IRLF JVo. Division Range Shelf Received THEORY OF FRICTION. A TREATISE ON THE THEORY OF FRICTION, BY JOHN H. JELLETT, B.D., SENIOR FELLOW OF TRINITY COLLEGE, DUBLIN; PRESIDENT OF THE ROYAL IRISH ACADEMY. DUBLIN: HODGES, FOSTER, AND CO., PUBLISHERS TO THE UNIVERSITY. LONDON: MACMILLAN AND CO. 1872. DUBLIN: PRINTED AT THE UNIVERSITY PRESS, BY M. H. GILL. PREFACE. THE Theory of Friction, considered as a part of Rational Mechanics, has hardly received the attention which it deserves. Even in the most complete systematic treatises, the space accorded to its discussion is small, compared with that bestowed upon questions in which the hypo- thesis of perfect smoothness has been assumed as the basis of the investigation. And it seems probable that many students have been led to regard the discussion of this force, less as a part of Rational Mechanics, than as a correction to be applied before the investigations of that science can be made practically useful. Such an idea, if it exist, is a complete mistake. The theory of friction is as truly a part of Rational Mechanics as the theory of gravitation. The force with which this theory is concerned is subject to laws as definite, and as fully susceptible of mathematical expression, as the force of gravity. And even if the imperfection of our analysis should render impossible the actual solution of the pro- blems which are presented to us in this theory, such an imperfection, which is purely mathematical, cannot affect the right of the theory itself to be considered as a part of Rational Mechanics. Neither is this right affected, even if it be shown that the laws usually as- signed to the force of friction are not mathematically true for any known substance. Rational Mechanics, Vi PREFACE. considered as a hypothetical science, takes no account of this fact, and for the reality and usefulness of the conclusions it is only necessary that these laws should be approximately true, as they certainly are. Through- out the present treatise the proportionality of the force of friction to the pressure is assumed. This law, though not mathematically correct, represents the facts with sufficient exactness to serve as the basis of a theory whose results approximate closely to the truth. The adoption of a more complicated law would have greatly enhanced the mathematical difficulties of the theory, yet without giving results mathematically coincident with facts. Here, as in all other applications of Rational Mechanics to actual phenomena, we must be content with approximations to the truth. The plan of the present work may be shortly stated as follows : Commencing with the two great classes of force, namely, moving and resisting forces, the Author has, in Chap. L, pointed out the important distinction between them, consisting in the fact that while forces of the first class are independent of the forces by which they are opposed, those of the second class vary in magnitude or direction, or both, with these opposing forces. Two kinds of resisting force are then especially noticed, namely, 1. Forces by which the geometrical conditions, connecting with each other the several parts of a material system, may be replaced, without making any change in its motion (or rest, if it be at rest). These forces are distinguished by the name " geome- trical forces," and their true character is pointed out. 2. Friction. The laws which govern this force are then given, and the action of a rough surface upon a particle PREFACE. Vll which rests or moves upon it is expressed by means of a geometrical conception which is (so far as the Author is aware) due to Mr. Moseley. The important dif- ferences between the friction of rest and the friction of motion are now noticed for the first time. In Chap. II. the Author has considered the problem of equilibrium, noticing especially the uideterminateness which often appears in questions where friction is one of the acting forces. This apparent indeterminate- ness is traced to its source, namely, the abstractions of Rational Mechanics ; and it is shown that if the con- ditions of the problem be stated as they really exist in nature, no such indeterminateness will appear. Various examples of the problem of equilibrium, both of particles and of solid bodies, are given. Chap. IIL treats of extreme positions of equilibrium, those, namely, in which the smallest diminution in the friction acting at some one or more points of the system will destroy its equilibrium. In such cases it becomes an important question : \Yhat points of the system are in extreme positions ? or, in common words, what points of the system are so situated, that any diminution of the roughness of the surfaces which are there in contact would cause the system to slip ? Proceeding to investigate this question, the Author has given some examples, in which it admits a simple solution. The general problem is then examined by the help of certain analytical Lemmas, and rules are given for its solution. But the great length of the process would make this solution very difficult, except for the simplest cases. A special section is devoted to extreme positions of a solid body. Vlll PREFACE. Chap. IV. treats of the motion of a particle, or sys- tem of particles. Here the indeterrninateness before noticed in statical problems appears again, and is par- ticularly considered. A special section is, on account of the importance of the question, devoted to the problem of initial motion, and another to the motion of a particle upon a surface which is itself moving. Chap. V. treats of the motion of a solid body, with special reference to the case in which the motion is a pure rotation round an axis fixed or variable, and some general propositions are given applicable to this case. The case of a body which rolls without sliding, and the initial motion of a solid body, are also specially con- sidered. Chap. VI. treats of the distinction, peculiar to the present subject, between necessary and possible equili- brium. It is shown, by a consideration of the difference between statical and dynamical friction, that there are certain positions of a system in which equilibrium must exist, and others in which it may exist, but does not necessarily exist ; and a method is given by which these two kinds of equilibrium may be distinguished from each other. This, although a question of equili- brium, and therefore appearing to belong properly to Statics, could not have been considered at an earlier part of the inquiry, inasmuch as, like the problem of stable and unstable equilibrium, it is solved by principles essentially dynamical. In Chap. VII. the Author has considered the prin- ciples on which we may attempt to remove the ambiguity so constantly occurring in problems in which friction is one of the acting forces. He has shown that this can PREFACE. IX be done only by rejecting the abstractions of Rational Mechanics, and considering the conditions of the pro- blem as they really exist in nature. The mathematical difficulties which obstruct such an investigation are very great, but the Author has been able to give one example in which the solution can be effected, and which will sufficiently illustrate the principle of the method. Chap. VIII. contains a number of miscellaneous problems, three of which have been discussed at some length, namely, the problem of the top, the problem of friction wheels, and the problem of locomotives. Sub- joined is a collection of questions for exercise, suited to the younger students. Walton's admirable volume rendered it unnecessary to multiply these to any great extent. Some of the principles contained in the present treatise are believed to be here enunciated for the first time, and the subject is, even in theory, very far from being exhausted. But if the Author shall have succeeded in so far attracting the attention of mathematicians to the properties of this remarkable force, as to secure for the theory a more ample discussion than has as yet been given to it, he will think that his labour has been well bestowed. The Author is deeply indebted to the Rev. Richard Townsend, University Professor of Natural Philosophy, riot only for his kindness in correcting the press, but also for valuable suggestions made during the progress of the work. He also desires to return his best thanks to the Board of Trinity College, for their liberality in aiding to defray the expenses of printing. TRINITY COLLEGE, DUBLIN, Feb., 1872. TABLE OF CONTENTS, CHAPTER I. DEFINITIONS AND PBINCIPLES. Sect. Page. 1. Moving and resisting forces, i 2. Second class dependent on the forces by which they are opposed, . . 2 3. Geometrical forces, how defined, 3 4. Geometrical forces dependent upon the abstractions of Rational Mechanics. Real nature of these forces, - . . . 4 5. Geometrical forces indeterminate, . , 6 6. Friction, its nature and species, 7 7 . Laws of friction, ........... 7 8. Cone of resistance, angle of friction, 8 9. Coefficient of friction ; statical and dynamical friction, .... 8 10. Important difference between statical and dynamical friction, ... 9 CHAPTER II. EQUILIBRIUM WITH FRICTION. I. General Principles of Equilibrium. 1. Positions of equilibrium not absolutely determinate, . . . . .10 2. Example heavy particle on .rough surface, 1 1 3. Principle which governs the direction of the forces of friction, . - . . n 4. Example two material particles connected by rigid rod, . . . .13 5. Principles deduced from this example, 15 6. Example best angle for draught, 15 II. Equilibrium of a System of Material Particles. 7. System of material particles situated on a rough surface; equations of equili- brium ; number of indeterminate quantities in solution, . . . .16 8. Extreme positions of equilibrium, . . . . . . . .18 9. Source of indeterminateness in a mechanical problem, - . . .19 10. System of material particles ; further conditions of equilibrium, ... 20 Xll CONTENTS. Sect. Page. 11. Case of two particles, . .' .23 12. Geometrical interpretation of conditions, 24 13. Example two heavy particles resting on rough inclined planes, and con- nected by a string, 25 III. Equilibrium of a System of Material Particles each of which is constrained to remain upon a Rough Curve. 14. System of material particles situated on a rough curve, .... 27 IV. Equilibrium of a Solid Body supported by one or more Rough Surfaces. 15. Solid body supported by two rough surfaces, and acted on only by gravity, . 28 16. The same, acted on by forces of any kind, 29 17. Example. Circular cylinder resting upon a rough inclined plane ; sustaining cord free, and supporting a weight, 32 18. The same; sustaining cord passing over a pulley, 34 19. Bodies in contact throughout a finite area, . . . . . '35 20. Conclusion derived from the consideration of incipient motion, . . 35 21. Example Heavy body resting upon a rough inclined plane, ... 36 22. General case in which the forces acting on a solid body have a unique result- ant, 37 23. Case of a cylinder resting within another of the same radius, ... 38 24. Cylinder resting upon two inclined planes, 39 V. Equilibrium of several Bodies. 25. System of three heavy bodies, of which two rest upon a horizontal plane, . 40 26. Pyramid composed of equal spheres, 42 27. Solid bodies connected by geometrical equations ; determination of geometri- cal forces, 45 28. The same ; determination of the forces of reaction, . . . . . 47 29. Example Two cylinders connected by a string, resting upon an inclined plane, 49 30. Example. Carriage kept at rest on an inclined plane by locking two of the wheels, . .".*-. . . . . . . ' . ~". 51 VI. Equilibrium of a Flexible String upon a Rough Surface. 31. General theory. Examples, . . 52 CONTENTS. Xlll CHAPTER III. EXTREME POSITIONS OF EQUILIBRIUM. Sect. Page. 1. Extreme positions defined, . . . . . . . . .58 2. Characteristics of extreme positions of equilibrium, 59 3. Examples. Beams connected by a hinge ; beam resting on or against a rough cylinder, . . . . . . . . . . .61 4*. System of material particles resting upon rough surfaces ; conditions of ex- treme position ; analytical lemmas, 64 5*. Consideration of a particular case, 73 6*. Geometrical signification of the differential coefficients, .... 75 7*. Case of three particles, . . . . . . . . . -77 8*. Extreme positions of a solid body, . . . . . . . .81 9*. Example. Solid body resting upon three surfaces, 82 10*. Conditions obtained necessary, but not sufficient, 89 CHAPTER IV. MOVEMENT OF A PARTICLE, OR SYSTEM OF PARTICLES. I. Movement of a Single Particle upon a Fixed Rough Surface. ' 1 . Fundamental equations, 91 2. Equations obtained by changing the independent variable, . . .91 3. Example Heavy particle on an inclined plane, 94 II Movement oj a System of Particles under the Influence of Friction. 4. Fundamental equations. Incompleteness of solution, .... 98 5. Solution incomplete only as regards quiescent particles, . . . .100 6. Mathematical conclusions deduced from this discussion, . . . . 101 7. Physical conclusions, 101 8. Mode of determining at any given instant the moving and the quiescent particles, . ....'. 102 III. Initial Movement of a System of Particles. 9. Possible indeterminateness of thia motion, 103 10. Case where the supporting surfaces are smooth, 104 11. Case where the supporting surfaces are rough, 105 12. Values of the geometrical forces, ........ 105 13. Initial movements dynamically impossible, 108 14. Indeterminateness still remaining, . . . . . . .109 15. Example Two particles resting on a rough inclined plane, and connected by a string passing through a small fixed ring, . . . . .109 I\ r . Motion of a Particle upon a Moving Surface. 1 6. General theory. Examples, . . . 115 Xiv CONTENTS. CHAPTER Y. MOTION OF A SOLID BODY. I Motion of a Solid Body on a Rough Plane. Sect. Page. 1 . Fundamental equations. Two kinds of motion, 121 2. Mode of determining which of these kinds of motion really takes place, . 1 23 3. Example Motion of a heavy sphere on a horizontal plane, . . . 124 4. Case of pure rotation, . . 127 5. Equation of motion round instantaneous axis, . . ',' ' . , : 128 6. Locus of axis for which this equation is true, . . . . . 129 7. Case of initial motion, . . . . . . . . . 131 8. Example Motion of cylinder on inclined plane, . . . . 132 9. Motion of pure rolling Theorem respecting accelerating force at any point, 134 10. Determination of resistance of plane at point of contact, .... 135 11. Example Rolling sphere, . . , . . . 138 II. Initial Motion of a Solid Body resting upon one or more Fixed Rough Surfaces. 12. Determination of the resistance of the supporting surfaces Case in which there is slipping at the point of contact, . . . .,_,-. 139 13. Case of pure rolling, . . . , , . . . . 142 14. Examples Initial motion of cylinder on inclined plane, . . . 143 15. Case in which the plane is vertical Apparent paradox, .... 144 1 6. Example Particle attached to one end of a rigid rod, the other end resting against a wall, . . . . . . .... . 145 17. Explanation of seeming paradox, . . . ... . .' 146 1 8. Example Cylinders resting on inclined plane, and connected by a cord, . 147 CHAPTER VI. NECESSARY AND POSSIBLE EQUILIBRIUM. 1. Definition of necessary and of possible equilibrium, . . . . 153 2. Statical and dynamical friction, . 153 3. Restriction upon directions of forces of dynamical friction, . . . 154 4. Example Three particles connected by rigid rods, . . . . 155 5. Example Rod laid on two rough rods, . . . " . J . . 156 6. Direction of force of statical friction, . . . . __ . . ; 157 7. Force of friction takes the best direction for equilibrium, . . . 158 8. Criterion of possible equilibrium, 159 9. 10. Criterion of necessary equilibrium, ...... 159, 160 CONTENTS. XV Sect. Page. 11. Example Heavy particle connected by a rigid rod with a fixed point, and resting against a wall, 162 12. Case of a solid body, 166 13. Example Rectangular drawer, 168 14. Example T square resting on door, . 170 CHAPTER VII. DETERMINATION OF THE ACTUAL VALUE OF THE ACTING FORCE OF FRICTION. 1. Source of indeterminateness in questions of equilibrium Real nature of the phenomenon, 1-74 2. Example Heavy body resting upon horizontal plane, . . . . 175 3. Example Beam resting on two rough surfaces, 175 4. Example Heavy body resting on an inclined plane, and attached by a string to a fixed point, . 176 CHAPTER VIII. MISCELLANEOUS PROBLEMS. I. Problem of the Top, 1. Statement of problem Determination of forces, i8r 2. Differential equations, . . . 183 3. Integral of these equations, 184 4. Axis rapidly becomes vertical, x . . 185 5. Transformation of differential equations, 187 II. Friction Wheels and Locomotives, 6. Friction wheels Three periods of motion, 188 7. Locomotive with a single pair of driving wheels, . ' . . . . 198 8-13. Locomotive with linked driving wheels, . . . . . . 201 III. Questions for Exercise, . . . . 212 NOTES, . .218 % The articles in Chap. III. marked * may be omitted by the junior Students. THEORY OF FRICTION, CHAPTER I. DEFINITIONS AND PRINCIPLES. i. THE forces which we find in nature appear to belong to two different classes. The former of these, comprising, among others, gravity, the elasticity of gas, and muscular force, are capable of producing sensible motion, and may therefore be termed moving forces. The second class, comprising the resistance of fixed obstacles, the resistance which fluid media oppose to the pas- sage of bodies through them, and friction, are not capable of producing sensible motion, and show their effects only by the apparent destruction of the motion which has been caused by forces of the former class. These may be termed resisting forces. Modern research has indeed made known to us that the de- struction is only apparent. That which we term the destruction of motion by friction or resistance is in reality but the conver- sion of the sensible motion of the moving body into the vibration of the resisting obstacle, or into that other species of motion which we denominate heat. Still, however, the distinction be- tween these two species offeree is real. Resisting forces are incapable of producing sensible motion, and if they do not ac- tually destroy it, they do convert it into a species of motion, which is, to sight, insensible. It must be remembered, however, that the application of the term "moving forces" to those of the first class does not imply that these forces always act in producing rather than in destroy- B 2 DEFINITIONS AND PRINCIPLES. ing motion. Thus, when a ball is thrown vertically upwards, the velocity with which it leaves the hand is gradually destroyed by the combined forces of gravity, and the resistance of the air. Both forces are then in this case resisting or destroying forces. But there is an essential difference between the principles upon which these forces act in destroying motion. Gravity destroys the upward motion of the ball, because it has a tendency to generate motion in the opposite direction a tendency which will, after a certain time, produce a visible effect in causing the ball to descend. But the resistance of the air has no such tendency. It is simply a destroying force, diminishing the up- ward motion of the ball while ascending, and diminishing its downward motion after it has begun to descend. 2. There is another important distinction between these two classes of force. Forces of the first class are independent, both in direction and in magnitude, of the forces by which they may be opposed. Thus, for example, the weight of a body is a per- fectly definite force, not in any wise dependent upon any other force which may be used to overcome it. If the bpdy be at rest, the force which its weight equilibrates must have a certain definite direction and magnitude. If it be in motion, the quan- tity of movement which is produced or destroyed by gravity is wholly independent of the other forces which may act upon it. But it is otherwise with the forces which we have denomi- nated " resisting forces." These are not independent of the opposing forces. The forces which they are severally capable of equilibrating are not definite, but are capable of being varied, sometimes within very wide limits, without disturbing the equilibrium of the system. In the case of the resistance of a medium, the quantity of movement destroyed by this force does depend upon the force by which the opposite movement is pro- duced. In the case of a fixed obstacle, the resistance is always equal and opposite to the force by which it is sought to over- come it. And, as we shall presently see, the force of friction depends not only upon the nature of the surfaces which are in contact, but also upon the other forces which act upon the system. The distinction here indicated may be expressed by saying DEFINITIONS AND PRINCIPLES. 3 that forces of the first class are in their nature determinate, every such force being, for any given position of the system, capable of assuming but one value. Forces of the second class, on the other hand, are in their nature indeterminate , every such force being for a given position of the system capable of assuming an infinite number of values. We shall see further on that these forces are in certain cases capable of being determined by the equations of equilibrium or motion, and in other cases wholly indeterminate, unless their values be specially given. 3. The resistance of a fixed obstacle belongs to a class offerees which we may call, using the abstractions of Rational Mechanics, " geometrical forces." When a material particle belongs to a system, that is to say, when it is connected by certain geometri- cal relations with a number of other particles, we know that neither its motion, if it move, nor the conditions of its equili- brium, if it be at rest, are the same as they would be if the particle were independent of all others. The effect of the forces which act upon it is modified by the connexions which exist between it and the other particles of the system to which it be- longs. Now, this modification may be always conceived to be produced by a force. So far, therefore, as questions of either equilibrium or motion are concerned, the geometrical relations may, without in any way altering the mechanical phenomena, be replaced by a system of forces, provided that these forces be suitably determined. These are what we have termed the "geometrical forces," which may be defined to be, forces which are mechanically equivalent to the geometrical relations of the system. The methods of ordinary mechanics furnish, as is well known, a fixed rule for determining the line of direction of each of these forces, these lines of direction depending only upon the nature of the geometrical relations which the forces replace. But the magnitudes of these forces are not determinate in the same way, depending as they do not only on the nature of the geometrical relations, but also upon the forces which act upon the different parts of the system. Thus, for example, let it be supposed that the system con- sists of a single particle, and that the geometrical condition tq which it is subjected is, that it shall always be found in a given 4 DEFINITIONS AND PRINCIPLES. plane. Then we know that the force which is equivalent to this geometrical condition must be in line of direction perpendicular to the given plane, and in magnitude equal to the force which acts upon the particle resolved along that perpendicular. The actual direction of the geometrical or replacing force is immediately opposed to this component. The principle contained in this example is general. The line of direction of the force which is equivalent to a geometrical con- dition depends solely upon that condition ; the choice between the two directions contained in that line, as well as the actual magnitude of the force, depends also upon the other forces which act upon the particle. 4. Strictly speaking, the forces which we have termed " geo- metrical forces" have no real existence in nature. Their existence depends upon the abstractions of Rational Mechanics, which are never mathematically true. No system of really existing bodies will perfectly fulfil geometrical conditions. Thus, when we speak of a material particle as being in all its positions situated in a certain plane, it would be necessary to the perfect fulfilment of this condition that the supporting surface should be geometri- cally plane, and absolutely unyielding, neither of which condi- tions is fulfilled by any substance which we find in nature. The forces which are really brought into play by the connexions of a system depend upon the elasticity of the bars and strings and supporting surfaces, by which these connexions are established, and this elasticity is developed, not by the fulfilment, but by the violation of certain geometrical conditions. Thus, for example, if the system be composed of two par- ticles, connected by the geometrical condition that the distance between them shall be invariable, the methods of Rational Me- chanics require us to replace, at each particle, this condition by a force whose line of direction passes through the two particles. But if we attempt to realize the condition by a connecting bar, the force which we thus introduce into the system, namely, the force of elasticity, is not brought into play until this condition is violated by the lengthening or shortening of the bar. The validity of the conclusions of Rational Mechanics is not, however, in general impaired by this distinction, and it is pos- DEFINITIONS AND PRINCIPLES. 5 sible to give a definition of the force developed by any of the connexions of a system, which is applicable either to the prob- lem considered in the abstract, or to the problem as it really exists in nature. In either case, the force in question is one which resists the violation of the geometrical condition. Whe- ther, therefore, the condition be perfectly fulfilled or not, this force will enter similarly into the statical or dynamical equa- tions. Thus, for example, whether we regard the tension of a string as a force resulting^from its inextensibility, or as an elastic force developed by an actual extension of the string, it is in either case a force which resists extension, and will enter simi- larly into the equations of equilibrium or motion of any system of which this string forms a part. It is not of course true that the complete solution of such a problem is the same, whether the abstractions of Rational Mecha- nics be or be not assumed to be mathematically exact. The geo- metrical forces and the forces of elasticity do indeed enter simi- larly into the equations of equilibrium or motion, which have therefore the same form in either case. Hence it is plain that the equations which may be obtained by eliminating the geometrical forces in the one case, and the forces of elasticity in the other, are absolutely identical for both cases. But the remaining equa- tions are different. These are, in the abstract problem, the geo- metrical equations themselves, which are here supposed to be true. In the physical problem, where these equations are not rigorously true, the remaining equations of the problem are derived from the supposition that the forces of elasticity, which here take the place of the geometrical forces, are given functions of the co- ordinates. But although the solutions of the rational and physical prob- lems respectively are not mathematically identical, it is easy to see that the solution of the rational problem is very approxi- mately true for the physical problem. For the equations by which the rational problem is solved may, as we have seen, be divided into three groups, i. Equations obtained by eliminat- ing the geometrical forces between the equations of equilibrium or motion. 2. The geometrical equations themselves. ,3. The equations which give the actual values of the geometrical forces. 6 DEFINITIONS AND PRINCIPLES. Equations of the first group are strictly true in the physical as well as in the rational problem. Equations of the second group are, in all physical problems to which it is customary to apply the methods of Rational Mechanics, very approximately true. Hence the positions of the particles and the relations between the acting forces necessary for equilibrium, depending altogether upon equations of the first two groups, are very approximately the same for the physical and the rational problems ; and the same is true for the movements othe particles, if the problem be dynamical. The remaining group of equations which in the rational problem determine the magnitudes of the geometrical forces, will, in the physical problem, determine the small exten- sions or contractions of the bars, strings, &c., which are necessary for the development of the equivalent elastic forces. The definition which the principles of Rational Mechanics give of the geometrical forces of a system, although depending upon an unreal abstraction, is thus in general very approxi- mately coincident in its results with the definition which would be derived from actual experiment. We shall show further on that this principle is liable to certain exceptions. 