars on and o?i' be erected at the middle points of the equal consecutive sides wtwt' and m'tn" (FS,g. 176) of the polygon having an infinite number of sides : the angle tm'm,", formed by one of these sides with the prolongation of the other, W\\\ be the angle which we have represented by •». But the angles n and n' being right anglesj we have non' ■\- ii'm'n' = 180° = tm'm" -\- nm!iï ; and therefore, trn'm" = » = non' = 2nom'. The angle nom' being infinitely small, its sine may be re- garded as equal to the arc which measures it ; but this sine , , m'ît' m'n' . j , i IS expressed by , or , smce 7io and mo may be con- m'o 710 sidered equal ; hence, 2m'n mm' û/= = . 710 710 If we now return to the consideration of the curve which is the limit of the polygon, the side mw' becomes the element of the curve, and 7io the radius of curvature : the preceding relation will therefore be transformed into ds 0,= — , y y denoting the radius of curvature. Let ç> denote the intensity of the incessant force which is due to the normal component of the velocity : this intensity being in general expressed by the quotient of the element of the velocity, divided by the element of the time, and the ele- ment of the velocity being represented in the present instance by V . sin «, we shall have V .sin a or, since the infinitely small arc may be substituted for its sine, this expression becomes MOTION UPON A CURVED SURFACE. 229 Vu replacing « by its value found above, we have vds v^ ^ = -^' or^=— . yat y The normal pressure resulting from the other forces may be determined by the parallelogram of forces, and this pressure must then be combined with that due to the velocity, in order to obtain the total pressure, 452. Let it be supposed, for example, that the material point describes a plane curve, and that the incessant forces are directed in the plane of this curve : let these forces be reduced to their resultant R, and denote by 6 the angle formed by the direction of the resultant with that part of the normal which lies on the concave side of the curve : the component of the resultant in the direction of the normal will be ex- pressed by R cos ^, and will act in the same or in a contrary direction to the pressure arising from the velocity, according as 6 is obtuse or acute. The pressure arising from the velo- city being always directed /rowi the centre of curvature, the entire pressure will be expressed by N=— — RcosO; y this pressure will be exerted /rom the centre of curvature so long as the quantity N is positive, and towards the centre in the contrary case. Of the Motion of a material Point when compelled to move itpon a Curved Surface. 453. When a material point which is compelled to move upon a curved surface is subjected to the action of inces- sant forces, these forces, and that resulting from the velocity of the point, will exert a pressure against the surface, which will be counteracted by the resistance of the surface. If we denote this resistance by N, the material point may be re- garded as moving freely in space, provided we include the components of the force N in the general equations (180), which express the circumstances of motion of a point under 20 230 DYNAMICS. the influence of incessant forces. Let «, /3, and y represent the angles formed by the direction of the force N with the co-ordinate axes ; its components in the directions of these axes will be expressed by N cos «, N cos /3, and N cos y : con- sequently, the equations of the required motion will be - — =X4-N cos « ^=:Y + NC0S/3 ^=Z+1S cosy dt^ (204). The angles «, /2, and y will become known when the equation of the surface L=0 is given, for we have, (Art. 62), dx cos ct— ± ^m^m^it) cos /3=± dy dL dy cos y= ± dL d^ ^{Èï-m'HÈ) the double signs prefixed to the values of the cosines, indi- cate that they may refer to the direction of a force which tends to pull the point, either along the normal to the surface, or along its prolongation. If we put, for brevity, 1 ± — - — =Y, ^{Èï^Èï^Èy dy the preceding equations will become .dL cos «=V dx' COS/S: dy^ COS»/— Y M. dz MOTION UPON A CURVED SURFACE, 231 these values substituted in equations (204), reduce them to dt'' dz (205). If N be ehminated between these equations, V will likewise disappear, and we shall thus obtain two relations, which, taken in connexion with the equation of the surface L=0, will determine the co-ordinates of the moveable point in functions of the time. 454. As an example : — Let it he required to determine the circumstances of the motion of a material point on the surface of a sphere : let the origin of co-ordinates be as- sumed at the centre, the plane of x, y being horizontal, and the co-ordinates z being reckoned positive downwards ; these co-ordinates will then be affected with the same sign as the force of gravity. The equation of the sphere being lj=x^-\-y^-\-z'—a^=Q (206), we obtain by differentiation, dl.=2xdx-^2ydy^-2zdz=0 (207), and consequently, g-=2x, ^=2y, -=2z. y , 1 ,\. ~v'(4a^ +4^2+42^)-"^ 2a' or, COS«=±-, 008/3= ±?^, cosy=±- (207 a). a a a Again, the force of gravity being the only incessant force acting on the material point, we have X=0, Y-0, Z=g; these values reduce the equations (205) to 232 DYNAMICS. ^=±N?, !^=±N^, '^. = ±^-+e (208). at- a at- a at^ a The positive signs should be taken together, and evidently correspond to Hke signs in the vahies of the cosines of «, /î) and y ; a similar remark is apphcable to the negative signs. We ehminate ± N between the two first of these equations, by miihiplying them respectively by y and i\ and taking their difference ; we thus obtain, after muhiplying by dt^ yd^x—xd^y _^ or ^(y^-^~^^y) -o . dt ' dt ' whence, by integration, dt being regarded as constant, ydx-xdy=Cdt (209). A second relation between the variables may be found by multiplying each ot the equations (208) by the differential of the variable which it contains ; the sum of these products will give dx.d-ix-\rdy .d^'i/-\-dz .d^z , N, , , , , , . , , dP ^ -{xdx-\-ydy-\-zdz)-\-gdz ; and since the quantity included within the brackets is equal to zero, by equation (207), the preceding result will be reduced to dx .d~x-\-dy .d^y-\-dz . d^z _ , W^ ~^ ^' multiplying by 2, and integrating, we have ^.^^l±^p:^=2gz+G' (210). If two of the three variables .t, y, and z be eliminated between the relations (206), (209), and (210), ihe result will be an equation which, being integrated, will give a relation between the third co-ordinate and the time / : this result will evidently be independent of the normal force, which has already disap- peared from these three equations. 455. The equations (207) and (209) being squared, give x^dx-+ 2xydxdy + y"" dy^ =z^ dz^^ y^ dx^ — 2xydxdy + x^ dy- =C^ dt^ . The sum of these equations being taken, the middle terms of the first members will disappear, and we shall have MOTION UPON A CURVED SURFACE. 233 substituting for {x^ +y") its value deduced from equation (206), there results , , , , , C^dP-ifZ^dz^ dx^' -\-dy= — ; and eliminating dx'^ -\-dy^ between this equation and (210), we find dt= — (211). V'[(a^-z^)(2^z + C')-C^l ^ ^ The integral of this equation, which can only be obtained by- approximation, will give the value of z in functions of the time. 456. To determine the expressions for the other co-ordinates in functions of the time, we will suppose ft to represent the approximate value of z determined from the integration of the preceding equation : if this value be substituted in equation (210), we may, by combining the resulting equation with that designated as (209), obtain two relations, the first between x £md t, the second between y and t : but as the variables in each of these equations would not be separated by this pro- cess, we adopt another method of determining the values of x and y in functions of t. Let KC=x, DC=y, iiiD^z {Fig. 177) be the three co- ordinates of the point tn on the surface of the sphere ; if for a given value of z, the angle CAD, formed by the projection AD of the right line AM with the axis of x, were known, the corresponding values of x and y might be readily determined in functions of z. For, the angle CAD being denoted by 6, and the radius Am by a, we shall have KD=^{a^ —z^)] and the triangle ACD right-angled at C, will give AC=AD . cos CAD, CD=AD . sin CAD ; or, x=^y/{a^—z'') Xcos Ô, y=^{aP- -z^) Xsin (212). These two equations establishing a relation between x, y, and z, may be considered as replacing the equation of the sphere, which can be obtained by taking the sum of their squares. An additional variable 6 is here introduced, but the number of relations is likewise increased by unity. 234 DYNAMICS. From the equations (212) we obtain by differentiation, ...(213): zdLz dx=—s'med6^(a'' — z") — ,cos Ô zdz di/=cos eds^ia' — z^) r- rMn 6 multiplying the first of equations (213) by the second of (212), and the second of (213) by the first of (212), and taking their difl'erence, we obtain ]/dx—xdy=~-{a^—z''){sm''ô + cos^6)d6', or, pdx —xdy={z^ — a^)d6. This value substituted in equation (209), gives {z''—a^)d6=Cdti and consequently, z^ -a" ' or, replacing dt by its value deduced from equation (211), we obtain , a.C .dz -\z^-a^^)^[{a--z')[2gz+C')-C^] This equation, being integrated by approximation, will deter- mine the value of 6 : we thence deduce the values of cos ê, and sin 6, which substituted in equations (212), determine the values of x and y in functions of z, and consequently in functions of the time t. 457. The equation (210) proves that the velocity is inde- pendent of the normal pressure ; for, we deduce f om that equation, or, *» ='°^+*^> v^=2gz-^0: and consequently, v=^{2gz-[-C'). To determine the value of the normal pressure, we must recur to equations (208) : these being multiplied respectively by X, y, and z, and added, give MOTION UPON A CURVED SURFACE. 236 ^,J = ±-{x'-^y^+z')+gz (214). But the diflerential of equation (207), xdx-\-ydi/ + zdz=0, being taken, we find xd''x+yd''i/-i-zd''z_ dx^ ■\- dy'' -\-dz^ __ j, dT^ df^ and this value substituted in equation (214) gives, after re- placing .r» -^-y^ +2;2 by «3, or, a a 458. If the moveable point be supposed situated at any instant below the horizontal plane passing through the centre of the sphere, the ordinate z will be positive, and the value of ± N becomes negative : and since N, which denotes the intensity of a force, is by hypothesis an essentially posi- tive quantity, the inferior sign must be taken in order that — N may be essentially negative. Hence, it will be neces- sary to take the inferior signs in equations (208), and also in equations (207 a). The resistance of the surface will there- fore be directed towards the centre, or the material point must be regarded as situated upon the concave surface of the sphere. When the material point rises above the horizontal plane of a-, y, the ordinate z will become negative, and the quantity — tj3 — gz may then become positive. In such case, the superior signs must be taken in equations (207 a) and (208), and the resistance of the surface will be exerted from the cen- tre, or the body must be supposed to be on the convex surface. The pressure exerted against the surface will be equal and directly opposed to the resistance which it offers, and will therefore be represented by ^— without reference to the sign of z. If the moveable point be retained upon the surface of the sphere by an inflexible thread connecting the point with the centre, this thread will experience a tension so long as v^ -\-gz is positive ; but, on the contrary, there will be a tendency to compress the thread whenever v^ -{-gz becomes negative. 236 DYNAMICS. Of the Motion of a material Point on the Arc of a Cycloid. 459. Let a material point M [Pig. 178) be supposed to move fram rest on the arc of a cycloid, under the influence of the force of gravity : the initial velocity being by hypothesis equal to zero, the equation (198) is reduced to v2 =2gZj or whence we deduce dt= ^ Let the origin of co-ordinates be assumed at the point E, the absciss ED of the point M' being denoted by w, and the absciss EC of the point of departure by A : we shall then have CD=EC-ED; or, z—h — u. This value being substituted in the preceding equation givesi dt=^—--^^ (215). This equation contains three variables ; we must therefore eliminate one by means of the equation of the cycloid. For this purpose, let 2a represent the diameter BE of the gene- rating circle, and x and o/ the co-ordinates AP and PM' of the point M', reckoned from A as an origin ; the equation of the curve will then be "^^TW^^ ^^^«>- But if s denote the arc AM', we shall have the relation ds=^{dx'' +dj/''); or, MOTION UPON A CYCLOID. 237 substituting in this equation the value of -7- deduced from the relation (216), we find or, ^='y\^^y (^^^- But from an inspection of the figure, we have 2a — y=u ; and hence, dy= — du. By substituting these values in (217), we obtain ds=^ — du\^/ — . V u The difierentials of 5 and u have contrary signs, since the first is a decreasing function of the second. The preceding value of ds will reduce equation (215) to du g' ^{hu—u^) 460. This equation may be integrated by the formula dx dt=-y/^. ,/^ ^, (218). lis eq / y/o"^ ^3^ =arc (versed sine ~x) ; for by making x=—, this formula becomes — 7x r=arc I versed sine =— | (219) ; ^{2az—z') \ a F v /> and consequently, by referring the integral of equation (21 8) to this formula, we obtain I^-k/-- . arc f versed sine= ^\-\-C (220). ^ g \ \h} To determine the constant, we remark, that since the time is reckoned from the instant when the body is at the point M, we must then have /=0, and «=EC=A; this supposition reduces the equation (220) to 238 DYNAMICS. 0= — \./ — • 3-rc (versed sine =2)4-C. ^ g But the arc whose versed sine is equal to 2, being a semi- circumference, if we denote by v the semi-circumference of a circle whose radius is unity, the preceding equation will become This value will reduce equation (220) to i=>v/— fw— arc (versed sine =-t-|. This expression gives the time of descent to the point M', the absciss of which is equal to u. To obtain the entire time of descent to the vertex E, we make w=0, and the value of t is then reduced to '-^\/~t g This value of the time being independent of the height h of the point of departure, we conclude that the time necessary for a material point to deseend to the vertex E of the cycloid, under the influence of the force of gravity, is constantly the same, whatever may be the position of its point of departure. Of Oscillatory Motion. 461. Let OBC {Fig. 179) represent a continuous curve, intersected at the points O and C by a horizontal line, and supposed to contain no angular points that might occasion a loss of velocity to a body or material point moving upon it. Let the tangent BT at the point B be supposed horizontal, the co-ordinate plane of x, y being likewise horizontal. If the co-ordinates z be reckoned positive downwards, we shall have the following equations to determine the circum- stances of the motion of a material point sliding along the curve under the influence of gravity : OSCILLATORY MOTION. 239 To determine the velocity of the moveable point, we proceed as in Art. 433 : multiplying these equations by 2da:, 2dy, and 2dz respectively, and adding, we find 2dxd^ X + 2dyd^ y + 2dzd'' z 2gdz', dt^ and by integration, ——de ^^^+^' or, Replacing-— by its value v^, there will result If V denote the velocity at the point O, when ;$r=Oj the pre- ceding equation will become and consequently, by substituting this value of C, we shall obtain 'V^=Y^+2gz (221). 462. Since the ordinates increase from the point O to the point B, it appears from equation (221) that the motion will be accelerated while the material point is describing the arc OB, and that its velocity will be a maximum at the point B : the ordinates decreasing beyond this point, the velocity of the moveable point will likewise be diminished. This diminution must be such that the material point will have at the point m!. the same velocity as it previously had at the point m^ situated in the same horizontal plane ; for the vertical ordi- nates of these points being equal, their values substituted in equation (221) will necessarily give the same values for the two velocities. The velocity diminishing constantly with the arc Om, we shall find on the prolongation of this arc a point A at which this velocity will have been equal to zero ; and the moveable point may therefore be considered as moving from rest at this point. If through the point A a horizontal line AA' be drawn, intersecting the second branch of the curve at A', the 240 DYNAMICS. velocity at A' will likewise be equal to zero. Thus, the motion will cease at the point A', and the action of gravity, causing it to descend from A' to B, will augment the velocity in the same manner that it was before diminished. At the point B the velocity will again become a maximum, and the moveable point will then ascend to the point of departure A, its motion being retarded in the same manner that it was before accelerated in descending from A to B. The same efiects being repeated by the action of gravity, the point will continue to oscillate indefinitely. If the arcs AB and A'B are similar, the times of describing them will evidently be equal. When the oscillations of a body or material point are all performed in equal times, they are said to be isochroiial. 463. Let B'OBO' {Fig. 180) represent a curve returning into itself, and symmetrical with respect to a vertical axis passing through the points B and B' at which the tangents are horizontal. If the material point descend from a point O, with an initial velocity such that upon arriving at B it can ascend from B to B' on the second branch of the curve, it will descend a second time on the arc B'OB, the force of gravity restoring the velocity lost during the ascent on the arc BO'B'. The same effects being repeated, the body will continue to revolve indefinitely. Of the Simple Pendulum. \ 464. The simple pendulum is composed of a material heavy point M {Fig. 181), suspended by an inflexible right line MC devoid of weight, and oscillating about a point C. In this motion the point M describes the arc of a circle about C as a centre, and the velocity of M will be given (Art. 444) by the equation v^ = V2 -[■2gz (222). ds Replacing v by its value —, we find dt= ^^/\ , (223). SIMPLE PENDULUM. 241 The origin being acsiimed at the point of departure, z will represent the ordinate M'P' {Fig. 182) of the point M', at which the material point is found after the lapse of a certain time, and V* will represent the square of the velocity which the body has at the point M, where xr=0. If h denote the height due to this velocity, we shall have the relation and the equations (222) and (223) therefore become -v/[2=^(A+.)], *=;7i^$+;)] (224) 465. To express the quantity z in functiors of the co-ordi- nates of the circle described with the radius CM, we demit the perpendiculars MB and M'D on the vertical line CE, and denote by a the radius CE, by h the vertical distance EB, and by x the absciss ED of the point M' referred to the point E as an origin ; we shall then have - = BD = 6— .r. And by introducing this value into equations (224), they be- come .=^[2^(, + 6-.)], d.=_^^^*__^ (223). From the first of these relations we obtain the velocity of the material point at the point M', corresponding to the absciss X ; the second, being integrated, will determine the time em- ployed by M in descending to the point M', To effect this integration, we must eliminate one of the variables contained in the second member, by m-eans of the relations ds=^iclx^-\-dij^) (226), y^=2ax—x^ (227). The latter being differentiated, gives ydy={a — x)dx ; and consequently, dy^=i^^IIpldx'. This value being substituted in equation (226), we find a 21 242 DYNAMICS. or, replacing y^ by its value given in equation (227), and re- ducing, we obtain , , /a^ , adx , adx . adx as=dx^ / — = ± = ± — -r r==t — ft^ r— t 5 V y2 y ^(2ax—x') ^[(2a—x)x]^ whence, ~~ ^[{2a-x)x]^[2ff{hi-b-x)]' The negative sign is here prefixed to the second member, be- cause the co-ordinate a; is a decreasing function of the time t. 466. If the initial velocity be supposed equal to zero, we shall have h=0] and if at the same time the arc through which the oscillation is performed be supposed extremely small, we can neglect x in comparison with 2a, and the value of dt will be then re- duced to , adx ^~^{2ax)^[2g{b~x)]- This equation may be put under the form dt=—l\/- X .^f"" , , (228). The value of i will be immediately obtained by an integration of the formula dx :^{bx—x^) which, by a comparison with (219), gives a=^b, z=x: and by substituting these values in equation (219), which is f:77^) (^«>^ / r- =arc I versed sine = — | , J^(2az-z'') \ a/' ^{2az-z'') it becomes /! :.^£- — -=arc (versed sine =^ | ^{bx-x') \ \b} =arc (versed sine=-^ j . But, in general, the cosine of the arc corresponding to the SIMPLE PENDULUM. 243 versed sine c and radius unity, being equal to 1 — c, we shall have arc (versed sine =— j=arc (cos=l — —\ )• / b =arc| cos= — V b This value of the expression (229) being substituted in equa- tion (228), we shall find ^=-i\/-Xarc(cos=^::^Wc (230). 467. The constant may be determined from the consider- ation that when t=0, x=b ', these values reduce the equation (230) to 0=-!^/- Xarc(cos=_l)-f C. If «• denote the semi-circumference of a circle whose radius is unity, we shall have (Pig: 174) arc (cos = — 1) = ar c BC A = sr ; and consequently. By substituting this value in the equation (230), we obtain The integral being taken between the limits ar=6, which cor- responds to ^=0, and x equal to any assumed absciss, will make known the time of descent from the point M {Fig: 182) to the point M' corresponding to the assumed value of x. 468. When we wish to obtain the time of descent to the lowest point E, we make x=0, in the preceding expression ; and since the arc whose cosine is unity is equal to zero, we shall have <=iTv/-^ (232). ^ g- 469. When the material point arrives at the point E, it Gt2 244 DYNAMICS. will have acquired its maximum velocity ; for the velocity being expressed by it will evidently be a maximum at that point of which the ordinate z is the greatest. Thus, in virtue of the velocity acquired at E, the moveable point will describe the arc EN ; and since this arc changes its sign in passing through zero, we find for the expression of the time requisite for the point to arrive at N' ^=i\/|[-+arc(cos =1-^)] (233). If from this expression we subtract that given by (232), which expresses the time of descent from the point M to the point E, there will remain I S/— X arc / cos= 1 — — V an expression Ibr the time of ascent from E to N': this time is equal to that employed in descending from M' to E, as may be proved by taking the difference between equations (231) and (232). Finally, when the material point shall have arrived at the point Nj situated in the horizontal line passing through M, we shall ]iave j:— 6, and the expression arc |cos=l — —I will then become arc (cos =— l)=5r; thus, the equation (233) will be reduced to ^ g Such will be the value of the time required by the moveable point to describe the whole arc MEN. This time being de- noted by T, we have T='^\/- (234). The velocity of the material point upon its arrival at the point N will be equal to zero ; for, since the initial velocity was supposed equal to zero, we have A=0; SIMPLE PENDULUM. 245 and this value, taken in connexion with that of x=b, reduces the equation v=^[2g{k-^b—x)] to v=0. The motion of the material point being entirely destroyed when it arrives at the point N, the force of gravity will cause it again to descend, and since the circumstances of the motion are precisely similar to those presented when the point commenced its motion at M, a second oscillation will be performed in the same time, and a similar motion will continue indefinitely. 470. The equation (234) being independent of the quantity b which expresses the vertical distance MK, it follows that if the point of departure had been taken at M', instead of at M, the time of oscillation would have been the same ; and consequently, that if several material points depart from the different points M, M', M", (fcc, they will all perform their oscillations in the same time. It should be recollected, how- ever, that this result has only been obtained on the supposition that the arcs described are extremely small. 471. These oscillations of equal duration are called iso- chrmial. But if the length of the pendulum be supposed varia- ble, the time of vibration will likewise vary : for, if I and V represent the lengths of two pendulums, whose oscillations are performed in the times T and T', we shall have g ^ g hence, T : T' : : v^Z : v^Z' (235). Thus, if the time of oscillation T of one pendulum be accu- rately known, we can determine by the preceding proportion the length V of a pendulum which shall vibrate in an arbi- trary time T'. 472. To ascertain with greater precision the time of a single oscillation, we will represent by N the number of oscil- lations made by the pendulum whose length is I in a time 6, and by N' the number of oscillations of the pendulum I' in the same time « : we shall then have t=Vf' '^-Vl' 246 DYNAMICS. T=^, andT'=^, (236). By means of these values, the proportion (235) is reduced to N'^ : N= : : ^ : l\ whence, When the number of oscillations made by a pendulum of a given length, in a given time, has been ascertained from observation, we can calculate the length of the pendulum which will oscillate in a second of time. If an error be committed in observing the time Ô, this error will he greatly reduced by being divided by the number of Dscillations, and if this number be taken large, the effect of the error upon the time of a single vibration may be regarded as insensible. 473. It is jXi this principle that the length of the seconds pendulum, which makes 86,400 oscillations in a mean solar day,in vacuo, and at the latitude of New- York, has been found ■equal to in. ft. 39.10168=3.25847, nearly. 474. To determine the value of g, the measure of the in- tensity of the force of gravity, we employ the equation (234)j which gives and by making in this equation in. ft. T=l", Z=39.1G168, and «-=3.1415926, or,. a-2 =9.8696046; sire find in. ft. ^=385.9183=32.1598. 475. If g and g' represent the intensities of gravity at dif- ferent places, and I and V the lengths of two pendulums which oscillate in the times T and T', we shall have T=-V'^-, T^'s/!:' g' ^ g" CENTRIFUGAL FORCE. 247 ffom which we deduce T:T':: v/|: V^l (237)- Let N and N' represent the numbers of oscillations made by these pendulums in the time 6 ; T and T' will be given in functions of 6 by equations (236), and their values being sub- stituted in the proportion (237), will give, after reduction^ If the same pendulum be used at the two places, Z and V will be equal to each other, and the preceding proportion will become whence,, 1 -1 •• /I- /L- N ■ N' ■ ' V ^ ■ V ^' N' Of the Centrifugal Force. 476. If a material point be supposed to move around a fixed centre C, describing the curve LMK {Pig. 183), and if, upon its arrival at the point L, the connexion with the centre be suddenly destroyed, the material point will, in virtue of the law of inertia, continue to move in the direction of the tangent LT. But if we conceive the point to be compelled to describe the curve, it will leave the tangent, and will after a certain time arrive at the point M. The arc LM being sup- posed indefinitely small, the angle LCM will be so likewise, and the lines LC and MC may be considered as parallel. Thus, replacing CM by the parallel CM, and constructing the parallelogram LDMN, it appears that the material point, if free, would describe the side LD, while by its connexion, with the fixed centre it is caused to describe the diagonal LM ; the effect of the force which draws the point towards the centre has therefore been to move it through the space MD. The point may be supposed to be retained on the curve LMK, either in virtue of a force of attraction which is con- stantly directed towards the centre C, or by the resistance 248 DYNAMICS. opposed by the curve regarded as material ; or, finally, by being connected with the point C, by means of a cord of variable length. Whilst the point is describing the elementary arc LM, we can regard it as moving upon the equal arc of the osculatory circle, and can suppose it to be retained on this arc by means of a thread of an invariable length, attached to the centre of the osculatory circle. Moreover, since this thread will ex- perience a tension only in consequence of the resistance offered by the material point to the force which tends to deflect it from the tangent, this tension or the resistance op- posed by the point will be precisely equal to the force which causes it to deviate from the tangent. This resistance is exerted in the direction of the radius of curvature, and its constant tendency is to remove the material point from the centre of curvature. Hence, it is called the centrifugal force ; and the force which constantly urges a body towards any fixed centre is called a centj'ipetal force. The centrifugal force evidently corresponds to the quantity represented by — in Arts. 451 and 452. 7 477. To determine directly the expression for the centri- fugal force, we replace the infinitely small arc LM by the chord of the osculatory circle at the point L {Pig. 184). Then, the versed sine LN will represent the space through which the point would be drawn in virtue of the action of the centrifugal force, during the time occupied by the point in describing the arc LM. From the known property of the circle, we have LN : LM : : LM : LE ; or, by substituting the arc for its equal the chord, hence, and by substituting for ds its value vdt, we find LN^"?^ (238). 2y CENTRIFUGAL FORCE. 249 A second expression for the value of LN may be obtained in the following manner. The time required to describe the arc LM will be represented by dt, since this arc is itself denoted by ds ; hence, dt will likewise represent the time in which the material point would be caused to describe a space equal to LN under the influence af the centrifugal force alone. Moreover, the centrifugal force acts incessantly, and during the infinitely short time dt, its intensity may be considered invariable. If, therefore, we regard this force as constant, and denote its intensity by/, the circumstances of motion of the point, under the influence of this force, will be expressed by the equations dv „ ds _ di~^' ir'''' and by integration, But the space LN being that which corresponds to the time dt, if in the preceding equations we make LN=5^ t will be- come dt ; we shall thus have LN = ^c?PX/. This value of LN being substituted in equation (238) gives, after reduction, y 478. If the material point be supposed to have a circular motion, — as, for example, when a stone is whirled round in a sling, V will become the radius of the circle described,, and the expression for the centrifugal force will then be /=^ (239). Let h represent the height due to the velocity v ; the follow ing relation will then subsist (Art. 401), eliminating v^ between this equation and that which pre- cedes, we obtain f^2h 250 DYNAMICS. from which we conclude, that the centrifugal force is to the force of gravity J as twice the height due to the velocity is to the radius of the circle described by the material point. 479, If a semicircle EAF {Pig. 185) be supposed to re- volve about its diameter EF=2R, the point A, the middle of arc EAF, will describe a circumference equal to 25rR ; if this motion be performed uniformly in the time T, with the ve- locity V, we shall have the relation vxT=2^R; and by eliminating v between this equation and (239), we find /=^ (240). In like manner, if/' represent the centrifugal force of a point which describes uniformly the circumference of a circle whose radius is R', in the time T', we shall have T' and consequently. From this proportion we immediately conclude, that when the radii R atid R' are equal, the centrifugal forces will he in the inverse ratio of the squares of the times of revolution ; and that when the times are equal, the forces will be directly as the radii. 480. The effect of the centrifugal force at the equator, caused by the revolution of the earth upon its axis, can now be estimated. For, the equatorial radius of the earth being 20920300 feet, we replace R by this value in equation (240), substituting at the same time the values of sr and T. But we have, approximatively, «■=3.1415926, ^='=9.8696046. The time T is determined from the consideration that the earth performs a revolution upon its axis in 0.997269 days, the day being composed of 86400 seconds. Thus we shall have T=0.997269 x 86400" ==86 164". CENTRIFUGAL FORCE. 251 Substituting this value and that of R in equation (240), there results /=0?Ï112 (242). 481. Having found the value of /, we can determine the intensity G of the force of gravity which would be observed at the equator if the earth were immoveable. For, since the force / is directly opposed to the force G, a portion of the latter will be destroyed by/; and hence, if g denote the in- tensity of gravity as determined by observation, we shall have ^=G-/; or, substituting in this expression the value of/ given by equa- tion (242), and that of ^, which at the equator is 32.0861 ft., we find G-32.086H- Oil 12=32.1973 (243). To determine the relation between the centrifugal force and the force of gravity, we divide equation (242) by equation (243), which gives / 0.1112"- 1 , ,„,,^ ^=321973^-^2-89 "^"^^^ ^^^^^' 482. The proportion (241) will furnish a solution to the following problem : To find the time in which a revolution of the earth should be performed, in order that the centrifugal force at the equa- tor may he equal to the force of gravity. Let T' represent the required time of revolution, and/' the corresponding centrifugal force ; we shall then have, by the nature of the problem, /'=G, and R'=R ; these values substituted in the proportion (241) reduce it to * * r£â ' fjvâ • whence we obtain 252 DYNAMICS. f If the fraction -- be now replaced by its value (244), we shall G find T T v/2S9 17' Thus, if the earths rotation were seventeen tinier more rapid tliati it actually is, the centrifugal force at the equator would be equal to the gravity. 483. To find the diminution of the gravity produced by the centrifugal force at any other point on the earth's surface, it will be necessary to determine the effect of the centrifugal force in the direction of the vertical BZ {Fig. 185) drawn through the point under consideration. For this purpose, we will regard the earth as spherical, it being nearly so : the latitude of the point B being then repre- sented by the arc AB, it will be measured by the angle BOA=ZBC=^^. Denoting by R the radius AO of the earth, and by R' the radius BD of the parallel of latitude passing through B, we shall have R'=R cos OBD ; or, R'=R cos ^. Let the centrifugal force at the point B, which is exerted in the direction of the radius DB, be represented by the line BC, and resolve it into the two components B6 and Be. The force BC will, by Art. 479, be expressed by -^- — , and the component /' in the direction of the vertical BZ, which is represented by B6, \vill be given by the relation /=->p,-Xcos^: and by substituting in this relation the value of R', we shall obtain /--Tj^XCOS^'r/.. 4 3f? The factor -=^ represents the centrifugal force / at the GRAVITATION. 253 equator ; this equation may therefore be transformed into tlie proportion /:/:: 1 : cos- ^] from which we conchide, that the dimhiations of gravity at different places on the earth's surface, arising from the action of the cetitrifu gal force, are proportional to the squares of the cosines of the latitudes. 484. The latitude of New-York beinor 40° 42' 40", its cosine will be 0.7580 ; and by multiplying- the value of/ (242) by the square of this number, or by 0.5746, we find /=o!b639. If G' represent the value of the force of attraction, or that which the observed gravity would have in the latitude of New- York, if the earth were immoveable, the gravity actually observed being denoted by g', we shall ha-'e, as in Art. 481, G'=g'+f. The observed gravity g', in the latitude of New- York, being 32!l598, we find, by substituting this value and that of/' in the preced- ing equation, G'=32?i598+o'.0639=32!2237 (245). Of the Systeiyi of the World. 485. In discussing the properties of the centre of gravity, we have already had occasion to consider that remarkable force exerted by the earth, in virtue of which all bodies are solicited in directions perpendicular to its surface. The ex- istence of this force was not entirely unknown to the ancients : Anaxagoras, and his disciples, Democritus, Plutarch, Epi- curus, and others, admitted the existence of such a force ; and similar opinions were entertained by Kepler, Galileo, Huy- gens, Fermât, Roberval, ''-\-dr^ (258) and by substituting this value in equation (254), we obtain, 'l^^^Èl=h-2f^dr (259). 502. To determine the equation of the curve described by the moveable point, we eliminate dt between equations (257) and (259) : the first of these gives - r-dçi dt = ; a this value, introduced into the second, transforms it into a^r^dç)'^ +a^dr r*d- -^-dr"^ , o/»t)j /oai\ —dt"' S3— =''-2-^^'- (^•'l'- The integral of the first of these equations is, by Art. 435, 2 . sector LAm=a^ (262) ; consequently, by making ^=1, we shall find that a is equal to twice the sector described in a unit of time. The same result may be obtained from the equation r'f^^a (263); for d/p being the infinitely small arc described in the time dt^ = 6— 2/Rrfr; GRAVITATION. 263 by a point on the radius vector whose distance from the centre of attraction is equal to unity, rd(p will be the arc described in the same time with the radius r ; hence \r . rdt, be denoted by 1 ; the force exerted by the sun upon a body placed at the distance k will then be expressed by the mass M of the sun, or, in other words, by the number of units which its mass contains : but the mass of a planet attracted by the sun being denoted by m, this planet will exert an attraction upon the sun, which will, for a similar reason, be expressed by its mass m : moreover, since the two forces M and m tend to cause the approach of the two bodies, their effect upon the relative motion of the bodies will be the same as if the force M + w were concentrated in the sun, and acted on the planet at the distance k. When this distance varies and becomes equal to r, the intensity of the force will like- wise vary. Let R denote its intensity at the distance r ; the assumed hypothesis will give the proportion M+7;ï:R::l:i; le- r^ whence, R=^!1M+^) ^2g5)_ Such is the value of the attractive force which, acting at the distance r, will cause the bodies to approach each other. 505. The value thus determined corresponds to that of the incessant force which we have hitherto represented by R : we therefore have /RA.=yiii(M&. putting, for brevity, /v2(M+m)=M' (266), the preceding equation will be reduced to 'Wdr fRdr=fl ■ ■ (267) ; but since the quantities M and m and the distance k remain invariable, the quantity M' will be constant ; the equation (267) may therefore be readily integrated, and will give GRAVITATION. 265 —M' replacing /Rc/r by this value, and 6— 2c by b', the equations (259) and (260) will become r^d^'+dr' 2M' V ^^^^+7^ ^^^^^' , adr ^^r^{b'r2-a'+2Wr) ^ ^' 506. To determine the value of the constant b', or its equal 6— 2c, we observe that the equations (258) and (268) give, by comparison, * dx^-^-dy^ j'^dp^ ^dr^ __ 2:vr dt^ dt^ ^ + r ' and since ^{dx"" ■\-dy'') is equal to ds, the element of the curve, it appears that the quantity — i- — - — is equal to (ds\ ^ — j , or equal to the square of the velocity estimated in the direction of the tangent to th^^ curve ; thus, denoting this velocity by v, the equation (268) will become 2M' ,;2=:6' + f^ (270). If V represent the velocity at a given instant, and x the cor- responding value of the radius vector, the equation (270) will contain but a single unknown quantity b\ whose value will result 507. The constant a may also be determined in functions , by replacing of the initial velocity ; for, by replacing -— in the formula dV ) by its value — deduced from equation (264), we shall obtain v2=^+^ (271). dt= ^r= "^ ^ 23 266 DYNAMICS. The quantity dr represents the infinitely small difference ml {Fig. 189) between two consecutive radii Am and Kn] and by regarding the triangle mnl as rectilinear, and right-angled at /j we shall have 7nl—mn . cos mnl, or, dr=ds . cos wmZ; (Is substituting this value of dr in (271), and changing — into v, we shall find v^=v'^ cos'^ nml-i — . But if * denote the value of the angle nml, when v and r are transformed into V and a, we shall have the relation Y''=Y^ .cos' a-\-~: whence, «2=^272(1— cos^ «)=A2V2sin"'«; and consequently, a=A. V.sin«. 508. Having determined the constants which enter into equation (269), we proceed to integrate it, for the purpose of discovering the nature of the trajectory described by the ma- terial point. To facilitate the integration, make r= -, and the equation (269) will then become ■~~^[b'-{a''z=' -2M'z)] ' or, adz d^= making az =p, and 6'-t— — =Aa, a a' the preceding equation will be reduced to GRAVITATION. 267 , _ —dp and by integrating, we find ^+ constant = arc |COS=^). Replacing p and A by their values, suppressing the common factor a, and denoting by 4- the arbitrary constant, we obtain / a^z—W \ whence. =cos( {Fig. 190), formed by the radius vector with the primitive axis AC, be supposed successively equal to 1°, 2°, 3", &c. and if the variable angle be reckoned from the axis AB, which forms with the axis AC an angle CAB=^^, the angle included be- tween the radius vector km and the axis AB, will be succes- sively equal to 1°+^^ 2°+>/., 3°+^/.,&c.; or, in general, to 510. The angle - In like manner, for a second planet m', which performs its revolution in the time T', in an ellipse whose semi-axis major is denoted by D', we shall have, since the mass of the sun remains invariable, 3 T/=-~?^l—^ (283) : but the masses of the planets being extremely small when compared with the mass of the sun, we may neglect the quantities 7Ji and m' in comparison with M ; and the equa- tions (282) and (283), being then compared, will give the proportion T : T' : : D^ : D'% or T^ : T'^ : : D^ : D'^ ; the squares of the times of revolution will therefore be pro- portional to the cubes of the greater axes of the orbits de- scribed, or to the cubes of the mean distances of the planets from the sun. 517. The inverse problem may also be resolved, and the law of gravitation deduced, from the elliptical motions of the planets. For this purpose, we must adopt the hypothesis that the equation (260) refers to an ellipse : but the polar equation of the ellipse being of the form Cr cos ç>=B^ — Ar, its differential will give *^^r^[(C2-A^)r»-B* +2AB'r]' 272 DYNAMICS. The condition of identity between this equation and equation (260) requires that we should have — 7/Rrfr= AB2 =a constant, or -fRdr ^^""^tant . differentiating, and suppressing dr, there remains Tj _ constant which proves that the force varies in the inverse ratio of the square of the distance. Of the Motions of Projectiles. 518. If an impulse be communicated to a material point in a direction oblique to the surface of the earth, the point being at the same time solicited by the force of gravity, it will describe a trajectory, the nature of which it is proposed to investigate. To determine the circumstances of this motion, we will denote by Aa-, Ay, and Az the three co-ordinate axes , the axis Ajz being supposed vertical. The force of gravity will then tend to diminish the co-ordinates z which are reckoned positive upward, and if its intensity be supposed constant, we shall have X=0, Y=0, Z=--. These values being substituted in the general equations (180) reduce them to d^x_f. ^''y_n d^z_^ 'dF ' JF ' dF~~^'' the first two of these equations being multiplied by dt, and integrated, give dx dy , dt ' dt ' the constants a and b represent the velocities of the material point in the directions of the axes of x and y respectively. These velocities distinguish the motion under consideration from that which takes place when the point is projected ver- tically, their values in the latter case becoming equal to zero. If tiie preceding equations be multiplied h^ dt, and again integrated, we shall obtain PROJECTILES IN VACUO. 273 x=at-\-a\ y=ht + h'\ and eliminating t between these relations, there results hx . ab' — a'h a a This equation appertains to a right line EC {Fig- 191), situ- ated in the plane of.r, y, and the trajectory ELC will therefore be contained in a vertical plane. 519. Since the trajectory described is confined to a vertical plane, it will only be necessary to consider the two co-ordi- nate axes of X and y, the former being supposed horizontal and the latter vertical ; we therefore employ the two equations 'dF ' dr-~ '^' Multiplying by dt^ and integrating, we find %=a, 'i^-St^r. (284). If we multiply again by dt^ and integrate, we shall obtain .r=a^ + a', y— — \gt^ +ht-\-h' (285). To determine the constants, we suppose the time to be reckoned from the instant at which the material point leaves the origin of co-ordinates ; whence, .r=0, y=0, and ^=0; this supposition gives a'=0, 6'=0; and the equations (285) are thus reduced to x—at^ y — — \gt'^-\-ht. Eliminating t between these two equations, we find y--a^-'^4' ('«")• The equations (284) indicate that the constants a and h express the values of the horizontal and vertical compo- nents of the velocity at the instant from which the time is reckoned, or when ^=0. If, therefore, V denote the initial velocity, and a the angle formed by the direction of the initial impulse with the axis of x^ the componeYits of this velocity will be S 274 DYNAMICS. V . COS « parallel to the eixis of x, V . sin « parallel to the axis of y ; whence, a=V cos «, 6=V sin «. These values reduce equation (286) to y=^ tang .-^g^-^ (287). 520. This equation appertains to a parabola, having its origin at the point A {Ptg. 192), the vertex being situated at a point C, above AB, and the curve extending indefinitely below AB ; for. the equation (287) being of the form i/=mx — iix'^, by making y =0, we shall obtain for the abscisses of the points at which the curve intersects the axis of x, x=0, and x= — . n 7/1 But every value of :r less than — will give a positive value for 7/, whilst every value greater than — will give y a nega- tive value. For, if we multiply by nx both members of the inequality m x<—, n we shall obtain nx^^mx, the condition which is obviously necessary, that the ordinate y may be positive. In like man- 771 ner, it may be shown that when .t>— , the value of y will n become negative. 521. If /i denote the height from which a body must fall to acquire the initial velocity V, we shall have (Art. 401) V=^(2^A) (288): by means of this value, the equation (287) is reduced to x^ y=x . tang » — — . — (289). ^ ^ 4/i cos^' « ^ ' 522. The distance from the origin A to the point B, at which the curve intersects the axis of a-, is called the range. PROJECTILES IN VACUO. 275 To determine its value, we make y=0, and the corresponding value of .r, which is not zero, will express the range. Thus making y=0, in (289), we have 3;=0, and a:=4A. tang «. cos^rtj the second value of x gives, by reduction, x=-4Ji . sin a. . cos « ; and consequently, range =4/i . sin a . cos u, (290) ; or, replacing 2 sin « . cos » by its equal sin 2<«, we have range =2/i . sin 2* (291). This equation may be employed in the construction of tables which shall express the ranges corresponding to different velocities, and different angles of projection. 523. The greatest positive ordinate will express the maxi- mum elevation of the moveable point above the axis of x. To determine its value, we make -^=0; or, ax -^ =tang «■—— =0 ; dx ^ 2h cos=^ a from which we deduce x=2h . cos2 « . tang a, or, x=2h . cos « . sin «6 ; and consequently, the absciss of the highest point of the tra- jectory will be equal to one-half the range. Replacing x by 2/t . cos « . sin « in equation (289), we find for the maximum elevation of the moveable point, y=/t . sin^ ct. 524. The projectik may be impelled in two different direc- tions, so as to produce the same range. For, let «' represent an angle equal to the complement of « ; the equation (290) will give the value of the range, 4A . sin « . cos «=4/i . sin « . sin « . But if the projectile be thrown in a direction forming an angle *' with the axis of x, the range will be expressed by Ah . sin «'. cos x=Ah . sin «' . sin «. S2 276 DYNAMICS. The identity of these expressions for the ranges corresponding to the angles » and <*', evidently proves that the ranges will be equal when the two angles of projection are complements of each other. 525. To determine the angle of projection which corres- ponds to the greatest range, we remark that the range is in general expressed by 2h sin 2«, and that this expression will become a maximum when the angle 2« is equal to 90° ; hence it follows that a projectile in vacuo will have the greatest range upon a horizontal plane when the angle of projection is equal to 45°. The supposition of 2<«=90° gives sin 2»=! ; consequently, the expression for the range then becomes equal to 2h ; or the range corresponding to the angle of 45° is equal to twice the height due to the velocity of projection. Let this range be denoted by P ; we shall have h = \V (292). To determine the value of the coefficient h, the projectile may be thrown in a direction forming an angle of 45° with the horizontal plane, and the corresponding range may then be measured. If this range be represented by P, the value of h will immediately result from equation (292). In fire- arms, the coefficient h serves £is a measure of the force of the powder, since the extent of the range evidently depends on the intensity of the force of projection. 526. The quantity h having been determined by taking the mean result of a large number of experiments, we substi- tute its value in equation (289), which will thus become 'y=x tanga- 2P cos- « If we represent by P' the range corresponding to an angle «', the equation (291) will give F=2^sin2«' (293); or, .replacing h by its value iP (292), we find P'=Psin2«'. This relation will determine the range P' corresponding to the angle «', when the value of the maximum range has been previously ascertained ; and, in general, we can calcu- PROJECTILES IN VACUO. 277 late the range P' which corresponds to an angle «', from, a knowledge of the range P" given by any other angle «" ; for, since P'=P sin 2«', P"=P sin 2«", we obtain, by division, F _ sin 2cc' ^ P" sin 2<*" ' if, therefore, the range P" corresponding to the angle »" be determined by measurement, the value of P' corresponding to X will result immediately from the preceding equation. 527. The value of h (292), being substituted in equation (288), will give, for the value of the initial velocity, V=v/(%) = v/(32ift.xP). If, for example, the range corresponding to an angle of 45"^ were equal to 1000 feet, we should find V=:^(1000 ft.x32ift.) = 179.3ft., nearly. 528. If, on the contrary, the initial velocity and angle of projection were given, we might determine the range : for example, let the initial velocity be supposed equal to 200 feet per second, and the angle of projection 15" ; we first determine h from the following formula, deduced from (288), V2 (200 ft.')* A= — = ^-— — ^ = 6^1 7 ft • ^ 2g 64ift. ^■^■^"•' and the range P' will then become, (293), F=2x621.7ft.xsin30°=261.7ft. 529. The problem may also be presented under the follow- ing form : — Having given the initial velocity and the co-ordi- nates a:'=AB, and y'=BC, of a point C {Fig. 193), it is re- quired to determine the angle of projection such that the trajectory may pass through a given point C. The equation V= v^(2g-A) will determine the value of h ; and since the co- ordinates x' and y' should satisfy the equation (287), we shall have by substituting i/ and y' for x and y, y'=x' tang«----^^l— (294). ^ ^ 4A . cos* cc In this equation the quantity « is alone undetermined : mak- ing tang ««=z, we have 278 DYNAMICS. 1 1 COS «=- sec* ^(1+taiig^ct) ^{i^z')' and by substituting these values in equation (294) we find j/=x'.z-^{l+z=) (295). This equation being resolved with reference to z, will give two values which determine the two angles of projection corresponding to the directions in which the projectile should be thrown in order that it may strike the point C ; we select the greater of these two angles when we wish to crush the object upon which the projectile falls, £is the vertical velocity at the point C will then be the greatest. It may occur, that instead of the line CB, we have given the angle CAB subtended by the object CB. Let this angle be denoted by ^ ; we shall have CB=:r' tang^=y'; this value of y', being introduced into equation (295), trans- forms it into • x' ta.ng^) \/-/>v/(l+P=')-logb + v^(l+P')]+B 541. We can also express the arc s in functions oip ; for the equation (307) gives Taking the logarithms, and reducing, we obtain 'mV-cos^ <* 1 {' [B— _p%/l+p2— log (p + v/l+jo*)] 2711 542. To obtain the equation of the trajectory, it would be necessary to integrate equations (308) and (309) : these inte- grations cannot be effected except by the aid of series. Never- theless, by employing equations (308) and (309), the curve may be constructed approximatively by points. For this purpose, we will write those equations under the form, dx= is usually called the acceleratiJig force, and F is called the waving force. When F is given, the value of ^ can.be determined by simply dividing by M, the mass moved. 549. It has been shown. Art. 163, that if g represent the force of gravity, P the weight of the body, and M its mass, we shall have MEASURE OF FORCES. 291 eliminating M between this equation and the preceding, there results g and if the incessant force