5. The distinction here indicated between the external forces which act upon a system and those which, following the abstrac- tions of Rational Mechanics, we have called the " geometrical" forces, is, as we shall hereafter find, of the greatest importance. For the present, it will be enough to observe, that forces of the first kind are properly reckoned among the data of a mechanical problem. That is to say, the intensities and directions of these forces are given functions of the co-ordinates of the several par- ticles. But it is otherwise with the geometrical forces. These are not, like the former, given functions of the co-ordinates. Their lines of direction are indeed deducible from these quantities by a known rule ; but their intensities are themselves, unless their values be specially given, among the unknown quantities of the problem, to be determined by the equations of equilibrium or motion, if this be possible. We shall hereafter find cases in which, although the positions of the particles and the external forces acting upon them are fully known, the geometrical forces remain still indeterminate. DEFINITIONS AND PRINCIPLES. 7 6. The other species of resisting force, which we propose to consider now, is Friction, or the resistance which we experience when we attempt to move one body along the surface of another against which it is pressed. Of this force there are two kinds, the difference between which depends upon the nature of the re- lative motion which we produce, or endeavour to produce. (i.) If we attempt to make the one body slip upon the other ; in other words, if we attempt to give to the point of the one body which is in contact with the second a movement along the surface of the second, we find that the attempt is resisted by a force which is called " friction of the first kind." (2.) If we attempt to make the one body roll upon the other, that is to say, if the movement which we give, or attempt to give it, be a movement of rotation round an axis passing through the point of contact, and situated in the com- mon tangent plane, this movement is resisted by a force which is called " friction of the second kind." We shall for the present confine our attention to the first of these two species. 7. The laws of friction of the first kind, as ascertained by ex- periment, are as follows : (i.) The line of direction of this force is always situated in the common tangent plane to the two bodies which are in contact with each other. (2.) If the particle of the body or system which is in con- tact with the other be in motion along its surface, the force of friction is always directly opposed to this motion. (3.) If the particle of the body or system, which is in con- tact with the other, be at rest, the force of friction is directly opposed to the tangential component of the resultant of all the other forces, external and geometrical, which act upon the particle. (4.) If the particle be in motion, as in (2), and if Pbe the normal pressure existing between the two bodies, and F the force of friction, then where ju is a numerical coefficient depending solely upon the nature of the surfaces in contact. 8 DEFINITIONS AND PRINCIPLES. (5.) If the particle be at rest, as in (3). the force of friction may have any value from o to juP. The actual value which it has in any given case, and which may be termed the effective force of friction, is necessarily equal to the tan- gential component spoken of above, which component must therefore not exceed /j.P. 8. The action which a rough surface exercises on any particle or body which is in contact with it may be regarded as being compounded of two forces at right angles to one another, namely, (i). The reaction of the surface, as that term is commonly used. This force is directed in the normal to the surface, and is equal and opposite to the force which we have above denominated P. (2). The force of friction, F. This force must be di- rected in a line touching the surface, and may have any value from o to juP. Let it be supposed that these two forces are compounded into one, which will then represent in magnitude and direction the complete action of the rough surface. Then it is evident that the magnitude of this force will be and that its direction will make with the normal to the surface an angle which may have any value from o to tan " 1 ^u. This law may be geometrically expressed as follows : Conceive a cone of revolution to be described having its ver- tex at the given point, and of which the normal to the surface is the axis. Let the semi-angle of this cone be tan - l p. Then it is plain from the laws above stated that the complete resistance of a rough surface is limited by one only condition, namely, that its line of direction cannot lie outside the cone so described. This cone may be termed the cone of resistance. The semi-angle of the cone, being the angle whose tangent is equal to the nu- merical coefficient /*, is usually termed the angle of friction. We shall in general denote it by the symbol e. 9. The numerical coefficient /*, which is called the coefficient of friction, depends, as has been said, upon the nature of the surfaces which are in contact. Experience has shown also that DEFINITIONS AND PRINCIPLES. 9 it has different values for the cases of rest and motion. The force with which a rough surface resists the commencement of motion is, cceteris paribus, greater than that with which it resists the continuance of motion once commenced. Hence, inasmuch as the force is in both cases represented by /*P, it is evident that in must be greater when the particle is at rest than when it is in motion. This is expressed by saying that the coefficient of statical friction is greater than the coefficient of dynamical friction. 10. There is another and a more important difference be- tween statical and dynamical friction. When a material particle moves upon a rough surface, the force of friction is, as we have seen, directly opposed to the motion of the particle. If, there- fore, this motion be governed by any geometrical condition, which determines its line of direction on the supporting surface, the line of direction of the force of dynamical friction is also de- termined. Thus, for example, if the supporting surface be a plane, and if the particle be attached by a rigid, inextensible rod to a fixed point, the motion of the particle will necessarily be in a determinate circle. It is plain, therefore, that the line of direction of the frictional force which acts upon the particle, while in motion, is a tangent to this circle. More generally, if a system of particles, whose positions are connected by certain geometrical relations, and each of which rests upon a rough surface, be in motion, we are not at liberty to assume, for the direction of the force of friction at each point, any line in the tangent plane at that point. For it is plain, from what has been said, that a system of possible lines of direction for the forces of friction must be coincident with a system of possible lines of motion of the several particles. Whatever limi- tations, therefore, are imposed upon these latter directions by the geometrical conditions of the system, the same limitations are necessarily imposed upon the directions of the forces of dy- namical friction. But the same limitation does not hold with regard to the forces of statical friction, whose directions are not necessarily coincident with any system of displacements governed by the geometrical conditions. For the present, itwill be sufficient to notice this important dis- tinction, which we shall discuss more fully in a subsequent chapter. CHAPTER II. EQUILIBRIUM WITH FRICTION. I. General Principles of Equilibrium. i. WHEN the forces which act upon the particles of which a system is composed are definite functions of the co-ordinates of these particles, it is known that such a system will have in general a finite number of positions of equilibrium, separated from each other by finite intervals. For, let n denote the num- ber of the particles of which the system is composed, and in the number of equations of condition : then we know that the num- ber of equations of equilibrium will be 371, which, with the ra equations of condition will give 371 + 771 equations in all. But this is also the number of the unknown quantities, namely, 371 co-ordinates of the particles, and m geometrical forces, of which the intensities only are unknown. Hence the proposition is evident. If, however, any one or more of the forces be not determi- nate functions of the co-ordinates, the number of the unknown quantities will exceed the number of equations, and there will be in general an infinite number of positions satisfying the con- ditions of equilibrium, disposed in one or more groups, in each of which these positions succeed one another continuously. If, for example, any of the forces which we have termed geometrical, that is to say, the forces which are mechanically equivalent to the geometrical relations of the system, were indeterminate in direction as well as in magnitude, it is plain that the number of the unknown quantities would exceed the number of the equa- tions, and therefore that the problem would be indeterminate. It is true, indeed, that no force which is, in this sense of the word, a geometrical force, is ever indeterminate in its direction, which can always be deduced by a well-known rule from the equation which it replaces. But the resistance of a rough sur- EQUILIBRIUM WITH FRICTION. t 1 1 face, which is intimately connected with one of the geometrical conditions of the system, is indeterminate both in magnitude and direction. The former of these is wholly indeterminate, and the second is restricted only by a certain limitation, namely, that it cannot lie outside a certain cone. Wthin this cone, the direc- tion of the resistance is, like its magnitude, wholly indeterminate. We may therefore expect, as we shall, in fact, find it to be true, that when the system includes a rough surface, the positions of equilibrium will not be absolutely determinate, but will succeed each other continuously in one or more groups. 2. Thus, for exanple, if a material particle, acted on solely by. gravity, be placed upon a fixed smooth surface, we know that it will only remain at rest if the position chosen for it be such that the tangent plane at this point is parallel to the horizon ; for only at such a point can the force of gravity, which is neces- sarily in the vertical, be neutralized by the reaction of the sur- face, which is necessarily in the normal. But it is otherwise if the surface be rough. Here the reaction of the surface is not necessarily in the normal, but may make with that line any angle not exceeding a certain fixed limit. If then, round each of the points which are characterized by a horizontal tangent plane, we describe a curve along which the tangent plane makes with the horizon an angle equal to the limiting angle, each of these curves will in general include a space throughout which the tan- gent plane is inclined to the horizon at an angle less than the given limiting value. Throughout the whole of this space it is possible, without violating any of the conditions by which the force of friction is limited, to assign to that force such a magni- tude and direction as shall neutralize the force of gravity, and thus allow the particle to remain at rest. Throughout the whole of this space, therefore, equilibrium is possible. 3. We have seen that the force of friction is in its own nature indeterminate, both in direction and in magnitude. But as abso- lute indeterminateness cannot really exist,* we must seek for some principle which willenable us to discover the actual direction * It is necessary here to distinguish carefully between the different senses in which the word "determinate" may be used. i. A force may be dependent solely ^ 1 2 EQUILIBRIUM WITH FRICTION. and magnitude which belong to this force in any given problem. This principle, which we obtain from experience, and which has been already enunciated in other terms, is as follows : The force of friction which acts at any point of a system will, if possible, assume such a magnitude and direction as will keep that point at rest. The same principle is true for the normal resistance of the surface, and therefore for the force compounded of these two, which we have denominated " the resistance of a rough surface." We have, therefore, for the determination of this force, the fol- lowing rule, which may also be called a principle of equilibrium. Let all the forces, external and geometrical (excluding the action of the supporting surface), which act upon the given point be com- pounded into one, R. Then if the line of direction of this force lie within or upon the cone of friction, the point in question is in equilibrium, and therefore the resistance of the rough surface is equal and opposite to R. Thus, for example, in the case of the gravitating particle which we have just considered, the region of possible equilibrium is shown by this principle to be also a region of actual equili- brium, the resistance of the rough surface being, throughout this region, equal and opposite to the force of gravity. If it be re- quired to determine the force of friction separately, this is readily done by resolving the complete resistance of the surface perpen- dicular and parallel to the tangent plane. The latter component is the force of friction. More generally, whatever be the force acting upon the particle, it will be in equilibrium, if the force make with the normal an angle not exceeding the angle of friction. Comparing this case with that of a particle situated on a smooth surface, we see that, whereas in the latter case two equations are necessary upon the positions of the several particles of the system, so that for any given posi- tion of the system it can have but one determinate direction and one determinate intensity. Such a force is. for example, the attraction of a centre. 2. A force may require for its determination not only a knowledge of the positions of the several parts of the system, but also of the other forces which act upon them. Forces belong- ing to the second class may be called indeterminate in their own nature ; but no force is absolutely indeterminate. Such a supposition would plainly be inconsistent with the universality of law. Vid. inf. 9. EQUILIBRIUM WITH FRICTION. 13 for the equilibrium of the particle, there is required in the for- mer case only a single condition, not an equation. The physical reason of this difference is, that whereas the resistance of a smooth surface must necessarily act in one determinate line, and can therefore only equilibrate an external force acting in the same line, the resistance of a rough surface may, within certain limits, assume any direction, and will, within those limits, rteces- sarily assume that direction which is requisite for equilibrium. The only condition of equilibrium is, therefore, that the applied force should not be without the prescribed limits. 4. If all the forces acting upon the particle (excluding the re- sistance of the rough surface) were of the kind which we have called external forces, their values would be necessarily de- terminate functions of the position of the particle.- In this case, the resultant of these forces being also determinate, the prin- ciple stated above would enable us to assign the direction and magnitude of the resistance of the surface. But if any of these forces be geometrical, only their directions are given by the geometrical conditions which they replace. There is nothing, so long as we adhere to the abstractions of Rational Mechanics, to determine their intensities, which are therefore themselves among the unknown quantities of the problem. In such a case, therefore, we may naturally expect, as we shall in fact find it to be the case, that the solution furnished by these abstractions will not always be complete, some one or more of the unknown quantities remaining indeterminate. Thus, for example, let the system be composed of two ma- terial particles, P, Q (Fig. i), acted on solely by gravity, con- nected by a rigid weightless rod, and each of which rests upon a rough plane. Conceive a vertical plane to be drawn through the connecting rod. Let the plane of the paper represent this plane, and let AP, A Q be the sections of the supporting planes. Let G be the centre of gravity of the two particles, and G V a vertical. Let the cones of resistance be constructed at P, Q, and let Pp, Pg, Qp, Qq, be the sections of these cones made by the plane of the paper, which we shall suppose to cut them both. Then if, as in the figure, 6rFlie between the intersections, p, q of Pp, Qp, and P^, Qq respectively, it is always possible to find in the vertical EQUILIBRIUM WITH FRICTION. a point v such that vP, vQ, shall lie within the respective cones of resistance. In fact, in Fig. i any point from r to r will satisfy the condition. Hence, if vg be taken to represent the sum of the weights of P and Q, and if gs, gs f be drawn parallel to Pv, Qv, respectively, the system will be in equilibrium if the resistances at P and Q be represented in magnitude and direction by vs' y vs. respectively. But from what has been said, Chapter I., it is plain that these lines are possible directions for the Fig. i. forces of resistance, and as the magnitudes of these forces are unlimited, the proposed representation is possible. If, there- fore, the vertical through the centre of gravity of P and Q lie between p and q the position is one of possible equilibrium, and the resistances at P and Q are represented by vs, vs. Since, moreover, the point v has been assumed, within certain limits, arbitrarily, the resistances at P, Q, are not absolutely determinate. If the vertical GVp&ss through either of the points p or ^, equilibrium is still possible, but the resistances are no longer indeterminate. Thus, let 6rFpass through p (Fig. 2). Then it is plain that Pp, Qp, are the only possible lines of direction of the forces of resistance. For if we connect the points P, Q, with any point on GV other than p, one, at least, of the connecting lines will lie outside the cone of resistance. If then we repre- sent by pg the sum of the weights, and draw the parallels gs, gs', the resistances at P and Q will be represented in magnitude and direction by the determinate lines ps 9 ps'. If GrFlie beyond^? or q, it is impossible to connect P and Q with any point in the vertical, without causing one, at least, of the connecting lines to fall outside the cone of resistance. In this case the position is not one of equilibrium. EQUILIBRIUM WITH FRICTION. 1 5 If either pair of the lines Pp, Pq, or Qp, Qq, coincide ; in other words, if the vertical plane through the connecting rod touch either of the cones of resistance P, it is easy to see that equilibrium is possible, if the vertical 0F(Fig. 3) lie between p and q. In this case the only possible directions of resistance are Pv, Qv, and these forces are perfectly determinate. If the vertical plane through the connecting rod lie wholly outside either cone of resistance, equilibrium is manifestly impossible. Fio . 5. We learn from this example i. That a system in which friction is one of the acting forces may have an infinite number of positions of equilibrium lying within certain limits. 2. That the resistance actually developed at each point of the system cannot in general, as in the case of smooth surfaces, be deter- mined simply by the equations of equilibrium, even if the posi- tion of the system be given. 3. That in certain cases this indeterminateness disappears, the equations of equilibrium enabling us to assign the actual values to the resistances. 6. As another example, let it be proposed to determine the best angle for draught on a rough inclined plane. Let W be the weight of the body, P the power applied to the rope, 9 the angle of inclination of the power to the inclined plane, and a the inclination of the plane. Then, if the body be just on the point of being dragged up, the equations of equilibrium are F, R = TFcos a - P sin 0, F = R tan where R is the pressure on the plane and F the force of friction. Eliminating F and 7?, we find P cos (0 - c) = JFsin (a + e) (i) It is evident from this equation that P is a minimum when = s. Hence the best direction of draught makes with the inclined plane an angle equal to the angle of friction. Ifa + e<9o, h there will always be a gain of power in dragging the body up 1 6 EQUILIBRIUM WITH FRICTION. the plane, rather than in lifting it vertically. If a + f = 90, it is easy to see that the direction of the force will be vertical. If a + e > 90, it might seem that P is still less than W, and that therefore there would be still a gain of power in dragging the weight up the plane. But if in the value of R we make 6 = e and P = JFsin (a -f e), we shall have R = W cos (a + c) cos e, (2) which is negative if a + e > 90. It would be, therefore, impos- sible to drag the weight up the plane by a force applied in this direction. In fact, it is easy to see that a force so applied will give with the weight a resultant down^ not up, the plane. Hence we infer that if a + c < 90, it is better to drag the weight, and that if a + f > 90, it is better to raise it vertically. For, in order that the force and the weight may give a resultant up the plane, it is easily seen that we must have a + 9 < 90. Hence, if a -f e > 90, we must have < c. Also, since a + < 90, we must have a + c - 90 < e - ; whence sin (a 4- c) > cos (9 - E), and therefore from equation (i) P > TF, or there is a loss of power in dragging the body up the plane. We now proceed to consider generally how many indeter- minate quantities remain in any case in which the positions of the particles are given. II. Equilibrium of a System of Material Particles. PROP. I. 7. A number, n, of material particles, each of which is situated on a rough surface, are connected by certain equations of condition, and are acted on by given forces. Given the posi- tion of the system, to determine the conditions necessary to its equilibrium, and the force of resistance actually developed at each point, if it be determinate. Let #1 y^ z ly # 2 yz, z& &c., be the co-ordinates of the several particles, and let L = o, M = o, &c,, EQUILIBRIUM WITH FRICTION. 1 7 be the equations of condition. Let Xi Y l Z be the components of the external force at the point x\ y^ z lt and X\ Y\ Z\ the com- ponents of the effective force of resistance at the same point. Then if the particle x\ y l z l be in equilibrium, we have v v , ^ dL dM 9 A! + .A ! + A -j + JU -j + &C., = O, dx l r dx l #*; dL dM YI+ FI + X-T-+/I -j +&c., = o, (3) yi o>y\ dL dM Zi + Z } + \-j- + u. + &c., = o. dzi azi Similar equations will hold for the remaining particles. Conversely, we know from the principle which has been enunciated above (Art. 3) that the forces of effective resistance will, if possible, take such directions and magnitudes as to keep the system in equilibrium. We have seen, further, that the only restriction upon the resistance of a rough surface is that its direction shall not make with the normal to the supporting sur- face an angle greater than the angle of friction. Hence, if a b t GI be the direction cosines of the normal to the supporting surface, and if we assume P 2 ( v , dL dM \ f .dL dM V &?"( A! + A -T- + u. -= + &c. + Yi 4 X -7- + u. + &C. V dx l * dx l / \ dy l * dyi ) ( .dL dM \ 2 + [Z l + A -5 + u -5 + &c. ) \ dz l n dz l / the required condition will be dM \ . /_. _ dL dM \ u+&c.} + bi[Y l +\ +u-j- + &c.) * dx l J l \ dy,, $ dy, ) l + \ -i-ju-j- + &c. J > R^ cos d, (4) where ei is the angle of friction. Similar conditions hold at the other points of the system. If these conditions, which are n in number, be satisfied, the system is in equilibrium. The components X\ Y\Z\, &c., of the seve- D 1 8 EQUILIBRIUM WITH FRICTION. ral forces of resistance are then completely determined by the equations of equilibrium (3) in terms of the quantities X, ju, &c., which are proportional to the intensities of the geometrical forces, and are in number equal to the equations of condition. There is nothing in the conditions of equilibrium to determine these quantities, nor even to establish any equations among them. They are restricted in their values by the conditions (4), but remain, so far as conditions of equilibrium are concerned, inde- terminate and independent. The forces of effective resistance at the several points of the system, which are, as we have seen, functions of X, JJL, &c., are, therefore, in general, indeterminate. The entire conclusion may be stated as follows : If a number, n, of material particles, each of which rests upon a rough surface, and is acted on by a given force, be con- nected by m equations of condition, and if the entire system be in equilibrium, in a given position, the values of the forces of effective resistance, and of the geometrical forces, uill in general contain m indeterminate quantities. It must be observed in applying this principle, that the equa- tions of the supporting surfaces are not reckoned among the equations of condition. 8. If any one or more of the particles be on the point of slipping, the number of the indeterminate quantities is reduced. If, for example, the condition of the system be such that the point #, ?/! Zi would slip if its supporting surface were ren- dered in any degree smoother, the friction developed at this point would have its greatest value, and consequently the line of effective resistance would be most oblique to the normal. The line of effective resistance, therefore, would lie upon, not within, the cone of resistance, and we should thus have an additional equation. The number of indeterminate quantities is therefore diminished by one for each particle which is in such a position, which may be termed an extreme position of equilibrium.* Thus, for example, in the problem discussed above (Art. 4), there is one geometrical condition, and consequently, in general, one indeterminate quantity in the solution. But we have seen * Vide Chap. III. EQUILIBRIUM WITH FRICTION. 19 that if the vertical plane through the connecting rod touch either of the cones of resistance, or if the vertical through the centre of gravity pass through either of the points p or q, the solution becomes determinate. This is in accordance with the general principle just enunciated. 9. Before proceeding further, we must say a few words in explanation of the indeterminateness which appears in the solu- tion of this and similar problems. When we find a mechanical problem to be indeterminate, the inference to be drawn is this, that the data are insufficient. In the present case we shall see that the conditions of the problem are not complete unless the values of the forces which we have termed " geometrical" be given. This will appear most readily by considering these forces as they really are,, namely, the limiting values of the forces of elasticity which are produced by the extension or contraction of the bars or laminae by which it is attempted to realize me- chanically the geometrical conditions.* If the amount of this extension or contraction be supposed to be indefinitely dimi- nished, the corresponding force of elasticity remaining finite, the limit thus attained is the force which we have termed " geo- metrical." Now, while the extension or contraction continues finite, it is plain that the problem will not necessarily be de- terminate, unless the elastic force, which is developed by the extension or contraction, be given. If this force be given, as well as the positions of the several particles, each particle may be considered as isolated from the others and acted on by a system of forces which are known. The equations of equilibrium will then be sufficient (and no more than sufficient) to determine the un- known quantities, namely, the resistance of the supporting sur- face in magnitude and direction. But if the elastic force be not given, the particle is acted on by a force of unknown intensity, and therefore the equations of equilibrium are insufficient to deter- mine all the unknown quantities. The same will necessarily hold when we pass to the limit, and thus substitute the geo- metrical force for the force of elasticity. Unless the intensity of this force be given, the data of the problem are insufficient. In * Chap. I., 4. 20 EQUILIBRIUM WITH FRICTION. fact, the intensity of this force might be varied more or less largely without disturbing the equilibrium. Thus, for example, if the system consist of a single particle which is subject to the geometrical condition that its distance from a fixed point shall be invariable ; this condition is mechani- cally realized by connecting the particle with the fixed point by a rigid rod. If this rod were perfectly inextensible, the condition would be perfectly realized. The geometrical or equivalent force would then be a force directed along the rod and absolutely undeter- mined in magnitude. But we know that there is in nature no such thing as an inextensible rod, and that the force which is really introduced into the system by such a connection is de- veloped by the actual extension or contraction of the rod. The true conception, therefore, of the force which we have termed geometrical is, that it is the limit to which the elastic force de- veloped by the extension or contraction of the rod tends, when the coefficient of elasticity is indefinitely increased. The data of the problem into which such a force enters are not then necessarily sufficient, unless we know the actual amount of the extension or contraction of the rod. This extension or con- traction is not necessarily determined by the conditions of equilibrium. For it might happen that the equilibrium would not be disturbed by replacing the connecting rod by x ano- ther in which the extension or contraction should be quite different. If, therefore, we have no other information than that afforded by conditions of equilibrium, we must in such a case look upon the force of elasticity as indeterminate. This con- clusion being independent of the actual value of the coefficient of elasticity, will necessarily hold also for the geometrical or replacing force, which is, as we have seen, the limit of the force of elasticity. This force cannot, therefore, in general be deter- mined by the equations of equilibrium. PROP. II. 10. Let a system of material particles be connected as in Prop. I., and let it be supposed that the equations (3) are satis- EQUILIBRIUM WITH FRICTION. 2 I fied. To determine what further conditions are necessary for the equilibrium of the system. We have seen (p. 17) that for each particle of which the system is composed we have the condition ^r -v dL dJIf n \ , ( T7- x U-U VUJZL n \ X l + \-+u~~+&C.) + bi { Y 1 -f \ + n - + & C . a#i fltafj / \ 2/i dy l J , dL dJf Zi + X -j -1- ju -r + &c. > Iii cos retaining the notation of Prop. I. Hence if we assume ( f v \ dL \ f dL 0! = { ! [ Xi + \ -r 4- &C. +61 ^ -f \ ^ + &C. ( V fai J \ fyi cos / . dL \ 2 ) i + ^ -j- + &c - dz i J ) a = (a 3 (A: 2 + &c.) + &c.) 2 - f(^ 2 + &c.) 2 + &c.) cos 2 2 ; 03 = &c. ; the required conditions will be each one of the quantities 1? 2 > &c., being a function of one or more of the coefficients X, ju, &c., which, as we have seen in Prop. I., remain completely undetermined by the equations of equilibrium (3). We have then to inquire whether it be pos- sible to assign to these coefficients such values as will satisfy these conditions. Let X, /u, v, &c., be a system of values satisfying the con- ditions 0! > O, . 02 > O, ..... O. Let it be supposed that X, ju, v, &c., are varied continuously until we arrive at a system of values which causes one of these func- tions to vanish. Let 0! be the vanishing function. Then the 22 EQUILIBRIUM WITH FRICTION. system of values of X, /*, v, &c., at which we have arrived, is such as to satisfy the conditions 01 = 02 > O> ..... 0n > O. Restricting now the values of A, ju, v, &c., to such as satisfy the condition fa = o, let these quantities be again continuously varied. Then, unless the relation established by fa = o be such as to make all the other functions 2 , 03> &c., essentially positive, we shall at last arrive at a system of values which will cause a second function, 2 , to vanish. This system satisfies the con- ditions 0! = O, 02 = 03 > O, ..... R > O. Restricting now the values of X, p, v, &c., to those which satisfy the conditions 0i = o, 2 = o, let them be varied as before. Then, unless the remaining functions are essentially positive, we shall, as before, arrive at a system which causes a third function, 03, to vanish. Continuing this process, we shall at length arrive at a system which causes a certain number of the functions to vanish, these functions satisfying the condition that every system which causes them to vanish renders the remaining functions positive or zero. Conversely, if the conditions (4) be fulfilled, it will always be possible to find a number, p, of the functions 0, &c., such that if the equations 0i = o, 02 = o, ..... P = o (5) be satisfied, the remaining functions P + 19 P + 3 . . . . n shall be essentially positive. If p = m, the foregoing equations are sufficient to determine the quantities A, /*, &c. In this case the remaining functions, P + 1? &c., will be constants depending upon the acting forces and the positions of the particles. The con- ditions of equilibrium expressed in terms of the given quantities of the problem will therefore be besides a number of conditions denoting that the equations (5) shall have at least one system of real roots. EQUILIBRIUM WITH FRICTION. 23 If p < m the quantities X, ju, &c., are not completely deter- mined, and the remaining functions, 0p + i, &c., will contain a certain number of these quantities. The conditions (5) will then be fulfilled independently of them. ii. To take a simple case of the general proposition, let it be supposed that the system is composed of two particles connected by a single geometrical equation. We shall have then C 2 , where dL dL dL\ fdL* dL* dL* ! + 0\ + G! - ( -H -= + - COS dtfi dy l dzj \d^ dy? d dL dL dL y, dzj d = (aA + k Y l 4- c^)* - (XS + Y, 2 + Zf) cos 2 l , with similar values for A 9 , B 2 , C- 2 . Then (i). If A l > o, A z > o, the conditions fa > o, ^ 2 > o can always be satisfied by taking X sufficiently large. (2). Let A! < o, A 2 < o. Then unless B^ - Aid > o, i? 2 2 -A z d>o> the conditions 0! >o, 2 > o, cannot be fulfilled by any value of X whatever. If ^ - AiC\> o, J5 2 2 - A Z C 2 > o, let the roots of the equations fa = o, 2 = o be Xi, X'i, and X 2 , X' 2 re- spectively. Then since 1? 2 > are evidently negative for X = oo, the conditions ^>i > o, 2 > o can only be satisfied by values lying between Xi and X'i, and also between X 2 and X' 2 . Hence if we suppose that X a < X'i and X 2 < X' 2 , the required conditions will be Xi o, A z < o. Then it is evident that the required value of X must, as before,lie between the roots X 2 X' 2 , and must not lie between the roots Xi X\ ; we must have therefore either X' 2 < Xi or X 2 > X'i. Similar conclusions hold for the case A l < o, A z > o. 24 EQUILIBRIUM WITH FRICTION. (4). If C\ > o, C z > o the conditions o 2 > o can always be satisfied by making A = o. 12. To interpret these several cases geometrically, let the normal to the supporting surface at either point #! yi ZH the direction of the applied force J?i, and the direction of the geometrical force be projected Fig. 4. upon a sphere by radii parallel to these lines. Let these radii meet the sphere in the points N, R, L respectively (Fig. 4). Then putting _ 1 ~ dxf dy? + d^ we have, as is easily seen, A l = ?? (cos *LN- cos 2 fl ), C t = ^(cos *RN- cos 2 fl ), #1 = FA (cos LNcos EN- cos LR cos 2 l ). The condition AI > o is therefore equivalent to LN< 1} denoting that at the point Xi y\ z l the line of direction of the geometrical force lies within the cone of resistance. The condition C\ > o is equivalent to ^^V< d, denoting that at the point x^ yi Zi the line of direction of the applied force lies within the cone of resistance. If, therefore, as in (i), AI > o A z > o, denoting that at each point the line of direction of the geometrical force lies within the cone of resistance, it is plain that the intensity of this force may always be taken sufficiently great to bring the lines of di- rection of the complete resultants within the cones of resistance. This is, as we know, sufficient for equilibrium. If, as in (4), C\ > o (7 2 > o, denoting that at each point the line of direction of the applied force lies within the cone of resist- ance, it is plain that the intensity of the geometrical force may be taken sufficiently small to bring the complete resultants within the'cones of resistance, and thus to secure equilibrium. In fact, in this case each particle would be in equilibrium if separated from the other. EQUILIBRIUM WITH FRICTION. To interpret the condition B l 9 -A^d>o 9 we have from the foregoing values of A^B^ d BS - A& = V^Rf cos 2 fl (cos *LN+ cos - 2 cos LN cos RNcos LR- sin 2 Z7iJcos 2 E l ). But if NP be perpendicular to LR, it is easily shown that cos 2 LN+ cos *RN- 2 cos LN cos RNcos LR = sin *LR cos 2 JVP. Hence we have B? -A&** VfRf cos 2 fl sin 2 Z# (cos *NP- cos 2 l ). The condition Bf -A 1 C l >o is, therefore, equivalent to NP < e^ But it is easily seen that this condition is necessary to equili- brium. For the arc NP measures the inclination of the normal to the plane containing the directions of the external force and the geometrical force, and therefore the minimum angle which a line situated in this plane can make with the normal. But the line of direction of the total force acting upon the particle, being the resultant of the external and geometrical forces, is neces- sarily situated in this plane. If, therefore, NP were greater than 1} the total acting force would necessarily make with the normal an angle greater than ci ; equilibrium would, therefore, be impossible. It is evident that if either A\ or C\ be positive, jE?! 2 - A\ Ci will also be positive. For if either LN or RN\)Q less than EI, NP must necessarily be less than s lm 13. As an example of this principle, we shall consider the case of two ma- terial particles, A, B (Fig. 5), resting upon two rough inclined planes, and connected by a string passing through a fixed smooth ring. Fig. 5. Let W, W be the weights of the particles, J'the tension of the string, and c, e' the angles of friction of the planes. Let also F y F' be the effective forces of friction reckoned upwards, E 26 EQUILIBRIUM WITH FRICTION. and JV, N 1 the normal reactions. Then the equations of equili- brium are Tcos - W sin a + F= o, Tcos 0' - W sin a' + F' = o, T sin d - FFcos a + N= o, Tsin 0' - W cos a + N' = o. It is further necessary to equilibrium that F, F' respectively should lie within the limits = e. F' = N' tan e'. Substituting the values of F, F\ N, N' from the equations of equilibrium, we have at the limits , . cos(0c)' cos(0 ') Let a > c, a' > e'. Then it is easily seen that the least value of Tin each case will correspond to the upper sign.* Hence, according to the principle stated (p. 21), equilibrium will be possible if either of the following conditions be fulfilled. cos (0-s) cos(0W) cos(0W) cos (0 + e) ' Hence it is evident that the ratio of the weights must lie within the limits sin (a' - c') cos (9 - c) sin (a' + e') cos (0 + e) sin (a + i) cos (0' -f e')' sin (a - e) cos (0'- e) ' Unless in which case hoth parts of the string lie at the same side of the vertical. It is easy to modify the reasoning so as to suit this case, as also the case in which either a or a is less than . EQUILIBRIUM WITH FRICTION. 27 III. Equilibrium of a System of Material Particles each of which is constrained to remain upon a Rough Curve. 14. A slight modification of the geometrical construction of p. 8 will be sufficient to adapt to the case of rough curves the reasoning which has been applied to the case of rough sur- faces. In the case of surfaces, the normal reaction has a de- finite direction, while the direction of the force of friction is in- definite. In the case of curves, on the other hand, the normal reaction is indefinite in its direction, which may lie anywhere in the normal plane, while the force of friction must be directed in the tangent to the curve, which is a definite line. The geo- metrical condition by which the resistance of a rough curve is limited is therefore evidently this : that its direction shall not make with the tangent an angle less than the complement of the angle of friction. The geometrical construction, therefore, by which we replace that of p. 8 is this : Let a cone of revolution be described round the tangent to the curve at any point, the semi-angle of the cone being the complement of the angle of friction. Then it is evident by the same reasoning as before that the resistance of a rough curve is limited by one only condition, namely, that its line of direction shall not lie within this cone. With this modification the methods of Prop. I. and II. are strictly applicable. Thus the equations of equilibrium (3) are true here without any change in their form. The condition (4) applied to this case may be written thus : 1 + A +&c V bi i Zi + A -7 + &C. ) < Hi COS where a l bi c\ are the direction cosines of the tangent to the curve, the other letters having the same signification as in Prop. I, The rest of the discussion is the same as before, 28 EQUILIBRIUM WITH FRICTION. IV. Equilibrium of a Solid Body supported by one or more Hough Surfaces. 15. The principles upon which this problem is discussed are identical with those already laid down. At each point of sup- port a force acts, whose intensity is unlimited, and whose line of direction is subject to one condition only, namely, that it shall not lie outside a given cone. In order, therefore, to determine the conditions of equilibrium, we must consider whether it be^ possible, consistently with this limitation, to assign to these forces such magnitudes and directions as shall keep the body at rest. Before considering the question generally, we shall examine the following particular case : PROP. II. A solid body of any form is supported by two fixed rough surfaces, and is acted on by the force of gravity alone. To de- termine the conditions of its equilibrium, and the resistance developed at each point of contact, if it be determinate. In this case the body is kept in equilibrium by three forces, namely, the weight, acting at the centre of gravity, and the two resistances, acting respectively at the two points of sup- port. In accordance with a well-known principle, the lines of direction of these three forces must lie in the same plane, and pass through the same point. Hence it is evident that the plane containing the centre of gravity and the two points of support is necessarily a vertical plane, inasmuch as it contains the line of direction of the force of gravity. Let this plane be represented by the plane of the paper, the shaded parts of the figure representing the sections of the body and the two supporting surfaces. Let G (Fig. 6) be the centre of gravity, and G Fa vertical. The rest of the discussion of the pre- EQUILIBRIUM WITH FRICTION. 29 sent question is similar to that of p. 14. If 80, SO', S'O, SO' be the sections of the cones of resistance, equilibrium is pos- sible, if 6rFlie between and 0', the effective resistances at S and S' being in this case indeterminate. It is also possible, if G V pass through or O', the resistances being then determi- nate. All the other conclusions of Art. 4 hold equally here. We now proceed to consider the general case of a body sup- ported by two surfaces. PROP. III. 1 6. A solid body of any form, supported by two fixed rough surfaces, is acted on by forces of any kind. To determine the conditions of its equilibrium, and the resistance developed at each point of contact, if it be determinate. Let X Y Z be the components of the force acting at any point of the supported body, and assume X, = J Xdm, Y, = J Ydm, Z, = J Zdm, and xY dm. Let R, JR' be the effective resistances at the points of support, and a j3 y, a' j3' 7' the cosines of the angles which their directions make with the co-ordinate axes. Then if x y z 9 x' y z be the co-ordinates of the points of support, the equations of equilibrium will be L, + R (p z - yy) + E Q3V - yY) = o, M, + R ( y x -az)+R (y'x - a'z) = O, (6) N l + R (ay - j3) + R' (ay - ]3 V) = o. If the position of the body be given, these equations contain six unknown quantities, namely, /?, R\ and four of the six cosines. It would seem, therefore, as if there were sufficient equations to determine the forces of resistance both in magnitude and direc- tion. We shall show, however, that there are in reality but five independent equations between these unknown quantities, the sixth being a relation between the given quantities of the prob- 30 EQUILIBRIUM WITH FRICTION. lem, necessary to the equilibrium of the body. For if the equa- tions of moments be multiplied respectively by x - x, y - y, z - z, and added, we have L, (x-x 1 ) + M^y-y) + N 1 (z- z'), -f (an/ - x'y) (Ry + Ry') + (zaf - xz') (Eft + Rft') + (yz e - zy') (R a + R'a) = o; or, substituting from the equations of forces for R and R 1 L^x-x^+M^y-y^ + N^z-z) = (xij- yx') Z, + (zx - xz') Y, + (yz' - zy) X,. From this equation, which is wholly independent of the forces of resistance, it is easy to see that if the external forces acting upon the body be resolved into a single force passing through any point in the line SS' (Fig. 4), and a moment with regard to this point, the plane of this moment must pass through the line SS'. This conclusion is evident in itself. Neither of the forces It, R' has a moment round the line SS': if, therefore, the act- ing forces had a moment round this line, the system could not be in equilibrium. Thus, for example, in the problem of Art. 15, we have seen that the vertical plane through the centre of gravity, G 9 must pass through the line SS'. This might be otherwise expressed by saying that the vertical line through G must intersect SS' ; or, in other words, that the force of gravity has no moment round this line. We learn from this discussion that inasmuch as there are but five equations among the six quantities required to determine the forces of resistance, one of these quantities must remain in- determinate. If, however, any additional condition be given, as, for example, if it be given that at either of the points S, *S', the force of friction has its extreme value, the problem becomes determinate. It is easy to see that each of the forces R, R' must lie in a certain determinate plane. This may be shown by eliminating a ]3' y' R R between the equations (6), when it will be found that the result is an equation of the form Aa + Bfi -f Cy - o- (8) EQUILIBRIUM WITH FRICTION. 3 I Similarly for a' /3' 7' we should have A'a -f Bp + Cy = o. (9) But it may be proved geometrically as follows : Let S' be taken as the centre of moments. Then since R has no moment with regard to AS', it is plain that the resultant moment of the applied forces with regard to this point must be equal and op- posite to the moment of R with regard to the same point. Hence it is evident that It must lie in the plane of this resultant moment. Similarly R' must lie in the plane of the resultant moment of the- applied forces with regard to the point S. It is evident from p. 30 that both these planes pass through 8S>. A third equation between a |3 7, a' j3' 7', will be obtained by eliminating R and R' between the first three equations (6). We have then 03/ - 7/3') *i+ (y* - ay 9 ) Y l -f (a/3' - /3a') Z l = o. (10) This equation denotes that J?, R\ and the resultant force are all parallel to the same plane. This is evident in itself, inasmuch as these three forces ought to equilibrate when transferred to the same point. It remains to consider whether any relation among the given quantities of the problem can be derived from the condition, that neither of the forces R, R' can make with the corresponding normal an angle exceeding the angle of friction for that surface. This relation may be investigated as follows : Let a sphere be described round any arbitrary point, and let the several lines and planes of the problem be projected on this sphere by parallel lines and planes through its centre. Let S (Fig. 7) be the projection of SS', and let 8M 9 SM', be the projections of the planes of the resultant moments of the applied forces, taken respectively with regard to the centres 8' and S". Let also F!OQ the projection of the resultant force, and N t N' the projections of the normals to the supporting sur- faces measured in each case outwards. Then if R,R' be the projections of the forces R,R', we know 32 EQUILIBRIUM WITH FRICTION. that R lies in the circle SM and R in the circle SM. Also if , be the angles of friction, we have RNnot> e , ^'JV'not > '. Hence if c < JVP or c' < N'P' equilibrium is impossible. If nei- ther of these inequalities be true, let NE = NE, = E, and N'E' Then R,R must be situated respectively in the arcs EE lt E'E\. Moreover, it is plain from equation (10) that R, R, F lie in the same great circle. Hence, in order that there may be equilibrium, it must be possible to draw through F a great circle cutting the two arcs EE 19 E'E\. The discussion of the special cases of this problem is analo- gous to that of Art. 4. If we draw the circles FE, FE lt and if these circles intercept a finite part of the arc E'E'^ the friction actually developed at the points of contact is indeterminate. If the points F, E\, E, or F, E, E^ lie in a great circle, the system is in an extreme position of equilibrium, and the actually deve- loped friction is determinate. We shall now give some examples of the general problem. Example i. 17. A circular cylinder rests in a horizontal position upon a rough inclined plane, and is supported by a string coiled round its middle section, and supporting a weight. To determine the con- ditions of equilibrium. EQUILIBRIUM WITH FRICTION. 33 Let the plane of the paper (Fig. 8) be that of the middle section. Then the cylinder is acted on by three forces ; namely, i . The attached weight, W' t acting in the vertical line HW'\ 2. The weight of the cylinder itself, TF, acting in the vertical line CV\ 3. The reaction of the rough plane R, acting at the point T. It ri s- 8 - is evidently necessary for equilibrium that this force should also be vertical. The angle which R makes with the perpendicular to the plane is therefore TCV (= inclination of plane). Hence it is evident that the inclination of the plane must not exceed the angle of friction. Moreover, the resultant of W and W must pass through T. Hence it is easy to see that, if a be the incli- nation of the plane, IF sin a = W (i - sin a). If these two conditions be fulfilled, the system will be at rest. We may inquire, further, i . Whether the cylinder can ascend or descend without revolving ; 2. Whether it can revolve with- out ascending or descending. 1 . It is plain that the cylinder cannot ascend without revolv- ing, inasmuch as such a movement would raise the centre of gra- vity of the whole system. In order that it may descend without revolving, the accelerating force down the plane must exceed the maximum force of friction, and the moment of all the forces round the centre of gravity must vanish. Hence, evidently, a > e, and p (W + W) cos a = W. 2. If the cylinder revolve without ascending or descending, the accelerating force on the centre of gravity must vanish, and the attached weight must exceed the maximum force of fric- tion. Hence, as in the case of complete equilibrium, the reac- tion, R, must act in the vertical. Since, moreover, the force of friction has its maximum value, R must make with CT an angle equal to the angle of friction. Hence the required conditions are = , (W+W) sin < W. F 34 EQUILIBRIUM WITH FRICTION. The reaction, which is in all these cases determinate, is given here, as in the case of complete equilibrium, by the equation, E = W+ W. Example 2. 1 8. A cylinder is kept in equilibrium, as in Ex. i, except that the sustaining cord passes over a fixed pulley. To find the con- ditions of equilibrium. The acting forces in this case are I. The weight of the cylinder acting in the vertical CV\ 2. The attached weight acting in the line VP ; 3. The resistance of the plane acting at T, which must, for equilibrium, be directed in the line VT. We must have, therefore, as one condition, VTR not > e. Fig. 9. If this condition be fulfilled, the attached weight will be deter- mined by the equation, W x sin TVP = weight of cylinder x sin TVC. There is another condition necessary to equilibrium, which is independent both of the attached weight and of the coefficient of friction. The point of intersection V must not be situated above I. For, if it were, it would be impossible that the reaction JR, should equilibrate the resultant of the two weights, inasmuch as its direction would in that case lie in the salient angle between the directions of these forces. Hence _/Q, making with the horizon an angle equal to the inclination of the plane, is the limit of the possible positions of the cord. It is easy to see that this limit can only be attained with an infinite coefficient of friction. If the cord, instead of sustaining a weight, were attached to a fixed point, the conditions would be reduced to two ; namely, that VTR should not exceed the angle of friction, and that the cord should make with the horizon an angle greater than the inclination of the plane. The foregoing equation would in this case serve to determine the tension. EQUILIBRIUM WITH FRICTION. 35 The reaction of the rough plane, which is determinate in this case also, is given by the equation, 7? sin TVP = weight of cylinder x sin CVP. 19. If two bodies be in contact, not at a mathematical point, but throughout a finite portion of surface, we must suppose the force of friction to act at each point of the surface of contact. But in "such a case it is known that the abstractions of .Rational Mechanics do not enable us to determine the actual pressure at each point, even if the surface be smooth. This pressure is really governed by some unknown law depending upon the act- ing forces, the form of the surface of contact, and the molecular constitution of the bodies. In the investigation of the laws of equilibrium or motion, however, we are in general, in the case of smooth surfaces, enabled to evade this difficulty. Thus, for example, when a heavy body rests upon a smooth horizontal base, we say that there will be equilibrium, if the vertical through the centre of gravity fall within the contour of the base. The validity of this conclusion does not in anywise depend upon the law regulating the pressure at each point. For we know that the action of gravity upon a rigid body is equivalent to a single vertical force passing through the centre of gravity. Jf this line fall within the contour of the base, the single force may, in the first place, be transferred to the point in which its direction in- tersects the plane of the base. This force is necessarily equili- librated by the reaction of the plane ; and the same will be true if this single force be replaced by a pressure acting at each point of the base. Whatever be the law by which its intensity is governed, provided that its resultant passes through the pro- jection of the centre of gravity, the pressure at each point of the base will be equilibrated by the reaction of the base, and there- fore the whole system will be in equilibrium. 20. We may arrive at the same conclusion by another method, which we shall find useful in our present subject. If the system be not in equilibrium, it must begin to move. Now, if the above- mentioned condition be fulfilled, it is easily seen that there can be no movement. For inasmuch as the whole force is vertical, 36 EQUILIBRIUM WITH FRICTION. there can be no horizontal movement ; and inasmuch as the re- reaction of the fixed plane is a resisting force, there can be no upward movement separating the base from the supporting plane. The only possible movement, then, is a movement of ro- tation, which is necessarily round a tangent to the contour of the base. But if we assume that such a movement has commenced, the contact of the body with the supporting plane will be at once limited to a single point through which the axis passes. The only force, then, which has a moment round this axis is the weight of the body, which, if its line of direction fall within the contour of the base, will necessarily tend to destroy the movement which we have supposed to have commenced. As this destruction must be absolutely simultaneous with the commence- ment of the movement, it is plain that no such movement is pos- sible. The same principle may, in certain cases, be applied when friction is one of the acting forces. We must consider, in the first place, what movements are geometrically possible, and then seek to determine what conditions are requisite, in order that the tendency of the forces, which would exist if any one of these movements had actually commenced, may be to destroy, or, at least, not to augment it. If these conditions be fulfilled, there is necessarily equilibrium. If they be not fulfilled, we cannot always say that equilibrium does not exist. For if any of the forces which act upon the body be geometrical, and if it be pos- sible to assign to their intensities such values as would produce equilibrium, then equilibrium is possible, even though these values be different from any which could exist if the system were in motion. We shall consider this point more fully when we come to treat of the difference between necessary and possible equili- brium. 