' — 11. The quantities of motion due to these velocities being such, by the principle of D'Alembert, as to produce an equi- libriinn, we shall have M{v—2i) + M'{v' -zi)=0 ; whence we deduce for the velocity after impact, _Mv + M'v' ''~ M+M' • When the bodies move in opposite directions, v' will become negative. 575. As a second example, let it be required to determine the circumstances of motion of two bodies M and M', which PRINCIPLE OF d'aLEMBERT. ; 305 rest on two inclined planes AB and AC {Fig. 201) having a common altitude, and are connected by a thread MEM', pass- ing over a fixed pulley. If the vertical line M^, drawn through the centre of gravity of the body M, be supposed to represent the intensity of the force of gravity ; the component of the force in the direction of the plane will be represented by MR ; this component will alone tend to urge the body down the plane : its value will be expressed by AD §• Xcos RMg-^^ . cos BAD=^— — . AB In like manner, the component of gravity, which tends to cause the descent of the body M' on the plane AC, will be expressed by g-j-^- AC Let the lines AD, AB, and AC be denoted by h, I, and V respectively ; the incessant forces exerted upon the bodies will then be gh J gh T' ""■* T- But if we suppose the motion to take place in the direction M'EM, and the velocities to be reckoned as positive in this direction, the force j^, which is opposed to the motion, must be regarded as negative ; and the incessant forces will there- fore be expressed by it, and _f . The general expression for the value of an incessant force being dv

dt : hence, the velocities imparted to the bodies in the time dU when they are unconnected, will be expressed by u 306 DYNAMICS. and the quantities of motion due to these velocities will be Ug!^di, -WffydL But the bodies being supposed connected by a thread of inva- riable length, if M should descend through any distance on the plane AB, M' will necessarily ascend through an equal distance on the plane AC ; or, in other words, the velocities of the bodies at any instant will be equal to each other. De- noting by V their common velocity at the end of the time t, the eifective velocities communicated to them in the succeed- ing instant dt, will be expressed by dv, and the effective quantity of motion imparted in the same time, will there- fore be (M.-\-M')dv. By the principle of D'Alembert, this quantity of motion when applied in a contrary direction, wil produce an equilibrium with the quantities of motion impressed on the bodies : hence, the sum of these quantities of motion will be equal to zero, or -{M+M)dv+Mg^dt-^^dt=-0 (325) : from which we deduce and by integration, ■ "=4+^-^'+° (^'^"'^ or, if we denote by G the coefficient of ;, we shall have v=Gt-\-0 (326). Let X represent the distance OK of the body M from the point O, the origin of the spaces, at the end of the time t ; the general expression for the velocity gives dx and therefore, ''' 'dt at PRINCIPLE OF d'aLEMBERT. 307 from which, by integration, we obtain ar=iG^2+o^ + C' (327). The formulas (326) and (327) indicate that the circumstances of motion in this system are precisely similar to those which attend the fall of heavy bodies ; the only difference consisting in the value of the incessant force, which in the latter case is denoted by g, and in the former by G. 576. If the planes AB and AC be supposed to become ver- tical, the case will be reduced to that of two weights con- nected by a cord which passes over a fixed pulley : the quan- tities A, Z, and I' are then equal, and the equations (325 a) and (327) may then be reduced to M— M' M— M' ''=ffw^^'+''' "=ra''*^''+'''+''' • • • <'''''^- 577. These formulas will serve to explain the principle of Atwood's machine, which is employed for the verification of the laws of constant forces. This machine consists essentially of, 1°, A fixed pulley, over which passes a very fine flexible thread, having its ex- tremities attached to two equal brass basins ; 2°. A vertical graduated scale with a moveabls stage to maxk the space passed over by the descending basin ; and, 3°. A seconds pendulum, by means of which the time of descent may be accurately observed. When the two basins are loaded with equal weights, they will sustain each other in equilibrio ; but if an addition be made to either, it will immediately preponderate, aîid will produce a motion uniformly varied. Moreover, by rendering the difference M— M' of the weights M and M' attached to the extremities of the thread, very small in comparison with their sum M + M', the space described and the velocity ac- quired in a given time which result from equations (327 a) may likewise be rendered small, and the observations will thus become susceptible of great accuracy. For the purpose of observing the velocity acquired at the end of any time, we give to the additional weight placed in the descending basin the form of a flat bar, and the basin being allowed to pass through a sliding ring attached to the U2 308 DYNAMICS. vertical scale, the bar may be removed at any instant during the descent. The equality of the weights in the two basins being restored by the removal of the bar, the motion becomes uniform with the velocity acquired at the instant when the bar was removed. By comparing the spaces described, the velocities acquired, and the times elapsed, we find that when the basins move from rest under the influence of a constant force, the velocities are constantly ])roj)ortional to the times, and that the spaces are proportional to the squares of the times. 578. For a third example, let it be required to investigate the circumstances of motion of two weights M and M', which are attached to cords passing around the respective circum- ferences of a wheel and of its axle. If we suppose the body M to prevail, and reckon the veloci- ties positive in the direction of its motion, the force of grav- ity will impress upon the bodies M and M', in the instant dt^ which succeeds the time t, the velocities gdt and — gdt ; and the quantities of motion impressed will therefore be Mgdt, and —Wgdt. But if V and v' represent the velocities of M and M' at the ex- piration of the time t, the effective velocities communicated in the succeeding instant dt will be expressed by dv and dv'. Thus, denoting by R and r the radii of the wheel and axle, we shall have Masses. Impressed velocities. Effective velocities. Distances from the axis. M . ... gdt dv R, M' . . . —gdt dv' r. The effective quantities of motion, being applied in directions contrary to those of the motions assumed, will sustain in equi- librio the quantities of motion impressed ; and since the equi- librium is maintained through the intervention of the wheel and axle, it is necessary that the sum of the moments with reference to the axis should be equal to zero : hence, we obtain WRgdt—Wrgdt-MRdv—'m:rdv'=() (328). This equation containing the two unknown quantities v and r', it will be necessary to discover a second relation between UNIFORM MOTION ABOUT AN AXIS. 309 them. For this purpose, we remark that the velocities v and v' bear to each other the constant ratio of R : r ; thus, we have V : v' ::B. : r ] or, r and by differentiating, V —V- R T dv'=—dv R substituting this value in equation (328), we find MRgdt—M'rgdt—MKdv-M'^dv=0 ; or, by reduction and transposition, MR'dv-^-M'r^dv^MR^gdt-M'Rrgdt; whence, , MR2-M'Rr ,, ^^^=mrm:mv^^^'' Denoting by K the constant coefficient of df, this equation becomes dv=K.dt ; and by integration, Replacing v by its value — -, and performing a second integra- tion, we find These results indicate that the motion is uniformly varied, the circumstances of the motion being similar to those of a body falling under the influence of the force of gravity. Of the Motion of a Body about a Fixed Axis. 579. When an impulse is applied to a system of material points connected together in an invariable manner, and sub- jected to the condition of turning about a fixed axis, which we will suppose to pass through the point A {Fig. 202), per- pendicular to the plane of the figure, the several particles tw, 310 DYNAMICS. m\ m", &c. will describe circles won, rr^o'i}!^ m"o"n", ', p", &.c. the angles formed by the directions of the forces m'v', vi'v", &c. with the planes o'm'?i', o"m"ii", (fee, the quantities of motion im- pressed will become mv cos (p, m!v' cos , UNIFORM MOTION ABOUT AN AXIS. 311 —m'r'u, —}7i"r"u, &c. are situated in planes perpendicular to the fixed axis. The conditions of equilibrium between these forces will evidently be the same as those which arise when the forces are situated in the same plane ; if, therefore, the forces be regarded as situated in the plane of the figure, the conditions of equilibrium will require that the sum of the moments of the forces which tend to turn the system in one direction about the point A, shall be equal to the sum of the moments of those which tend to produce rotation in a contrary direc- tion ; or, that the algebraic sum of the moments shall be equal to zero. But the quantities of motion — mr, &c. being derived from the common motion of the system, they will tend to turn it in the same direction ; and since these motions take place in the circumferences of the circles mno, m'u'o', m"n"o", &c., the radii r, r', r", &c. will represent the perpendiculars demitted from the point A upon their respect- ive directions ; consequently, the sum of the moments of the effective quantities of motion, when applied in opposite direc- tions, will be expressed by — mr'^61 — mr'^u — rn'r'^u — &Lc. = — «(mr^-j-TOr'^-f-OT V'^-i-&c.). Let the quantity within the brackets be denoted by :s{mr^) ; the sum of these quantities of motion will then be repre- sented by — û)2(mr2). To determine the value of the sum of the moments of the impressed forces, mv . cos (p, mJv' . cos ', m"v" cos ", . p, m'v' cos . p) — al.(mr^ ) =0. This equation gives the value of the angular velocity ^^Hmv.cosç.p) and the motion of the body about the fixed axis will there- fore be uniform. 580. When the forces mv^ m'v\ m"v", (fcc. are exerted in planes perpendicular to the axis, the angles "=0, cos «=: — (OOi). 582. It may happen that the velocity v has been impressed iipon only a limited number of the particles ?n, m\ m", &c. : then, M will no longer represent the entire mass of the system, but merely the sum of those particles upon which the velocity has been impressed ; and Q, will express the perpendicular demitted from the centre of gravity of this part of the system upon the plane AK. The quantity ^(mr^) is called the moment of inertia : the method of determining its value will be explained in the next section. 583. It is frequently necessary to consider the effects pro- duced upon the fixed axis by the application of an impulsive force to any point of the system. For this purpose, let the axis of rotation Kz {Fig. 205), be assumed as the axis of z, and resolve the impulsion P, which is supposed to be applied at a point O, into two components P' and P", which shall be respectively parallel and perpendicular to the plane of x, y. liCt the axis of y be then assumed parallel to the direction of P', and denote the co-ordinates of the point O by a, è, and c : since the force P may be applied at any point in its line of direction, we can always suppose the point of application O to be contained in the plane oi x^z'. this supposition gives 6=0. Instead of regarding the axis as fixed, let such forces be introduced as may be necessary to retain it. These forces will be equal, and directly opposed to the impulsions expe- rienced by the axis, and may in general be reduced to three forces respectively parallel to the axes of ^, y, and z. Let X, 27 314 DYNAMICS. Y, and Z represent the impulses communicated to the axis, and call AB=«, AC=/3. The particle m will describe a circle parallel to the plane of X, y, and its velocity in the direction of the tangent ml will be expressed by ret (Art. 579) : the cosines of the angles formed by this direction with the axes of x and y respectively, will be — and — — ; hence, the effective quantity of motion of r r the particle m will be mrai, and its components in the direction of the axes of x and y will be mi/a and —mxa : the same remarks apply to the other particles m', m", m"\ &c. But, by the principle of D'AIembert, an equilibrium will subsist between the effective forces and the force P, the latter being applied in a contrary direction ; thus, we shall have Forces. Components parallel to axes of Co-ordinates of points of application parallel to X y z X y z — P Pcos^ — Psin^. ... a c, X X Y Y ^, Z Z 0, niru myu — mxu x y z, m'r'a) m'y' a — mx'u x' y' z\ a=0, X«+«2;(my2;) + P sin ^a=0, Y/3+P cos (331 b). Y^= '-. 586. Although the axis will receive no impulse at the in- stant of impact, yet the motion of rotation will immediately give rise to centrifugal forces which will exert a pressure upon the axis. Of the Moment of Inertia. 587. The momert of inertia being the sum of the products formed by multiplying each material point of a system by the square of its distance from a fixed axis, it has been repre- sented in the preceding section by 2(wir"). In this expres- sion, we may replace the particle m by dM., the element of the mass ; and the moment of inertia will then result from the integration of an expression of the form/r^rfM. 588. For example, let it be required to determine the moment of inertia of a material right line CB {Pig. 206), with reference to an a\is AZ perpendicular to the plane CAB. Let AB=/i represent the perpendicular demitted from the point A upon the right line, and BP=;r the distance of a point P assumed arbitrarily on this line, from the point B : we shall have PA2=A2^.'r2. This expression being multiplied by the diflerential of the mass, the integral of the product will express the moment of inertia. The volume, in the present case, being a right line, the element of the volume will be represented by the infinitely small difference dx between two consecutive abscisses BP=^ and BV=x-\-dx] and the element of the mass rfM will therefore be expressed by dx multiplied by the density D, or by Ddx. Thus, by multiplying h- -^-x^ by Ddx^ and inte- MOMENT OF INERTIA. 317 grating, we obtain for the expression of the moment of inertia of the right hne, In the present disposition of the figure, the integral should be taken between the limits of the point B, where x=0, and the point C, at which x=a] the moment of inertia thus becomes ('"«+Ï)d- In effecting this integration, we have regarded the line as homogeneous, or the density D as constant : but if the differ- ent parts of the line be supposed unequally dense, the quan- tity D will be variable, and may in general be regarded as a function of x. The form of this function will depend on the law according to which the density is supposed to vary. 589. When the body is homogeneous, it is frequently con- venient to regard the density as equal to unity ; and the factor D is then replaced by 1, in the general expression for the moment of inertia. Having determined the moment of inertia of a body whose density is equal to unity, we can determine that of a similar body whose density is equal to D, by simply multiplying the former moment by the density D. In the succeeding examples, we shall regard the density as equal to unity. 590. As a second example, we will determine the moment of inertia of the area of a circle CBD {Pig. 207), ivith refer- ence to the axis AZ passing through its centre, and perfen- dicular to its plane. Let m represent a point in the plane of the circle, at a dis- tance mA=x from the fixed axis : the areas of the circles described with the radii x and x-\-dx will be expressed respectively by vx^, and 7r(x-\-dxy ; and the difference between these areas, by neglecting the infi- nitely small quantities of the second order, will be 25rx . dx:. This expression will represent an elementary ring, every point of which will be at the distance x from the axis : hence, by multiplying this element by x^, we shall obtain 27rx^dxfoi the differential of the moment of inertia. Taking the integral 318 DYNAMICS. from a?=:0 to x='>' we shall find i^rr* as the moment of inertia of the area of a circle whosk radius is denoted by r. 591. Let it be required to determine the iiioment of inertia of a sphere with reference to an axis passing through its centre. If the sphere be cut by a plane EE' perpendicular to tlie fixed axis AB {Fig. 208), the section will be a circle whose centre will be fuund at the point D. Denote by x the absciss AD of this section, and by y the ordinate DE; or the radius of the section. The moment of inertia of the area of this circle taken with reference to the axis AB, will be expressed (Art. 590) by and if this expression be multiplied by dr=DD', the product, l^y'dx, will express the moment of inertia of the elementary volume EE'F'F bounded by parallel planes drawn through the con- secutive points D and D'. The integral of this expression, being taken between the limits :r=0 and a:=AB=2r, will give the moment of inertia of the entire sphere. But by the property of the circle, we have y 2 =2rx — x^ ; and therefore, f^7ri/*dx=^^f(2rx—x''ydx =7rf(2r^x^ —2rx^-\- ^x*)dx ; or, f^^y'dx=^x^{"fr^-irx+-f\x^)-\-C. The constant C will be equal to zero, since the moment is zero when x=0: and by making x=2r, we obtain for the moment of the whole sphere, _*_«■/• ^. 15' • These examples are sufficient to explain the manner in which the determination of the moment of inertia is reduced to a simple problem of the integral calculus. 592. When the moment of inertia of any body with refer- ence to an axis passing through its centre of gravity has been determined, its moment with respect to a parallel axis is readily found. MOMENT OF INERTIA. 319 For let GF and CK {Pig. 209) represent two parallel axes, the first of which passes through G, the centre of gravity of a body : let the origin be assumed at the point G, the line GF being the axis of z. Through a point m, assumed arbi- trarily within the limits of the body, let the plane mKF be drawn, parallel to the plane oi x^y \ this plane wilt cut the axes GF and CK at two points F and K, and the distances of the point ni from these axes will be represented respect- ively, by the right lines mK and mF, which we shall denote by r and r'. From the point m let the perpendicular mE be demitted upon the plane of a;, y ; the triangles ECG, mKF will be equal in all respects, and the sides of the former may there- fore be substituted for those of the latter. Denote by a, and j3, the co-ordinates GD and DC of the point C, X and y, the co-ordinates GP and PE of the point E, a, the distance between the axes : we shall have GC2=GD2.fDC=, GE2=GP2-fPE2, or, «2 =^2 4-^2^ ^'2 ^j.2 J^y2 (332). Again, the right line CE passing through points whose co-ordinates are x and y, « and /3 ; the value of CE— r will result from the equation or, by developing the terms of the second member, r2 = r2 -f y2 _2«.r— 2/3y+«2 4.^2 j and reducing by means of equations (33,";), we obtain multiplying by dM and integrating, we have fr''dM=fr'^d^l-2ccfx(M—2i2fydM.+a^fdM (333). The expressions fxdM. and fydM. which enter into this equa- tion, are equal to zero ; for, let x and y represent the co- ordinates of the element dM of the mass M ; the moments of this element with reference to the planes ofx, z and y, z will be ydM and xdM. : hence, the co-ordinates x, and y, of the centre of gravity of the mass M will be determined by the equations M.x=fxdM, My=fydM.. 320 DYNAMICS. But in the present instance, the centre of gravity is situated in the axis of z ; and the co-ordinates x, and y, are therefore equal to zero : hence, /r^M=0, fy(m.=0. Reducing equation (333) by means of these values and sub- stituting M for its equal /c/M, we shall obtain fr-dM^fr'^dM + Ma^ (334). The expression y?''2rfM being the moment of inertia with re- ference to the axis passing through the centre of gravity, we conclude that when the value of this moment has been found, that of the moment of inertia /r^cZM, taken with reference to a parallel axis, may be immediately determined, by adding to the former the product of the mass of the body by the square of the distance between the two axes. The equation (334) may be written under the form y-..-\-rda) ; and the effective quantity of motion of the particle m will be {7'ù) + rdw)m. The same remarks being applicable to the other particles which compose the system, it is necessary that the quantities of motion impressed, or 2,[[ru-{-(p cos S'.dt)m] should, by the principle of D'Alembert, sustain in equilibrio the effective quantities of motion l,[(ru-\-rd being substituted in equation (336), we obtain dç^_/gpdM . dt fr^dM ' or, since g is constant, dco^gfi/dM dt .fr^dM^ The expression ydW represents the moment of the elementary mass dM taken with reference to the plane oî x, z\ if, there- fore, we denote by y, the distance of the centre of gravity of the entire mass M from the same plane, we may replace fydM. by My,, and the preceding equation will then become d^_ gWy, ^^ dt-J^l ^'^*^^^- and since //'^6?M expresses the moment of inertia with refer- ence to the axis C2;, this moment may be represented (Art. 592) by M(Â:2+a^). Substituting this value in equation (337), we find ^ = , ^y' (338). 596. It has been shown (Art. 592), that the quantity a in the expression M(a- + k"^ ) represents the distance CG (Pig. 209) between the axis CK and the parallel axis GF passing through the- centre of gravity. But, by the motion of the system, the centre of gravity describes a circle having its radius CG=a {Fig. 213), and its plane arCL perpendicular to 324 DYNAMICS. the axis CK ; hence, the ordinate DG will represent the quantity y,, and we shall have from the property of the circle, Again, if s denote the arc described by the point G, the velo- ds city of this point will be expressed by — - : but this velocity at will also be expressed by a» (Art. 579). Hence, we shall have ds and consequently, ds «= — —' adt The values of a and y^ being substituted in equation (338), convert it into ade kr-^w" 597. If we multiply each member of this equation by 2ac?5, the first member will become an exact differential, and we shall obtain by integration, %' ""' ""' ^y^FS^^*^^^"^^'""^''^ ^^^^^' The integral of the second member can only be obtained after eliminating one of the two variables which it contains : this may be effected by means of the equations ds=^{dx^- -\-dy;-), y=y/{2ax,—x;-) ; and by proceeding as in Art. 465, we find , _ — adx, ^(2ax,—x,^) ' substituting this value in equation (339), we have V2 = — / , ^^ dx, ; whence, by integration, ^^^_2a2^ (340). To determine the value of the constant C, let EB=6 repre» COMPOUND PENDULUM. 325 sent the value of .x\ at the instant when v~0 ; the supposition of w=0 and x=b gives C = 2ar-gb . k^-+a ' and the equation (340) will therefore become v^, or dp -'^"/) ; whence, dt— ■ 7 ^ ds dt=- This equation can be readily integrated when the oscillations are performed through very small arcs, as usually happens ; for, by replacing ds by its value — -!_ obtained on the y/{2ax) supposition that x, may be neglected as exceedingly small in comparison with 2a, in the expression » — CiUiX I ^~ y/(2ax,—x,^)' th^ equation (341) becomes jdx^ which may be written under the form di=-.^(^^l±^) X ,f - , , (342). '^ \ ag J ^[{b-x)x] 598. By comparing this equation with the equation (228), it will appear that they differ only by the constant factor, / ( k"^ _L^2\ which in the former is \ \/ \ j , and in the latter -\/ — Hence, the integral of (342) may be immediately obtained from that of (228), the constants being determined by the same condition, that when jf =0, x,=b. Consequently, if we denote by I the length of a simple pendulum, or if we replace — in equation (228) by - -, and determine I by the con- dition 2§ 326 DYNAMICS. g «^ the simple pendulum and the compound pendulum will per- form their oscillations in the same time. The preceding equation gives a Thus, by means of this formula we can always find the length of the simple pendulum which will perform its oscillations in the same time as a given compound pendulum. 599. If, at the distance I from the axis of suspension AB, a line EF {Fig. 214) be drawn parallel to the axis AB, this parallel will enjoy the property, that all points contained in it will perform their oscillations in the same time as though they were unconnected with the other points of the body. When the line EF is contained in the plane passing through the axis of suspension AB and the centre of gravity of the body, this line is called the aads of oscillation^ and its several points are called centres of oscillation. 600. The axes of suspejision and oscillation are recipro- cal ; that is to say, if we take the axis of oscillation EF {Fig. 214) as a new axis of suspension, the corresponding axis of oscillation will coincide with the original axis of sus- pension. To demonstrate this property, we resume the expression for CDj the distance between the axes of suspension and oscilla- tion given in Art. 598, l^ a^+k^ .343) a If we then assume the line EF as an axis of suspension, and represent by V and a' the corresponding distances of the centres of oscillation and gravity from this axis, we shall have by the nature of the centre of oscillation, n'- -4-Z-2 1'-= ^ ^ (344). a' And since the equation (343) indicates that the distance I exceeds a, it follows liiat the centre of gravity will be situated COMPOUND PENDULUM. 327 between the axes of suspension and oscillation. We shall therefore have the following relation, a-{-a'=l, or, a'=l — a. By means of this value, the equation (344) becomes l,^ (l-a)'-\-fc' .345)^ I— a Again, from equation (343) we have 7 ^' l^a= — ; a and the value of I' may therefore be changed into I' or, by reduction, l'='L + a=l a consequently, when the line EF is taken as the axis of sus- pension, the axis of oscillation KH is situated at a distance MX from the line EF, precisely equal to that which separates the axes AB and EF. 601. The equation (343) gives a{l—a)=k^ ; and by replacing I — a by its value a', we have aa!-=k^ : but the value of /j^^ which is dependent on the moment of in- ertia taken with reference to an axis passing through the centre of gravity, and parallel to the axis AB, will remain constant so long as the direction of the axis remains unchanged : hence it appears that if the body be caused to oscillate about any axis parallel to AB, and at a distance from the centre of gravity represented by a, the corresponding axis of oscillation will be found at a distance a' from the centre of gravity ; thus the value of a-\-a\ or the length of the equivalent simple pendulum, will be the same as when the oscillations were per- 328 DYNAMICS. formed about the axis AB. A similar remark is applicable to all those axes parallel to AB which are situated at a distance a' from the centre of gravity. If, therefore, the body be sus- pended successively from any number of axes parallel to AB, and at a distance from the centre of gravity equal to a or a', the times of oscillation about such axes will be equal to each other. These parallel axes of suspension about which the oscilla- tions are performed in equal times, will evidently be found in the surfaces of two cylinders having a common axis pass- ing through the centre of gravity. 602. The expression for the distance I between the axes of suspension and oscillation may be put under the form Wi Ma ' and since this value is precisely equal to that which was obtained for the distance of the centre of percussion from the axis of rotation (Art. 585), it appears that the centre of per- cussion, when it exists, will be found upon the axis of oscillation. Of the Motions of a Body in Space when acted upon by Impidsive Forces. 603. In the preceding sections, the circumstances of motion of a body retained by a fixed axis have been alone discussed ; it now becomes necessary to consider the motions of a body in space when unconnected with fixed objects. IjCt m, ni', m", (fcc. represent material points composing a system whose several particles are unconnected, and let v, v', v", &c. represent the velocities respectively impressed upon these particles in directions parallel to each other : it is required to determine the motion of the common centre of gravity of the system. If a plane be passed through the primitive position of the centre of gravity parallel to the common direction in which the impulses are applied, the sum of the moments of the particles m, m', rii'% (fcc, taken w\û\ reference to this plane, Mall be equal to zero at the commencement of tlie motion ; PERCUSSION. 329 and it is likewise evident that this sum will remain equal to zero during the motion, since the distances of the bodies from the assumed plane remain invariable. Hence, the motion of the centre of gravity will be confined to this plane ; and since the same may be said of any other plane drawn through the primitive position of the centre of gravity and parallel to the direction of the motions, it follows that the centre of gravity will continue in each of these planes, or in their line of inter- section ; and we therefore conclude that the motion of the centre of gravity of such a system is rectilinear , and parallel to the direction of the m,otions of its several parts. Let a plane be drawn perpendicular to the direction in which the bodies move, and represent the distances of the several bodies from this plane at the commencement of the motion, by S, S', S", &c. : their distances, at the expiration of the time t, will be expressed by S+î)^, S'-f v'/, S"+D'7, (fee. If a and x, represent the distances of the centre of gravity of the system from the perpendicular plane, at the commence- ment of the motion, and at the end of the time t, we shall have, by the property of the centre of gravity, wS -{- m'S' -j- m"S" + «fee. = (m 4- ^W'' • I- W 4- : the superior sign applies to those points which are situated upon the same side of the centre of gravity as the point O ; and the inferior sign to points situated on the opposite side. 610. If we consider the motion of the point O for an ex- ceedingly short interval of time, the path Oih described by this point, whilst the centre of gravity describes the line GG', may be regarded as a right line : thus, the line OGC will assume the position AG'C, the point C remaining at rest during this interval. This point is called the centre of spon- taneous Isolation : its position may be determined by the con- dition that its velocity of rotation shall be equal to that of translation : indeed, whilst the point C would be carried for- ward over the line CC by the motion of translation, it would FREE MOTION OF A SYSTEM, 333 be moved backward through the same distance by the motion of rotation : this condition will give the absolute velocity of the point C V — aa>=0 ; whence, and we therefore have OC=OG + GC=»+a-=» + — ; P from which we conclude, that the centre of spontaneous rota- tion will coincide with the centre of percussion, if the axis of rotation he supposed to pass through the point O. 611. When the plane passing through the direction of the impulse and the centre of gravity divides the body into two portions which are not symmetrically situated with respect to this plane, it will usually occur that the axis about which the body revolves will not retain an invariable position. For, the rotatory motion of the body will develop in each particle a centrifugal force, producing a pressure upon the axis ; and unless these pressures are such as to destroy each other, the direction of the axis will necessarily be changed. Of the Motions of a System in Space when acted upon by Incessant Forces. G12, We will next investigate the circumstances of motion in a system whose different particles are acted upon by inces- sant forces. Let the force acting on a particle m be resolved into three components X, Y, Z, respectively parallel to three rectangular axes ; that acting on m' into the three X', Y', Z', (fee. Let a, 6, and c represent the variable co-ordinates of the centre of gravity referred to the fixed axes, and let three axes be drawn through the centre of gravity, parallel to the fixed axes, and moveable with the system in space. Then, if x, y, z, x', y\ z\ &c. denote the co-ordinates of the points m, m\ m", &c. referred to the moveable axes; a+a:, 6-fy, c + 5?, a-\-x', h-\-y\ c-\-z', &c. will express the co-ordinates of the same points when referred to the fixed axes. 334 DYNAMICS. 613. The velocity of the particle m in the direction of the axis of X, at the expiration of the time t^ will be expressed by dla-\-x) da-\-dx v=—^ = : dt dt ' and in the succeeding instant dt, this velocity would receive the increment Xdt, by the action of the incessant force X, if the praticle m were entirely free ; but in consequence of the connexion existing between the diiferent parts of the system, the effective velocity communicated to the particle m in the time dt, will be expressed by , , da-\-dx dv—d ; . dt and the velocity destroyed in the particle m, by the connexion of the parts of the system, will therefore be Xdt-d ^^. dt The same remarks being applicable to the velocities parallel to the axes of y and z, we shall have for the quantities of motion destroyed in the particle m, parallel to the three axes, m\ m Similar expressions may in like manner be obtained for the quantities of motion lost by the other particles ; and we shall therefore obtain, for the sum of the quantities of motion lost parallel to the axis of x, 2[m(xdt-d ^^'^)] (346); or, by completing the differentiation indicated, regarding dt as constant, we have In like manner, the sums of the quantities of motion lost in directions parallel to the axes of y and z, will be expressed by .[„(y..-'^/L^)] (34r), FREE MOTION OF A SYSTEM. 335 4.(z..-*£±*f)] (348) The quantities of motion (346), (347), (348), or the forces capable of producing them, being such as to destroy each otlier, they must satisfy the general equations of equilibrium (66) and (67), which appertain to a system of forces having various directions and applied to different points of a body. The equations (66) indicate that the sum of the components parallel to each of the axes will be equal to zero ; we shall therefore have for those components parallel to the axis of a: ;Kx<._^ii^^)]=0; (349 a) : (349 b). or, by multiplying by dt, and changing the form of the expres- sion, we have Q={mX-{-m'X'-\-m"X"-\-iSLC.)di^ —d^ a(m+m'+m"+&c.) —{md^x+m'd'x'+m"d^x"-\-&c.) (349). But, by the nature of the centre of gravity, mx+m'x'-\-m"x"-{-ôcc.=0 ) my-\-m'i/'-\-m"y"-\-ôùC.=0 ) ' ' and by differentiating twice, we find md'x+m'd^x'-{-m"d^x"-\-ôùC.=0 md'i/+m'd''y'-\-m"d' y"+&c. = The first of these values being substituted in (349), and the mass of the system being denoted by M, there will result Md^a=(mX+m'X'-\-m"X"+&c.)dt', or, M^=2(mX): dt^ ^ ' the same being true with respect to the components parallel to the axes of y and z^ we shall obtain, for the three first equations expressing the circumstances of motion of the system, M^=.(»X) 1 i M^-2(mY) \ (350). MÇ^=2(mZ) 336 DYNAMICS. These equations serve to determine the motion of the centre of gravity of the mass M; for when integrated, they will ex- press the velocities -r > ;r j ;t- ^^f the centre of gravity, par- allel to the three axes. 614. The equations (350) make known a remarkable property of the centre of gravity. For, let the particles m, m\ m'\ (fee be supposed concentrated at their common centre of gravity, and let the forces wiX, ?/iY, mZ, w'X', m'Y', m'7i\ &c. be applied directly to that point, parallel to their original directions. These forces may be reduced to three, MX,, MY,, MZ„ the values of which will result from the equations MX=2(mX), MY=2(mY), MZ=2(mZ.) Eliminating the second members of these equations by means of equations (350), we have ^=^» 1^'=^» '^-^ (^^'>- But when the forces MX,, MY,, MZ, are applied to the centre of gravity regarded as a material point whose mass is M, the circumstances of its motion are expressed by the equations (180), which are precisely similar to the equations (351) ; hence, we conclude that the centre of gravity of the system has the same motion as though the forces were applied directly to that point. 615. To determine the circumstances of motion of the several particles m, ra\ m", cfec. with respect to the centre of gravity, we resume the equations (67), which express the conditions that the forces have no tendency to turn the sys- tem about either axis : that this may be the case, it is neces- sary that the sum of the differences of the moments of the components parallel to any two of the axes, as x and y, taken with reference to the corresponding planes of y, z and a-, z, should be equal to zero. But if we consider the particle m, the distance of the component X, which acts upon it, from the plane of a-, z will be equal to y-f6, the co-ordinate of the point m, parallel to the axis of y : in like manner, the distance of the force Y from the plane of y, z will be expressed by x+a: we shall therefore have, for the difference of the moments, TU FREE MOTION OF A SYSTEM. 337 The same remarks being applicable to the particles m\ m", &c., we shall obtain a similar expression for each. By- placing the sum of these expressions equal to zero, as in equation (67), performing the multiplications, and reducing by means of equations (349 a) and (349 6), we shall obtain 62(mX) -M6|^+2(myX) -2 (^y^) — a2(mY)+Ma^-2(ma:Y)+2^w^:^•^^ =0. This equation admits of simplification ; for, if we multiply the first of equations (350) by 6, and the second by a, and take their difference, we shall have 62(mX)-a2(mY)— Mô^ + M«— =0. This relation reduces the previous equation to 2(myX)-2(m;2;Y) -2 (^y^) +2 {jnx^^ =0 ; whence, 1 (m î^!ty^) =x[«(Yx-Xy) A]. The integral of the first member, taken with reference to the time L is ;(m-!^^^^): and by adopting the same process with reference to the other two axes, putting, for brevity, l.[mf(Yx-^Xy)dt]='L, j\mf{Zx—Xz)dt\='^, ^mf{Zy-Yz)dt\=^, we shall obtain the three equations of motion ^/ ri^-^\ ^ |. (351a). Y 29 J 338 DYNAMICS. The equations (351 a) are independent of the co-ordinates of the centre of gravity, and would undergo no change if forces were appUed at that point sufficient to destroy its motion of translation, since such forces would not enter into the ex- pressions L, M, and N ; thus, the motion of rotation about the centre of gravity, determined by these equations, is pre- cisely similar to that which would take place if the centre of gravity were immoveable. Hence we conclude, that whe?i arvy body is acted upon hy incessant forces applied to its several particles, the body will receive two motioiis : one of translation, in virtue of which its centre of gravity 2vill be transported in space as though the forces were apjjlied directly to that point ; and a seco?id, of rotation about the centre of gravity, as though that point were absolutely at rest. General Equations of the Motions of a System of Bodies. 616. Let Z, /', l", &.C. represent the velocities lost or gained by the several material points which compose a system, in consequence of the mutual connexions of its parts ; the cor- responding quantities of motion lost or gained will be ml, m'l', m'l", «fcc, and, by the principle of D'Alembert, these quantities of motion, when impressed upon the particles m, m', m", &c. are such as will produce an equilibrium : hence, they must fulfil the conditions of equilibrium expressed in equations (66) and (67). The components of these quantities of motion, or the forces capable of producing them, estimated in the directions of three rectangular axes, will be ml cos», 7w/cos/3,' ml cos y components of ml. m'l' cos », m'l' cos /3', m'l' cosy' components of m'l'. m"l" cos »", m"r cos /3", m"l" cos y" components oîm"l". (fee. (fee. &c. (fee. We shall therefore have for the equations of equilibrium, 2(mZ. cos «)=0 i 2(mZ.cos(s)=:0 V (352). 1{m.l . cosy)=0 J FREE MOTION OF A SYSTEM. 339 ^[ml{x COS /3 — y cos a)] =0 i ^ml(z COS »—x COS y)] = > (353). 2[m% cos y — 2; cos /3)] =0 ^ 617. If the system is retained by a fixed point, the three equations (352) cease to be necessary ; the equations (353) being alone sufficient, provided the origin be placed at the fixed point. 618. When there are two fixed points within the system, we connect them by a right line, and assume this line as one of the co-ordinate axes, z for example ; the first of equations (353) will then be sufficient to ensure the equilibrium (Arts. 132 and 133). 619. The velocities lost or gained are here indicated by the letters /, l', I", &.c. ; but to express these quantities in functions of the incessant forces which solicit the several material points, we shall first consider the particle m, and suppose that the forces acting upon this point have been reduced to three, X, Y, and Z, respectively parallel to the co-ordinate axes. The velocity of the particle m, parallel to the axis of x, at the expiration of the time t, will be expressed by -j- (Art. 430) ; and at the end of the time t-{-di, this velocity will become -^J^d — ; this will be the expression for the effective dt dt' ^ velocity of the particle m. But if the particle m were perfectly free, the incessant force X would communicate to it in the time dt, a velocity repre- sented by ILdt (Art. 391), and the velocity of m at the expira- tion of the time ^4-<^^j would be expressed by — +Xd/; (XZ hence, the velocity lost or gained by the particle m will be equal to dx , ■^j. (dx jdx\ and by reduction, we shall find that Xc?/— rf^— will express the velocity lost or gained by the particle m, in the direction of the axis of x. This velocity being multiplied by the mass m, gives Y2 840 m DYNAMICS. (x..-.^), for the quantity of motion lost or gained by m, in the direc- tion of the axis of x : we shall therefore have ml . cos ct=m { Xdt r— Ï V dt / (354). In like manner, by considering the velocities lost by m, in directions parallel to the axis of y and z^ we shall find ml. cos ^=m I Y dt — -^j (355). 7nl. cosy=m(Zdt—-—\ (356). Similar expressions may be obtained for the quantities of motion lost or gained by the particles m', m", «fcc. ; and by including their sums under the sign 2, the equations (352) and (353) may be reduced to .(.|f)=.(.X)' 2(m^) =2(mY) [ (357). 2(.|£)=.(.z)_ de ^ ^ iK5^!^:=^^^=2[m(X;2-Z:r)] . dt^ ^^^(y^y-^^'^J^2[m(Zy-Yz)] CLZ (358). Such are the most general forms of the equations expressing the circumstances of motion of a system. 620. The expressions Ya:— Xy, X2; — Zx, 7,y—Yz, &c. become equal to zero under the following circumstances : 1°. when the incessant forces acting on the particles m, m', m", (fee. are equal to zero ; 2°. when all the forces are directed towards the origin of co-ordinates : 3°. when the forces are such as arise from the mutual attractions of the différent parts of the system. FREE MOTION OP A SYSTEM. 341 In the first case, the incessant forces being equal to zero, their components must likewise be equal to zero ; and hence X=0, Y=0, Z=0, X'=0, &c.: the second members of equations (358) will therefore dis- appear. 621. The second members will likewise disappear, when the forces are directed towards the origin of co-ordinates. For, it has been shown (Art. 436), that when the fixed point towards which the forces are directed does not coincide with the origin of co-ordinates, if we represent by a, 6, and c the co-ordinates of this point, and by jo, p', /?", &c. the distances of the several particles from the fixed point, the components of the forces P, P', P", (fcc, in the directions of the co-ordi- nate axes, will be expressed by p p p p p p P^ P'^-=^, P"f^,&c.; p p p but, by hypothesis, the origin coincides with the fixed point towards which the forces are directed, and we therefore have a=C, h=% c=0: hence, the preceding expressions are reduced to Vx V'x' V"x" P"'- , &c.. Py py P"y" &c., Vz Vz' F"z" &p v' p' P"' And by substituting these values of the components for X, X', X", Y, Y', Y", Z, Z', Z", &c. in the expressions Yx—Xy, Xz-Zx, Zy-Yz, YV-Xy, &c (359), we shall find each of these expressions equal to zero. Con- sequently, when the incessant forces which act upon the several particles are constantly directed towards the origin, 342 DYNAMICS. the expressions (359) become equal to zero, and the second members of equations (358) will therefore disappear. 622. The same consequences may be deduced when the material particles are subjected only to their mutual attrac- tions. For, by putting the second members of the equations (358) under the following forms : m(Y.-r— X2/) + m'(YV-Xy)+(fcc. ^ m(X2;-Za:)+m'(X';s'-ZV)+&c. > (360), m(Zy — Yz) + m'{Z'y'-Y'z')-[-&.c. ) and considering the material points two by two, it is evident that the moving force exerted by the point m upon m' is equal to that exerted by m' upon m. Hence, if X, Y, Z, X', Y', Z', &c. represent the components of the incessant forces P, P, P", (fee, we shall have w'X'= — wîX, m'Y'= — mY, m'7/= — wiZ, &c. : eliminating X' and Y' by means of these values, the first of the expressions (360) becomes mYix-x')-mlL{y—y') (361) : but the force whose components are X, Y, and Z being denoted by P, and the distance between the points m and m! by /?, the cosines of the angles formed by the direction of the force P with the co-ordinate axes, will be represented respect- ively, by x~x' y—y' z—z\ p P p and we shall have ,x-—x -^ j^y y „ -ç^z z X=pl— ^, Y=P^— ^, Z=P P p P Substituting these values in the expressions (361), we obtain mV .- — —{x—x')—mV. iy—y) ] a quantity evidently equal to zero. In like manner, it may be proved that the other terms of the expressions (360) destroy each other ; it therefore follows, that when the material particles m, rn', m'\ &.c. are subjected only to their mutual attractions, the second members of the equations (358) will disappear ; and since this result is inde- FREE MOTION OF A SYSTEM. 343 pendent of the position of the origin, that point may be selected arbitrarily. 623. When either of the three cases just considered presents itself, the equations (358) will reduce to dt' ' I,['m(zd^x—xd^z)]_f. df^ ' 2[m{yd^z-zd^p)] _^ dp The quantities included within the brackets being exact dif- ferentials, these equations may be written under the form 2[*w . d{xdy — ydx)\_ç. ~~dP " ' 2[m . d(zdx — xdz)] _ „ ^[m.d{ydz—zdy)] _ dt^ And by multiplying by dt, and integrating with respect to the time, denoting the arbitrary constants by a, a', and a", we shall have 'S\m{xdy—ydx)\=adt ^ 'l[m,{zdx—xdz)\=a'dt \ (362). '2[m{ydz —zdy)'\ = a"dt ^ 624. To understand the signification of these integrals, draw the three rectangular axes Ax, Ay, and kz {Fig. 217), and call AP=:r, VQi—y : let AQ, the projection of the radius vector Am on the plane of x, y, be denoted by r, and the angle formed by AQ with the axis of a; by tf ; the infinitely small arc Q,Q,' described with the radius r will be expressed by rdè ; the right-angled triangle APQ, gives a;=r.costf, y=r .svni; and, by differentiating, we obtain dx = —r . sin 6 .d6 + cos ê . dr, dy=r . cos ê . dê+sia ê . dr. Substituting these values in the expression xdy —ydx, we find xdy - ydx =r^d6=2x^r'Xrd6=2.a.Te3L Q AQ,' ; and therefore, 344 DYNAMICS. m{xdi/—ydx)=2m{a.re2iCiAQ,'). By forming similar products for the other masses m', m", COS /3 a" cos y= . The angles «, /s, y are constant ; and hence we conclude that the position of the principal plane remains invariable during the motions of the several particles of which the system is composed. General Principle of the Preservation of the Motion of the Centre of Gravity. 629. In discussing the circumstances of motion of a sys- tem of material particles, acted upon by incessant forces, it was proved that the centre of gravity of the entire system has the same motion as though the several forces were applied directly to that point. Thus, denoting by x„ y„ and z, the variable co-ordinates of the centre of gravity, we shall have, as in Art. 614, MX,=s(mX), MY,=2(mY), MZ=2(mZ) (366). 346 DYNAMICS. and, '^=^" ^■=^" ^'=^ (^«'^- 630. If the material points which compose the system be subjected only to the action of forces arising from their mutual attractions, the equations (367) will reduce to ^'^/^n d^y.-a ^'^/_n. these equations being integrated give dx, dy, , dz, —!-=a, -~=h, -r^=Cj dt dt dt and by a second integration we find x, = at-\-a\ y,~bt + b', z=ct-{-c/. eliminating t, we have x—a'=-{z—c'), y—h'=-{z—d). c c These equations appertain to a right line in space, and the motion of the centre of gravity will therefore be rectilinear. This motion will also be uniform ; for we have the velocity of the centre of gravity expressed by y(dxldyldzl\ which is evidently a constant quantity. 631. If the masses m, m', m", (fee. be subjected to the action of constant forces whose directions are parallel to a given line, we may adopt this line as one of the co-ordinate axes, z for example, and the equations expressing the circumstances of motion of the centre of gravity, then become ^=0, ±^1=0, t^=Z; df" ' dt^ ' dt"" ' and it may then be proved, as in Arts. 518 and 519, that the trajectory described by the centre of gravity is a parabola. 632. Finally, it may be shown that if two or more of the bodies composing the system impinge against each other during the motion, the velocity of the centre of gravity will remain unchanged. For, by the nature of the centre of grav- ity, we have FREE MOTION OP A SYSTEM. 347 Mx^—l{mx), My=-S.{my), M.z,=:L{mz): differentiating with respect to the time t^ we obtain M^-l;=^(J-^\ M^l^=^(Jy\ M^-§^=4m^-pi. dt \ dt)' dt \ dt )' dt \ dt) And if we denote by a, a', a", &c. the velocities of the parti- cles m, m', m", (fee. before the colhsion, and by A, A', A", ^^^ - j 3.nd the components of the force a will be r r X=-A-, Y=-x^, Z=-A-; r r r the negative signs are prefixed to these components because they tend to diminish the co-ordinates of the particle dM.. By substituting these values in equation (388), we shall obtain for the diîîerential equation of the surface of the fluid t(xdx+ydy + zdz)=0 (390). r Suppressing the common factor , and integrating, we find 358 HYDROSTATICS. an equation appertaining to a spherical surface ; hence the surface of the fluid will be spherical. 647. If the radius of the sphere be very great in compari- son with the extent of the surface, as is the case when we consider a small portion of the earth's surface, the curvature will be insensible, and the surface may therefore be regarded as a plane. 648. The integration of equation (390) was effected imme- diately in consequence of equation (388) becoming, in that example, a particular case of the theorem demonstrated in Art. 436, relative to forces directed to fixed centres. It is by virtue of this theorem that equation (388) will always be integrable in such cases as refer to the equilibrium of fluids resting upon fixed surfaces. 649. If, in equation (386), we replace the quantity within the brackets by its equal d[F{x, y, z)], we shall obtain dp=T>xd[F{x,y,z)]] or, by division, ^=4F(:r, i/,z)] (391). But d[F{x, y, z)] being by hypothesis an exact differential, -^ must likewise be an exact differential ; hence, D will con- tain no variable except/? ; this condition may be expressed by the equation Ty^fp (392). If the pressure ^ be supposed constant, the density D will be likewise constant, and (391) will reduce to d[F{x,7/,z)]=0. The integration of this equation will reproduce that already found in Art. 645, the properties of whic^ have been dis- cussed. 650. The fluid being still supposed incompressible, but heterogeneous, the density D will be variable ; and in order that the pressure p may be determinate, the quantity DQidx+Ydy+Zdz) must be an exact differential : but if ELASTIC FLUIDS. 