2 1 . As an example of the principle here stated, we may consider the common problem of a heavy body resting upon a rough in- clined plane. In this case, the base upon which the body rests being supposed finite, there are two kinds of movement possible ; namely, either a rotation of the body round a tangent to the contour of the base, or a slipping of the base itself upon the sup- porting plane. The former of these movements becomes im- EQUILIBRIUM WITH FRICTION. 37 possible, as in the case of a smooth body, if the vertical through the centre of gravity fall within the base. With regard to the movement of slipping, we know that this movement cannot take place at any point where the direction of the acting force falls within the cone of resistance. If then, as in the preceding case, we conceive the weight of the body to be transferred to the point where the vertical through the centre of gravity intersects the base, and then resolved into a system of parallel forces acting at every point of the base, each of these forces will evidently make the same angle with the perpendicular to the supporting plane. Hence, whatever be the law which governs the intensity of these forces, the condition of equilibrium, necessary and sufficient, is, that their common direction (i. e. the vertical) shall not make with the normal to the supporting plane an angle greater than the angle of friction. This is the same as the condition for a single particle. 22. The same principle is applicable to any case in which the forces acting on the body may be replaced by a single force. The conditions of equilibrium in such a case are that this unique resultant shall pass through some point in the plane of contact, and shall not make with the normal to this plane an angle greater than the angle of friction ; the former of these conditions excluding a motion of rotation, and the second excluding a mo- tion of slipping. As an example of this principle, we may consider the case of a cube resting upon a rough inclined plane, four of its edges being horizontal. Let it be supposed that it is supported by a string attached to the middle point of the upper edge (to which it is perpendicular), passing over a pulley, and sustaining a weight. Let it be required to determine the least coefficient of friction which is consistent with equilibrium. In this case it is evident that the two acting forces namely, the weight of the cube, and the tension of the string lie in the same plane passing through the centre of the cube, and perpen- dicular to its horizontal edges. These forces have, therefore, a single resultant. Let the plane of the paper (Fig. 10) represent the plane of the two forces. Then if, as in the case of the cube A B CD, a 38 EQUILIBRIUM WITH FRICTION. vertical through the centre of gravity intersects the line P C beyond C, it is always possible to determine the ratio of W to the weight of the cube such that the resultant of these forces may be perpendicular to the inclined plane. Hence, however small the coefficient Fl S- I0 - of friction may be, equilibrium is always possible. If, on the other hand, as in the case of the cube A' B' C D\ the vertical through the centre intersect P H at a point between F and #, it w evident that of all directions of the resultant intersecting the base A If, that which is least oblique to the inclined plane passes through A'. The minimum coefficient of friction, therefore, which is consistent with equilibrium is represented by the tangent of the inclination of this line to A' B'. 23. The power which we have, in these cases, of evading the difficulty arising from the fact that the body rests upon more than three points, depends, as we have seen, upon the constancy of the inclination of the acting force at each point to the support- ing surface. The same principle, however, may be extended to the case in which the acting force, though not making a con- stant angle with the supporting surface itself, does make a con- stant angle with that curve on the surface in which the corresponding point of the body must necessarily move. Thus, for example, if a solid homogeneous cylinder, acted on only by gravity, rest within a fixed hollow cylinder of the same radius, the whole system being inclined to the horizon, the force at each point is manifestly indeterminate, and the vertical is in- clined to the supporting surface at an angle which assumes all values from o to 90. But it is plain, from considerations of symmetry, that the only movement which such a system could take will be at every point parallel to the axis of the cylinder. The same reasoning as before, therefore, will suffice to determine the condition of equilibrium, which is, that the axis of the cylinder shall make with the horizon an angle not exceeding the angle of friction. We are thus enabled in certain cases to solve a problem of EQUILIBRIUM WITH FRICTION. 39 equilibrium by considering what movement or movements could take place if the equilibrium were broken, and then determining the conditions requisite in order that no one of these movements maybe possible. 24. As another example, let the supporting surface consist of two inclined planes, whose line of intersection is horizontal, and whose slope is at the same side of the vertical, the upper plane being the steeper. Let a circular cylinder rest in a horizontal position on the two planes, being supported by a string coiled round its middle section, passing over a pulley and sustaining a weight, W. To determine the conditions of equilibrium. Here, if the equilibrium be broken, the motion may be any one of the three following: i. The cylinder may descend the lower plane ; 2. It may ascend the upper plane ; 3. It may re- volve round its axis without either ascending or descending. Let the accompanying figure (Fig. n) represent the middle sec- tion of the cylinder, 1A, IB being sections of the supporting planes, P the pulley, W the weight, PS the string, and VCV a vertical. Then, if the cylinder be just on the point of descending the lower plane, there will evidently be no pressure on the upper plane, and the cylinder will be kept in equilibrium by three forces ; namely, i . The weight W acting in the line VP ; 2. The weight of the cylinder itself W, acting in the line V'C ; 3. The reaction of the plane 1A, which must necessarily act in the line V'T. Hence, if CTV be greater than the angle of friction for the plane IA, equilibrium is impossible. If C TV be not greater than the angle of friction, the minor limit of the weight W is evidently given by the equation, W sin TV'P = W sin TVC. If the cylinder be on the point of ascending the upper plane, we must have, for a similar reason, CT'F'not greater than the 40 EQUILIBRIUM WITH FRICTION. angle of friction for IB, and the major limit of W is then given by the equation, W sin TVP = IF sin TVC. If CT' V be greater than the angle of friction for IB, while CTV is less than the angle of friction for 1A ; and if we sup- pose the weight W to be gradually increased from its minor limit, the equilibrium will be broken by a rotation of the cylinder round its axis without either ascent or descent. To determine the value of W for which this takes place, draw 772, T'R, making the angles CTR, CT'R equal respectively to the angles of friction for the planes IA, IB. Then, if the cylinder be on the point of of slipping at T and 7", the reactions of the two planes must act in the lines 772, T'R respectively. The result- ant of these reactions must, therefore, pass through R. But since the cylinder is kept in equilibrium by this force, and the weights W, W, acting respectively in the lines VC, V'P, it is evident that the resultant of the reactions must pass through F', and must, therefore, act in the line V'R. In this case, therefore, the major limit of W is given by the equation, W sin R V'P = W sin RV'C. V. Equilibrium of several Bodies. PKOP. IV. 25. If a heavy body rest in a given position upon two others which themselves rest upon a horizontal plane, to de- termine the conditions of equilibrium, of the system, and the friction actually developed at each point of contact, if it be deter- minate. Consider, in the first place, the upper body. This is kept in equilibrium by three forces ; namely, its own weight, IF 2 , and the two forces of resistance R, R' acting at the points of contact. These three forces must (as in Art. 15) be situated in the same vertical plane. Again, the lower bodies are kept in equilibrium respectively by the forces W it - R, R, and JF 2 , - R', R 3 . Each of these systems of three forces must, therefore, also be situated in a EQUILIBRIUM WITH FRICTION. vertical plane. But inasmuch as the systems W^ ft, R, and W lt - R, RI, are both situated in vertical planes and have a com- mon line, namely, the line of direction of E and - R, the planes of these two systems must coincide. So also with the systems W Z9 R, R', and W z - R', R 3 . Hence it is evident that the three centres of gravity and the four points of contact must be situated in the same vertical plane, which will also contain all the forces of the problem. Let this plane be represented by the plane of the paper, and let the accompanying figure (Fig. 12) represent the section of the system made by this plane. Fig. 12. Let 6ri, ft, ft, be the centres of gravity of the three bodies, and ftFj, ftF 2 , ft F 3 vertical lines. Let T } V l be the line of direction of R lt Then it is evident that F 1 CiF 2 will be the line of direction of R, F 3 (7 3 F 2 the line of direction of^', and T 3 F 3 the line of direction of R 3 . All these directions, therefore, are known if any one of them be known. Moreover, if the direc- tions of these forces be known, the forces themselves are known. For since each of the bodies is kept in equilibrium by a system of three forces, we have sv-B-'fri-? sin S 3 F 3 C 3 sn sm JF, sn sn sn (II) 3 sin T 3 F 3 C 3 Sn These six equations involve five unknown quantities ; namely, , R 3 , /?!, R ; and one of the angles which determine the direc- tions of the forces. Eliminating these, we have one equation G 42 EQUILIBRIUM WITH FRICTION. among the given quantities of the problem, which, with the conditions respecting the centres of gravity and points of con- tact, form the only conditions of equilibrium which can be expressed by equations. Thus, for example, let the centres of gravity, ft, G 3 be situated vertically over the respective points of contact. Then, inasmuch as two of the three forces which keep each of the lower bodies in equilibrium pass through the lower point of contact, 7\, the third force namely, the pressure at the point Ci must also pass through this point, and must, therefore, be directed in the line CiT^ In this case, therefore, the additional condition of equilibrium denotes that the lines T^C^ T^C 3 intersect the veitical G t V 9 in the same point. If the three bodies be spheres, this is necessarily true, as is easily seen. The other conditions are, that no one of the directions found from the preceding equations for the forces of resistance shall make with the corresponding normal an angle greater than the angle of friction. It is evident that if the vertical plane, which, as we have seen, contains all the forces, make with any one of the normals an angle greater than the angle of friction, equilibrium is im- possible. PROP. V. 26. A number of rough equal spheres are piled one upon another so as to form a pyramid, the lowest layer resting upon the ground. If there be no pressure between spheres belonging to the same horizontal layer, to determine whether the pressures at the several points of contact be determinate or indeterminate. Let n be the number of spheres forming one side of the base of the pyramid. Then we know that if jobe the total number of spheres contained in the base, and q the number of the remain- ing spheres contained in the pyramid n (n + i ) n(n - i)(n + i) P" -~' q = -- 1 ' 2 ' 3 EQUILIBRIUM WITH FRICTION. 43 The number m of distinct points of contact being thres for each sphere not contained in the base, and one (with the ground) for each sphere contained in the base, is given by the equation, m = 3? + P = I '2 Now, at each point of contact there is a force, namely the mutual action of the surfaces which is unknown both in maimi- o tude and in direction. Hence the number of unknown quantities introduced by these forces will be But the number of equations of equilibrium, being six for each sphere, will be n(n + i) (n -f 2). Hence, if this number be less than 3, the forces of reaction are indeterminate. The limiting value of n for which the forces can be determinate, is found by equating these numbers. This gives n = 4. If n have this value, the equations of equilibrium will be in number equal to the unknown forces of reaction. But we cannot infer that these forces are, therefore, determinate. For it maybe possible to eliminate all the forces of reaction from the equations of equilibrium, and to obtain thus a number of equations of condition, which, of course, give no assistance in determining these forces ; or (as in the present case) one or more combinations of the equations of equilibrium may be identically true. It is, therefore, necessary to examine separately the cases n - 2, n = 3, n = 4. For this purpose we shall somewhat generalize the question by supposing the bodies of which the pyramid is composed not to be spherical, the number in each layer being, however, the same. i. Let n = 2. There are, in this case, three bodies resting upon the ground, each of which is kept in equilibrium by three forces ; namely, gravity, the reaction of the ground, and the pressure of the single body which forms the apex of the pyra- 44 EQUILIBRIUM WITH FRICTION. mid. These three forces must, therefore, be situated in the same plane. Hence it is easily seen that, for each of the lower bodies, the plane containing the centre of gravity and the two points of contact must be vertical. This condition, which is equivalent to three equations, is evidently identically true if the bodies be spheres. Again, since for each of the lower bodies the reactions of the ground and of the upper body are situated in a known plane, and have a known resultant, they are functions of a single unknown quantity. Hence the reactions which sup- port the upper body are functions of three unknown quantities. Eliminating these unknown quantities from the six equations of equilibrium of the upper body, we have three additional equa- tions of condition. In this case, therefore, the equations of equilibrium are sufficient to determine all the forces of resis- tance, and to furnish besides six equations of condition, which must be satisfied if the given position 'be one of equilibrium. This is in accordance with the general theory, in which we have seen that the number of equations of condition must be at least 3/i 2 (n f i) , . n(n + i) (n+ 2)-- - ' =6 (if* -2). If the bodies be equal spheres, the equations of condition are necessarily fulfilled. In fact, it is easy to see that in this case there is no condition of equilibrium which can be expressed by an equation. For it is evident, as in Prop. IV., that the pressure of the upper sphere on each of the lower spheres is directed in the line joining its point of contact with the upper sphere and its point of contact with the ground. It is evident, also, that the lines of direction of these pressures or of the corresponding reactions of the lower spheres intersect the vertical through the centre of the upper sphere in the same point. Considerations of symmetry show that the resultant of these reactions is directed in the vertical. Hence it is plain, that unless the line joining the two points of contact of any one of the lower spheres makes with the corresponding radius an angle greater than the angle of friction, it will always be possible to assign such values to the reactions of the ground and the spheres as will keep the whole system in equilibrium. EQUILIBRIUM WITH FRICTION. 45 2. Let a = 3. In this case, according to the general theory, the equations of equilibrium are sufficient to determine all the forces of reaction, and to furnish besides, equations of condition. The base of the pyramid consisting, in this case, of six bodies, each of the three bodies which are situated at the corner, is, as in (i), kept in equilibrium by two pressures and its own weight. Hence it is easy to see that for each of these bodies the plane containing the two points of con- tact and the centre of gravity must be vertical. This is, as be- fore, the geometrical signification of three of the six equations of condition. 3. Let n = 4. In this case the formula (12) gives the number of equations of equilibrium equal to the number of the unknown quantities ; namely, the intensities and directions of the reactions at the several points of contact. These unknown quantities are not, however, determinate, inasmuch as it is pos- sible to form from the equations of equilibrium three equations not containing any of these unknown quantities. These equations express, as in the two previous cases, the geometrical condition that, for each of the bodies situated at the corners of the base, the centre of gravity and the two points of contact lie in the same vertical plane. PROP. VI. 27. A number of solid bodies are connected by certain geo- metrical equations. To determine the geometrical forces corre- sponding to any one of these equations. It is. known that the position of a solid body in space depends upon six quantities, which may conveniently be assumed to be three co-ordinates fixing the position, in space, of some certain point in the body ; and three angles, fixing the directions, in space, of any system of lines, which are fixed in the body. These angles may be taken in a variety of ways ; 46 EQUILIBRIUM WITH FRICTION. for convenience of physical interpretation, we shall suppose the angular position of the body to be fixed as follows : Conceive a system of rectangular lines, x y z, having a fixed direction in space, and a second system, x y' z, having the same origin as the first, and a fixed direction in the body. Let three planes be drawn through xx, yy', zz\ respectively ; and let < be the angle between the first plane and xy, the angle between the second plane and yz, and i// the angle between the third plane and zx. Then it is easily seen that these three angles fix the angular position of the body. It is known that the most general movement which can be given to a solid body may be conceived to be made up of a movement of translation common to all its points, and a move- ment of rotation round any one point considered as fixed. Now, if the common origin of the axes xy z t x y z be taken to be the centre of rotation, xyz being the co-ordinates of this point with regard to axes having a fixed origin and parallel to the first system of axes, it is plain that the components of a small motion of translation are #, Sy, 2, and the components of a small motion of rotation S<, $0, Si//.* Let L = o be one of the geometrical equations. Then it is plain that, in general, L = { (x, y, z, 0, 0, i//, a?!, y l} &c.). Then if X, Y, Z^ X\ t &c., be the components of the result- ant force for the several bodies, and Jf, JV, P, M lt &c., the components of the resultant movement ; we have, by the prin- ciple of virtual velocities, * It must be observed that $$, 59, Sty do not denote the actual rotations round the axes a; y z during the time St. In the case of a motion of translation, a move- ment parallel to one axis causes no movement parallel to either of the other axes. But if a solid body be made to revolve round one axis, it will, in so doing, also re- volve round each of the other axes. When we say, therefore, that 8$, 89, 8$ are the components of the motion of rotation, it is only meant that the complete rotation can be effected by causing the body to revolve successively round the three axes through these small angles. Vid. Routh's Rigid Dynamics, Chap. V. EQUILIBRIUM WITH FRICTION. 47 Zlz + Xxi -f &c. + J% + NSB + Pty + MJfr + &o., k dL . dL .. dZ . dL + X -- &c + - S + &r -f -- + -7- #! + &C. ) + &C. = O ; dx l ] X being an indeterminate multiplier. Hence it is evident that the equation L = o may, so far as the body (#, y, z, 0, 0, i//) is concerned, be replaced by a system of forces whose resultant force is determined by the equations, Y' \ dL V \ dL 7' \ dL . .A = A -: J. = A = > ^ = A =- , C?^P (fy C?2 and whose resultant moment is determined by the equations, A-\ dL B-\ dL C-\ dL ~ A ^' { ~dO> A 5f It is evident that the replacing system will be equivalent to a single force if dL dL dL dL dL dL = HM d + dy dO + dz d^ ~ This is most commonly the case in practice. In general, we infer from the foregoing discussion that, as in the case of material particles, the forces which correspond to one geometrical equation are functions of a single unknown quan- tity X. PROP. VII. 28. A number of solid bodies connected by given geometri- cal equations rest upon fixed rough surfaces, and are acted on by given forces. To determine the forces of reaction at the several points of contact, so far as they are determinate. Let n be the number of the solid bodies, and p, p lt &c., the numbers of the supporting surfaces for each of these bodies re- spectively. Let also m be the number of the equations of con- dition. Then, as each point of contact introduces a force of 48 EQUILIBRIUM WITH FRICTION. reaction unknown both in intensity and direction, the number of such unknown quantities will be 3 (p + p + &c.). Moreover, as we have seen in Prop. VI., each equation of condition, when replaced by the equivalent system of forces, in- troduces one unknown quantity. Hence the total number of unknown quantities is 3 (p + p l + &c.) 4 m. But the number of equations of equilibrium, being six for each body, is altogether 6n. If, then, 3(p + pi + &c.) + m > 6n, the equations of equilibrium are insufficient to determine the forces of resistance. Even if this condition be not fulfilled, these forces are not necessarily determinate. For it may be possible to eliminate all the unknown quantities from some group of the equations of equilibrium, and thus to obtain a number of equa- tions of condition among the given quantities of the problem, necessary for equilibrium, but not giving any help towards the determination of the unknown forces of resistance. Thus, for example, let it be supposed that one of the solid bodies rests upon a single fixed surface, and that the number of equations of condition involving the position of this body does not exceed two. Then the six equations of equilibrium of this body cannot contain more than five unknown quantities ; namely, the in- tensity and angles of direction of the single force of reaction, and the two (or one) quantities introduced by the geometrical equations. In this case, therefore, there must be at least one equation of condition among the given quantities of the problem, derived from the equations of equilibrium, and giving no help in the determination of the reactions of the other bodies. This indeterminateness will necessarily occur if any one of the solid bodies rest upon more than one fixed surface. For sup- pose it to rest upon two. Then the six equations of equilibrium of this body will contain six unknown quantities ; namely, the intensities and angles of direction of the two unknown reactions, which do not enter into any of the other equations. Now, it appears from Prop. III., that it is possible to eliminate these reactions from the equations of equilibrium, and thus to obtain a single equation denoting that the plane of the resultant mo- EQUILIBRIUM WITH FRICTION. 49 ment of the remaining forces which act upon the body is parallel to the line joining the points of contact. 'Even then if the geo- metrical forces were all determined by the other equations, there would be but five equations to determine the forces of reaction at the two points of contact, thus leaving one unknown quantity undetermined. Example i. 29. Two cylinders act upon a rough inclined plane, and are connected by a string coiled round each of them at right angles to the axes which are horizontal. To find the position of equili- brium and the friction actually developed at the points of contact. Assuming that the cord is coiled round the middle section of each cylinder, we may suppose the entire system to be reduced to these sections, and to be re- presented as in Fig. 13. Now, it is plain that the system of geo- metrical forces corresponding to the relation established between the two bodies by the cord is in this case equivalent to a single force ; namely, the tension. This is in accordance with the general equation of condition (13). For, if we take the axes of x and y respectively parallel and perpendicular to AA' 9 and the axis of z perpendicular to the plane of the paper, it is evident that the motion of translation will be parallel to #, and the motion of rotation round the axis of z. Hence, in this case, whence it is plain that equation (13) is satisfied for both bodies. Let Tbe the tension of the cord, W 9 W the weights of the cylinders, and/,/' the effective forces of friction at A, A'. Then resolving parallel to AA' 9 we have TFsin a + jPcos|3 -/= o, W sin a- T cos /3 -/ = o. (14) \ Again, taking moments round the axes of the cylinders, we have H 50 EQUILIBRIUM WITH FRICTION. whence T (i - cos |3) = T^sin a, 7 T (i + cos )3) = W sin a. Dividing these equations one by the other, we have W tan 2 i0 = ^,, which determines the position of equilibrium. Again, substi- tuting/or/ for Tin the equations (14), we have FT) sin a. Let R, R be the normal reactions at A, A'. Then resolving perpendicular to A A' we have R + Tsin /3 - IT cos a = o, R - Tsm j3 - W cos a = o ; whence R = W cos a - sin a t (fF cos a -sin V WW'\ 2fji ( W cos a + sin a V WW.) It is easy to verify that the limiting value of a obtained from this condition cannot exceed the angle of friction. The greatest value which /3 can have, corresponding to the case in which the cylinders are in contact, is given by the equation, Hence, if the cylinders be not in contact Wr < W'r. The minor limit of the ratio of Wto W is derived from the equation R = o, giving W -W tan V If the cylinders be in contact the question is modified by the friction which they mutually ex- ert upon each other. EQUILIBRIUM WITH FRICTION, Example 2. 30. It is required to keep a carnage at rest upon an inclined plane by locking two of the wheels. To determine whether this can be done more effectually by locking the fore-wheels or the hind-wheels. Assuming that the carriage is symmetrical with regard to its fore- wheels, and also with regard to the hind- wheels, we may, for the purposes of the present question, suppose the whole weight to be equally divided between two points, each of which is situated in the plane of one fore- wheel and one hind- wheel. It will then be sufficient to consider the equilibrium of one of the systems composed of these two wheels and the at- tached mass. Let, then, the plane of the paper (Fig. 14) represent the plane of the two wheels, and G the point at which the weight is concentrated. Let TT' be the inclined plane, GN a perpendicular on TT 9 and G V a vertical. Then the system is kept in equilibrium by three forces ; namely, the weight act- ing in the line GV, and the two reactions of the rough plane at the points T, T'. Now, it is plain that the reaction at the point of contact with the unlocked wheel is perpendicular to the inclined plane. Hence, if the fore-wheel be locked, the three forces act in the lines TP', T'P', PV, and the system will slip as soon as the inclina- tion of the plane is such that CTP' > c. For a similar reason, if the hind- wheel be locked, the system will slip when CPT' > e. If, therefore, CTP > CPT, it is better to lock the hind- wheels; and if CTP' < CPT', it is better to lock the fore-wheels Now, TT TT tan CTP = rj , tan CPT TNcota- GN' It is, therefore, better to lock the fore-wheels, unless TN- 5 2 EQUILIBRIUM WITH FRICTION. indicating that the vertical through the centre of gravity cuts the inclined plane above the middle point of TT. The extreme inclination of the plane for which equilibrium is possible is found by putting CTP' = t, or CPT = ei according as the fore-wheels or hind-wheels are locked. In either case the inclination is less than the angle of friction. For it is evident from the figure that each of the angles CTP, CPT', is greater than a (= TPP = T'P'P). VI. Equilibrium of a Flexible String. PROP. VI. 3 1 . A flexible string rests upon a rough surface, and is acted on by forces of any kind. To determine the conditions of equilibrium and the effective force of friction at each point, if it be deter- minate. We shall find it convenient in the present case to consider the resistance of the rough surface as a single force compounded of the normal reaction and the effective force of friction (Chap. I., Art. 8). Let Xds, Yds, Zds be the rectangular components of this force acting upon an element of the string, ds. Let 6 be the transverse section, and OX^ds, 9 YicZs, QZids, the components of the acting force. Let also Tbe the tension, and x y z the co-ordinates of any point in the string. Then we have by the ordinary method, Now, let it be supposed that the position of the string is given. Then, since x y s are known functions of s, the foregoing equa- tions contain the four unknown quantities, 7\ X, Y, Z. These EQUILIBRIUM WITH FRICTION. 