359 X-dx -\-Y dy ^'Zadz be likewise supposed an exact differential, it will appear, as in equation (392), that the density will be always a function of the pressure. Thus the pressure and density will become constant together, and will remain in- variable for all points situated in a level stratum. We conclude, therefore, that a heterogeneous fluid mass cannot remain in equilibrio, unless it be disposed in such man- ner that each of the level strata shall be of equal density throughout. The law of variation in the density in passing from one stratum to another, will depend on the manner in which D is expressed in functions of x^ y, and z : and since the nature of the function is entirely arbitrary, the law of the density will likewise be arbitrary. Application of the General Equations of Equilibrium to Elastic Fluids. 651. The characteristic property of an elastic fluid is its power of sustaining compression, and subsequently regaining its original volume and elasticity, when the compressing force is removed. Thus, a fluid which is elastic exerts in addition to the pressure due to the forces which act upon it, an eflîbrt arising from the elasticity of its particles. It has been ascertained experimentally, that this effort, which is called the elastic force of the fluid, is proportional to its density, so long as the temperature remains invariable. Thus, if .we suppose the temperature to remain constant, and represent by P that pressure exerted upon the unit of surface which is necessary to produce a certain density assumed as the unit, this density will be doubled when the pressure becomes 2P ; trebled when the pressure becomes 3P, (fee. ; and hence, if the density be expressed by D, the corresponding pressure will be PD. This pressure being denoted by p, we shall have p^VJy (393) ; the quantity p represents, as heretofore, the pressure exerted upon the unit of surface. 360 HYDROSTATICS. 653. By combining equation (393) with the equation dp=\y{lidx^-Ydy+Zdz\ there results dp _K.dx-\-Yd'y-\-'Ldz tOQA\ 7 p ^"^^^^ ' and by integration, we have _ PlLdx + Ydy-\-Zdz ç, 653. The temperature being supposed constant throughout the mass, and the nature of the fluid particles everywhere the same, the quantity P will be constant, and may therefore be placed without the integral sign : thus, by representing the constant C by log C, we shall have or, if we denote by e the base of the Naperian system, this equation will reduce to fÇ%.dx+Ydy+Zdz) log^=loge ^ +log C : reducing, and passing from logarithms to numbers, we find /(Xrfj+Yrfy+Zdz) p=C'e P This value being substituted in equation (393), we obtain /(Xrfj+Yrfy+Zdz) Ce ^ P The pressure and density being both functions of the quan- tity /(Xc^:r+Yc?y+Z^2;), they will become constant at the same time ; and hence, the density of the fluid throughout each level stratum will remain invariable. The value of the density in any stratum results immediately from the pre- ceding equation. 654. It should be remarked, that in the case of elastic fluids, the equation :Ldx+Ydy+Zdz=Q cannot be deduced from the hypothesis of ^=0: for, if we suppose ^=0, the equation will give D=0 ; and hence, we perceive that it would be necessary that the density of the PRESSURE OF HEAVY FLUIDS. 361 fluid should be likewise equal to zero ; a supposition which would destroy the existence of the fluid. We conclude, therefore, that in an elastic fluid, the pressure cannot be equal to zero at the surface ')f the fluid, as is the case with incompressible fluids. Thus, a mass of elastic fluid cannot be in equilibrio unless contained in a close vessel, or extended indefinitely in space. Of the Pressure of Heavy Fluids, 655. It is now proposed to examine the circumstances of equilibrium in fluids whose particles are acted on by the force of gravity. For this purpose, let it be supposed that a vessel is placed upon a horizontal plane, and filled with water, or other heavy fluid, to a certain height. The surface of the fluid, as has been demonstrated, will assume a horizontal posi- tion ; let this surface be assumed as the plane of a-, y, and let the co-ordinates z be reckoned positive downwards ; the force of gravity being the only force exerted upon the fluid parti- cles, we shall have X=0, Y=0, Z=^; and the equation (386) will become dp^Dgdz. The density of the fluid and the intensity of gravity being supposed constant, the integration of this equation will ^ive p='Dgz-\-C (395). To determine the value of the constant C, we make z—0, and since the pressure p is equal to zero at ihe same time, we deduce C=0 : thus the equation (395) is reduced to p=T>gz (396). 656. If a horizontal plane be drawn below the surface of the fluid, every point in such plane will have a common ordi- nate z ; and the pressure p—Dgz will therefore be constant throughout this plane. 657. Let h represent the distance between the surface of the fluid and the horizontal base of the vessel ; the pressure supported by the unit of surface of the base will be determined 362 HYDROSTATICS. by equation (396), in which we replace z by A, and thus obtain p=T)gh (397). Let J)' represent the pressure supported by the entire base, which is supposed to contain h units of surface : the quantity p will be contained b times in p' : we therefore have p'=bp (398), and by substituting for p its value given in equation (397), we find p'=Tighb (399). But bh represents the volume of a prism whose base is 6, and height h ; and by multiplying this vokime by the density D, we obtain bliD for the mass of the prism : therefore bg/iD will express the weight of such prism ; and hence, it appears that the base b supports a pressure equal to the weight of the column of fluid which rests immediately upon it. 658. The pressure p', exerted by the same fluid, being dependent only on the base b and height h, it follows that the pressures supported by the bases of diflerent vessels will be equal, whatever may be the forms of the vessels, provided their bases, and the heights of the fluid above them, be respectively equal. 659. To determine the lateral pressure exerted against the sides of the vessel, let da represent the element of this sur- face, and z the distance of the element from the surface of the fluid ; the pressure j) (referred to the unit of surface), which is supported by the element da, will be determined by equa- tion (396) : this value being substituted in equation (399), and the area b being replaced by da, we obtain T) . gz . da> for the expression of the entire pressure on the element da. A similar expression may be obtained for the pressure upon each element ; and since the pressures will be exerted in par- allel directions when the side of the vessel is supposed plane, we shall have, for the total pressure exerted against the side, p'=fDgzda. The second member of this equation contains two variables, one of which must be eliminated before the integration can PRESSURE OF HEAVY FLUIDS. 363 be efiected. This elimination is readily accomplished when the figure of the surface )4-C. The integral being taken between the limits v=0, and v=l, we obtain p'=T>gb(al-{-ll^ cos (p). 661. To determine the point of application of the resultant of all the pressures exerted upon the rectangle, we remark, in the first place, that this point must be situated upon the line 364 HYDROSTATICS. EH, which bisects the sides AB and CD. We next regard the pressures exerted upon the different points of the surface ABDC as parallel forces, and determine their moments with reference to a vertical plane passing tlirough the horizontal line CD : the pressure sustained by the element abfe being Dgzbdv, its moment will be expressed by DgzbdvXv . sin (p ; and by denoting the distance EG of the point of application of the resultant from the line CD by v,, the principle of mo- ments will give p'v, sin 4)= sin - Having found the pressures exerted upon the base and upon each of the sides of the vessel, we combine these pres- sures, and determine their resultant : such resultant will express the entire pressure produced by ihe fluid. 662. We will next consider a body immersed in a homo- geneous heavy fluid : the pressure exerted by this fluid against any portion of the surface of the body may be deter- mined by the method for finding the pressure against the sides of a vessel ; but when it is required to consider the total pressure exerted against tiie surface of a body immersed PRESSURE OF HEAVY FLUIDS. 365 in a fluid, we commonly employ the following theorems, the truth of which will be demonstrated. 1°. The pressures exerted upon the surface of a body en- tirely immersed in a fluid have a single resultant ^ which is vertical and directed ujjwards. 2°. The resultant of all the pressures is equal in intensity to the weight of the fluid displaced. 3°. The line of direction of this resultant passes through the centre of gravity of the displaced fluid. 4°. The horizontal pressures destroy each other. To establish the truth of these propositions, let us suppose a vessel ADE {Fig. 222) to be filled with a heavy fluid in equilibrio, and let a portion of this fluid KL be conceived to become solid, its density remaining unchanged : the state of equilibrium will not be disturbed by this change. But this solid is urged downwards by a force equal to its weight, applied at its centre of gravity. This force can only be destroyed by the resultant of all the pressures exerted by the fluid against the solid ; hence, it follows that these pressures must have a single resultant equal in intensity to the weight of the displaced fluid, and that this resultant must be applied at the centre of gravity of the displaced fluid, and be directed vertically upwards. Moreover, as the direction of the result- ant is vertical, the horizontal pressures will mutually destroy each other. When a body is partially immersed in a fluid, an equilibrium cannot subsist unless the centres of gravity of the body and of the fluid displaced be situated upon the same vertical line : this condition will necessarily be fulfilled when the body is entirely immersed, provided it be homogeneous ; since its centre of gravity will then coincide with that of the fluid displaced. The buoyant eflbrt exerted by the fluid being directed along a line which passes through the centre of gravity of the dis- placed fluid, that point is called the centre of buoyancy. 663. Let V represent the volume of fluid displaced, and v' that of the body ; D the density of the fluid, and D' that of the body : the weights of the volume of displaced fluid, and 366 HYDROSTATICS. of the body will be respectively Dgv and D'gv'. If the body be supposed to rest in equilibrio, we shall have Dgv=:D'g-v' ] and if we suppose it to be entirely immersed, the volumes v and v' will be equal, and the densities D and D' must likewise be equal, in order that tft«^ equilibrium may be preserved. But if the weight of the body bc.lelsg than that of the fluid displaced, w^e shall have ■*>.. .. and the body will be urged upwards by a force equal to the difference Dgp—'D'gi'. If, on the contrary, we should have T>gvgv. Of the Equilihrium, Stability, and Oscillations of Floating Bodies. 664. The propositions demonstrated in Arts. 662 and 663 establish two principles which serve as tlie basis of the theory of floating bodies ; these principles are, l'^. When a body is partially or totally miTnersed in a fluid, an equilibrium cannot subsist unless the centre of gravity and centre of buoyancy be situated upon the same vertical line. 2°. If an equilibrimn be maintained, the weight of the body will be equal to that of the fluid displacef. The latter principle is frequently employed for the purpose of estimating the weight of a ship either with or without her cargo. For this purpose, we measure the capacity of the part immersed, and allow a weight of one ton for every 35 cubic feet which it contains. By taking the difference of the weights of the vessel with and without the cargo, the weight of the latter may be obtained. We can also arrive at the same result, by simply measuring the additional portion of the vessel immersed, when the cargo is introduced. EQUILIBRIUM OF FLOATING BODIES. 367 665. The horizontal surface of the fluid is called the flane of floatation. 666. If V denote the volume of fluid displaced, D its den- sity, and g the intensity of the force of gravity, the weight P of the body ABC {Fig. 223), which floats upon the surface of the fluid, and is partially immersed, will be equal to Dgv. 667. When the floating body and fluid are both homogene- ous, the centre of gravity of the part immersed will coincide with the centre of buoyancy. 668. The fluid and body being homogeneous, the centre of gravity G {Pig. 223) will be situated above the point O, the centre of buoyancy. For let g be the centre of gravity of that portion of the body which lies without the fluid : then, the centre of gravity G of the entire body will necessarily be situated upon the line ^O, and between the points g and O ; hence, it will be found above the point O. 669. But if the floating body be heterogeneous, it may happen that the centre of gravity of the entire body will lie below the centre of buoyancy. For by supposing the density of the lower part of the body to be very much greater than that of the upper portion, the centre of gravity of the entire body may be situated extremely near the lower surface : but the position of the centre of buoyancy depends only on the figure of the part immersed, since the density of the fluid is supposed uniform, and it may therefore be situated at a greater distance from the lower surface of the body than the centre of gravity of the entire mass. Hence we conclude, that the centre of gravity of the float- ing body is sometimes situated above, and sometimes below, the centre of buoyancy. 670. When the body is but partially immersed, the weight of the immersed portion is less than that of the fluid dis- placed, and the equilibrium is maintained by the weight of that portion of the body which lies without the fluid : this weight is equal to the difiîsrence of the weights of the fluid displaced and of the part of the body immersed. If the weight of the body be increased, it will sink to a greater depth, until the weight of the additional quantity of fluid displaced shall be equal to the weight added. 368 HYDROSTATICS. G71, Let us now suppose that a body floating upon the surface of a fluid {Fig. 224) is deranged in a very sUght de- gree from its position of equilibrium, by the application of any force, and let us examine whether the body will tend to return to its original position, or, on the contrary, to deviate farther from it. Let ADB represent the immersed part of the body before derangement, and a6D that immersed after de- rangement : we suppose the new position of the body to be such, that the weight of the fluid displaced shall still be equal to the weight of the body, or that ABD = a6D. The centre of gravity G may be regarded as fixed during the rotation^ since the forces will tend to turn the system about that point, as though it were immoveable. The centre of buoyancy will not retain its position O, but will be found nearer to the portion CB6, which, by the rotation, has become immersed in the fluid : and if we suppose, for the sake of symplifying the question, that the body is divided symmetrically by the plane ABD, the centre of buoyancy will obviously be found in this plane after the derangement. Let o represent the centre of buoyancy in the deranged position, and through o and G let perpendiculars oi and Gk be demitted upon the line ab. If an equilibrium subsist, the weight of the body and the up- ward pressure of the fluid will be equal and directly opposed. The first condition will necessarily be satisfied, since we have supposed the volume of fluid displaced to remain un- changed; the second condition will be fulfilled when the points i and k coincide with each other : but if this coinci- dence should not take place, the point i may fall either to the right or to the left of the point k. In the first case, the pres- sure of the fluid applied at o and acting upwards, will evi- dently tend to restore the body to its primitive position,, or to render the line DG vertical. But if the point i should fall to^ the left of k, this pressure would tend to turn the body in a contrary direction about the point G, and would thus cause it to deviate farther from its original position. If the body, when deranged in a very slight degree from its position of equilibrium, should tend to resume its former posi- tion, the equilibrium is said to be stable ; but if, on the con- trary, it should tend to depart still farther from this position, EaUILIBRIUM OF FLOATING BODIES. 369 the equilibrium is called unstable ; when the body neither tends to return to its original position, nor to deviate farther from it, the equilibrium is said to be one of indifference. 672. By examining the directions of the pressures before and after derangement, we shall find that the lines OG and oi perpendicular to AB and ab respectively, are inclined to each other, and being contained in the same plane, they will intersect in some point m, {Fig. 225). This point is called the metacentre ; and it appears from Art. 671, that when the point G is situated below w, the extremity k of the perpendicular GA; will fall to the left of the point i, and the equilibrium will be stable ; but if the point G be situated above the point w, the extremity k of the perpen- dicular Ok will fall to the right of the point ?', and the equi- librium will become unstable. If the points i and k coincide, the equilibrium becomes one of indifference. 673. Let it now be required to determine the position of the metacentre. This point being found upon the line con- necting the centre of gravity and centre of buoyancy in the primitive position of the body, it will be sufficient to determine its distance from the point G, or the point O. For this purpose we remark, that when the body is slightly inclined, the line AB {Fig. 226) which represents the profile of the plane of floatation in the primitive position, assumes a position inclined to the new plane of floatation ab in a certain angle «, the portion ACa being at the same time withdrawn from the fluid, and the portion BC6 being immersed. Hence, the immersed portions of the body in the two positions will be, aCBD+ACa in the primitive position, aCBD + BC6 ..... after the derangement. But if y, g, and g' represent the respective centres of gravity of the volumes aCBD, aCA, and èCB, the centre of gravity O of the volume ABRD will be found by dividing the line gy in the inverse ratio of the volumes aCBD and ACa ; and in like manner, we may find the centre of gravity o of the volume aèBD : thus, we shall obtain the proportions vol aCBD : vol aCA ; : O^ : Oy (400), vol aCBD : vol 6CB ::og' :oy (401) ; Aa 370 HYDROSTATICS. but the second terms of these proportions are equal to each other : for, the floating body being supposed to displace the same quantity of fluid after it has been deranged as it did in its primitive position, the volumes ABRD and abRD will be equal to each other ; and if from these equals we subtract the common part aCED, there will remain the volumes aCA and BC6 equal to each other. Hence, we deduce from the proportions (400) and (401), Og- : og' : : Oy : oy ; which proves that the lines gy and g'y are cut proportionally by the right line Oo, which line is therefore parallel to gg'. But, the derangement of the body being, by hypothesis, extremely slight, the line gg' may be considered as nearly coincident with the primitive plane of floatation ; and since Oo is parallel to gg', this line may be regarded as parallel to the same plane, 674. To determine the value of Oo, we deduce, from the proportion (400), vol aCBD+vol aCA : vol aCA : : O^+Oy : Oy, or, vol ABRD : vol aCA : : gy : Oy. But the similar triangles gg'y and Ooy give gy.Oy:: gg' : Oo ; and by comparing this proportion with the preceding, we obtain vol ABRD : vol aCA :: gg':Oo (402) ; whence, vol_aCAx^' .^„. ^'- vol ABRD ^^^^^' 675. Having determined the value of Oo, we can readily obtain that of Om (Fig. 227) ; for, the lines Om and om being respectively perpendicular to CA and Ccr, the angles at C and m will be equal ; and since these angles are exceed- ingly small, we may regard the triangles ACa and Oîno as similar and isosceles : hence, we shall obtain the proportion Aa : Oo : : Ca : viO ; and therefore, ^Q^OoxCa Aa ËQ,lîlLlBRltJM OP t^LOATING BODIES. 371 6r6i To obtain the analytical expressions for Oo and mO, we remark, that the plane of floatation AB {Pig. 228), which limits the immersed part of the body in its primitive position, is replaced by the plane ab after the derangement : these two planes, being intersected by a vertical plane perpendicular to their common intersection, will exhibit the section ACa rep- resented in Fig. 226 ; and if we continue to draw other par- allel vertical planes, we shall divide the solid included between the planes KAL, KaL {Pig: 228) into an infinite number of elementary laminae parallel to the plane ACa. But it is evident, that when the plane KAL, which in the primitive position of the body coincided with the surface of the fluid, shall have been detached from the surface, revolving around the line KL, each right line in this plane, as CA, will have described the sector of a circle ; so that the sections of the solid included between the planes ALB and oLb {Fig. 228) by the system of parallel vertical planes, will be repre- sented by the sectors ACa, A'C'a', A"C"a", &c. {Fig. 229). But if we assume the line of intersection KL as the axis of x, and place the axis of y in the plane KAL, the ordinates y will be the perpendiculars AC, A'C, A"C", (fcc. The infinitely small angle formed by the planes KAL and KaL being every- where the same, let the arc described by a point at the distance unity from the line KL be expressed by fy^'dx (406). Such will be the analytical expression for the second term of the proportion (402). 372 HYDROSTATICS. To determine the value of the third term, we remark that the Une C^ {f'ff- 22(3) being the distance of the axis KL {Pig, 229) from the centre of gravity of the sohd KaLA, we shall determine this distance, by dividing the sum of the mo- ments of the elementary solids by the volume KaLA. If we consider the elementary sector ACa {Fig. 229), the centre of gravity g of this sector will be found upon the radius CR=CA {Fig. 230), at a distance from the point C (Art. 184) expressed by chordAa arc Aa but the angle C being supposed extremely small, the arc Aa may be regarded as equal to the chord ; and since CR is equal to CA or y {Fig. 228), the preceding expression will give |y for the distance of the centre of gravity from the axis KL. Multiplying the elementary solid \ay^dx by this distance, the moment of this solid with reference to this axis KL will become \uy^dx : thus, we shall have f^uy'^dx=i\\.ç. sum of the elementary solids, f^iiy^dx=X\ve sum of the moments of the elementary solids : and from the property of the moments, the distance Cg of the centre of gravity of the small solid CAa {Fig. 226), or KALa {Fig. 229), will be expressed by ^ f^^^y'dx ' the quantity « being constant, this expression may be re- duced to ^ 2jrdx ^ ^fy'dx Ç)77. The value of C^ will result from the integrations here indicated ; and that of C^' {Fig. 226) may be obtained in a similar manner ; but, if the floating body be symmetrical with respect to a vertical plane passing through the axis KL, as will always happen in the case of a ship, we shall have and therefore, KQtJILlBRIUM OF FLOATING BODIES. 373 The volume of the part immersed, which hkewise enters into the equation (403), can be calculated directly, when the figure of the vessel is supposed known. Let this volume be denoted by V, and let its value and those of the volume ACa and gg', given in equations (406) and (407), be substituted in equation (403) : we shall thus obtain and lastly, by substituting in equation (404) this value, and that of the arc Aa, given by equation (405), replacing Ca by y, we find 3V Such is the formula expressive of the distance of the meta- cenire from the centre of buoyancy. 678. When the floating body is homogeneous, and of such figure that its parallel sections will be similar, we may rea- dily determine the position of the metacentre, without the necessity of performing an integration. For let a^ represent the area of the section AEB (Fig. 231), which is supposed to have been determined by direct measurement, and let b repre- sent the half-breadth CA of this section : the half-breadths of the sections A'E'B', A"E"B", «fee. will be represented by C'A', C'A", &.C. or by the ordinates y of the curve KAL. These sections being by hypothesis similar figures, they will be proportional to the squares of their homologous sides ; and hence, we shall have section AEB : section A'E'B' : : AC^ : A'C'^, or, «2 : section A'E'B' : : b^ x y^\ whence, section A'E'B'=^. b^ The distance CC between two consecutive sections being denoted by dx, we shall have a^y'^dx ~~b^ for the expression of the elementary solid. 32 374 HYDROSTATICS. 679. Let g represent the centre of gravity of the section AEB, which, in consequence of the symmetry of the figure, will be found on the vertical CE, The centre of gravity of this section having been determined, let its distance from the surface of the fluid be denoted by n : we shall then have, from the similarity of figures, , ( the distance of the centre of ffravitv of the ; ' } section A'E'B' from the surface of the fluid ) ' whence, Multiplying this distance by the elementary solid, we shall obtain for the moment of this solid, taken with reference to the surface of the fluid, ny a^y^ , and therefore the expression— -/ y ^^f a,- will represent the sum of the moments of the elementary solids taken with refer- ence to the surface of the fluid. This sum being equal to the product of the volume V of the solid immersed by the depth HG of its centre of gravity, if this depth be denoted by Gj we shall have whence, G = ; -jy^'dx. Y.b- But it has been shown (Art. 677) that the distance mO of the metacentre from the centre of buoyancy is given by the formula mu ^^ , and if we compare these two expressions, we shall find G:mO::3na-.2b': na^ . r ^ J 2 ^^ ly^dx or EQUILIBRIUM OF FLOATING BODIES. 375 whence, ™"=l^ w- 680. For the purpose of applying this formula, let it be required to find the metacentre of a rectangular parallelo- piped ML. Let AF represent the intersection of the body- by the surface of the fluid {Fig. 232), supposed parallel to the base NL. The depth AN, to which the body must be im- mersed in order that it may be sustained in equilibrio, will depend on the weight of the parallelopiped and the density of the fluid (Art. 664) : this depth may be considered as deter- mined by experiment : the quantity a", which represents the section BN, and which will be constant for all parallel sec- tions, will be determined immediately ; for we have a^=ABxCE. Again, the semi-breadth of the section being equal to |AB, there results 6 = iAB=AC; and since the centres of gravity of all the sections are equally distant from the surface of the fluid, the centre of gravity of the fluid displaced will be situated at the same distance ; so that we shall have W = G=:iCE. By substituting these values in formula (408), the distance of the metacentre m, from the centre of buoyancy will be found equal to ,^ SAC^» ^^=âÂB^E or, by reduction, 3CE For example, if the semi-breadth of the parallelopiped be supposed equal to 9 feet, ^ud the depth of the part immersed 4 feet, we shall find the height of the metacentre above the centre of buoyancy equal to 6| feet ; if, therefore, we subtract from this height, 2 feet, the depth of the centre of buoyancy, there will remain 4| feet, for the height of the metacentre 376 HYDROSTATICS. above the surface of the fluid. Hence, the centre of gravity of the parallelopiped should not be more than 4| feet above the surface of the fluid, if we wish the equiUbrium to be of the stable kind. 681. As a second example, let us consider a vessel whose vertical sections below the surface of the fluid are equal right- angled isosceles triangles, such as AEB {Fig. 233). If the perpendicular EC be demitted upon the base, the triangle ABC will likewise be isosceles, and the height EC will therefore be equal to one-half the base AB : thus, the quantities which enter into the formula (408) will be, in the present case, a2=area of the triangle AEB^AC^, ^=G=iCE, h=AC=CE; consequently, by substituting these values in formula (408), it will reduce to and if from this value we subtract that ot the distance of the point O below the surface of the fluid which is equal to ^CE, there will remain iCE for the distance of the metacentre above the surface of the fluid. Hence, in a prismatic vessel whose vertical sections are right-angled isosceles triangles, the metacentre will be found at a distance above the surface of the fluid equal to the distance of the centre of buoyancy below the surface. 682. If we suppose the body to be slightly deranged from a position of stable equilibrium, and conceive the resultant of all the upward pressures of the fluid to be applied on its line of direction, at the metacentre, we can determine the circum- stances of oscillation of this body about the centre of gravity, by a method entirely analogous to that employed in consider- ing the motion of the compound pendulum. For this pur- pose, let the origin of co-ordinates be placed at the centre of gravity, and let the proper value of y, be substituted in for- mula (337), which may be put under the form (338) (It k^'+a'"' OSCILLATIONS OF FLOATING BODIES. 377 This formula admits of simplification in the present case, from the consideration that the oscillations are performed about the centre of gravity ; and the general expression of the moment of inertia M.{k^-\-a'') is therefore reduced to Mk^ : hence, we obtain ^=^ (409). dt k^ ^ ^ This equation, when integrated, will serve to determine the angular velocity, and the time of performing a complete oscil- lation. 683. To determine the value of y,, which represents the perpendicular distance from the axis passing through the cen- tre of gravity, about which the oscillations are performed, to the line of direction of the upward pressure, we remark, that the distance of the metacentre from the centre of buoyancy O is expressed by 2fy=dx 3V * Let this distance be denoted by A, and the distance GO {Fig. 234) by B ; we shall then have or, since the point G may fall above O, we may likewise have Gm=A— B ; hence we may comprise the two cases under the double sign, by writing Gm=A±B. If the angle LmG {Pig. 234), formed by the vertical mh with the new direction of the line GO, be represented by 6, we shall have the relation GL=Gm sinfl; or, replacing the sine by the arc, since the arc is supposed extremely small, and substituting the value of Gm, this equa- tion will become GL=(A±B)<»; and by introducing this value of y, in formula (409), we shall obtain dcj_ g{A±B)ê di ¥^ 376 HYDROSTATICS. 684. But the angular velocity a being that which corres- ponds to the arc i described with a radius unity, this velocity will be expressed by — ; and since the arc « {Fig. 234) is a decreasing function of the time t^ cU should be affected with the negative sign ; hence, "=^' («o>- Multiplying the corresponding terms of these equati(Mis toge^ ther dt will disappear : and there will result Putting, for brevity, ^^^=E (411), and multiplying by 2, we obtain 2'Eêd6-^2a>dco=Q. Integrating, we have whence, Substituting this value in equation (410), we obtain or, by reduction, d6 dt- and, by integration, t=—= arc ( cos = -'^ ) +C': ^E V ^G/ from which we deduce '-^=cos[(^-C')v/E]: and, finally. ,, ^C.cos[(^-CVE] ; v^ 685. When E is negative, tlie value of o becomes imagin- SPECIFIC GRAVITY. 379 ary, and the oscillatory motion cannot take place ; but in order that E may be negative, the first member of equation (411) must likewise be negative ; and consequently, A ± B=a negative quantity : this case occurs when B exceeds A, and is affected with the negative sign ; and since A± B represents the distance of the centre of gravity from the metacentre, it follows that the meta- centre will then be situated below the centre of gravity, and- the equilibrium will be unstable. On th& contrary, if A±B be positive, the metacentre will be found above the centre of gravity, the value of E will be positive, and the values of 6 and u will be real : thus, the oscillations can be performed, and the equilibrium will be of the stable kind. 686. The time of oscillation being determined by a method entirely similar to that employed in investigating the circum- stances of motion of the compound pendulum, we may con- clude that this time will be independent of the extent of the arc through which the oscillations are performed, provided the arcs be extremely small. Specific Gravity — Hydrostatic Balance — Hydrometer. 687. Let P represent the weight of a body M : if this body be immersed in a fluid, the buoyant effort exerted by the fluid will tend to support the body, and the force P' necessary to sustain it will be less than P, that required previous to the immer- sion, by a quantity equal to the weight of the fluid displaced. For example, if M be supposed a sphere of lead whose weight is equal to eleven pounds, and if it be found to weigh but ten pounds when immersed in water, we should conclude that the weight of an equal volume of water would be one pound ; and therefore tfiat the weight of lead was to that of water as eleven to one. 688. The specific gravity of any substance is the ratio between its weight and the weight of an equal volume of some other substance assumed as the standard. Thus, in the preceding example, if water be adopted as the standard of comparison, the weight of the sphere of lead 380 HYDROSTATICS. being eleven times greater than that of an equal volume of water, the specific gravity of lead will be represented by the number 11. The density of a body has been defined (Art. 161) to'be the ratio between the quantity of matter contained in the body and that contained in an equal volume of some other substance assumed as the standard ; and since the weights of bodies are proportional to the quantities of matter which they con- 'iain, it follows that the ratio of the weights of two bodies will be equal to the ratio of their quantities of matter. Hence, the number expressing the specific gravity of a body will be the same as that which expresses its density, provided we refer the density and specific gravity to the same sub- stance as a standard. In practice, it is usual to adopt water as the standard in determining the specific gravities of solids and incompressible fluids ; and for the purpose of rendering the comparison more exact, the water is first deprived, by distillation, of any im- purities which it may contain. The specific gravities of gases and vapours are generally referred to that of atmo- spheric air. 689. The dimensions of all bodies being more or less aiFected by changes of temperature, it becomes necessary to adopt a standard temperature, at which experiments for the determination of specific gravities may be performed. A convenient temperature for this purpose is that corresponding to 60° of Fahrenheit's thermometer, it being easily obtained at all times : and the tables of specific gravities are usually calculated for this temperature. When circumstances will not permit the experiments to be performed at the standard temperature, the results obtained must be reduced to this tem- perature, by introducing a correction for the change of vol- ume which the substance would undergo if reduced to the standard temperature. This correction is readily applied when the law of dilatation has been previously ascertained. 690. If we wish to determine the specific gravity of a fluid, as olive-oil, we may immerse successively the same solid in water and in this fluid ; we shall thus be enabled to deter- mine the weights of equal volumes of the two fluids ; and a SPECIFIC GRAVITY. 381 comparison of these weights will give the specific gravity of the oil. For example, if the sphere of lead weighing eleven pounds have its weight reduced to 10.085 lb. when immersed in oil, the weight of the fluid displaced would be equal to 0.915 lb. ; and since the weight of an equal bulk of 0.915 water was found equal to 1 lb., we shall obtain -^ — = 0.915, for the ratio of the weights of equal bulks of the two fluids : this number will therefore represent the specific gravity of oil. From the preceding remarks, we may infer that if two bodies of unequal volumes, suspended from the arms of a balance, sustain each other in vacuo, the equilibrii^m will not be maintained when the bodies are similarly suspended in the atmosphere ; the weight of the larger body being most sup- ported by the buoyant eflbrt of the atmosphere. 691. The instrument usually employed for determining with accuracy the specific gravities of bodies, is the hydro- static balance. This consists merely of a delicate balance, having a small hook attached to one of its scales, by means of which the body can be suspended, for the purpose of deter- mining its weight when immersed in a fluid. The body is connected with the hook by a hair or slender thread, whose weight is inconsiderable. When we wish to determine the specific gravity of a solid, we place it in the scale to which the hook is attached, and add weights in the opposite scale until an equilibrium is produced. The weights thus added will represent the weight of the body in air. The body is then attached to the hook and im- mersed in water ; and the weight necessary to be placed in the opposite scale to produce an equilibrium will give its weight in wçiter : the difllerence between the weights in air and water will be equal to the weight of an equal volume of water, and by comparing this difference with the weight in air, we shall obtain the specific gravity of the substance under consideration. This process is slightly inaccurate ; since the buoyant efforts exerted by the atmosphere upon the body when im- mersed in it, and upon the weights introduced into the opposite scale, have been neglected. But as the density of 382 HYDROSTATICS. the atmosphere is very small, this omission will not affect the results materially. When the given substance is soluble in water, we deter- mine its specific gravity with reference to some fluid in which it is insoluble, and then compare the specific gravities of the two fluids. If the body be lighter than water, we can con- nect it with a heavier body, which will cause it to sink. Then, having the weights of the heavier and lighter bodies, and that of the compound in air, and having ascertained the loss of weight sustained by the heavier body and the com- pound when immersed, we can readily deduce the weight of the fluid displaced by the lighter. The specific gravity of a fluid may be determined by weighing successively the same body in this fluid and in water, and comparing the weights of the equal volumes displaced. Or it may be ascertained by weighing the same vessel when filled with water, and with the fluid under con- sideration ; these weights, being diminished by that of the vessel when empty, will give the relation between the specific gravity of the fluid and that of water. 692. The hydrometer is an instrument usually designed to determine approximatively the specific gravities of fluids. It is composed of a cylinder of glass or metal, to the lower extremity of which a cup is attached loaded with shot or mercury, and terminated at top by a slender graduated wire. When the hydrometer is plunged into a fluid, the weight with which its lower extremity is loaded causes it to assume a vertical position, and it sinks to a greater or less depth, according to the specific gravity of the fluid. Hence, that division on the graduated stem which corresponds to the surface of the fluid will serve to indicate the specific gravity of the fluid. For example, if the hydrometer be immersed in distilled water whose temperature corresponds to 60° Fahrenheit, the surface of the water will intersect the stem at a certain division, which we shall suppose to be that marked 10 : if plunged in wine, it will sink deeper, say to the 11th, 12th, or 13th division ; and if in brandy, to a still greater depth, SPECIFIC GRAVITY. 383 the division indicated being dependent on the quantity of alcohol which the brandy contains. The use of this instrument evidently depends upon the principle, that when a body is immersed in a fluid, a portion of its weight equal to that of the fluid displaced will be sup- ported by the buoyant effort of the fluid : thus, the heavier the fluid, the less the depth to which the hydrometer will sink. 693. The hydrometer, as improved by Nicholson, will serve to determine the specific gravities of solids or liquids. The instrument consists of a hollow copper ball A {Fig. 235), to the lower part of which is attached a brass cup of sufficient weight to maintain the hydrometer in a vertical position when immersed in a fluid. The upper part of the ball carries a slender wire D, which supports a small dish C des- tined to receive the weights. The weight of the hydrometer is such that the addition of 500 grains in the dish C will just sink the instrument in distilled water, at the temperature 60°, luitil the svirface of the water intersects the stem at its middle point D. If, therefore, a body be placed in the dish C, and weights be added until the point D shall correspond to the surface of the water, the difference between 500 grains and the weights added will express the weight of the body. The body being then transferred to the lower dish B, it will be found necessary to place additional weights in the dish C, in order to sink the hydrometer to the same depth: these additional weights will be equal to the loss of weight sus.- tained by the body when immersed. Hence, the specific gravity of the solid may be readily determined. When we wish to determine the specific gravity of a fluid with this hydrometer, we immerse the instrument succes- sively in distilled water and in the given fluid, and ascertain the weights necessary to be added in each case to the dish C, in order to sink it to the same level. Then, the known weight of the instrument added to the weights introduced into the upper dish will give the weight of the fluid dis- placed. Thus, we can compare the weights of equal volumes of the two fluids. 384 HYDROSTATICS. Of the Pressure a?id Elasticity of Atmospheric Air. 694. The weight of the atmosphere was first recognised by Gahieo. Torricelh, his pupil, demonstrated the existence of this weight by the following experiment. Let AB (Fig. 236) represent a glass tube, 3 feet in length, filled with mercury, closed at the lower extremity and open at the upper : let the finger be applied to the open extremity, and let the tube be inverted, and its open extremity plunged in the basin of mer- cury : on withdrawing the finger, the mercury will be found to descend in the tube, leaving a certain portion of it BE {Pig. 237) unoccupied. If the experiment be tried with tubes of different lengths or diflîerent diameters, the height of the column of mercury sustained in the tube will be found, in each case, to be about 29 or 30 inches above the level of the fluid in the basin. This column of mercury is sustained by the pressure of the atmosphere, arising from its weight; which pressure, being exerted upon the surface CD, is suf- ficient to counterbalance the weight of the column. If the experiment be performed with fluids of different densities, the heights at which they will be supported will be found to differ : thus, if the fluid be water, whose density is to that of mercury as 1 to 13i, the height of the column will be found equal to 30in. Xl3|=34 feet, nearly; the weight of such column being equal to the weight of the column of mercury. 695. The operation of the common siphon is also to be referred to the pressure of the atmosphere. The siphon is a bent tube having its two branches of unequal lengths. The shorter branch EF {Pig. 238) being plunged into the fluid contained in the vessel ABCD, and the air being withdrawn from the siphon, the pressure of the atmosphere exerted upon the surface BC will cause the fluid to rise in the siphon ; and if the height of the point F be less than that at which the atmospheric pressure can sustain the given fluid, it will pass into the longer branch, and will be delivered at the point C. The current having commenced in the siphon, it is maintained in consequence of the superior PRESSURE AND ELASTIClTV OF AIR. 385 M'-eight of the fluid in the longer arm overcoming, in part, the pressure of the atmosphere at the point C, and thus permitting the equal pressure of the atmosphere exerted upon the surface BC to force the fluid up the shorter branch. Hence, it is obvious that the point C nmst always be below the surface of the fluid in the reservoir ABCD, in order that the siphon may be effective. 696. Air is an elastic fluid, which is susceptible of being compressed into spaces which bear to each other the inverse ratio of the forces applied. This may be established experimentally as follows : Let A'BCE {Fig. 239) represent a curved tube closed at E and open at A' : let mercury be introduced into the tube until it shall stand at the same level CC in the two branches : the air contained in the space CE will then be of the same density as the exterior air. If mercury be now poured into the tube until the part ABCD be entirely filled, the length AB being equal to 30 inches, the column of air DE will be found reduced to one-half its original bulk CE : if mercury be again intro- duced until it extend from A' to d, the length A'h being equal to 60 inches, the volume of air will be found reduced to a space Et^ = iCE. This experiment establishes the law of compressibility ; for, before the introduction of the mercury, the air contained in the space CE, being pressed by the weight of the atmo- sphere, must support a pressure equivalent to 30 inches of mercury. When the same volume of air is caused to sustain the additional pressure of a column of mercury AB=30 inches, it is reduced to one-half its original bulk ; and by the further addition of 30 inches, the air is reduced to one-third of this bulk. Thus, it appears, that the spaces occupied by the same mass of air are inversely proportional to the pressures applied ; and since the densities of the air are inveisely pro- portional to the spaces occupied by the same mass, it follows that the densities will be in the direct ratio of the pressures. If the mercury be withdrawn from the tube, the air will expand and occupy the same space as it did previous to compression. 33 386 HYDROSTATICS. Of Pumps for raising Water. 697. The pump is a machine employed for the purpose of raising water. There are three principal kinds of pumps, viz. the sucking pump, the lifting pump, and the forcing pump. The sucking pump, represented in Fig. 240, consists of two tubes ABDC and DCHL, of unequal diameters, connected together ; the first of these is called the sucking pipe, and the second the body of the pump. Within the body of the pump, an air-tight piston MN, having a valve opening upwards, is moved through the space MH, which is called the play of the piston. At the lower extremity of the body of the pump, a second valve ^-, called the sleeping valve, is placed, which likewise opens upwards. The lower extremity AB of the sucking pipe being im- mersed in a reservoir containing water, and the piston MN being raised from the position MN to HL, the air contained in the space CN will expand and fill the space CL, its density and elastic force being both diminished : at the same time, the air contained in the pipe AD, having a density equal to that of the exterior air, will, in virtue of its elasticity, exert upon the valve A:, a stronger pressure than that arising from the elasticity of the rarefied air contained in the space CL : hence, the valve k will be forced open, and the air contained in the interior of the pump will acquire a density that is uni- form throughout, but less than that of the exterior air : then the pressure exerted upon the surface of the water AB being less than that exerted by the atmosphere upon the surface at other points of the reservoir, the water will rise in the suck- ing pipe to the level A'B', such that the weight of the column A'B , together with the pressure of the rarefied air contained in the pump, shall be equal to the pressure of the exterior air. The densities of the air in the body of the pump and in the sucking pipe having become equal, the valve k closes by its own weight. The piston being then depressed from the position HL to MN, the air contained in the space CL will be compressed PUMPS. 387 into the space CN, and its density and elastic force will become greater than those of the air contained in the sucking pipe : the pressure on the upper surface of the valve k being now greatest, this valve will continue closed during the de- scent of the piston, and will intercept the communication be- tween the sucking pipe and body of the pump : hence, the density of the air in the sucking pipe will remain unchanged, and the water will retain the level A'B'. When the piston shall have regained the position MN, it will have compressed into the space CN, not only the quantity of air originally con- tained in CN, but likewise that portion which was introduced into the body of the pump from the sucking pipe. The den- sity of the air contained in the space CN will therefore exceed that of the exterior air, and its elastic force will open the valve I : the air contained in CN will thus be restored to its original density. The piston being raised a second time, the air in MD will be again rarefied, a portion of that con- tained in A'D will pass into the body of the pump, and the equilibrium will be restored by the water rising to a new level A"B". The same operation being repeated, the water will rise through the valve k into the body of the pump, will pass through the valve I in the piston, and will finally be delivered by the spout Q,R. 698. We will next examine the mechanism of the lifting pump. In this pump, the piston MN {Fig. 241) is situated below the fixed valve /r, and being depressed from the posi- tion MN to HL, is supposed to pass below the surface a'h' of the water contained in the reservoir : the piston contains a valve opening upwards, through which the water passes, regaining its level a'h'. The piston being then elevated, the column of water a'L, which rests upon its superior base, being prevented from returning through the valve, will be raised through a height equal to the play of the piston, and will oc- cupy the space «N : at the same time, a vacuum being formed below the piston, the water will be compelled to follow the piston in its motion by the pressure of the atmosphere on the surface of the water in the reservoir. But the air contained in the space a'D being comprpssed by the elevation of tlie Bb2 388 HYDROSTATICS. piston, its elastic force will become greater than that of the exterior air, and the valve k will open, restoring the air below k to its original density. The circumstances will then be the same as before the first stroke of the piston, with the ex- ception that a portion of water has passed above the piston. When the piston is again depressed, the column of water aN, which rests upon it, will also descend, and the air contained in the space Co will therefore be rarefied. The descent of the water will continue until the elastic force of the rarefied air contained between the valve k and the surface of the water, together with the weight of the column of water raised, shall be equal to the pressure of the atmosphere : the valve in the piston will then open, and an additional quantity of water will pass above the piston. By repeating the process, a certain portion of water will pass above the piston at each stroke ; and reaching the valve k, will pass into the body of the pump, and may be delivered at any height. 699. The forcing pump is a combination of the sucking and lifting pumps. In this pump, the piston MN {Pig. 242) is without a valve, but the lateral pipe HE is provided with one at I, opening upwards ; and there is a sleeping valve at L, as in the sucking pump. The piston being raised, the water rises into the space MCDEF, for the reasons assigned in describing the sucking pump ; when the piston is depressed, the water is forced through the valve I into the tube HG ; and by con- tinuing the process, it may be delivered at any height. 700. If the dimensions of the sucking pump be improperly chosen, it may happen that the water will rise only to a cer- tain height. For the purpose of discovering in what cases this will occur, we shall simplify the question, by supposing the pump to be of uniform bore throughout. Let the water be supposed to have been raised to the level ZX {Pig, 243), and the piston to move through the space ML : call a=LN, the play of the piston, 6= LB, the height of the piston at its greatest elevation above the surface of the water contained in the reservoir, .T=the distance LX. When the piston is raised from the position MN to HL, the PUMPS. 