53 equations are, therefore, sufficient to determine any three of these quantities in terms of the fourth, which must remain inde- terminate. This result is in strict accordance with that obtained for the case of a system of material particles (Art. 7). We saw there that, if the position of the system be given, the forces of effective resistance at the several points are fully determined by the equations of equilibrium in terms of the intensities of the geometrical forces which remain indeterminate. Now, in the present case there is but one geometrical force ; namely, the tension T. Hence, in accordance with the princi- ple of Art. 7, the force of effective resistance at each point of the string should be determined in direction and magnitude in terms of r l\ which should remain indeterminate. We have seen that this is precisely the case. If the string be at each point about to slip, the direction of effective resistance must be on the cone of resistance. We have thus an additional equation, and the problem becomes completely determinate. This is in accordance with the results obtained in Art. 8. To express the condition that the direction of the force of effective resistance can never lie outside the cone of resistance. Let a ]3 7 be the direction cosines of the normal to the support- ing surface, and the angle of friction. Then we have, as in Art. 10 : (aX + )3 Y + yZy- > cos 2 (X 2 + Y 2 + Z 8 ), where the values of X Y Z are to be substituted from equations 15. We shall now proceed to consider some particular cases of the general problem. i. Let the string be acted on by forces at its extremities only. Let a b c be the direction cosines of a perpendicular to the osculating plane to the curve. Then we know that adx + bdy + cdz = o, ad?x + bd*y + cd*z = o. Hence, if the equations (15) be multiplied respectively by a, b t c, and added, we shall have aX + bY -\- cZ = o. 54 EQUILIBRIUM WITH FRICTION. The reaction of the rough surface, therefore, lies in the oscu- lating plane to the curve formed by the string. Now, let it be supposed that the string is about to slip at each point. Then, if the cone of resistance be described at any point of the curve, the direction of the force .of reaction must lie upon this cone, and must, therefore, coincide with one of its intersections with the osculating plane. If, therefore, the curve of the string be given, the direction of the reaction is also given at each point as a function of e. Let R be the intensity of this force, and Z, m, n t its direction cosines. Then, if we eliminate R between any two of the equations (15) ; as, for example, be- tween the first two, we shall have dx , dy\ dT / d 2 x . d*y -r - I-T T- + m -T5- l^n. ds dsj ds \ ds* ds z But, since #, y, z, 1 9 m, n, are known functions of s, this equa- tion is of the form, dT= T(s)ds-, whence T= TQ being the value of the force of tension at one extremity of the string, which is evidently equal to the applied forc e at this point. This force must evidently be tangential. If J\ be the applied force at the other extremity, we must have T, = T eW (8)ds - This is the relation which must exist between the forces which act at the extremities, if the string be about to slip. The geometrical meaning of 0(s) is readily found by taking the tangent to the curve as the axis of #, and the radius of ab- solute curvature as the axis of y. We have then dx _ dy _ d 2 x _ d z y _ i Ts = '' ^ = ' "^ = ' 5? = " p' (> being the radius of curvature. Moreover, since the force of reaction lies in the plane of xy, if be the angle which the direction of this force makes with the radius of curvature, we have cos 6 = M, sin = + I ; EQUILIBRIUM WITH FRICTION. 5 the ambiguity of sign resulting from the fact that the force of reaction maybe directed in either of the two lines of intersection of the osculating plane with the cone of resistance. Assuming the tangent to be measured in the same direction with s, it is easily seen that one of these lines makes with it an acute angle, and the other an obtuse angle ; the upper sign in the value of sin 9 corresponding to the former of these cases, and the lower sign to the latter. Substituting these values in the value of 4>(s), we have tan 00) = + Hence, putting ds = pc?w, the value of T becomes 7'= T showing that T augments or diminishes according as the direc- tion of the force of reaction makes an obtuse or an acute angle with the direction of s. This is evident in itself. If the string be about to slip at each point in the direction of its length, the effective force of friction is directed in the tan- gent to the curve. Since, therefore, the complete reaction, R 9 of the surface is compounded of the normal reaction and the effective force of friction, it is evident that the plane passing through R and the tangent to the curve, which is the osculating plane, is normal to the surface. The curve, therefore, must be a geodetic line. It is also true, conversely, that if the string form a geodetic curve, the friction at each point is necessarily directed in the tangent. Moreover, since the radius of curvature coincides in direc- tion with the normal to the surface, 9 will denote the angle which R makes with the normal ; and since the string is about to slip, this angle must be at every point equal to the angle of friction. We have then, tan 9 = n, and T= T Q e+n O - o). A familiar instance of the general theorem is found in the case of two weights attached to the ends of a thin string coiled 56 EQUILIBRIUM WITH FRICTION. round a rough horizonal cylinder at right angles to the axis. Neglecting the thickness of the string, we may suppose it to be coiled n times round the cylinder. We shall have then, o>! - w = (an + i) TT, and T, = T e^ m + l ^. Hence, if W^ W l be the suspended weights, the extreme ratios of Wi to W consistent with equilibrium are given by the equations, W l = V" + * - (2n These results are true for a cylinder of any form if it have no singular lines. 2. As another example, we shall consider the case of a heavy chain wrapped round a rough horizontal cylinder. Let the axis of # be parallel to the axis of the cylinder, the axis of z vertical, and the axis of y perpendicular to the other two. Let the chain be wrapped round the cylinder in the form of a helix, each spire being in contact with the two adjacent spires. Let a be the radius of the cylinder, and b the width of the chain. Then the equations of either edge of the chain will be y 2 + z* = a 2 , z = y tan -- ; or, assuming iirx = 6o> and 2wm = 6, y = a cos W, z = a sin w, x - mw, ds = dw \/d* + m*. Now, if we conceive a section of the chain bounded by two curves parallel to either edge, it is evident that the equations of equilibrium of this section will be the same as those of any other similar section, and may therefore be taken for the equa- tions of equilibrium of the chain itself. Introducing then the foregoing values into the equations of equilibrium (15), and putting n z = a 2 + w 2 , we have (since X 1 = o, Y l = o t Z l --g) t 7m / 7 rri \ m + /i 2 X = o, a f sin w -r- + Tcos w ) - n 2 Y= o, aw \ aw J (16) a ( cos CD -= -- Tsin w ) + n 2 (Og - Z) = o. V aw I EQUILIBRIUM WITH FRICTION. 57 If the chain be about to slip, we have the further condition that the reaction of the cylinder must lie upon the cone of resistance. This condition gives the equation, X z + Y* + Z z = sec 2 e (Zsinw - Fcos o>) 2 , (17) in which the values of X, F, Z t are to be substituted from the equations (16). These equations admit of integration, if we neglect the width of the chain in comparison with the radius of the cylinder, in which case the curve of the chain becomes q p, a circle. We have then m = o, n - a. Hence, from the first equation (16) X= o. This .condition enables us to reduce the equation (17) to the form (Fsin & + ^Tcos o>) 2 = tan <2 c (Fcos w.- ^sin o>) 2 ; whence Fsinw + Z"coso) tan e (Fcos w -/Tsin w) = o ; substituting for Fand Zfrom the equations (16), we have i rr\ + Oga cos ) = d(t) the upper or lower sign being taken according as the slipping is towards or from the origin. Assuming that the slipping is about to take place from the origin, and integrating the equation upon this hypothesis, we have T=ce (ai e - Oga sin (26 - w), c being an arbitrary constant. To determine c, let /be the length of one of the vertical parts of the chain. We have then at the first point of contact with the cylinder, T= 9gl, w = o; whence c - 9g(l + a sin ac). Let /' be the length of the other vertical part of the chain. Then the greatest value of I' consistent with equilibrium is found by the equation I' = (1 + a sin 2f) e (2ra + 1 \ tan e + a sin 2*, the chain being supposed to be wrapped n times round the cylinder. If /and I' be increased indefinitely, this result agrees, as it ought, with that in page 56 for the cord. I CHAPTER III. ON EXTREME POSITIONS OF EQUILIBRIUM. i . A MATERIAL particle may be said to be in an extreme position of equilibrium when any diminution in the coefficient of friction of the surface on which it rests would cause the equilibrium of the system to be broken. Thus, for example, in the case of an isolated particle, the position of equilibrium is extreme, if the acting force be directed along a side of the cone of resistance. For, in this case, if the coefficient of friction were diminished by any quantity, no matter how small, the cone of resistance would be contracted, the acting force would fall outside, and therefore (p. 12) the equilibrium would no longer exist. Again, a finite body which rests on more than one surface may be said to be in an extreme position of equilibrium, if any one of the points of contact be in an extreme position ; that is to say, if the equilibrium would be broken by the diminution of the coefficient of friction of the supporting surface at that point. An important class of questions arises with regard to these positions. When a number of rough bodies are in contact with each other, it is not in general necessary to the rupture of equi- librium that slipping should take place at every point of contact. It frequently happens that, so far as the geometric conditions of the system are concerned, many different kinds of motion are equally possible. Thus, for example, if a double ladder be placed in an upright position on a horizontal plane, and if its two sides (supposed to be of unequal length) be gradually drawn out from each other till equilibrium be broken, the initial motion may be one of three kinds; namely, slipping at either end ac- companied by rotation round the other end, or slipping at both ends simultaneously. Similarly, if a beam be placed with one extremity on the ground and the other resting against a wall, and if its position be gradually changed till equilibrium is broken, the initial motion may either be slipping at both extremities simultaneously, or slipping at the upper extremity combined with rotation round the lower. If the beam be so placed that I ON EXTREME POSITIONS OF EQUILIBRIUM. 59 an indefinitely small diminution of the coefficient of friction at the extremities would cause motion of the second kind, and could not cause motion of the first kind, the upper end alone is said to be in an extreme position. If the beam be so placed that such a diminution might * cause motion of the second kind, both ends are said to be in extreme positions. 2. Instead of an indefinitely small diminution of the coefficient of friction we may substitute an indefinitely small change in the force, external or geometrical, which acts upon the particle ; this change being so made as to render the resultant more oblique to the normal. For the effect of both kinds of change is the same namely, to throw the line of direction of the acting force with- out the cone of resistance, and therefore to cause the equilibrium to be broken. Such a change may be produced by an indefinitely small shock, or instantaneous force communicated to the par- ticle. If, therefore, we use the word " disturbance" to de- note such a force, we may say that the position of a particle is extreme if an indefinitely small disturbance would cause it to slip. But in thus enunciating the definition, it must be carefully observed that ** disturbance" is quite distinct from " displace- ment" or " velocity." For, as we shall hereafter see, there are cases in which an indefinitely small velocity communicated to a particle not in an extreme position, as here defined, may cause the particle to pass into a state of finite motion. It becomes then an important question, in the case of a system which is in an extreme position of equilibrium, to decide which of the several geometrically possible movements may be expected actually to happen if the system experience a very small disturbance. We have to determine at which of the points of contact slipping may take place ; or, in the terms above ex- plained, which of these points are in extreme positions of equi- librium. Now, one characteristic of an extreme position of * I say " might cause ;" for even though both ends be in extreme positions, it does not follow that both extremities will actually begin to move. But if both ends be in extreme positions, an indefinitely small disturbance in the system may cause motion at both ends. Whether the actual initial motion be at one or both ends will depend upon the nature of the disturbance which has been communicated to the system, and must be determined dynamically. We shall return to this sub- ject in a subsequent chapter. 60 ON EXTREME POSITIONS OF EQUILIBRIUM. equilibrium is this, that the effective resistance of the surface at that point shall be directed along a side of the cone of resistance. For, unless the line of direction of the effective resistance were so placed, a small diminution in the coefficient of friction would not throw this line without the cone, and therefore (p. 13) the equilibrium would still be preserved. If the system be of such a nature that in its positions of equili- brium the effective forces of resistance are determinate, at least in direction, this characteristic is sufficient, and there is no theoretic difficulty in determining whether any given position of equilibrium be an extreme position, and also at what point or points the motion of slipping may be expected to happen, if the equilibrium be broken in the manner above described. For we have only to determine the angles of effective resistance at the several points of contact in functions of the given quantities of the problem. Then, if we find that at any one or more of these points these angles have their maximum values, we conclude that the points at which such values occur are in extreme positions, and that a motion of slipping may be expected to occur at any or all of these points, if the system receive a small disturbance. If the directions of the effective forces of resistance at the several points of contact be not determinate, the problem is one of somewhat greater difficulty. In all cases, however, whether these directions be determinate or indeterminate, the two fol- lowing principles are sufficient for its solution : 1. If any point of contact be in an extreme position of equili- brium, the angle of effective resistance at that point must have its extreme value tan' 1 ^. 2. Any value less than this at that point must be inconsistent with some of the conditions of equilibrium. Hence we infer for the case where the directions of effective resistance are indeterminate, 1 . That, at every point of contact which is in an extreme position, the value tan " l p must be an admissible value for the angle of effective resistance. 2. That at that point this value must be the least which will satisfy the condition of equilibrium. If, then, a system contain a number of points of contact. ON EXTREME POSITIONS OF EQUILIBRIUM. 6 1 some of which are, and some of which are not, in extreme positions, the relations of such a system must be satisfied by the following system of values assigned to the angles of effective resistance : i . For those points which are in ex- treme positions the values tan ~ y^, tan" 1 ^, &c. 2. For the points which are not in extreme positions, a system of values, each of which differs by a finite quantity from the corresponding extreme value. Moreover, it must be impossible, for the first class of points, to diminish any one of the values so assigned to the angles of effective resistance at these points, without increasing some one or more of the others. For, as we have seen, any such diminution must be inconsistent with some one or more of the conditions of equilibrium. It must/therefore, be incon- sistent either with some one of the geometric relations, in which case the value in question is evidently an absolute minimum, or with the condition that no one of these angles shall exceed the corresponding extreme values tan "^i, tan ~ 1 ^ &c. If, therefore, the value in question be not an absolute minimum, it must, at least, be impossible to diminish it without at the same time in- creasing the angle of effective resistance at some other point which is also in an extreme position. If one point only of the system be in an extreme position, the value of the angle of effective resist- ance at that point must evidently be an absolute minimum. 3. Before entering upon the general problem, we shall consider some particular cases which admit of easy geometrical solutions: i . Let two beams, connected at their upper ends by a smooth hinge, be placed with their lower extremities resting on a rough horizontal plane, and let it be required to determine whether the equilibrium be on the point of being broken ; and, if so, in what way this will happen. Let AC, BC (Fig. 15) be the beams; their plane, which is necessarily vertical, being that of the paper. Then the beam AC is kept in equilibrio by i. The weight acting in the vertical which passes through its centre of gra- ^ Jf ~ ~%T~g j " vity, E. 2. The reaction of the Fig. 15. rough plane acting at A. 3. The reaction of the hinge acting at C. 62 ON EXTREME POSITIONS OF EQUILIBRIUM. The direction of these forces must, therefore, pass through the same point. Let pE, Ap, Cp be the lines of direction of these forces, and qF, Bq, Cq the corresponding directions for the second beam. Then, since the hinge is smooth, the lines Cp, Cq, lie in directum. Now, considering the two beams as forming a single system, this system is kept in equilibrium by i . The weight of the system acting in a vertical through the centre of gravity, G. 2. The reaction of the rough plane acting in the line Ap. 3. The reaction of the rough plane acting in the line Bq. These forces must pass through the same point, V. The determination of the reactions at the points A, C, B, is therefore reduced to the following geometrical question : To draw a line through C intersecting the verticals pE, qF'mp and^, so that the lines Ap, Bq may meet on the vertical gG. This is evidently done by making Ar f Ap, Vq\ _ Aa x gb Br \ Vp, Bq) Bb x ga The directions of the reactions at A, B being thus ascertained, if either of these directions make with the corresponding normal an angle equal to the limiting angle for that plane, the system is about to slip at that point ; and if the angle exceed the limit- ing angle, the position is not one of equilibrium. Thus, for example, let the beams be uniform, and of the same thickness, and resting upon a rough horizontal plane. Then it is easy to see that if Ac > Be, then Ag > Bg ; and, there- fare, AVg> BVg. Hence the system will slip at the extremity of the longer beam. 2. As another instance, let us take the case of a beam, AB, B'A (Fig. 1 6), resting with one extremity on the ground, to which it is attached by a smooth hinge, and supported by a rough circular cylinder (also resting on the ground), to whose axis the beam is at right angles. Let the coefficients of friction be the same for the Fig. 16. beam and the ground, and let it be required to determine whether either of the points G jTbe in an extreme position of equilibrium. ON EXTREME POSITIONS OF EQUILIBRIUM. 63 It is evidently sufficient to consider sections of the cylinder and beam made by a plane perpendicular to the axis of the former. These are represented in Fig. 16. We may then con- sider the cylinder to be kept in equilibrium by three forces, namely i. The weight directed in CG. 2. The reaction of the ground acting at G. 3. The reaction of the beam acting at T. As these forces must, for equilibrium, pass through the same point, it is evident that the reaction of the beam must pass through 6r, and must therefore be directed in the line TG. Hence it is evident that the reaction of the ground must lie somewhere within the angle TGC, and must, therefore, make with the normal CG an angle less than TGC, and therefore less than GTC, the angle which the reaction of the beam makes with the normal CT. Hence G cannot be in an extreme position. The other point, T> will be in an extreme position if G TC be equal to the angle of friction. This conclusion may be other- wise stated by saying that, if the cylinder be slowly pushed towards A, equilibrium will be broken by slipping at TJ which will happen when GAT = double the angle of friction. 3. As another instance, let us take the case of a beam resting against a rough cylinder, and on a rough horizontal plane, upon which the cylinder also rests, its axis being at right angles to the beam ; and let it be required to determine whether or not the equilibrium is on the point of being broken ; and, if so, in what way this will happen. Let the plane of the paper (Fig. 17) represent a section of the beam and cylinder at right angles to the axis of the latter, and passing through the centre of gra- vity of the beam. Then, so far as the geometrical relations of the system are Fig. 17. concerned, the equilibrium may be broken by slipping at any of the points A, T, R. It is evident, as in the fore- going example, that slipping must take place at R before it takes place at T. It remains to consider whether it will take place first at R or at A. Now, the obliquity of the reaction at R = CRT = CTR = TVG. And the obliquity of the reaction at 64 ON EXTREME POSITIONS OF EQUILIBRIUM. A. = A VG. But it is easily seen that, if the beam be uniform, TVG > A VG. Hence the rupture of equilibrium will be by slip- ping at R, rolling at T 1 , and rotation of the beam round A. PROP. I. 4. A number of material particles are situated each on a rough surface, connected with each other by known relations, and acted on by given forces. To determine whether any given number of these particles be in extreme positions of equilibrium. Let n be the number of the material particles, and let them be denominated i, 2, 3, .... n. Let the particles i, 2, 3, . ... be in extreme positions of equilibrium ; that is to say, in such positions that, if the coefficient of friction of the surface on which any one of these particles rests were diminished, the equilibrium would necessarily* be broken ; the position of these particles must then satisfy the following conditions : : i . It must be possible, consistently with the conditions of the system, to assign to the angles of effective resistance at the points 1,2, . . . . k, their extreme values. 2. It must not be possible, consistently with these conditions, to diminish the value of any one of these angles without in- creasing that of one or more of the others. We shall now proceed to consider how these conditions are mathematically expressed. Let m be the number of the equations of condition, and RU JR Z Rm the geometrical forces resulting from these equations. Let AI, A 2 . . . . A n be the resultants of the given forces at the points i, 2, 3, ... n respectively, and let us denote by the symbols .or, .... ATS. .... the angles between the directions of A\ and R\ and of fit and R^ . . . . Then if u lt u z . . . . u n be the cosines of the angles which the directions of effective resistance at the several * The word "necessarily" is important. For, the position of the system being given, the values of the geometrical forces may he such that rupture of equilibrium would follow a diminution in the coefficient of friction at any one of the points 1,2, . . . A, while at the same time these forces are capable of values which would permit such a diminution without any rupture of equilibrium. This point will be more fully considered when we come to treat of " Necessary and possible equilibrium." ON EXTREME POSITIONS OF EQUILIBRIUM. 65 points make with the normals to the respective surfaces, we have, denoting these normals by jV"i, N 2 . . . . normal component of total force total force, V 'A ! the number of the geometrical forces. ^ R z , &c., which enter into the value of u^ being the same as the number of the equa- tions of condition which involve the co-ordinates of the point i. Similar values are, of course, found for w 2 , u$, &c. Each of these cosines is, therefore, a function of a certain number of the quantities JRi, JR 2 , &c., which are, so far at least as conditions of equilibrium are concerned, indeterminate. We may represent them generally by the equations Wi = 0i(^l,^2 - ), U t = fa (Hi ....), &C. Denoting then by a\ 9 a z ..... the cosines of the angles of friction, the first of the above-mentioned conditions requires that the equations Ml = 01, Ma = Oa, . . . . Mjfc = Ok, should be satisfied by real values of Ri\ R^ &c. Again, differentiating the values of u^ w 2 , &c., with respect to R^ R^ &c., &c., we have &c., &c., duk - &c. The second of the conditions requires that the cosines w 1( w 3 . . . w, ; should be incapable of being all simultaneously increased. 66 ON EXTREME POSITIONS OF EQUILIBRIUM. Hence it must be impossible to assign to dR^ dR z .... such values as will make all the increments du lt du^ .... &c. duk t positive. Before proceeding to consider how these conditions are to be fulfilled, it will be necessary to premise the following Lem- mas : PROP. II. Let ati t # 2 , &m be a series of variables limited by one only restriction, namely, that no one of them shall be negative. Let u\ t u Zt u 3 . . . u n be a series of functions of the form &C. 4- A^mXm, &C., &C., , , , &c. + ^,arm, where n > m. To determine the conditions requisite in order that it may be possible to assign such values to x l # 8f . . . . x m as will make Ui t M 2 , . . u n all positive. 1. It is plain that no series of coefficients such as A\ t , ^i, i, . . . Ai tm mthe same horizontal line can be all negative. For if this were the case, no system of positive values assigned to ,r, t # 2 . . . . x m would render Ui positive. 2. If any such series of coefficients be positive, it is un- necessary to take any account of the function Ui. For inasmuch as any system of positive values assigned to x lt x%, &c., would make u positive, the required condition must be independent of this function. 3. If any vertical series, as A lt lt A Z n t &c., be all positive, it is always possible to make all the functions positive by taking the corresponding variable x l sufficiently large. 4. If any such vertical series be all negative, we may always take BI = o, and this will obviously be the most favourable value. Hence in the first and third cases we are enabled to give an ON EXTREME POSITIONS OF EQUILIBRIUM. 67 immediate answer to the question, while in the second and fourth the subsequent operations may be simplified by leaving out of consideration either one of the functions or one of the variables. Let us suppose that in the first p of the functions Ui, u%, &c., the coefficient of x { is positive, and in the remaining functions negative. Then the first p functions will be positive if -4 2 ,i #1 > - (^2,0 + A Zt 2 #2 + &c.), &c., &c. ; A p>l Xi >- (A pt0 -f Ap^Xz + &c.). In the remaining n - p functions we must have - A p + i; i a?i < A p . 