389 air which was previously contained in the space ZN will occupy the space ZL, and its elasticity will therefore be diminished in the ratio of LX to NX ; so that if R represent the elastic force of the air contained in the space NZ, and R' the elastic force of the rarefied air contained in LZ, we shall have LX : NX : : R : R' ; or, X : X — a ; : R : R' : whence, ,r — a R'=R: X But the air contained iw. the space NZ being of the same density with the exterior air, its elastic force will be properly measured by the weight of a column of water whose base c is equal to the surface MN, and whose height is equal to 34 feet. Let this height be denoted by h ; the density of water being supposed equal to unity, and the force of gravity being denoted by g^ we shall have R=cA^. This value, substituted in the preceding equation, gives T,, x—a , R'= dig. But it is evident that when an equilibrium subsists, the elastic force of this rarefied air, together with the weight of the column of water BZ, must be just sufficient to counterbalance the pressure of the atmosphere, which tends to produce the ascent of the water. The weight of the column of water ABXZ will be expressed by ^c X BX, or gc X ^—x) ; and the pressure exerted by the atmosphere will be expressed by the column gch ; hence, we shall have, in case of an equilibrium, ~ gch-\-{h—x)gc=gchy X or, by suppressing the common factor gc^ X — a -h + b — x=h. But, if it were required that the water should rise above the level ZX, it would then be necessary that the atmospheric 390 HYDROSTATICS. pressure should exceed that arising from the weight of the column ZB, and the elastic force of the air contained in the space ZL : we shall consequently have -h-\-h — x, and r the respective capacities of the barrel, pipe, and receiver, and by d the original density of the air, we shall have the proportion h-\-v-{-i^ : p-^-r :: d'. d-^ =density after the first double o-f-^-fr ' stroke. In like manner, b-{-v-\-r:'p-\-r::d~ : di-^ p=densitv after the ^ ^ b+p+r \b+p + r/ ^ second double stroke. And generally, ,+p+r :f+r : : d (j|±^,)""' : d (j^J' = density after the nih double stroke. For the purpose of illustrating the rate of exhaustion, we will suppose that the capacity of the barrel is one-fourth of the sum of the capacities of the receiver and pipe ; then, we shall have 394 HYDROSTATICS. fc=i(p+r)=i(6+;'+r); and the density after the first double stroke will be d-^ =d—=^d. b-\-p+r 5b ' Thus, by the first double stroke of the piston, one-fifth of the air contained in the receiver and pipe will be withdrawn, and the quantity remaining will be four-fifths of the original quantity. The density after the second stroke will, in like manner, be four-fifths of that after the first, or || of the ori- ginal density ; and after the third, the density will be reduced to -fYjj or nearly one-half It thus appears that every three strokes will reduce the density nearly one-half; and conse- quently, that after twenty-seven strokes, the air would be reduced to about one-five-hundredth of its original density. 709. The preceding calculation is based upon the suppo- sition that the relative capacities of the barrel, pipe, and receiver have been accurately ascertained, and that the mechanical construction of the pump is perfect, neither of which conditions is strictly fulfilled : and as it is frequently necessary to know the precise degree of exhaustion that has been attained, it becomes important to have a gauge, or index, by the aid of which we may ascertain the density of the remaining air at any moment. The instruments commonly employed for this purpose are, 1°. The barometer gauge, which consists of a straight glass tube about thirty-two inches in length, and open at both extremities. The tube is placed in a vertical position, its upper extremity communicating with the receiver of the pump, and its lower being immersed in a basin of mercury. When the process of exhaustion has been commenced, the air in the tube being rarefied, the pressure of the atmosphere upon the surface of the mercury in the basin will cause the mercury to rise in the tube, and the height at which it stands will indicate the difference between the exterior and interior pressures. These pressures are in the direct ratio of the densities of the air. The principal inconvenience of this gauge arises from the necessity of having a barometer with which to ascertain the pressure of the exterior air at the same time. AIR-PUMP. 395 2". The short barometer gauge is formed of a tube eight or ten inches in length, open at one extremity, and filled with mercury. This tube being inverted, and immersed at its open extremity in a basin of mercury, the pressure of the atmosphere upon the surface of the mercury in the basin will retain the tube entirely full. This apparatus being placed under a receiver which communicates with that of the pump, and the rarefaction being commenced, the short tube will remain full until the density of the air in the receiver has been so far reduced that its elastic force is insufficient to sup- port a column of mercury of a length equal to that of the tube. The mercury in the tube will then fall, and its height at any moment will indicate the pressure of the air within. This gauge is evidently unfit for use when only a moderate degree of exhaustion is required. 3°. The siphon gauge is composed of a short bent tube, having two parallel branches, one of which is closed, and the other open. The closed branch being filled with mercury, and the tube being placed with the bend downwards, the mercury will be supported in that branch by the pressure of the exterior air. The tube is then placed beneath a receiver, and acts upon the same principle as the short barometer gauge, the bend in the tube serving as a substitute for the basin of mercury. This, also, is only applicable when a con- siderable degree of rarefaction is required. 710. The working of the piston being opposed by the pressure of the atmosphere on its superior surface, and this difficulty constantly increasing as the rarefaction proceeds, it has been found advantageous to adapt a second barrel to the pump, whose piston shall descend whilst that of the first barrel ascends, — and the reverse. The rods of the pistons have the form of a rack whose teeth engage in those of a wheel which is turned by a winch. The pressures on the pistons are thus caused to oppose each other, and the pump works with much greater ease. The rapidity of the exhaus- tion is likewise doubled by this arrangement. 711. If the construction of the pump be such as to require the lower valves to be opened by the elasticity of the air remaining in the receiver, the operation of the pump will evi- 396 HYDROSTATICS. dently cease whenever the rarefaction has been carried so far that the weight of the lower valve is sufficient to overcome the elastic force of the air within. To obviate this inconve- nience, the- lower valves are opened and closed by the motions of the piston, as shown in Fig. 245, which represents a sec- tional view of one of the most approved pumps. The dis- position of the several parts has been somewhat altered, for the purpose of exhibiting them more clearly. A represents the glass receiver resting upon the ground glass plate BC, and communicating by the cavity DFG with the tvvo pump barrels VR and V'R'. The receiver likewise communicates by the cavity svy with the barometer gauge yz, immersed in the vessel of mercury M, and with the siphon gauge vx. E is a stopcock for cutting off the communication between the receiver and the barrels when the exhaustion has been effected, and E' a second stopcock for re-admitting the external air. In the best pumps, the barrels are made of glass, to prevent the corrosion which would take place by the action of the oil with which the pistons are lubricated to render them air-tight : for similar reasons, the pistons are sometimes made of steel. The racks L and L' of the pistons are worked by the wheel W, which is turned alternately to the right and left by the winch H. The lower valves Y and V are metallic, and have the form of a conic frustrum. To the back of the valve is attached a slender rod VR, which passes through an air-tight hole in the piston P, and carries near its upper extremity a small projection or shoulder. When the piston is raised, the friction of the valve-rod which passes through it causes the rod likewise to rise, opening the lower valve V: but this upward motion is soon checked by the shoulder coming into contact with the top of the barrel, and the rod then slides through the hole in the piston. Again, when the piston is depressed, it carries with it the valve-rod RV, closing the valve at the bottom of the pump, and the descent of the piston is then continued by sliding along the rod. 712. The valves of the pistons are variously constructed. In some instances they are metallic, resting upon a metallic bed ; and in others, they are composed of strips of oiled silk, AIR-PUMP. 397 bladder, or parchment, stretched across an opening in the pis- ton, and ahernately allowing and preventing the communica- tion between the air beneath the piston and the exterior air. During the ascent of the piston, the valve remains closed by the stronger pressure of the atmosphere on its upper surface, and when the piston descends, the compressed air beneath it will force open the valve. This latter condition will always be fulfilled, whatever may be the degree of exhaustion, provided the piston can be forced into actual contact with the bottom of the barrel. 713, The pistons are usually composed of two metallic plates, which carry between them a packing of leather soaked in oil. The distance between these plates can be varied by means of a powerful screw ; and by the application of a proper degree of pressure, the packing is caused to fit the barrel with accuracy. 714. By the aid of the air-pump we are enabled to exhibit many of the most important properties of atmospheric air : 1°. The weight of the air may be shown by screwing a vessel provided with a stopcock to the air-pump, and ex- hausting the air from within it. The weight of the vessel will be diminished by about ^\ of a grain for every cubic inch of air that has been withdrawn. 2^. The pressure of the atmoi^nhere is rendered evident by the difficulty with which the receiver is removed from the plate of the pump after the air within it has been withdrawn. A small strip of bladder being stretched across the moutJj of an open receiver, and the air exhausted from beneath, the bladder will be ruptured by the pressure of the exterior air. Two brass hemispheres, being ground so as to fit accurately to each other, and attached to the pump, cannot be separated without great difficulty after the air has been exhausted from the space enclosed by them. The pressu e of the atmo- sphere is found to be equivalent to about 15 lb. for each square inch of surface exposed to its action. 3°. The elasticity of the air may likewise be shown by various experiments. If, for example, a bladder containino-a small quantity of air be enclosed in a receiver, from which the air can be extracted, the elasticity of the air contained iii 34 398 HYDROSTATICS. the bladder will cause it to distend when the exterior pres- sure is removed ; and on the re-admission of the air into the receiver, the bladder will again collapse. If a light glass bulb, having an opening in its lower surface, b« loaded with weights so that it will just sink in a vessel of water when the bulb is partially filled with water ; upon withdrawing the air from the receiver in which the vessel of water has been deposited, the portion of air contained in the bulb will expand, expelling a portion of the water through the orifice in the bottom of the bulb. The bulb and weight will thus be rendered specifically lighter than water, and will consequently rise to the surface of the fluid in the vessel : upon re-admitting the air into the receiver, a portion of water will be forced into the bulb, and it will again sink. 4°. The resistance of the air to the motion of bodies may be exhibited by allowing two bodies of very unequal den- sities to fall in the exhausted receiver of the air-pump, and in the same receiver after the re-admission of the air. When the bodies fall in vacuo, they will reach the bottom of the receiver at the same instant ; but when the receiver contains air, the denser body being least retarded by the resistance which the air offers, it will fall through the height of the receiver in much less time than that required by the rarer body. Many other experiments may be contrived to illustrate the properties of air, but it is unnecessary to notice them in this place. Of the Barometer. 715. The barometer is composed essentially of a bent tube ABC [Fig. 246), closed at A, and open at C, and filled with mercury throughout the portion NMBEF. The air is sup- posed to have been exhausted from the space AMN, and the column of mercury included between the planes MN and DFE is supported by the pressure of the atmosphere upon the surface FE. This column is usually about thirty inches in length, when the barometer is placed at the level of the ocean. BAROMETER. 716. This instrument serves to indicate the changes which are constantly taking place in the pressure of the atmosphere ; for, when the pressure becomes greater, the length of the column of mercury which it can sustain is necessarily increased, and the mercury therefore rises in the tube AD : but if, on the contrary, the pressure of the air should dimin- ish, the length of the column will undergo a corresponding diminution. The pressure of the atmosphere at any point being that due to the weight of a column of air extending from that point to the top of the atmosphere, it follows that this pressure will decrease as we ascend above the earth's surface, and consequently, that the height of the mercurial column will diminish. 717. This principle has been employed to determine the difference of level of two places situated at unequal distances above the surface of the earth. For the purpose of investi- gating a formula which shall be applicable to this object, we shall denote by //.' the height of the mercurial column at the lower station, h the height of the mercurial column at the upper station, D' and D the corresponding densities of the atmosphere at the two stations. Then, if we suppose the axis of z to be vertical, the general equation of equilibrium of heavy fluids as obtained in Art. 655, will be d2i=T)gdz* Let the origin be assumed at the lower station, and let the co-ordinates z be reckoned positive upwards ; then, as we ascend in the atmosphere, the pressure arising from the weight of the superincumbent strata will diminish, and the * This result may be obtained directly by considering a column of the atmo- sphere, whose base AB {Fig. 247) is the unit of surface : the pressure sus- tained by this base is measured by the \\'eight of the column of air ABDC extending to the top of the atmosphere ; and the elementary pressure dp will be represented by the weight of a column having the same base, and a height equal to dz. The base of this elementary column being equal to unity, its volume will be expressed by 1 Xdz, or dz, and its mass by Ddz : thus, gDdz represents the weight which will measure the elementary pressure dp. This result will obviously be independent of the particular form given to the base AB which has been assumed as the superficial unit. 400 HYDROSTATICS. density of the air will undergo a corresponding decrease. Thus, the pressure p being a decreasing function of the alti- tude z, dp and dz will be affected with contrary signs : hence, the preceding equation should be written dp=—Dgdz (412). If the difference of level of the two places be but slight, the force of gravity g- may be regarded as constant : and hence we shall obtain, by integration, ^—IM («^>- But it has been shown (Arts. 651 and 696) that when the temperature is supposed constant, the pressure and density are proportional to each other ; hence, if P denote the pressure capable of producing a density represented by unity, we shall have p=PD; and therefore, dp=VdT): this value substituted in equation (41 3) gives __P /*dD ^- gJ D-' and by effecting the integration indicated, there results g- To determine the constant, we remark, that \vhen z=0, the density becomes that which we have supposed to exist at the lower station, and which has been denoted by D'. Thus, the preceding equation becomes 0=-?logD'+C; g- eliminating C between this equation and the preceding, we find z=^ (log D'-log D), or. p, jy But the densities being proportional to the pressures, they BAROMETER. 401 will likewise be proportional to the observed altitudes of the mercurial column : hence, h: h' ::D : D', or^=5.'; h U D' this value of -- being substituted in that of z, we obtam P, h' g ^A h' The logarithm of —, which appears in this expression, apper- tains to the Naperian system : if therefore, we represent by h' Log—, the tabular logarithm h the modulus, we shall have and, by substitution. h' h! Log—, the tabular logarithm of —, and by M the reciprocal of h h MLog|'=log^; MP h! g h 718. To determine the value of the constant P, which represents the pressure exerted upon the unit of surface, and capable of producing a density of air represented by unity, we remark, that the density D' at the lower station corres- ponds to the pressure exerted by the atmosphere at that point : this pressure is measured by the weight of a column of air whose base is the superficial unit, and whose altitude is equal to that of the atmosphere : but this column of air is equal in weight to the mercurial column whose height is h! ; if therefore D" denote the density of mercury, the mass of the column will be expressed by 1 X A'D", or /i'D" : and by multiplying this product by g^ we shall obtain the expression h'D"g, for the weight of the column supported at the lower station. Such will be the pressure capable of producing the density D'. To obtain the pressure P corresponding to the unit of density, we make the proportion D' : 1 : : h'jy'g : P ; whence, 402 HYDROSTATICS. substituting this value in the formula (414), there results ^=-ËF-Log- (415). 719. The intensity of the force of gravity being different at di^rent places on the surface of the earth, the weight of the same column of mercury will likewise vary when it is transported from one place to another : thus, if the force of gravity be denoted by g at on« station, and by (1 —^)g at a second, the mercurial column whose height is h' will become heavier or lighter at the second station than it was at the first, according as ^ is negative or positive. Let the quantity ^ be considered positive : then 1 — ^ will be positive, and less than unity, since the variations of gravity are exceedingly small. But a column of mercury whose height is h' becoming lighter at the point whose gravity is denoted by (1 — J)^, it will correspond to a less pressure of the atmosphere, and hence, the density of the air correspond- ing to this pressure will be less. The densities of the air being proportional to the pressures exerted, and these pressures being measured by the weights of the column of mercury whose height is h'. it follows that the intensities of gravity, which are represented respectively by g- and (1 — .-,4:). 730. This equation must be integrated with reference to z. We remark, however, that s will necessarily vary with z, but that the quantities w and -^, which represent the particular dt eliminating -^ by means of equation (420), there results DISCHARGE OF FLUIDS. 411 values of v and -— corresponding to the orifice, not being functions of the quantity xr, they may be regarded as con- stant in effecting this integration. 731. If we regard u and — as constant, it is obvious that at all the integrals will be taken with reference to z, and there- fore apply merely to the dimensions of the vessel. But, when these integrals have been obtained, we may regard ii and — (JLt as variables, and functions of t. 732. By effecting the integration, we obtain ^=K-^-4'/t-S^)+« (^^«)- The velocity u which enters into this equation is equal to dz the value — corresponding to the orifice, and will obviously be a function of the time. Consequently, as the quantity u has been supposed constant in the preceding integration, the time t must be constant likewise. Hence, the constant C will in general be a function of the time. 733. To determine this constant, let P represent the pressure sustained by the superior surface CD of the fluid {Fig. 249), the area of this surface being denoted by s'. If /dz — be taken in such manner that it shall be equal to zero when s becomes equal to s\ this section s' will correspond to an ordinate 2;'=0L, and the equation (425) will give, upon this hypothesis, C=V-Ti(gz'-^^\. This value being substituted in (425), we obtain ,=P + d[,(._V)-..^/^^+,«.C^-|;)J....(426). 734. This pressure is exerted at every point of the stratum whose distance from the plane AB is equal to z. If we wish to obtain the pressure Q, at the orifice, we denote by z" the 412 HYDRODYNAMICS. corresponding value of the ordinate z which will be equal to Ow, and observe that the section 6- will, at that point, be equal /dz — being then taken between the hmits z=z' and z=z'\ we shall obtain, by representing this inte- gral by N, and substituting these values in equation (426), 735. This equation makes known the pressure at the ori- fice : the first member expresses the difference between the pressures at the orifice and at the surface. Let these pressures be supposed equal, as is the case when they arise from the weight of the atmosphere: then, d — P will reduce to zero, the common factor D will disappear, and there Avill remain gi,z"-z')-m^^^-\-\^i^ (^-l) =0; but the area k of the orifice being always supposed less than the area s' of the superior surface, the fraction -^ will be less than unity ; if therefore, we wish to render the coefficient of u"^ positive, we may write this equation under the form gi^^"^^')-m^^-\u^ {}-Ç) =0 (427). 736. If in this eqiiation we introduce the vertical distance of the orifice below the surface of the fluid, making z"-z'=h (428), we shall have The quantity h, which represents the distance EP (Pig. 250), will be constant if the surface of the fluid be supposed to be maintained at the same height ; but it will be variable if the vessel be supposed to discharge its contents without being replenished. 737. In the latter case, if we make EO=a, P0=5r, and EP=r/j, we shall have the relation h = a-z (430) ; DISCHARGE OF FLUIDS. 413 and the equation (429) will become g{a-z)-m'^-^-iu^ (l-^) =0 (431). 738. If the surface of the fluid be constantly maintained at the same height, the quantity h will have a constant value, and the integral N, which will then be a function of constant quantities, will likewise be invariable. Thus, equation (429), containing no other variables than t and m, may be put under the form a — b~ cu^ =0 dt ' from which we deduce hdu dt—- a — ca^ This equation can be readily integrated by the method of rational fractions; for, if we make h=h'c^ and a=a'^c, the quantity c will become a factor of the numerator and de- nominator, and may be stricken out ; whence we obtain ,^ h'du dt= . The second member of this equation being resolved into factors, we shall have ,^ 2-^^r2^'^" a'-\-u a' — u which, being integrated, gives or, ^=^ log {a'+u)-^, log {a'-u)-hC', . b' , a'+u , ^ t=—- log — ■ hC. 2a' ^a'—u The constant C is determined by the condition that the ve- locity u is equal to zero at the same instant as the time t ; thus, the supposition of w=0, and i=0, reduces the preceding equation to ^'og 1+0=0; 414 HYDRODYNAMICS. or, C=0: whence, 6' , a'-\-u this equation will determine u, if we suppose the time Mo be given. 739. If we denote by e the base of the Naperian system, and pass from logarithms to numbers, we shall obtain 2a't a'-\-u -^r. -, =e a — u and by resolving the equation with reference to w, there results 2a't _ «te '''-!) . or replacing a' and h' by their values (Arts. 737 and 738), the expression for the velocity will become But, if the area of the orifice, which is denoted by A-, be sup- posed extremely small, the exponent of e, increasing with the time ^, will become exceedingly great after the expiration of a very short time. Hence, we may neglect unity in the nume- rator and denominator of the last factor, as very small with reference to the term which precedes it, and the value of u will then be reduced to vte) by neglecting k^ with reference to s'^. Thus, it appears that the expression */(^gh) is a limit which the velocity of the fluid at the orifice never attains, but to which this velocity becomes very nearly equal after the expiration of an exceedingly short time. DISCHARGE OF FLUIDS. 415 The value of the velocity being thus determined, we sub- stitute it in equation (425), and thence deduce the pressure on the unit of surface. 740. If the vessel be supposed to empty itself, the upper surface will be depressed as the fluid is discharged, and the quantity A, or (a — z) must therefore be regarded as variable in equations (429) and (431). The equation (431) will thus contain the three variables t, u, and z, and will consequently be insufficient for the solu- tion of the problem : but a second relation may be obtained by means of equation (420) in which we replace * by s', and thus obtain ku=s'^ (432). 741. This equation likewise contains three variables, and we are therefore unable to integrate it ; but it will serve to eliminate z. For this purpose, we differentiate equation (431), which gives —g- A;N- u-r\ 1 r- I =0> ^dt dV- dt\ 5'V ' dz and by eliminating — ^, we obtain (jLZ ~g—, — ^^N-- — ?«-—( 1—— ) =0. s' di'' dt \ s'V This equation, which can only be integrated by approxima- tion, makes known the relation between the time and the velocity. 742. When the orifice is supposed extremely small, the terms containing k may be neglected, and the equation (426) will be reduced to p=V-\.-Dg{z-z'); but z~z' is represented by On — OL {Fig. 249) ; and it will therefore express the distance of the point w, whose ordinate is equal to z, beneath the surface of the fluid. Hence, the pressure p exerted upon the unit of surface at the point n is equal to the pressure P at the surface of the fluid, plus the pressure arising from a column of a fluid whose height is equal to the distance of this point below the surface. 