1? o + ^ + i, 2 #2 + &c. -4p + 2, 1 2 # 2 + &C. &c., &c. ; - ^n.l-^i < A nt o + A njZ X 2 + &C. We have thus p inequalities of the form #1 > bi, #1, > b z . . . . x l > b ft and n - p inequalities of the form \ < b p + 1, x 1 bi t b p + 1 > 6 2 , . . . . b + i > b t b n > h b n > b p . We must have besides the further conditions, b p + 1 > o, b p + 2 > o . . . . b n > o. The remaining variables x t1 # 8 , . . . . ^, rt must, therefore, be such as to make the functions 68 ON EXTREME POSITIONS OF EQUILIBRIUM. b p + 1 - #1, b p + 1 - 6 2 , . . . . bp + i - b Pt b p + 2 - bi t b p + 2 - b z , . . . . b p + z - bp, &c., &c., &c. ; bn - bi t b n - &2, ...... ^n - &, all positive. Arranging these functions, which are evidently (/;+ i) (w-jt?) in number, and contain only m - i variables, we have (p + i)(n-p) conditions of the form B\ o, + -#1,2 #2 + &c. + B^ m x m > o, #20, + #2,2 #2 + &C. + B^ m X m > O, &c., &c , &c. These functions are to be treated in the same way as u lt w 2 , &c. Thus, if any horizontal series of coefficients be all negative, we infer that the problem is impossible ; and if any vertical series be all positive, we infer that the problem is always possible. Also, as before, if any horizontal series be all positive, or any vertical series all negative, the corresponding function or variable may be neglected. If we do not find either a negative horizontal series or a positive vertical series, the same process must be continued until such a series be arrived at, or until all the variables be eliminated. In the latter case it is plain that we shall have a number of conditions of the form L > o, M > o, &c. L y Jf, &c., being functions of the coefficients alone, without any of the variables, in the series for u lt u Zt &c. PROP. III. To determine the conditions requisite in order that it may be possible to assign to a? lf ,.-... x m any real system of values, positive or negative, which will render u^ u.^ . . . u n all positive. If n be less than or equal to m, it will in general be possible ON EXTREME POSITIONS OF EQUILIBRIUM. 69 to find an infinite number of such systems. For it is only neces- sary to equate u lt u 2t . . . . w n> respectively, to any series of positive quantities, a lf a 2f . . . . a n , and from these equations to determine x^ # 2 , - . . - x m . (If n be less than m, we may add m - n equations ad lib., so as to make up the number necessary for the determination of &i # 2| . . . . x m ). But it may so hap- pen that in the elimination of some of the quantities x lt # 2> .... the rest may disappear with them, leaving one or more equations of the form + &c. -f L n a n = o, or L&! -f L 2 u 2 + &c. + L n u n = 0. (i) Let the number of such equations be p, and let them be solved for u^ u z , . . . . u p ; we shall have then results of the form Wi Mp + ^ + ! -f- M p . 2 U p + 3 -f &C., W 2 = &C. &c., &c. It must, therefore, be possible to assign such positive values to Up+ i' t u p + 2 t . . . . u n as will render the functions Mp^Up^ + Mp + z u p + z + &c. &c. &c., positive. This case is reduced, therefore, to Prop. II. If n be greater than m, the elimination of #., t x Zt &c., can, of course, be always effected, giving one or more equations of the form (i). The remainder of this case is as before. PROP. IV. Let Wj, w 2 , . . . u m be any functions of the n variables aii x z . . . # and let a lf a 2 , . . . . a m be a given system of values of these functions. To determine the conditions requisite in order that it may be impossible to raise any of the functions above the given values without diminishing one or more of the others be- low their corresponding values. 70 ON EXTREME POSITIONS OF EQUILIBRIUM. If this augmentation be impossible, it is plain that some one or more of the differentials or C?M! s du x i + -= dx z + &G. + -= dx m * , 2 , f 2 ^ dxi + d# 2 + &c. + ^ dx m dxi dx 2 dx m &c., &c., &c., must be negative when MI, w 2 .... are replaced by a lt a z . . It must, therefore, be impossible to assign such values to (e lf dx^ &c., as will make any of the increments dui t du 2t &c., positive without making some one or more of the rest negative. The problem is, therefore, similar to that discussed in Prop. III., putting diii, du^ &c., in place of u lt w 2 , &c., and 2 &c cfo,-' in place of A^ A^ 3f &c. If, therefore, it be impossible to fulfil the conditions there spoken of, it will be impossible to augment all the quantities MI, M 2 , &c., beyond the given series of values. We infer, therefore, that if the augmentation be impossible, there must exist one or more equations of the form + L z du 2 + &c. = o, (2) and that it must be impossible to satisfy these equations by any system of positive values of the quantities du^ du z , &c. If the number of the functions w lt w 2 , &c-> be greater than that of the variables x\ t x^ &c. one or more equations, such as (2), can always be found by eliminating dx^ dx%, &c. from the equa- tions ON EXTREME POSITIONS OF EQUILIBRIUM. 7 I dui du^ f du^ = -= d$i + -r dx % -f &c. dx l dx 2 du, = ^ 2 <&, + &c. (3) &c. &c. If the number of the functions u^ &c. be less than that of y^ &c., the conditions requisite for the existence of one or more equations of the form (2) will be found by multiplying the equations (3) respectively by the indeterminate factors AI, A 2i &c., adding and equating to zero the coefficient of each of the increments dx^ dx^ &c. Eliminating then the n - i ratios, \z AS A n A? x; %' we have m n + i equations of condition among the quantities du, dui 2 &c tel ' &c., &c. These equations must hold for all values of x\ % x Zt &c., con- sistent with the conditions If this condition be fulfilled, there will be one equation of the form L^dui + L z du z + &c. = o. If there be two or more equations of this form, the existence of such equations will appear in the process of eliminating the ratios, in the same way in which the first appeared in the elimination 72 ON EXTREME POSITIONS OF EQUILIBRIUM. of a?!, ;r 2 , &c. Thus, for example, if the elimination of p - i of these ratios among the first p of the equations, , du^ du 2 AI j -- h A 2 -7 + (feC. = O, dx l dx l A,g + &c. =o, (4) &c. &c., cause all the rest to disappear, we shall have only p - i equations between the n coefficients X lf A 2 , ..... A n . These equations are of the form ^Ai + A 2 \ 2 + &c. = o ; (5) we must have, therefore, AIC?MI + X 2 du 2 + &c. = o, (6) for all values of A lt A 2 , &c., which satisfy the p - i equations (5). Hence, if we eliminate p - i of these coefficients from (6) by means of the equations (5), we may equate separately to zero the coefficients of the rest. We shall then have n - p + i equations of the form &c. = o. Some particular cases of the general proposition require to be noticed : i . If there be but one equation of the form Ltdui + L z du 2 + &c. = o, it is plain that this equation may always be satisfied by positive values of du lt du z , &c., unless the coefficients Z lf L 2> &c. have all the same sign. 2. The same is true even if there be more equations than one ; provided that these equations are capable of being re- duced to the form Lidut + L 2 du 2 + &c. + Lidui = o, Li^dui^ + &c. - o, (7) &c. &c. ON EXTREME POSITIONS OF EQUILIBRIUM. 73 no two of these equations containing the same differential. In this case the coefficients of each of the equations must have the same sign. 3. If one of the variables, as o?i, enter into one only of the functions, as MI, the given value a^ must be an absolute maxi- mum with regard to this variable. For if we make dx-i = o, dx$ = o, &c. dx m = o, we shall have dui = -j dxi, du-i = o, du 3 = o, &c. du n = o. dx l If now we take dxi of the same sign as ~ -, we shall have a a%i positive value of dui : unless, therefore, it will be possible to augment the value of u^ without any augmentation of w 2 , w 3 , &c. In this case, then, we must have implying that the function u } is an absolute maximum with re- gard to X-L. 4. The foregoing Lemmas enable us at once to solve the ques- tion of Prop. I. For if we find that, for a given position of the system, the equations !, u 2 = 2 > &c. Uk = Ok (8) can be satisfied by real values of B lt R*, &c. ; and if, moreover, we have ascertained, by the method of Prop. IV., that the system (8) is an extreme system, then we infer that the given position of the material points may be an extreme position of equilibrium. 5. We shall proceed to consider a particular case which will elucidate the general principle. Let the number of the geometrical forces R^ R 2 , &c., acting 74 ON EXTREME POSITIONS OF EQUILIBRIUM. at the points i, 2, 3, .... k be the same as the number of the points. We shall have then dui _,.. d dR > + &c !4 du z = ~ dR l + &c,, (9) &c., &c., du k = -7^ dR l + &c. CtjKi If all the geometrical forces act at every one of the points i, 2, ...... k ; or, in other words, if the co-ordinates of all these points enter into each one of the equations of condition, the coefficients d^ duj, du 2 o dR? dRj 6 5< dfi? & will be all finite. If any one of the forces, as R^ only act at the points i, 2, . . . . /?, we shall have du p + 1 du p + 2 " = = We shall suppose as the most general case, that every geome- trical force acts at every point. There is evidently no difficulty in introducing the limitations which result from the absence of any of these forces. We have seen that if the points i, 2, .... k be in an ex- treme position of equilibrium, the elimination of dR^ dR^ &c. ain.ong the equations (9), must give at least one equation of the form L^dui + L z du z + &c. + Luduk - o. Hence, according to the theory of elimination among linear equations, we must have the determinant, ON EXTREME POSITIONS OF EQUILIBRIUM. 75 du du k o. (10) ii. It may be well to investigate the geometrical meaning of any one of the differential coefficients, du, du & dut dR? dRj dft' ^We have (p. 65) U! = Nj. + ft cos ^"ift + ft cos where &c. &c. i^i cos F& + &c. COS J\35 + &C. Differentiating, we find rftl! COS ^V^j A^i + ^ COS i^ 2 COS ^JK 2 + &C. &C.) But since Vi is the resultant of all the forces acting at the point i , it is plain that i + R, COS 7?^ -f &c. COS R! + ^ COS F^ + &C. = FX COS FuBi. Substituting these values, we have % = 4 (cos JvVST- cos TW" cos Denoting by the symbol of JVi F n and the plane of the angle between the plane j, we know that 76 ON EXTREME POSITIONS OF EQUILIBRIUM. cos A 7 ! /^ - cos K^Vj cos T 7 !^ = sin F^sin F^ cos Hence ^ Wl sin Fffii sin V\R l cos , Similar expressions hold for the other differential coefficients. If one of the geometrical forces R\ act at one only of the points which are in an extreme position, it is plain from p. 73 that we must have du } dR^ Substituting the value given above, we must have one of the following conditions : The first of these equations would denote that the resultant of all the forces acting at the point i is directed along the normal. This is plainly impossible if the point be in an extreme position. The second equation would denote that the force jR x coincides in direction with the resultant of all the other forces. In this case it is plain that, so far as the points under consideration are concerned, the force Ri is indeterminate. For the slipping of any point is determined by the direction, not the magnitude, of the force acting at that point. But for the point i, since the direction of R coincides with that of the resultant of all the other forces, it is plain that no change in the magnitude of R l will affect the direction of the resultant of all the forces acting at this point. And since R^ does not act at any other of the points under consideration, none of the forces acting at these points will be affected in any way. The third equation signifies that the plane passing through the complete resultant and the normal is at right angles to the plane passing through the same resultant and R^. Hence, as the complete resultant is situated on the cone of resistance at the point i, it is evident that the second of the above-mentioned planes is a tangent to this cone. If only one point be in an extreme position, the value of i^ must be an absolute maximum. ON EXTREME POSITIONS OF EQUILIBRIUM. 77 Example. 7. Three material points, A, B, (7, rest, each on a rough sur- face, in given positions, and are acted on by given forces. The points A, (7, are connected with B by rigid weightless rods, A B) CB. To determine whether any, and if so which, of the points are in extreme positions of equilibrium. It is plain that, so far as the geometrical relations of the system are concerned, the equilibrium may be broken in any of the following ways :-- i . By the slipping of either or both of the extreme points A, C, the intermediate point B remaining unmoved. 2. By the simultaneous slipping of A and B, or C and B. 3. By the slipping of all three points A, B, C. We shall proceed to consider the conditions to be fulfilled in each of these cases : (i.) If the point A alone be in an extreme position, it is plain that we must have = It is plain from p. 76 that this is equivalent to supposing either that the acting force is directed along the line AB, and that this line is itself situated in the cone of resistance at the point A, or that the plane passing through AB and the direction of the acting force is a tangent to the cone of resistance at A. The line AB and the direction of the given acting force must, there- fore, lie outside the cone of resistance. Hence, if the acting force and the line AB be not coincident with the same side of the cone, the magnitude of the force R { may be determined as follows : Through AB draw a plane touching the cone of resistance ; this plane must contain the direction of the acting force, and the line of contact will be the direction of the resultant V\ of JK l and FI. Hence obviously A-*^' <"> sm V\Ri Let us next consider the conditions to be fulfilled at the 78 ON EXTREME POSITIONS OF EQUILIBRIUM. point B y which is supposed not to be in an extreme position of equilibrium. This point is acted on by three forces namely, i, a given force, F z ; 2, a force - 7?i directed along the line AB-, 3, a force - R 2 directed along the line CB. The second of these forces being determined by the equation (12), we may suppose B to be acted on by two forces namely, i, the resultant of F. 2 and - R lt which is a known force ; 2, a force - R% situated in the line CB. Hence it is evident that the plane passing through CB and the resultant of F z and - MI must cut the cone of re- sistance at the point B, and that one at least of the lines of section must lie within the angle contained by - R z and this resultant. Similarly the point C, which is acted on by the two forces F a and R z , not being in an extreme position, it is evident that the plane containing the directions of these two forces must cut the cone of resistance at (7, and that one at least of the sec- tions must lie within the angle F^R^. Moreover, it must be possible to assign such a value to R z that the resultant of F Z) -Ri, and - R z may lie within the cone of resistance at B, and that the resultant of F 3 and R 2 may lie within the corresponding cone at C. 1 . If the line of direction of the forces R z , - R z lie within both the cones, this condition may always be fulfilled by taking R z sufficiently large. 2. Let the directions of the forces R 2 , -R% be situated, the one without, and the other within the corresponding cone. Draw a plane through - R z and the resultant of F 2 and - R^. This plane will cut the cone of resistance at B in two lines, one, at least, of which will lie within the angle made by the directions of these forces. Let the angles which these directions make with this side of the cone be a, )3, the latter being the angle corre- sponding to - R 2 . Make a similar construction for the point C, and let the corresponding angles be a', ]3 r . Suppose the angle j3 to be situated within the cone at B, and the angle ]3' to be situated without the cone at C ; then, for the value R z = oo, it is plain that the resultant of the forces at C would lie without the cone of resistance, and that the resultant at B would, for the same value, lie within the corresponding cone. If, now, con- ON EXTREME POSITIONS OF EQUILIBRIUM. 79 tinually diminishing values be assigned to R z , the effect of this diminution will be to cause the resultant at B to move out- wards, and the resultant at C inwards, with respect to the cor- responding cones. Let P 2 be the resultant of F z and - R^ Then, when R z attains the value sing 2 sin (a + j3)' the resultant at B will in its outward motion have reached the surface of its cone. Again, when R% attains the value sng 3 sin (a' + 07 the resultant at C in its inward motion will have reached the surface of its cone. It is then necessary and sufficient for the fulfilment of the condition which we are investigating, that the first of these values should be less than the second. Hence, sina sing 2 sin (g + j3) 3 sin (a' + 0')' 3. If this line be without both these cones, and if the direc- tions of the given forces lie within both cones,* the condition may always be satisfied by taking E 2 sufficiently small. (2.) Next, let it be supposed that A and B are both in extreme positions, and that C is not in an extreme position. Then it is plain that, in accordance with the principle of p. 76, we must have du z SB,' ' The conditions required are, therefore, du z Ui = ai, u 2 = a z -7^- = o. a/i 2 The first two of these conditions are sufficient to determine ^ and E z - If then the points A, B, be in extreme positions, and if C be not in an extreme position, the substitution of the values * If this be not true, the problem is solved as in (2). 8o ON EXTREME POSITIONS OF EQUILIBRIUM. so found must satisfy the remaining condition, and the value of R z must make w 3 > a z . It is easy to interpret these conditions geometrically. From the equation (13) we learn that, unless the line BC be situated on the surface of the cone of resistance at B, the plane passing through this line and the resultant of all the forces at B is a tangent to this cone, the line of contact being, of course, the direction of the resultant. This direction is, therefore, known. Hence, of the three forces, F 2 , -Ri, -R^ acting at the point J3, we know the directions, the magnitude of one, and the direction of the resultant. These forces are, therefore, completely deter- mined. If now we compound the forces R^ F^ acting at A, the resultant of these forces ought to be situated on the cone of re- sistance at A. Moreover, since = -r~ dRi = ~ sin V\N\ sin V\R\ cos N V\R\ dRi (-4) duz = -j^ dRi = if- sin V t N t sin F 2 fi, cos A^ FjA, w 3 = 3 . (15) These equations must give real values for RI and R 2 . Elimi- nating these two quantities from the three equations (15), we have an equation of condition between the given quantities of the question. Moreover, in the equations du, . dB lt du, = dR l+ dB, A 3 aKi ciri\ drt z it must be impossible to assign any values to dR } , dR 2 , which * It must be remembered that the directions of ^ in these angles, respectively, are opposite. If we measure ^i in the same direction, the condition requires that the affections of these angles shall be the same. ON EXTREME POSITIONS OF EQUILIBRIUM. 8 1 will make du v du 2 , du 3 positive. Eliminating dRi, dR 2 between these equations, we have du 2 du z dRi 7 ~dR 2 , du z = -7- "-i + "7"~ du 3 . du^ dus dR, dR 2 Hence, it is plain that the coefficients of du lt du 2 , in this equa- tion must both be negative. Interpreting this as before, we infer that the dihedral angles Ni V^R^ N 3 V 3 R 2 must have the same affections with the angles N 2 V Z R^ N 2 V 2 R Z respectively, the directions of Ri,R 2 , being in the last two angles the same as in the first two. II. Extreme positions of a solid body. 8. When a solid body, resting upon a number of others, is disturbed either by an external shock, or by the diminution of the coefficient of friction at one or more of the points of support, equilibrium will, in general, be broken simultaneously at all the points of support. This rupture of equilibrium may happen^at each point in any one of three ways: i. By slipping of one body upon the other. 2. By rolling without slipping this will rarely happen unless both bodies be moveable. 3. By the sepa- ration of one body from the other at the point of contact. In the problems with which we are concerned, it is in gene- ral necessary to the rupture of equilibrium that at some one or more of the points of contact motion of the first kind should take place. For, if motion consisting altogether of the second and third kinds were dynamically possible, it would have been equally possible before the diminution of the coefficient of fric- tion ; equilibrium could not, therefore, have existed. If, then, the body be capable of being set in motion by such a diminu- tion, it must necessarily be about to slip at one or more of the points of contact. This we may express by saying that if the body itself be in an extreme position, some one or more of its points of contact must be in an extreme position, applying that expression to points which are characterized by a movement of M 82 ON EXTREME POSITIONS OF EQUILIBRIUM. slipping^ not to those which are characterized by a movement of rolling or of separation. The criterion of points which are, in this sense, in extreme positions of equilibrium is to be investigated in a manner per- fectly analogous to that which has been adopted for the case of a system of material particles. If the equations of equilibrium enable us to determine, at least in direction, the force of resist- ance at each point of contact, it is only necessary to inquire whether the obliquity of any of these directions to the corresponding normal be equal to, or less than the angle of friction. In the former case the point of contact is, and in the latter is not, in an extreme position of equilibrium.. If the equations of equilibrium be insufficient to determine these directions, the points of contact which are in extreme po- sitions must be determined by conditions identical with those of p. 64, namely (i.) At every such point the angle of friction must be a possible value for the obliquity of the force of resistance. (2.) It must be impossible to assign to the obliquity of the force of resistance at this point a value less than the angle of friction, without assigning to this obliquity at some other point an angle greater than the corresponding angle of friction. The remainder of the investigation is also analogous to that of the former case. The cosine of each obliquity is given by the equations of equilibrium as a function of a number of vari- ables, and it is necessary to consider whether it be possible to assign to these variables real values which will satisfy the equa- tions MI = i, ih = 2 ? "&c. ; and also whether it be impossible to augment simultaneously the entire system of values, ai, a 2 > &c., without causing some of the variables to become imaginary. Example. 9. A solid body, acted on by any given forces, rests upon three fixed rough surfaces. To determine the nature of the equili- brium, if the body be in an extreme position. ON EXTREME POSITIONS OF EQUILIBRIUM. 83 We shall suppose that the body actually rests upon the three surfaces, or, in other words, that the pressure at each point of contact is finite. We thus exclude from consideration all cases in which equilibrium is on the point of being broken by the separation of the heavy body from one or more of the support- ing surfaces. These cases being excluded, the rupture of equilibrium may take place in one of three ways i. By slipping at one point of contact combined with rotation round the other two points. 2. By slipping at two points of contact combined with rotation round the third point. 3. By slipping at all three points. This is expressed in the phraseology of the present chapter by say- ing that one, two, or all of the points of contact may be in ex- treme positions. With regard to the first of these, it is easy to show that, ex- cept in a particular case, the motion there supposed is geo- metrically impossible. Let A, B, C, be the points of contact, and let it be supposed that slipping takes place only at C. Then, since the motion is one of rotation round A, and also round B, it must be a motion of rotation round the axis AB. The motion at C must, therefore, be perpendicular to the plane ABC. But inasmuch as this motion is also a motion of slip- ping, its line of direction must be contained in the tangent plane at C. Hence it is evident that the motion here supposed is impossible, unless the normal to the supporting surface at C lie in the plane ABC. It is easy to see, also, that the system admits in general of one definite motion of the second kind. For, if the motion be one of rotation round A, and of slipping at jBand C, it must be pos- sible to draw through A. an axis of rotation AR, such that the planes ABU, ACR shall contain respectively the normals to the supporting surfaces at B and C. Conversely, if such a line can be drawn, the supposed motion is possible. But it is evident that the required line is obtained by drawing through A two planes passing through the normals at B and C respectively. The intersection of these planes is the required axis of rotation. The third movement is evidently possible in an infinite variety of ways. 84 ON EXTREME POSITIONS OF EQUILIBRIUM. Proceeding now to consider the problem statically, we shall take the plane of the three points of contact as the plane of xy. The co-ordinates of the points of application of the three forces of resistance are then x^y^, #23/20, #33/30. If now the compo- nents of these forces be X^Y^Z^ X 2 Y Z Z Z , X 3 Y S Z 3 , and if we denote the components of the resultant of the acting forces by X, Y, Z, and the components of the resultant moment by L, M, N, we have the six equations of equilibrium, = X 9 Y l + F 2 + 7 3 = F, Z l + Z, + Z 3 = Z, L, - Z&i - Z 2 x 2 - Z 3 x 3 = M, (16) &s - X l y l - Xty* - X 3 y 3 = N. Moreover, denoting by aijSiyi, a$*y^ a 3 /3 3 7 3 , the direction co- sines of the normals to the supporting surfaces at the points A, j>, C, respectively ; and by Ui 9 w 2 > u ^ the cosines of the angles which these normals make respectively with the directions of the forces of resistance, we have (a^ + ft F! + 7l ^) 2 = ^ (X? + ^ + ^ 2 ) (a 2 ^2 + ft F 2 4- 72 ^ 2 ) 2 = ^ (X? + Fa 8 + ^ 2 2 ) (I?) (a,X 3 + j3 3 F 3 + y,Z^ = Uj (X 3 2 t Y 3 2 + ^ 3 2 ). We have also Ui > or = cos d, u z > or = cos 2 > w 3 > or = cos c 3 , 1, c 2 , 3 being the angles of friction at the three points of sup- port. We now proceed to consider the several cases enumerated above. (i.) Let it be supposed that only the point Cis in an extreme position. Then we must have Wi > COS 1, W 2 > COS Z , W 3 = COS 3 . Moreover, since there is but one point in an extreme position, it is plain that, in accordance with the principles stated in p. 76, this value of u s must be an absolute maximum. Now, it appears, from the third, fourth, and fifth of the equations (16), that Zi, Z 2 , Z* are determinate ; and from the remaining three equa- tions (16) that of the remaining six components X\, X 2 , X 3 , ON EXTREME POSITIONS OF EQUILIBRIUM. 85 Yi 9 Fa, F 3 , three remain indeterminate. Hence, we have w 3 given by the third equation (17) as a function of the two independent variables X 39 F 3 . We must have, therefore, o. dX 9 dY 3 Hence, from the third equation (17), (18) (a 3 .Y 3 + j3 3 F 3 + 73^3) ft = u? F 3 . But we have seen (p. 83) that in this case the normal at C must lie in the plane ABC. We must have, therefore, y 3 = o. Hence, squaring and adding the equations (18), we obtain, without diffi- culty, from equation 3 (17), z, + y 3 2 +^, 2 = 3 2 (^ 3 2 + Yf), an impossible equation, inasmuch as w 3 < i . It is therefore im- possible that one only of the points of contact should be in an extreme position. (2.) Let it be supposed that B and C are both in extreme positions. We have then u z = cos c 2 , w s = cos 3 . Differentiating now the second and third of equations (17), recollecting that Z% and Z$ are constant, and putting Itf = XJ + Y 2 * + Z 2 *, R*u. A = a *X z + /3 2 F 2 + 7 ,Z 2 RJ = Z 3 2 + Y 3 2 + Z 3 2 , -R^ = a 3 ^ 3 + j3 3 F 3 + 73 Z 3 , we have Y 2 dY 2 ) Now, if B, C, be in extreme positions, it must be impossible to assign to d3f 2 , dY 2 , dX 3 , dY 3 such values as will render du z , du z both positive. Hence, in accordance with the principle established in p. 70, there must be, independently of these differentials, an equation of the form 2j z du 2 + L z du 3 = o, in which L z , L 3 have the same sign. Eliminating Xi, YI from- 86 ON EXTREME POSITIONS OF EQUILIBRIUM. the first, second, and sixth of equations (16), and differentiating the resulting equation, we have (y - yO dx z + (2/3 - 3/1) dX 3 - (x z - #o dY z - (x 3 - #0 dY z = o. If, then, we add to this the first equation (19) multiplied by L Z9 and the second multiplied by Z 3 , we may equate separately to zero the coefficients of dX z , dX Z9 dY z , dY 3 . We have then Eliminating Z 2 , L 3 from these equations, we find -R 2 fa 2 (^ 2 -^i)+ )3 2 (2/2-2/0) = W 2 {(#2-iBi) -2 + (2/2 - A (a z R 2 - u 2 X 2 ) +2/2-3/1 = 0, L 2 (fi z R z - u z Y z ) - (x z -a?,) = o (20) ^3) +3/3-3/1 = 0, L 3 (|3 3 /? 3 - u 3 Y 3 ) - (x 3 - #0 = o. (21) We have thus altogether ten equations namely, the six equations (16) ; the second and third of equations (17), and the two equations (21), to determine the nine unknown quantities, Xi Y } Zt, X 2 Y% Z z , X 3 Y 3 Z 3 . These quantities being eliminated, there remains a single equation of condition between the given quantities of the problem which must be satisfied if the points B> C are in extreme positions, the point A not being in an extreme position. It is further necessary that the values of X 2 , L 3 , obtained from equations (20), should have the same sign. To interpret the equations (21), let the symbols N 2 B, denote the angles which the line AB makes respectively with the normal and the force of resistance at B. Then a 2 (# 2 - #1) + /3 2 (3/2 - 2/0 = AB cos (x z - *!