416 HYDRODYNAMICS. It should be remarked, that this pressure is precisely that which would be exerted at the point n if the fluid were sup- posed at rest. 743. If the terms containing k in equation (429) be neg- lected as infinitely small, it will reduce to whence, u=^{2gh) (433): and we therefore conclude, that when a fluid escapes from an infinitely small orifice in the bottom of a vessel, the velocity will be the same as that acquired by a heavy body in falling through a distance equal to the height of the surface of the fluid in the reservoir above the orifice ; and since it has been shown (Art. 405) that a body projected vertically upwards will rise to a height equal to that through which it must fall to acquire the velocity of projection, it follows, that if by means of a curved tube, the jet of fluid be directed upwards, it will rise to the level of the surface of the fluid in the reservoir. 744. The expression for the velocity with which a fluid will issue from an extremely small orifice in the bottom of a vessel may be investigated in a more elementary manner, as follows. Let EF {Fig. 251) represent a very small orifice in the bottom of a vessel ABCD, which is filled with a fluid to the level AB, and let GF represent an infinitely thin stra- tum of the fluid directly above the orifice EF. Denote the height of this stratum by dh, the entire height of the fluid FI being represented by h. Then if the stratum of fluid GF be supposed to fall through the height HF under the influ- ence of the force of gravity, it will acquire a velocity v, ex- pressed by v = ^{2gy.YYL)^^{2gXdh). But if the stratum be supposed to descend through the same height, being urged by its weight and the pressure arising from the column of fluid GI, which is directly over it, the incessant force g', which is then exerted upon it, will be to the force of gravity, as FI to FH. Hence, we shall have ^~FH dfi DISCHAKGK OF FLUIDS. 417 Again, if u' denote the velocity acquired by tne stratum in descending through the space FH, when urged by the force g\ we shall have and by comparing this value with that of v^ we find V ^(2gxdh)' or, by substituting the value of — , and reducing, there results This expression is precisely the same as that which would be obtained for the velocity of a body falling freely through the height FI. 745. When the orifice, which is still supposed exceed- ingly small, is pierced in the vertical face of a vessel, the fluid will issue in a horizontal direction, and will describe the arc of a parabola, if the resistance of the air be neglected. The angle of projection denoted by a in equation (289), being in the present case equal to zero, we shall have tang «=0, cos ct=l: these suppositions reduce the formula (289) to an equation of a parabola whose axis is vertical, and whose vertex coincides with the origin of co-ordinates. 746. The distance to which the fluid will spout upon a horizontal plane situated at any distance below the orifice may be readily determined. For, let O {Fig. 252) represent an orifice in the vertical side of a vessel which is filled with a fluid to the level EF ; and let AB represent the horizontal plane upon which the jet is allowed to fall. Then, the quan- tity h will represent the distance OF, and the ordinate CD of the parabola OD will be determined by making y=OC : we thus obtain CD=:r = v'(4%)=V(OFxOC). But the expression y'(OFxOC) is equal to the ordinate OG of a semicircle described upon CF as a diameter. Hence, we derive the following rule : The horizontal distance to wliicli ajluid will spout from an orifice in the vertical side of Dd 418 HYDRODYNAMICS. a vessel, is equal to double the ordinate of a semicircle de- scribed upon the distance intercepted between the upper sur- face of the fluid and the horizontal plane iipon which the fluid falls ; this ordinate being drawn through the point lohich corresponds to the orifice. When the orifice is pierced at the middle of the hne CF, the ordinate OG will be a maximum, and the distance to which the fluid will spout will therefore be the greatest. 747. The velocity u having been determined, we can readily ascertain the quantity of fluid discharged in the time t. For this purpose, we remark, that whilst the stratum of fluid CD {Pig. 250) sinks to the level MN, a volume of fluid equal to that contained between the planes CD and MN must pass through the orifice. But if we represent by 5 a section of the vessel, and by dz the thickness of an elementary stratum, the integral fsdz taken between limits CD and MN will express the volume of fluid discharged. If this volume be denoted by Q, we shall have Q,=fsdz (434) : but the equation (420) gives sdz=kudt ; whence, by substitution, we obtain Q, =fkudt. The value of the quantity discharged may be deduced imme- diately from that of the velocity. For, if de represent the space passed over by the fluid filament in the time dt, upon leaving the orifice, we shall have udt=de: and if this expression be multiplied by k, the area of the ori- fice, we shall obtain kudt for the volume discharged in the time dt. Taking the integral fkudt, we shall find the quan- tity discharged in the time t. To effect the integration, we replace u by its value y/{2gh) given in equation (433) : we thus find Qi=k^{2g)f^h.dt (435). 748. Two distinct cases may now be presented, viz. when h is constant, and when h is variable. The first occurs DISCHARGE OP FLUIDS. 419 when the fluid in the reservoir is constantly maintained at the same height, and the preceding equation can then be integrated without difliculty, since the quantity h may be replaced by a constant a. Thus, we shall have Q,=kt^{2ga) + C. The constant C may be determined by the condition that the quantity Q, is equal to zero at the commencement of the time, or Q,=Oj and ^=0 ; hence, and the equation therefore reduces to a=A-V(2^«) (436). 749. If the orifice k be supposed circular, its radius being represented by r, we shall have k—vr^\ and the formula will become a=^^(2^)^rV« (437). The quantity '^^/{2g) will be the same for all problems which may be proposed, and its value may be immediately deduced, since we have ^=3.14159, ^=32.1.598. The quantity g being expressed in feet, the values of r and a must be expressed in units of the same kind, and the quan- tity discharged will then be expressed in cubic feet. 750. The time t must be expressed in seconds, since the second has been adopted as the unit of time in determining the value of g. 7o\. If the fluid be water, the weight of the quantity dis- charged may be determined by allowing 62ilbs for every cubic foot. 752. The formula (437) likewise serves to determine the time necessary for a given quantity of fluid to be discharged from an orifice in a vessel, when the fluid is maintained at a constant height ; for the formula gives t= ^_ (438). 420 HYDRODYNAMICS. 753. As an example, let the vessel be supposed cylindrical, the radius of its base being denoted by b ; and let it be re- quired to determine the time necessaiy to discharge a volume of fluid equal to that of the cylinder. In this case, the horizontal sections being all equal to xb^, the equation (434) will give Çi^firb^ dz ; and consequently, a^Trb^z+C. Taking the integral between the limits «=0 and Zz=a, there results (X=vb^a. This value substituted in formula (438) gives Trab^ t= or, by reduction. b^^a t= 754. If we suppose the fluid to be maintained at a height a' in a second vessel, and denote by Q,' the quantity dis- charged from an orifice k' in the time t, the equation (436), when applied to the present case, will give Ci'=k'^{2g). t^a' ; and by comparing this equation with (436), we can estabhsh the proportion Gl: a' :: k^{2g).t^a: k'^{2g).t^a' ; or, by suppressing the common factor t-^{2g), this proportion becomes d : Q,' :: k^a: k'^/a'. Hence it appears, that the quantities discharged in the same time, from orifices of different sizes, and situated at different depths, are directly proportional to the areas of those orifices and the square roots of their deptJis. 755. From the formula (436) we can deduce another con- venient theorem relative to the quantity of fluid discharged. For, let s represent the space through which a body would fall in the time t ; we shall have DISCHARGE OF FLUIDS. 421 or, Substituting this value for t in equation (436), we obtain and since ■^/{as) is equal to a mean proportional between tlie distances a and s, we deduce the following rule : The volume of fluid discharged from an oriflce k,in the time t,is equal to tioice the volume of a cylinder whose base is the area of the oriflce, and whose height is a mean projjortional between the depth of the oriflce beloiv the surface and the distance through which a body u'ouldfall in the time t. 756. Let the vessel be now supposed to discharge itself, without receiving an additional supply of fluid : the quantity h in equation (433) must then be regarded as variable, and being replaced by [a — s), that equation will become This value of u substituted in (432) gives fif — *'^^ ~k^[2^^^a-z)y or, dt^ ^'^^ (439). The quantity s' represents the section of the vessel which corresponds to the upper surface of the fluid. This section will be a function of the variable z, and may be eliminated by means of the equation of the interior surface of the vessel. Thus, the value of s' in terms of z being introduced into equation (439) will render that equation susceptible of inte- gration, and the relation between z and t will therefore become known. If we subtract the value of z thus obtained from the constant a, we shall obtain an expression for h in terms of t, which substituted in (435) will give, after integra- tion, a relation between the time t and the quantity dis- charged Q,. 757. Let us take, as an example, a vessel whose interior surface has the form of a paraboloid of revolution. T! is 36 422 HYDRODYNAMICS. surface being generated by the revolution of the parabohc arc AD {Mg. 253) about the vertical axis AB ; if we denote by a the distance AB between the orifice and the surface of the fluid in its primitive position, by z the distance PB, and by y the ordinate PM, we shall have the relation, y^=1){a—z) the equation of a parabola referred to its vertex A. Hence, if ?r represent the ratio of the circumference to the diameter, the area of the circle described with the radius PM will be expressed by jry^ =7rp(a — z) ; and consequently, s'=^p{a-z) (440). Let this value be substituted in (439), and we shall obtain dt=-, — ^— — X — -, ^dz ; or, by reduction, dt = :r^^'^---(a-zydz. 758. For the purpose of integrating this equation, we make a—z=x] whence, f{a—zYdz=—fx^dx=—%x^ + C'. replacing x by its value, we have f{a-zYdz = -%{a-z)^ + C', and consequently. The constant C is determined by making 2;=0andi=0; this supposition gives and the equation (441) can therefore be reduced to vp ■[a^—{a-zY]. 'k^{2gy To determine the quantity discharged in a given time, we find in this equation the value of ■.-.^{J-^>) t=-^^y.^ ''^ DISCHARGE OF FLUIDS. 423 and substitute it for h in formula (435) : we thus obtain the relation This equation may be integrated by a process entirely similar to that adopted in finding the relation between z and t. 759. Let it be required to determine the time in which the water contained in a vessel having the form of a right cyl- inder will be discharged through an orifice in the bottom of the vessel. Let h represent the radius of a section of the cylinder by a plane perpendicular to its axis : then, s'=?rb'', and the equation (439), when applied to the present case, wil. give Making a—z=.T, then integrating the transformed equation, and replacing a: by its value, we find The constant is determined as in the last example, by making z=0 and t=0 : whence we deduce The integral being taken between the limits z=0 and z=a, we find, for the time of emptying the vessel, '-iBm^" <^*^'- If we suppose, as in Art. 749, that the orifice is a circle whose radius is equal tor, we shall have k=7rr^ : this value reduces (443) to By comparing this result with that obtained in Art. 753, it will appear that the time necessary for the entire discharge of the fluid when the vessel empties itself, is double that in which an equal quantity of fluid would flow through the same orifice if the vessel were kept constantly full. 760. The formulas (442) and (443) will serve as a guide in 424 HYDRODYNAMICS. the construction of a clepsydra, or water-clock. This instru- ment consists merely of a vessel from which the water is allowed to escape through an orifice in the bottom, and the intervals of time are measured by the depressions of the upper surface. Thus, if we wish the clock to run 12 hours, we reduce the time o seconds, which g-ives 12 x (60) 2, or 12 X 3600 ; and by substituting this value of t in formula (443), we can then assume arbitrarily two of the three quan- tities A-, 6, and a. Let the values of k and h be assumed ; that of a, the height of the clepsydra, will then result from formula (443). To discover the manner in which this height should be divided in order that the superior surface of the fluid may be depressed through the several divisions of the scale in equal intervals of time, we deduce from equation (442) the value of {a—z), which is .A•V(2^)^^ -H^'^-'W) and by making t successively equal to 1 hour, 2 hours, 3 hours, &c., we can determine the corresponding values of a—z, which should be laid off from the bottom of the vessel. We can, however, readily discover the general law according to which the scale must be divided : for, since the vessel is supposed to discharge itself in 12 hours,if we make ^=12 hrs., we shall have a—z=0; and consequently, /• Â:(12hrs.)v^(2,^)_^. or, (12hrs.)^^'(2o-) 2^6=* ' ^ When ^=11 hrs,, we have t . V(2^) (llhrs.)V(2g-) _,, and therefore, a-z = {^/a-\i^ar- =^(j\y Xa. In like manner, when ^=10 hrs, we shall find a—z=:{j%y- xa. DISCHARGE OF FLUIDS. 425 Thus the successive values of a—z^ which correspond to the several hours, will bear to each other the same relations as the terms in the series (TV)^(T^3)^(^)^&c. These terms are to each other in the same ratio as the squares of the natural numbers 1, 2, 3, &.C. Hence, if we divide the whole height a into 144 equal parts, and lay off from the bottom of the vessel distances which shall be equal respectively to 1, 4, 9, &.c. of these parts, we shall obtain the points of division in the scale which will correspond to the upper surface of the fluid at the expiration of the several hours. The form of the vessel being prismatic, the figure of its base may be assumed arbitrarily. 761. When the surface of the fluid shall have arrived nearly at the bottom of the orifice, the quantity discharged will be influenced by the formation of a hollow tunnel, which is then found to be produced directly above the orifice : it is therefore advisable to employ only the first eleven divisions of the scale. 762. It usually occurs that the condition of the particles descending in vertical lines, and with velocities which are equal at every point of the same stratum, ceases to be ful- filled when the surface of the fluid has arrived within 4 or 5 inches of a horizontal orifice. The fluid particles then assume directions which are more or less inclined to the hori- zon, and the tunnel spoken of in the last article is then formed. When the orifice is found at a considerable depth, the upper surface of the fluid remains sensibly horizontal, and the tunnel above the orifice is no longer formed, in con- sequence of the greater velocity with which the fluid par- ticles near the orifice are compelled to flow into the vacancy which has been left by those immediately preceding them. 763. This tunnel becomes much less perceptible when the orifice is formed in the side of a vessel. But when the upper surface of the fluid has nearly attained the level of the orifice, a slight depression on the side of the orifice begins to be observed. 764. This tendency of the fluid particles towards the ori- fice, occasioned by their sustaining less pressure in that direc- 426 HYDRODYNAMICS. tion, gives rise to a contraction in the jet of fluid, which, in issuing from the orifice, assumes the form of a truncated pyramid or cone, whose greater base corresponds to the ori- fice. This diminution in the size of the jet is called the con^ traction of tJie vein. With a circular orifice, the smallest section of the fluid vein is found at a distance from the orifice equal to the radius of the orifice. Beyond this point the diameter of the section again increases, so that the entire jet has the form of two truncated cones which are united by their smaller bases. 765. The contraction of the vein likewise takes place when the orifice is pierced in the side of a vessel ; but if the orifice be large, and be placed at a short distance below the surface of the fluid in the reservoir, the jet will be found to be more contracted in the vertical than in the horizontal direction. 766. When a conical tube whose interior surface corres- ponds to the form of the contracted vein is adapted to an orifice pierced in a thin plate, the quantity discharged is found to be very nearly the same as though the fluid issued directly through the orifice. Hence, we may regard the vessel as coiitinued to the point at which the greatest contraction of tiie stream takes place, and consider the least section as forming the real orifice. It is proved by experience, that the quantity actually dis- charged may be deduced from that calculated according to the theory, by sinjply changing the value of the constant k. Thus, if we represent by MA: the area of the orifice which has been calculated from a knowledge of the quantity actually discharged, tne theoretic formula must be modified by substituting Wt for k : we shall thus obtain, for the actual discharge, 767. When the orifices are pierced in thin plates, the ratio M is found to be independent of the size of the orifice, and of its depth below the surface, provided that depth be not very small. Hence, if we represent by Q,' the quantity discharged DISCHARGE OF FLUIDS. 427 from an orifice k' at the depth a\ we shall have the pro- portion a : a' : : Wc^{2g) . V« : MAV(2^) • V«' i and we therefore conclude, that the quantities discharged from two such orifices are to each other as the products of the areas of those orifices, and the square roots of their depths. 768. The number M has been found by Bossut to be about 0.62, and the orifice k must therefore be multiplied by this fraction, in order that the quantity given by the formula may correspond with the results of experiment. Thus, the cor- rected expression for the quantity discharged will be Q = (0.62)AV(2^).V«- This formula is alike applicable, whether the orifice be pierced in the side or bottom of a vessel. 769. When the vessel is allowed to empty itself, the cir- cumstances of the discharge become very complicated after the upper surface of the fluid has fallen to within a short dis- tance of the orifice. If, however, we only consider the ex- penditure previous to the arrival of the upper surface within a few inches of the orifice, the same correction may be applied to formula (439), which will thus become -(0.62)Av(2^)v/(«-^)' and will serve to determine the time necessary for a given quantity of fluid to be discharged. 770. In applying the preceding correction to the theoreti- cal discharge, it has been supposed that the orifice was pierced in a thin plate : when a similar orifice is pierced in a thick plate, the quantity discharged is found to be consider- ably greater. Hence it occurs, that when the fluid is dis- charged through a thick plate, or through a cylindrical tube applied to the orifice, the coefiicient 0.62, which has been em- ployed in calculating the discharge through a thin plate, is no longer applicable. In this case the fluid adheres to the sides of the tube, and the contraction of the stream is in a great measure avoided. The lengths of such tubes, accord- ing to Bossut, should be at least twice the diameter of the orifice, in order that the contraction of the vein may be pre- 428 HYDRODYNAMICS. vented. There will however be a limit to the length, proper to be given to such tubes, since the friction of the fluid against the sides of the tube will necessarily increase with its length. 771. The quantities discharged by cylindrical tubes are proportional to the products of the orifices by the square roots of their depths, as in the case of apertures pierced in a thin plate ; but the coefficient M, by which the area of the orifice must be multiplied for the purpose of reducing the theoretical discharge to that given by experiment, has been found by Bossut to be about ||, or, more accurately, 0.81, when a short cylindrical tube is applied to the orifice. Thus, the formula (436), which serves to determine the quantity discharged from a reservoir in which the fluid is maintained at a constant height, will become, when corrected for the case of a cylin- drical tube, a=(0.Sl)^(2^).AV«; or, if we replace k by its value «-/-s, r denoting the radius of its section, the formula may be written a=(0.81)^(2o-).^r=V«- 772. When the vessel is supposed to empty itself by an orifice to which a cylindrical tube has been adapted, we can still employ the coefficient (0.81), provided we only consider the circumstances of discharge previous to the arrival of the upper surface of the fluid at such a level that the tunnel begins to be formed above the orifice. 773. By adapting tubes of diflferent forms to an orifice pierced in the side or bottom of a vessel, the quantity of fluid discharged is generally found to be more or less increased. The following table presents a view of the relative quan- tities discharged in some of the simplest cases. 1°. Theoretical discharge in a given time through an orifice pierced in a thin plate - - - - 1.00 2°. Actual discharge in the same time through the same orifice 0.62 3°. Discharge through a cylindrical tube, whose length is equal to two diameters of the orifice 0.81 MOTION OF WATER IN PIPES. 429 4°. Discharge through a conical tube having the form of the contracted vein, the larger base being regarded as the orifice 0.62 5°. Discharge through the same tube, regarding the smaller base as the orifice 1.00 Of the Motion of Water in Pipes. 774. Let AB {Fig. 254) represent a cylindrical pipe, by- means of which the water contained in the reservoir R is transferred to the reservoir R', and let it be supposed that the current has assumed a uniform motion : it is proposed to investigate a formula by means of which the quantity of water delivered at the point B, in a given time, may be estimated. Let CC'D'D represent an elementary stratum of the fluid included between two consecutive transverse sections. Then, since the motion is supposed to have become uniform, the forces which tend to accelerate the motion of the element CD' must be precisely equal to those which are exerted upon the element in a contrary direction. The force exerted upon CD', urging it in a direction from A towards B, is the com- ponent of the weight of this element, in a direction parallel to the axis of the pipe : and the forces which urge it in an opposite direction are, 1^. the difference of the pressures exerted upon the faces CC and DD' ; and, 2". the resist- ance arising from the friction of the fluid against the sides of the pipe. 775. lip denote the mean pressure, referred to the unit of surface, in the section CC, the corresponding pressure in the section DD' will be expressed by p-\-dp, and if we denote by a the area of the transverse section of the pipe, the entire pressures upon the sections CC and DD' will be respectively adp, a{p + dp). These pressures being exerted in contrary directions, the ele- mentary stratum CD' will be acted upon by a force equal to their difference adp. The resistance arising from the friction against the sides of the pipe will be directly proportional to the surface of the 430 HYDRODYNAMICS. fluid in contact with the pipe, and will likewise be dependent upon the velocity of the current. Hence, if v denote the velocity, c the circumference of the section, and s the distance of the section CC from the extremity A, the distance CD will be expressed by ds, and the resistance experienced by the element CD', in consequence of friction, will be cds .{v) may be expressed by two terms which are respectively proportional to the first and second powers of the velocity : thus, we shall have ç>{v)=bv-^b'v'' ; b and b' representing constant quantities. This value of k (445). The values of « and ^ may be regarded as known, since 432 HYDRODYNAMICS. they result immediately from those of a and 6, which are sup- posed to be determined by observation ; and the value of k will likewise be given when the length of the pipe, the differ- ence of level of its two extremities, and the difference of the pressures at those points are previously given. Hence, the velocity v in a pipe of a given diameter can be readily cal- culated. 779. The numerical values of « and /3 have been found by Prony to be «=0.00017, /3 =0.000106; and the preceding equation therefore becomes 0.00017t? + 0.000106^2 ^ ^DA. If we neglect the first term, which is generally admissible when the velocity v is not extremely small, the formula will reduce to v=48.56^(DA-). 780. Let Q, denote the quantity delivered at the point B in a second of time, and t the number 3.1416 ; we shall have and by substituting this value of v in equation (445), there results 4a , i6a= ,T., or, if we neglect the term containing the first power of v, and make — -=^'^, we shall obtain The numerical value of — is 38.12 ; and the formula there- 9>' fore reduces to a-38.12^(D^A). In this investigation the dimensions are supposed to be expressed in English feet. THE END. J'/.L" -.fi .;: ^/q 7 .■^ S IJ / / ■Jf>. ' / a" ■•/ (.'•' /■': c y J^- X ' i p^ Y 'HÎEttjlltltltliijitl ^^'t^ ^ y/.: .'iff. 1'.' J* )^_ C A (• .;/. y /I .1" V ; k... : / / ■■i7. t I.. 1 4^'. A 1) D U B ^T ^^-^ . >'>. Vr,.,l7,.;„„,.: Sr .XY v.^. ■•.t-l.\ w H, /iV H "■'^x f \ /■\ / V lA y^y J'/..1'.' -A 71. ?n ' c w //■J,. as. <9ff. Prtt4j?umttr$r .\t 'NY 1 -/./''/< / m. .94-. A E b ^y [ V-H /y. /} '.'' 220: 124-. K P,-,../7„-n,,,„- S.- XV J -m' fa ^ y -y -ft isfi. Tn K^ i / / • 1 L ^''•^^ / z^. ^^ s 7 "^^^ / jr^SB. ^ ^7^- /■: /-,„/i,.„,.,„ xy- J'/.'/.'/l I 'I (')''' 2JJ'. A„ (' 1» !•: M Y V. 1) r K .V II ^ f7 ' ,/■ ■i20. y i — ^ ■j:i(). <^ a? /•; 1 •>■,,./ hmnnu Sc-VY. L /'/..O'J' 2:',7- c!3 -v/>, ij: K Hr,„ih.,,.,«,< Si-yy V 'U A[ v - . L7 UNIVERSITY OF CALIFORNIA LIBRARY Los Angeles This book is DUE on the last date stamped below. AUG 2 î95St Form L9-10m-6,'52(A1855)444 QA i