> X z + (y z - yO F 2 = R 2 x AB cos R^ Substituting these values in the first equation (21), we find cos N%B = cos ^V 2 ^ 2 cos R Z B. Hence it appears that the planes of the angles N 2 R 2t R Z B are at right angles to each other. Since, then, the direction of RI lies upon the cone of resistance of which N t is the axis, it is ON EXTREME POSITIONS OF EQUILIBRIUM. 87 evident that this direction is found by drawing through AB a tangent plane to the cone of resistance at B, the line of contact being the direction required. A similar interpretation holds for the second equation (21). If, therefore, either of the lines AB, A C lie within the cone of resistance at B or C y respectively, this point cannot be in an extreme position. With regard to the condition which requires that Z 2 , L 3 should have the same sign, it is evident that the origin may be so assumed that y % - y and y z - y shall both be positive. Hence, from the first and third of equations (20), it appears that the signs of L 2 and L 3 depend upon the signs of the quantities or, using the foregoing notation to express the angles, upon the signs of the quantities cos N Z X - cos RzNz cos R z Xj cosN 3 X - cos R 3 N 3 cosR 3 X. To interpret this, let us conceive a sphere to be described round A 9 and the several lines and planes to be projected on it by parallel lines and planes through A. This projection is represented in the accompanying figure (Fig. 1 8), in which B, C x c Fig. 1 8. are the projections of the lines AB,AC; N Z ,N 3 the projections of the normals, and the small circles the projections of the cones of resistance. Then, if BR Z , BR Z , CR 3 , CR' 3 be tangents to the small circles, it is evident, from the above, that the projection of the force of resistance R^ must be either R 2 or R' z , and the pro- jection of Rs either R 3 or R' 3 . 88 ON EXTRKMK POSITIONS OF EQUILIBRIUM. Let X be the projection of the axis of x. Then cos N Z X - cos R Z N Z cos R Z X = sin R 2 N Z sin R z Xcos N Z R Z X, or = sin R'^N Z sin R' z Xcos N Z R' Z X. Hence it is evident that the sign of L z depends upon the affection of the angle N Z R Z X or N2R' Z X. But it is easily seen that of these two angles one is always obtuse, and the other acute ; the same being, of course, true of the angles N Z R 3 X, N 3 R 3 X. In order, therefore, that Z 2 , A may have the same sign, the forces of resistance must either correspond to the points R z , R$, or R' z , R' 3 . (3.) Let it be supposed that the three points^, B, C are all in extreme positions. We have then u^ = cos t^ u z = cos e 2 , w s = cos 3 . Differentiating the three equations (17), recollecting that Z^ ZK Z$ are constant, we have (a l dX l + /SidFO - u, (X l dX l + T l dY l ) a z dX z + j3 2 ^F 2 ) - u, (X 2 dX z + Y z dY z ) (22) X 3 + fcdYJ - u, (X,dX 3 + Y 3 dY 3 ). Again, differentiating the first, second, and sixth of equations (16), we have dX l + dX z + dX 3 = o, dY l + d Y z + dY 3 = o, (23) yidXi + y z dX z + y A dX 3 = XidYi + a; z dY z + x 3 dY 3 . Then, if all the points of contact be in extreme positions, it must be impossible to assign to the increments dX^ dX^ &c., such values that du^ du 2 , du^ may be all positive. Hence, in ac- cordance with the principle stated in p. 70, there must exist, for all values of these increments consistent with the equations (22), an equation of the form Lid^ + L. 2 du<2 + L 3 du 3 = o, in which L 19 L z , L 3 have all the same sign. Hence, if we mul- tiply the equations (22) and (23), respectively, by A Z 2 ,Z 3 , P, Q, and i , and add them, we may equate separately to zero the co- efficients of the differentials dX^ &c. We have then ON EXTREME POSITIONS OF EQUILIBRIUM. 89 + P+ yi = O, L. 2 (R 2 <*2 ~ ^2^2) + P + y* = O, - ^ = o, L 2 ( 2 ft - u 2 Y 2 ) + Q - x 2 = o, Eliminating jL x L 2 L A P Q from these equations, we have a single equation of condition which must be satisfied by the values of Xi X> X 3> YI Y 2 Y^ Zi Z 2 Z 3 which are obtained from the nine equations (16), and (17). Moreover, the values of LI L 2 L 3 ob- tained from the foregoing equations must have the same sign. These are the conditions which must necessarily be fulfilled, if the three points of contact be in extreme positions. 10. We have thus obtained for a system of particles and for a solid body the conditions which must be fulfilled, if any given point, or system of points, be in an extreme position. But we cannot infer conversely that, if these conditions be fulfilled, the given system of points, or body, is necessarily in an extreme position. It is true that, if the forces which are not determined by the equations of equilibrium have the values assigned by these conditions, a diminution in the coefficient of friction at any of the points under consideration would cause a rupture of equilibrium. But we cannot say that the forces which are un- determined by the equations of equilibrium must necessarily have the values corresponding to an extreme position of equili- brium. These forces may have any system of values consistent with equilibrium, the particular system which obtains in any particular case depending not only upon the position of the par- ticles or body and the forces which act upon them, but also upon other conditions, which may be termed " initial ;" namely, the forces which have been used in placing the particles or body in the given position. We shall consider this point more fully afterwards ; for the present it is sufficient to remark that the importance of these conditions will at once appear, if we state the question of equilibrium as it really exists in nature, rejecting the abstractions of Kational Mechanics, and replacing the forces which we have called "geometrical" by the elasticity of the pO ON EXTREME POSITIONS OF EQUILIBRIUM. rods, strings, &c., by which geometrical conditions are mechani- cally realized. In this way of considering the question, it becomes evident that two systems, identical in position and in the external forces which act upon them, may, nevertheless, differ widely in the values of the forces of elasticity exerted by these connecting rods, strings, &c. These forces will, within cer- tain limits, depend upon the states of compression or extension in which these rods and strings have been originally placed, or, in other words, upon the conditions which we have called " initial." CHAPTER IV. MOVEMENT OF A PARTICLE, OR SYSTEM OF PARTICLES. I. Movement of a Single Particle upon a Fixed Rough Surface. i . WE have seen (p. 7) that the force of friction, when acting on a particle in motion, has necessarily its greatest value, and is always directly opposed to the movement of the particle. Hence, if #, i/, z be the co-ordinates of the particle, ds the element of the trajectory, a, /3, y the direction cosines of the normal to the supporting surface, and X, F, Z the components of the acting force, the equations of motion are , dt z== N being the normal reaction of the surface. The unknown quantities are evidently the same as in the problem of the motion of a particle upon a smooth surface namely, #, y, z, N. The equations by which these unknown quantities are determined are also similar to the equations of the other problem namely, the three equations of motion, and the equation of the supporting surface. These equations are rarely integrable, even in the simplest cases, but some general properties of the movement may be deduced from them. 2. If the independent variable be changed from t to 5, the foregoing equations will become d 2 a dv dx ^ A7 / dx\ , d*y V* +V =X + Jy[a- [Jl-r-l V Z -f- -f &C. = F+ &C., ds* ds ds \ ds J ds z 2 d z z V* -f &C. = Z + QLC. ds z Let a, b, c be the direction cosines of a line, i, in space, and 92 MOVEMENT OF A PARTICLE. let these equations be multiplied respectively by a, 6, c, and added. We have then, / d*x ,&y d*z\ I dv A7 \ / dx dy dz v ( a 7. * & "33 * * "53 1 + f*.T + vN}(a- r +b-j-c. \ ds* ds* ds z j \ ds J \ ds d$ ds '(3) = aX + bY+ cZ + N(aa + 6/3 + cy). (i.) Let it be supposed that L is perpendicular to the plane of the normal section passing through the tangent to the trajec- tory. Then dx dy dz a + b -f + c = o, aa + op + cy + o. ds ds ds Also, if $ be the angle which the osculating plane to the trajectory makes with the plane of the normal section, and 9 the angle which the acting force, ./?, makes with the latter plane, we have, p being the radius of curvature of the trajectory, sin $ d z z ,d 2 y d*z . = a -T- + b -^r + c , R sin = aX + b Y + cZ. p ds z ds* ds z If these values be substituted in equation (3), it becomes v 2 - sin -i- R sin = o. (4) Hence, if = o, or, in other words, if the acting force lie in the plane of the normal section, we have = o, denoting that the osculating plane is normal to the surface, which is the well- known property of a geodetic line. This includes the case of motion in consequence of a primitive impulse only, without any acting force, as also the case in which the acting force is tan- gential ; as, for example, when the particle moves in consequence of a primitive impulse in a resisting medium. In both these cases the trajectory will be a geodetic line. (2.) Let L be perpendicular to the osculating plane to the trajectory. Then it is easy to see that the whole of the left- hand side of the equation (3) disappears, and that we have simply aX + b Y + cZ + N(aa + fy3 -f cy) = o, (5) MOVEMENT OF A PARTICLE. 93 denoting that the resolved part of the normal reaction perpen- dicular to the osculating plane is equal and opposite to the re- solved part of the acting force along the same line ; or, in other words, that the total force lies in the osculating plane. This is known to be a general property of curvilinear motion. (3.) Let L be the normal to the surface ; we have then dx _ dy dz d*x d*y d*z cos i, and, therefore, since s > y, i - cot a =- ^4 2 , where A is finite. Hence the velocity is constantly diminishing, and the particle will come to rest in a finite space. To find the equation of the trajectory, let dy = pdx, d*y = qdx z , d*y = rdx 3 . We have then p (I Substituting for p and v 2 in the first equation (9), we have t/ 2 + 2(j (y sin a - jus cos a) = g sin a ( ). (i i) Differentiating and reducing, we find 2/u.q 2 cot a = r 4W + c = -- + - . (14) 2 - n 2 + n The trajectory is represented by the system of equations (13) and (14), between which is to be eliminated. If tan a > ju, whence n< 2, it is plain from equation (14) that when y = oo, = oo, whence p = oo, or the trajectory is ultimately perpendicular to the horizontal trace of the plane, supposed unlimited. If n > i < 2, we have, for p = oo, 2cx + c' = o ; hence the ultimate value of#is finite, or the trajectory has an asymptot perpendicular to the horizontal trace of the plane. If n< i, the curve has no asymptot. The three arbitrary constants c, c', c" are readily determined from the initial position, direction, and velocity of the projection. Thus, for example, if the particle be projected horizontally with a velocity v', and if we take the origin at the point of projection, we find from (n), (13), and (14), _ , _ in _ 4 ~~^~ = i - n* ~ ^re- integrating the equation, - and determining the arbitrary constant from the conditions s = o, 9 = i, we find n 2 - 2n 405 = ------- + - (i c) 2 - n 2 + n n 4-n z n UPON FIXED ROUGH SURFACES. 97 Substituting the values of y and s in equation (10), we find without difficulty, (14) If fi cot a > i , we have already seen that the final velocity = o. If JUL cot a < i , it is plain from the foregoing equation that the final velocity is infinite. Ifjucota= i, the same equation shows that the final value of v is half the velocity of projection. To determine whether, in general, the velocity increase or diminish after projection, we recur to the second equation (9), from which we have 2vdv = g sin a (idy - nds). But from equations (12) and (13), we have (2-rc)0 2 -(2 + n)0- z - an) dO. The limit of the values of which make this positive is found by equating it to zero. This will give n , 2 + n . n tr = , whence p 2 - n If be the angle which the tangent to the trajectory makes with the horizontal trace of the plane, this equation gives sin = jj, cot a. Hence we infer that the velocity will diminish until sin ^> = ju cot a. If, therefore, the particle be projected downwards at an angle greater than this, it will continually in- crease. If it be projected upwards, or downwards at an angle less than this, it will at first diminish, and after attaining a minimum, continually increase. It is evident from the value of />, that this can happen only when n < 2. This agrees with what we have seen before (p. 95). 9 8 MOVEMENT OF A SYSTEM OF PARTICLES II. Movement of a System of Particles under the influence of Friction. PROP. I. 4. A system of material particles, each of which rests upon a rough surface, is connected by certain geometrical relations, and acted on by given forces. To determine the equations of motion. Let Xi y l z ly # 2 y 2 %*> &n y n z n be the co-ordinates of the par- ticles, and QI |3i 71, a 2 /3 2 y 2 the direction cosines of the nor- mals to the several supporting surfaces. Let, also, R\, R 2 , &c., be the normal reactions of these surfaces, and Xi YI Z^ X z Y z Z^ .... the components of the acting forces. Let L - o, M = c, &c., be the equations expressing the geometrical relations which subsist between the particles of the system. Then, if we denote by X\ Y\ Z^, X' z Y' z Z\, the components of the several forces of friction, the equations of motion will be j. i , 2 D m l -- = X l + X ! + aiRi +A! - - + A 2 -=- + &c. at axi axi ^ ^ V> mj. -7TT = Z l + Z x Ut ! i y72/v m z r? = X* + X' 2 + &c., &c. at 2 In addition to these equations, which are 3^ in number, we have the n equations of the supporting surfaces, and the geo- metrical equations L = o, M = o, &c. The number of these equations is the same as that of the co-ordinates BI yi z lt x z y z z^ . . . . x n y n ^n 5 the normal reactions E lt R z , . . . . R n , and the coefficients Xi X 2 , . . . . There remain, however, still to be de- termined the 371 components of the forces of friction, X\ Y\Z\, ..... X' n Y' n Z' n . We require, therefore, 371 more equations UPON FIXED ROUGH SURFACES. 99 for the complete solution of the problem. These equations will differ according as each particle is in motion or at rest. (i.) Let us suppose that the particle #, y l Zi is in motion. Then we know that the force of friction acting upon this particle has its maximum value, and that it is directed in the tangent to the path of the particle. Let ^ be the coefficient of friction, and dsi the element of the path. Then we have did dy } dz } f Xi = -/^i -^-, FI--/IA -7-^1 = -^,, (16) IW-i O<6i U/Si and three similar equations for every particle which is in motion. Thus, if every particle were in motion, we should have the 3^ additional equations which are necessary for the solution of the problem. (2.) Let the particle x^ y\ z l be at rest. Then the first three equations become , if X\ + oA + A! + A, + &c. = o. F 1+ Y\ + J3A + &c. = o (17) Z l + Z\ + jiRi + &c. = o. We have besides, since the force of friction always acts in the tangent plane to the supporting surface, ai X\ + ^Y\ + 7 iZ\ = o. (18) Now, since x l yi z } retain their original values, they are no longer unknown quantities. Hence, if the equations of motion of the other particles of the system be sufficient to determine the co- efficients Xi, X 2 , &c., the four equations just obtained will de- termine the four unknown quantities X\, Y\> Z\, R, and the solution will be complete. It remains to consider when the foregoing condition is fulfilled. Let p of the particles be at rest, and let it be supposed that one of the geometrical equations, as L^ - o, only contains the co-ordinates of quiescent particles, or that it is possible to form such an equation by eliminating the co-ordinates of the moving 100 MOVEMENT OF A SYSTEM OF PARTICLES particles among the equations of condition. Then it is evident that the 4p equations (which are, in fact, equations of equili- brium) corresponding to these p particles will contain 4p + i unknown quantities namely, the 4p quantities, . i Y \ and the coefficient Ai, none of which will enter into any other equation. One of these quantities, therefore, must remain inde- terminate. The mathematical reason of this is, as has been stated before, that the equation L\ = o, involving only quantities which are, in the present case, known quantities, gives us no additional information ; while, at the same time, it necessarily introduces the new unknown quantity Ai. The same conclusion will, of course, hold, even if the co- ordinates of the moving particles enter into all the geometrical equations ; provided that it be possible to form, by eliminating these co-ordinates, an equation or equations between the co- ordinates of the quiescent particles. For in these cases the number of the geometrical equations, considered as the means of determining unknown quantities, is reduced by the number of the equations which can be so formed, while the number of the unknown coefficients which are introduced by these geometrical equations remains the same as before. 5. We now proceed to show that the incompleteness of the solution, when it exists, only applies to the quiescent particles, and that for the particles in motion there are sufficient equations to determine their positions at any time, and also the forces, geometrical and frictional, which act upon them. Suppose that k of the geometrical equations only involve the co-ordinates of quiescent particles. Then, as we have seen, these k equations, involving only known quantities, give us no assist- ance in determining the unknown quantities, while, at the same time, they introduce the k new unknown quantities, Ai, A 2 , . . . A/t, which must, therefore, remain indeterminate. But it must be observed that no one of these coefficients enters into any of the equations which concern the moving particles. Every co- efficient which appears in these equations has been introduced UPON FIXED ROUGH SURFACES. IOI by an equation involving the co-ordinates of one or more moving particles, which, being necessarily an equation between unknown quantities, does assist in the determination of these quantities. 6. The mathematical result of the whole discussion may be thus stated. Every geometrical equation involving only the co-ordinates of quiescent particles augments the number of the unknown quantities by one, while it leaves the equations for the determi- nation of these quantities precisely as it was before. Every geometrical equation involving the co-ordinates of moving particles augments the number of the unknown quan- tities by one, but augments also by one the equations by which these quantities are determined, thus leaving the number of the unknown quantities unchanged. We have then, in the case of the moving particles, a sufficient number of equations for the solution of the problem namely, i. The dynamical equations, which are the same in number as the co-ordinates ; 2. The equations of the supporting surfaces, which are the same in number with the normal reactions; 3. The equations (16) which are the same in number with the components of the forces of friction; 4. The geometrical equations, which are the same in number with the coefficients \. 7. The physical result of the investigation is as follows: When a system of material particles, each of which rests upon a rough surface, is subject to the action of external forces, it will, in general, be found that of these particles some will be in a state of motion, and others in a state of rest. Everything connected with the moving particles namely, their positions, their velocities, and the forces, geometrical and frictional, which act upon them, is fully determined by means of the dynamical and geometrical equations. The geometrical and frictional forces which act upon the quiescent particles, will also be determinate, unless it be possible to form by elimination one or more equations between the co- ordinates of the quiescent particles only. If this be possible^ the geometrical force replacing every such equation will be inde- terminate in intensity. 102 MOVEMENT OF A SYSTEM OF PARTICLES 8. It remains now to consider whether we have any means of determining at any moment which of the particles are in motion. Let it be supposed that the initial velocities of the several particles are given, and that no one of these velocities is absolutely zero. Then the equations of the problem are, for every particle, those which correspond to a state of motion ; and therefore, as we have seen, sufficient for the determination of all the unknown quantities. Suppose that these equations have been integrated, so as to determine the several co-ordinates as functions of the time. We have then, by differentiation, the velocities of the several particles at any given instant. And so long as these velocities continue finite, the solution obtained continues to be applicable. Suppose now that the velocity of the point x l y^z l becomes at any instant zero. This particle being for the moment at rest, we must consider, in the first place, whether or not it will con- tinue at rest. In order that this may be the case, the equations ('?) (P- 99) must be satisfied. Now, unless there be (independently of the equation of the supporting surface) a geometrical equation involving only the co-ordinates x\ y z^ it is plain, from what has been said, that the coefficients Ai, A 2 , &c., are all determined, as functions of the time, by the other equations of the system. Then the four equations (17) and (18) (p. 99) are sufficient to determine the four quantities X\ Y\ Z'\, R^ If the values so found be substi- tuted in the equation, TT '2 i V '2 O. 7 1 ..2 7? 2 . -A-l + J- 1 + A\ fJL J3-1 > and if we find fj.< or = jui, the conditions of equilibrium are satisfied, and the particle remains at rest. If jUi - ju be finite, the particle will continue at rest for a time which cannot be in- definitely small. If fi = fi\ 9 the particle will remain at rest or commence to move again, according as -^ is negative or positive. If this co- efficient vanish, the result will depend on the sign of the lowest differential coefficient which does not vanish. In any of these cases it is evident that there will be a suffi- UPON FIXED ROUGH SURFACES. 103 cient number of equations to determine all the unknown quanti- ties of the problem. And it appears, from the foregoing dis- cussion (p. 101), that this will always be so, unless a group of particles, between whose co-ordinates there exists a geometrical equation, be at rest simultaneously. Suppose now that there be such a group consisting of p particles. Then, so long as any one of these particles continues to move, all the forces will be determinate. Moreover, at the moment when the p th particle comes to rest, all these forces are still determinate. But if we adhere to the abstractions of Rational Mechanics, there will be thenceforward nothing to determine the intensity of the geo- metrical force corresponding to the equation which connects the co-ordinates of these particles while they continue at rest. Thus every group of quiescent particles between which there exists a geometrical equation, will introduce into the solution one inde- terminate quantity. If the system start from a state of rest, the solution will, in general, be indeterminate for another reason namely, the im- possibility of determining which of the particles remain at rest, and which of them commence to move. But, on account of the importance of the problem of initial motion, we shall consider it separately. Ill Initial Movement of a System of Particles. 9. A system of material particles connected by given geo- metrical relations is, in general, geometrically capable of an infinite number of systems of movements, only those movements which are inconsistent with the geometrical relations being excluded. But we know that only one system of movements is dynamically possible ; that is to say, that if the dynamical conditions of the system be also fully given, the motion is determinate and unique. Now, the dynamical conditions are fully given, if we know the po- sitions of the particles, the forces which act upon them, and their velocities at any given time. Hence, if the forces which act upon the system be perfectly determinate, the initial motion of the system that is to say, the motion which succeeds to an original state of rest is also determinate and unique. 104 INITIAL MOVEMENT OF A But it is plain that this conclusion does not necessarily hold if any of the forces which act upon the system be in their nature ^determinate. If, all the other conditions remaining the same, any of the acting forces be capable of assuming different values, there would be, in general, for every distinct system of values assumed by these forces a distinct system of initial movements. Now, with regard to the forces commonly called external, these are, in general, perfectly determinate, depending entirely upon the positions of the particles of which the system is composed. But it is otherwise, as we have seen, with the geometrical forces. The intensities of these forces are not given functions of the co- ordinates, but distinct unknown quantities, to be determined, if they can be determined, by the equations of motion them- selves. So also with the forces of friction, which depend, as we have seen, upon the other forces of the system, and, therefore, upon the geometrical forces. Hence, it is plain that a system acted on by given external forces, and having a given initial position, may\)Q capable of more than one kind of initial motion. It becomes then an important question to decide in what cases this indeterminateness really exists, and in what cases such a system is capable of but one initial motion. 10. It is easily shown by the principles of ordinary dynamics that, if the supporting surfaces be smooth, the initial motion and all subsequent motions are perfectly determinate. For, let n be the number of the particles of which the system is composed, and m the number of the geometrical conditions by which they are restricted, the equations of the supporting surfaces being in- cluded in these relations. The unknown quantities of the problem are then i. The co-ordinates of the several particles, in number ^n. 2. The intensities of the geometrical forces, in number m. For the determination of these quantities we have the 3^ differential equations of motion, and the m geometrical equations. These equations, which are in number equal to the unknown quantities, will, when integrated, determine these latter in terms of the time and 6n constants, the constants being themselves determined from the ^n initial values of the co- ordinates and the 3% initial values of the velocities. If, as we here suppose to be the case, the system start from rest, the 311 SYSTEM OF PARTICLES. 1 05- initial velocities are severally zero, but the solution is not the less determinate. 1 1 . Let us now consider the case in which the supporting surfaces are rough. Here, as we have seen (p. 99), the equa- tions of the problem enable us to determine completely the condition of the moving particles, if we know that they are moving. But as these equations do not give us any means of deciding which of the particles are at rest and which are in motion, the initial motion of the system, in the sense in which that term is here used, remains still indeterminate. We shall now proceed to show, in the first place, that part of this am- biguity may be removed by the exclusion of systems of move- ments, which, though geometrically possible, are dynamically impossible. For this purpose we shall, in the first place, investigate the following proposition : PROP. II. 12- A number, w, of material particles, resting each upon a rough surface, are connected by known relations, and acted on by given forces. If k of these particles commence to move, the impressed velocities being either nothing or indefinitely small, the remaining n k particles continuing at rest, to de- termine the values of the geometrical forces, and of the forces of resistance for all the particles of the system. Recurring to the notation of p. 98, we have, for the k moving particles, equations of motion of the form, dLi dL z - = Xi + X\ + ttA + A! - + X 3 -y-i + &c. at* ax\ dXi .- r > + r - + 0'*' + x - f; + A i| 2 + &c - i _, n i z mi ~dp = + l + 7 * ~dz "*" d~ + and for the n-k particles which are at rest, equations of equili- brium of the form, p I06 INITIAL MOVEMENT OF A _r ._, r> \ dL\ -^ dL 2 O = JL* + i -f A A + ! + o A + 1 Jt/c + 1 + AI - - + A 2 - - + &c. UiJLj\ . U-Z*/2 _ - + X 2 ~ - + &c. (20) rjt T) \ JL/I ^ 2 p + ! -f- Z k + 1 + y* + * /*+ 1 + AI -= - + A 2 -j - + &c. dzk^i &*+\ Now, if we twice differentiate the first k equations of the supporting surfaces with regard to the time, and after the second differentiation, reject the terms containing the squares and pro- ducts of d#i dt> ~d?* since the initial velocities are zero, or indefinitely small, we shall have k equations of the form dui d z %i du-L d*yi du^ d z 2i 5i W + dyt ~d# + d^ ~d = ' * Substituting for the second differential coefficients from the equations of motion, we have the k equations, o = p- (X, + X\ -f ai^i + &c.) + 4* (F, f F, _4 && -f- &c.) ax\ ay\ (21) ' - Again, we know that when a particle is in motion the force of friction is directly opposed to the motion of the particle. Now, if the particle start without any impressed velocity, the initial motion must be in the direction of the tangential com- ponent of the initial force. Hence, in this case, if we compound into a single force all the acting forces, external and geometrical, and resolve this resultant perpendicular and parallel to the tan- gent plane, the force of friction will be directly opposed to the latter component. (It is obviously indifferent whether in thus compounding the acting forces, we include or exclude the force SYSTEM OF PARTICLES. 1 07 of friction itself, or the normal resistance.) Since, then, the line of direction of the force of friction lies in the tangent plane, we must have a l Z' 1 + /3iFi + 71^1 = 0. (22) Moreover, it is plain from what has been said, that the line of direction of this force lies in the plane containing the normal to the surface and the total acting force ; hence, if we denote by A, fi, C, the rectangular components of this latter force, it is easily seen that JB 7l ) X\ + (A 7l - CaJ T\ + (Ba, - Afr) Z\ = o ; or, replacing A, B, C by their values, as given in (19), omitting the normal resistance and the force of friction (which would evidently disappear from the equation of themselves), we have *L' + & c . ) 7l - f Z, + A, ^+ &c.V I T\ (23) ayi / \ azi j ) If the initial velocity be not zero, its direction will be given, and, therefore, the direction of the force of friction in the tan- gent plane. We shall, therefore, have, in addition to the equa- tion (22), a second equation of the form PX\ + QY\ + RZ\ = o. (24) P, Q, R being known coefficients. Again, since the particle is in motion, the force of friction has its greatest value namely, piR. We have then a third equation, sc; X/ 2 + F/ 2 + ZS = ^R,\ (25) It is plain, therefore, that for every moving particle we have the four equations (21), (22), (25), and (23) or (24). For each quiescent particle we have the three equations of equilibrium, 108 INITIAL MOVEMENT OF A (20). Moreover, the equation (22) expressing the fact that the force of friction is in the tangent plane, is true for every particle, whether in motion or at rest. In every case, therefore, whether of rest or motion, the number of equations is equal to the number of the unknown quantities JT'i, Y\ t Z\, R^ depending upon the forces of resistance. With regard to the geometrical forces, the number of un- known quantities which are introduced by them is, as we know, the same as the number of the connecting equations. Let L = o be one of these equations. Then, if this equation be twice differentiated, rejecting small quantities as before, and sub- stituting the values of the second differential coefficients, we obtain a linear equation between the unknown quantities of the problem. So also of all the connecting equations, provided, as in p. 99, that no one of these equations contain only the co- ordinates of quiescent particles. If there be no such equation, the number of equations is evidently the same as the number of unknown quantities, which are, therefore, perfectly determinate. 13. Know, on substituting the values ofX\, T\ 9 Z\ 9 JR for any of the quiescent particles, in the equation X'* + F' 2 + Z'* = fi 12 R\ we find a value for p greater than the coefficient of friction at that point, it is evident that the supposed movement is impos- sible. All such movements, therefore, must be rejected. Again, we must reject every system of movements for which the necessary effect of the forces acting on any one of the mov- ing particles is to diminish its velocity. For, it is plain that such a force could not set the particle in motion. Now, since there is no impressed velocity, if the acting forces, external and geometrical, be resolved perpendicular and parallel to the tan- gent plane, the initial motion, if there be any, will be in the direction of the resultant of these latter components. Unless, then, this resultant exceed the force of friction, the supposed movement is impossible. Now, we have seen that for a moving particle all the forces are determinate. Let V be the resultant of the forces external and geometrical (excluding the force of friction) resolved along the tangent plane, and R the normal SYSTEM OF PARTICLES. 109 reaction of the supporting surface. Then it is evident, from what has been said, that every system of initial movements must be rejected as dynamically impossible, which does not satisfy the condition, V- pR > o. 14. But, after every system of movements has been rejected except those which are, both geometrically and dynamically, possible, it will frequently be found that the question remains still indeterminate. We have seen, for example, that this will be, in general, the case, when there exists an equation of con- dition involving only the co-ordinates of quiescent particles. The cause of this indeterminateness has been fully considered before (pp. 19, 20, 89, 90). Example. 15. Two material particles, m l5 m 2 , rest upon a rough inclined plane at the points A, B, and are connected by a thin string passing through a small smooth ring fixed to a point, 0, in the plane. If the inclination of the plane be gradually increased till equilibrium be broken, to determine the nature of the initial movement. It is evident that the initial movement of the system may be one of two kinds, i. One of the particles may move, the other remaining fixed. 2. Both particles may move simultaneously. We shall consider these species of movements successively. (i.) If the particle m t move, while m 2 remains fixed, the motion will necessarily be in a circle described round 0. The force of friction at .4 is, therefore, perpendicular to OA, and has its maximum value. Let the plane itself be taken for the plane of xy, being the origin, and the axis of x parallel to the horizon. Let a be the inclination of the plane at the commencement of the motion, and let the angles AOY, BOYloQ represented by |3i, |3 2 , re- spectively, the positive direction of Y being downwards. Then if #1 2/i be the co-ordinates of m 1? we have the equations 1 10 INITIAL MOVEMENT OF A d 2 ^! mi -j = - Z 7 sin /3i + ^m^g cos a cos |3] (26) h ~T^ - ~ Tcos |3i - jum,^r cos a sin /3i 4- m } g sin a. u/c/ T being the geometric force (or the force of tension). Again, for the quiescent particle, m z , we have o = - T sin j3a + o = - Tcos j3 2 + Y' t + w 2 # sin a, where X' t , Y' z are the components of the effective force of fric- tion at B. Now, since the particle m l moves in a circle round 0, we have *i 2 + y\ = const. Differentiating this equation according to the method of p. 106, we have Substituting for the differential coefficients from (26), we find T = m\g sin a cos j3i. (28) Hence, from equations (27), JSTj = ??i,^sinacosj3i sin j3 2 , Y'z = migsm a cosj3icosj3 8 - m 2 g sina. (29) We have, therefore, X 2 ' 2 + Fz' 2 = g z sin 2 a (m^ cos 2 j3i - 2mim 2 cos |3i cos /3 2 Now, since the resultant of X 2 7 , Y 2 ' must not exceed the maximum force of friction, .m^ cos a, we have |u 2 m 2 8 cot 3 a >, or = mf cos 2 ]3i - 2mim 2 cos )3i cos j3 2 + m 3 2 . (30) Again, since the resultant of the tension and the force of gravity acting on the moving particle mi must exceed the maximum force of friction, we must have T 2 sin 2 /3i + (mig sin a - Tcos /3i) 2 > fj?m*g* cos 2 a ; SYSTEM OF PARTICLES. I I I or, substituting for Tfrom (28), ft 2 cot 2 a < sin 2 /3i. Hence, and from (30), we have, putting m\ - nrn^ sin 2 j3i > i - 2n cos j3i cos j3 2 + n* cos 2 ]8i, or (i -f n 2 ) cos 2 /3i- 2wcos |3i cos j3s < o. (31) It is evident that cos ^ and cos j3 2 cannot both be negative, inasmuch as in that case both particles would be above the fixed ring, and would, therefore, necessarily move as independent particles, this movement commencing as soon as ju cot a had fallen below the value i. We learn, then, from (31) that neither cos |3i nor cos j3 2 can be negative ; and, therefore, that both particles must be situated beloio the fixed ring. * Moreover, it appears from (31) that cos j3i must be less than cos]3 2 . Hence, we infer that if either particle move while the other remains at rest, the moving particle must be that for which the connecting string is most oblique to the axis of ?/, and, therefore, most oblique to the vertical. If the condition (31) be satisfied, the particle m } may commence to move by itself as soon as the inclination of the plane exceeds the value given by the equation tan a = jucos fBi. If cos pi im^z mf + m 2 2 cos 3 3 w^ + w 3 2 ' neither particle can move separately. (2.) If both particles move simultaneously, we have for the movement of m t the equations # mi -~ ufr Similar equations hold for the movement of m z . We have also, -~ - Tcos |3i 4- Y\ + m^g sin a. ufr I I 2 INITIAL MOVEMENT OF A from the geometrical condition of the inextensibility of the string, \/#i 2 + 2/i 2 + V ' % /urn^g cos a, V 2 > /mm 2 g cos a. Now, the equation (36) may be put under the form m z V* ( T - m } g sin a cos |3i) (V l - fjLm } g cos a) + mi V l ( T m^g sin a cos ]3 2 ) ( V 2 imm 2 g cos a) = o. But, inasmuch as the first and third factors in each of these pro- ducts are essentially positive, the remaining factors must have opposite signs. Hence, the value of T must lie within the limits m\g sin a cos |3i, and m 2 g sin a cos |3 2 . To determine the least inclination of the plane for which movement of both particles is possible. Considering T as a function of a, determined by the equa- tion (37), we have to solve the following problem : To find the smallest value of a which will satisfy the con- ditions Vi - fining cos a > o, F 2 - ]u.m 2 g cos a > o. Here it is plain that the limiting value of a is found by equating one of these quantities to zero. Let F, - jnm } gcos a = o. (38) Then, from equation (37) we have either Vz - nm z g cos a = o, or T - m 2 g sin a cos )3 2 = o. (39) (a). The former of these alternatives combined with (38) gives 1 14 MOVEMENT OF A PARTICLE T* - 2m l gTsm a cos |3i + m^g 2 - sin 2 a = ^m^g* cos 2 a (4o) T* - 2m 2 gTsm a cos j3 2 + ^ 2 2 # 2 sm 2 = ^ m ^9 z cos 2fl - Subtracting these equations one from the other, we have 2T(m 1 cosfi l - ra 2 cosj3 a ) =(w 1 2 - m 2 2 ) ( i - ju 2 cot 2 a) # sin a (41) Let mj > m z . Then, since the inclination of the plane is greater (or, at least, not less) than the angle of friction, i - ju 2 cot 2 a is positive. Hence f^cos /3i > rft 2 cos j3 2 - Again, multiplying equations (40) by m 2 2 , m^, respectively, and subtracting, we have (Wi 2 - ra 2 2 ) T = 2m 1 m z (m^. cos j3 2 - m z cos ]3i) g sin a. (42) Hence, evidently, ra^os j3 3 > m 2 cos |3i. The suppositions V\ - iim\g cos a = o, Vz- juw 2 <7 cos a = o, are, therefore, impossible, unless C08J3, > , < ,_ ( } cos p 2 m\ m 2 If the ratio of the cosines lie within these limits, the value of a is found from (41) and (42). We have then i-ju 2 cot 2 a= / i- -^ It is easily shown that the value of a obtained from this equation is less than that obtained for the case in which one particle moves separately. If, then, the ratio of the cosines lie within the limits (43), the inclination at which the movement of both particles is possible will be obtained sooner than the in- clination at which one particle may move separately. More- over, if the ratio of the cosines lie within the limits (32), in which case separate motion is impossible (it will, as is easily seen) lie within the limits (43), in which case joint movement of the kind here discussed is possible. UPON A MOVING SURFACE. 115 (&). With regard to the second alternative, T-m z = X (*' > y . *> 0' (44) 0, i, x being known functions. Let a?, y, 2 be the co-ordinates of the moving particle. Then, if a, /3, 7 be the angles which the tangent to the path of the par- ticle on the surface make with the axis of the co-ordinates, it is known that (45) where dt dtj*\dt dt) * \dt dt Let ^V be the normal resistance, and a, j3, 7 the cosines of the angles which it makes with the axis. Let, also, X, F", Z be the components of the acting force. Then the equations ofmo*.ion are (46) where the values of the differential coefficients of a?', y', s' are to be substituted from equations (44). Since, moreover, the co-ordinates #, y, z always denote the same point as the co-ordinates #', y', s', we may put x' - x, y - y, z 1 - z in the functions ^>, ^/> ^' ^ e nav thus three equations between the unknown quantities #, y, g, N. The fourth equa- tion is that of the supporting surface. Let F(x l1 y^ s x ) = o, be the equation of the surface, referred to axes fixed with regard to the surface. Then the equation of the same surface referred to the axes fixed in space, will be JF^tf-r + by + cz + e, ax -f b'y + c l z + e, ax + U'y + c"z + e") = o, a b c e, a') &c. being given functions of t. UPON A MOVING SURFACE. I I 7 Example i. A heavy particle is projected along a rough horizontal plane, which moves uniformly parallel to itself without rotation. To determine the movement of the particle. Since the only force acting is directly opposed to the move- ment of the particle on the plane, it is evident that the relative path will be a right line ; and, since the intensity of this force is constant, the velocity in this path will be uniformly diminished. The absolute motion of the particle in space is found by com- bining the uniformly retarded motion in this relative path with the uniform motion by which the path is carried parallel to itself. The path of the particle in space is evidently a para- bola. Example 2. A rough horizontal plane revolves round a vertical axis with a constant angular velocity. Required to determine the motion of a particle placed without initial velocity on the plane at any point other than the centre of rotation. Let w be the angular velocity of the plane, x y the co-ordi- nates of any point on it, referred to fixed axes in the plane, the centre of rotation being the origin. Then the components of the velocity of the point x y, considered a point in the plane, are to y parallel to #, and - w x parallel to y. Hence, the components of the relative velocity of the material particle are dx dy Tt'^'Tt^"- Since, then, the direction of the force of friction is directly opposed to the relative velocity of the particle, the equations of motion are ^~ 2 = ~ where i. the mass of the particle being supposed I I 8 MOVEMENT OF A PARTICLE If the co-ordinates be transformed, so that the axes, instead of being fixed in space, shall revolve with the plane, these equa- tions become d z x dy dx _4 2(U --u,* = - w7 - (48) 2 y = o. ds 2 \ ds y J ds ds Multiplying these equations by dx, dy, respectively, and adding, we have vdv x pgds - o> 2 (xdx + ydy] = o. Whence, by integration, v 2 + 2fjigs - w 2 (x 2 + y 2 ) = const. (50) But, since the initial value of the absolute velocity is zero, the initial value of v 2 is w 2 (x or = cos e. Substituting for a, /3, 7 from (53), we have a'tf't - X) + |3' (ft - F) -f y'tft - Z) ~ -xy or= COS> 120 MOVEMENT OF A PARTICLE UPON A MOVING SURFACE. Example 4. A material particle is placed on a rough horizontal plane, which is made to revolve round a vertical axis with a velocity commencing from zero, and continually increasing. To deter- mine the velocity at which the particle will commence to move on the plane. So long as the particle continues at the same point on the plane, its co-ordinates are given by the equations, x = a sin u, y = a cos w, z = o, where u <= J wdt. We have then in Prop. IV., (jj't = - ao> 2 sin u + a cos u r- dt . + . du) \L t = - aor cos u - a sin u -=-. at Also, X - o, F=o, Z= g, a'= o, j3' = o, y = i. Substituting these values in the condition (54), we find, without difficulty, H 2 g z du* -f- > or = w 2 + -r:-. a 1 dt 2 Hence, if the velocity increases more rapidly than is indi- cated by the law ao> = pgt the particle, will commence to move on the plane simultaneously with the commencement of the rotation of the plane itself. 121 CHAPTER V. MOTION OF A SOLID BODY. I. Motion of a Solid Body on a Rough Plane. i. THE problem which we propose to discuss here is in gene- ral so complex that it does not admit of a complete solution, except in some of the simplest cases. All that can be done in the case of the general problem is to state the physical principles on which the solution depends, and to deduce the general equations, from the integration of which it might be obtained. We shall, in the first place, investigate the problem in its gene- rality, and then consider some particular cases which admit of a solution more or less complete. Let the fixed plane be taken for the plane of xy. Let x\ y v z\ be the co-ordinates of the centre of gravity, and , r\ the com- ponents of the velocity of the point of contact of the moving body with the fixed plane, both referred to the fixed axes. Conceive the forces acting on the body (not including the resistance of the rough plane) to be reduced to a single force passing through the centre of gravity, and a single moment taken with respect to the same point. Let X^ Y^ Z^ be the components of the force, and L\, M^ NI the components of the moment. Let also X', Y f , Z' be the components of the resist- ance of the rough plane, all these components being referred to the fixed axes. Consider now the system of principal axes which pass through the centre of gravity. Let a, ]3, y, be the co-ordinates of the point of contact referred to these axes, and abc, a'b'c', a'b"c" the cosines of the angles which they make with the fixed axes xyz. Let p, (7, r be the components of the angular velocity of the body referred to the principal axes, and 0, <, i// the ordinary polar angles which determine the position of these axes with re- gard to the fixed axes. Then, taking the mass of the body*- i, we have the following equations : K 122 MOTION OF A SOLID BODY ON A ROUGH PLANE. ( i .) For the motion of the centre of gravity, (2.) For the rotation round the centre of gravity, A, B, C being the principal moments of inertia, Adp + (C-B) qrdt = (aL* + bM l + cN, + fiZ - y Y) dt Bdq +(A- C)prdt = (a'A + b'M l + c'N l + y X- Z) A (2) Cdr + (B - A) pqdt = (a'L, + V'M, + c"^, -f a F - j3^) dt, where X = aX' 4- a'Y' + a"Z', F = bX' + &c., Z = cX' + &c. (3.) We have, besides, the well-known equations, pdt *> sin sin Odifr - cos 0c?0 qdt = cos sin Od$ + sin 0d0 (3) rdt = d - cos 0dt//. Then, since aftc, a'6V, d'U'c" are all expressed in terms of 0, ^, ;//, these equations contain fifteen unknown quantities namely, tfi^Si, pqr> 00^, aj3y, X'Y'Z'. We require, therefore, six equations in addition to those given above. These are ob- tained as follows : Let u = f(x,y,z) = o be the equation of the moving body re- ferred to the principal axes. We have then, /(a, 0, 7) = o. (4) Moreover, since the tangent plane at the point a, y3, y, is the plane of b T + b 7n + b "T = ' (5) da df$ dy da dp dy Also, since z l is the perpendicular from the origin of the move- able axes upon the tangent plane at the point aj3y, we have du 2 du* du 2 du ^ du du ^ + ^ + rf? =a ^ + /3 ^ + r ^- We have thus four of the six equations which are required. MOTION OF A SOLID BODY ON A ROUGH PLANE. i 23 The remaining two will differ according to the nature of the motion of the solid body. If there be slipping at the point of contact that is to say, if the point of the body which is in contact with the supporting plane have a finite velocity parallel to the plane the force of friction will have its greatest value, and its direction will be im- mediately opposed to that of the resolved velocity. Now, the velocity of the point of contact resolved parallel to the axes of x and y, respectively, will be - q a ) (7) - r/3) + V (ra - py) + and the two equations sought for are o, X'* + '* = M 2 Z' 2 . (8) If, on the other hand, there be no slipping at the point of contact that is to say, if the point of the body which is in con tact with the supporting plane have no finite velocity parallel t that plane the required equations are evidently ? = O, r] = O. (9) In this case the motion is a pure rotation round an axis passing through the point of contact. 2. The principle to be employed in deciding between these two alternatives is perfectly analogous to a principle before enunciated in Statics. As in the statical problem, the force of friction will always, if possible, reduce the particle which is in contact with the rough surface to rest; so in the dynamical problem, the force of friction will, if possible, determine such a motion of the moving body, that the point of the body which is in contact with the supporting surface shall have no finite velocity parallel to the surface. The second alternative is, therefore, always true, if it be possible. The rule to be adopted is, therefore, as follows : Solve the problem, adopting the second alternative. Having 124 MOTION OF A SOLID BODY ON A EOUGH PLANE. obtained the values of X', P, Z', substitute them in the equa- tion X* + Y* = n*Z'*, (10) and thus determine the value of p. If, then, we find p not to be greater than p, the coefficient of friction, the solution is pos- sible, and therefore true. If the value found for p be greater than ju, the solution is impossible, and we must re-solve the problem, adopting the first alternative. It is evident that the nature of the motion may change from one of these species to the other. The times of such changes are determined as follows : Suppose that, having solved the problem on the second supposition, we find ju' 2 =/(0> where /(o) < ju. Then, if for any time ^ we find/(^) = ju 2 (unless this be a maximum value), the motion passes at this instant from the second to the first kind, and the problem must be solved anew on the first sup- position. Suppose that we obtain from this solution F-tV-JiXO. (") Then if for any value t% we find F(t z ) = o (unless this be a minimum value), the motion passes again to the second kind. Thus any number of changes may be determined. Example. 3. A homogeneous sphere is projected with given velocities of translation and rotation along a rough horizontal plane. To determine the nature of the movement and the moment at which it becomes one of pure rolling. Taking the given plane as the plane of xy, we have for the motion of the centre of gravity the equations d*x fry where X, !Fare the components of the force of friction. Also if u, r, w be the angular velocities round the axes of #, y, 2, re- MOTION OF A SOLID BODY ON A ROUGH PLANE. 125 spectively, k the radius of gyration round the centre, and r the radius of the sphere, . du 7 dv ~ r dw *-di~ Tr '*a- Xr > W = ' (I3) Let , 17, be the components of the velocity of the point of contact parallel to x and y, respectively. Then y, dx dy x **-""' = di + ru - (I4) Then we obtain from (12) and (13), putting & 2 + r* = n*k z , But, since the motion will, in general, be a mixture of slipping and rolling, Xn - Y?i = o, and X* + Y 2 = ^g\ Hence Substituting these values in the equations (15), we obtain readily r\dr\ = - fJigdt g 2 + rj 2 . Whence, by integration, + (- (e - ws*-). (17) Let v' be the initial velocity of the point of contact. Then, evidently, c = v' 2 , and the moment at which pure rolling com- mences is given by the equation It is easily seen that this motion will continue indefinitely. For, at the moment when pure rolling commences, we have g =7 o, r\ = o ; whence X = o, Y - o ; and, therefore, from equa- tions (15), 126 MOTION OF A SOLID BODY ON A ROUGH PLANE. Again, differentiating equations (12), and equating to zero , 17, and their first differential coefficients, we find that the second differential coefficients also vanish, and so of all higher coefficients. We must have, therefore, generally, = o, q = o, showing that pure rolling continues indefinitely. Multiplying the equations (15) by j, , respectively, and subtracting, we find o ; whence, by integration, ,j = m = tan a (suppose). We have then, JT =- ju# cosa, Y = -ju<7sina. Substituting these values in (12), and integrating, we have x = at -- figt 2 cos a, y = bt -- pgt 2 sin a, a, b being arbitrary constants, and the origin being taken at the initial position of the point of contact. Eliminating t between these two equations, we find (ay - bx) (b cos a - a sin a) = fig (x sin a - y cos a) 3 . ( 1 8) Hence, the path of the point of contact, and, therefore, of the centre of the sphere, is in general a parabola. The parabola will become a right line if a sin a = b cos a. This condition gives, evidently, arj - bo = o, denoting by r? , ? , the initial values of , r\. Substituting for , r\ Q their values, a - rv Q , b + ru , we find au + bv = o. Hence, if the rotation originally communicated to the body be resolved into two one round a horizontal, and one round a vertical axis the motion of the centre will be aright line, if the MOTION OF A SOLID BODY ON A ROUGH PLANE. 127 horizontal axis be at right angles to the direction of the initial motion of the centre. If the initial motions of translation and rotation be communi- cated to the sphere by a single impulse as in the case of a billiard ball the motion of the centre will be necessarily rectili- near. For, if we take the origin, as before, at the initial position of the point of contact, and the axis of x parallel to the direction of the impulse, we shall have 5 = o, u = o, and, therefore, from (14), T)Q = o ; whence a = o. The equation (18) becomes, there- fore, y = o, denoting that the motion of the centre is parallel to the axis of x. 4. We proceed now to consider specially the case in which the motion of the solid body is a pure rotation round an axis, fixed or variable. This kind of motion may be thus defined The motion of a solid body is a pure rotation, if it be possible at each instant to assign a line such that if a plane be drawn through this line and any point of the moving body, the motion of that point shall be at that instant perpendicular to the plane so drawn. The condition of pure rotatory movement may be otherwise expressed, as follows : It is known that the most general movement which can be given to a solid body may be conceived to be made up of the two following namely, i , a motion of translation common to every point in the body ; 2, a motion of rotation round a point, which may be chosen arbitrarily. Now, the motion will be a pure rotation when (and only when), this decomposition having been effected, the motion of translation is at right angles to the axis of rotation. This may be readily shown from the expressions given by Lagrange for the displacements of the different parts of a solid body. These are of the form $x = A + Qz - Ry ty = B + Itx-Pz (19) $z = C + Py - Qx, 128 MOTION OF A SOLID BODY ON A ROUGH PLANE. where A, B, C, are the components of the motion of translation, and P, Q, H, are proportional to the index cosines of the axis of rotation. Now, if the motion be a pure rotation, the movements of all the particles must be at any given instant parallel to the same plane. Hence it must be possible to determine the co- efficients a, ]3, 7 so as to satisfy the equation, a$x -f j3Sy + y<$z = o, (20) independently of #, y, z. Substituting the values of &&, Sy, 2, we find thus, ^a+3/3 + C r = o,:? = | = |. (2.) From these equations it is easy to see that AP+BQ+CR=o, (22) which equation evidently expresses the proposition enunciated above. We shall now proceed to give some general propositions connected with this kind of motion, which will be found useful in our subsequent investigations. PROP. I. 5. If the motion of a solid body be at each instant a pure rotation, the equation of rotation round the instantaneous axis will be ^Irf = *L; (23) where dO is the elementary angle described by the rotating body, /the moment of inertia round the instantaneous axis, o> the velocity of rotation, and L the statical moment of the acting forces round the instantaneous axis. This is easily proved from the general equation of vis viva, MOTION OF A SOLID BODY ON A ROUGH PLANE. I 29 or for a solid body, For, since the motion is a pure rotation, we have for each ele- ment dm, p being the perpendicular from dm on the axis, also, if ^ be the angle which a plane passing through the mole- cule, x, y, z, and any given tine, makes with a fixed plane, it is easily seen that the statical moment, round this line, of a force acting on the molecule will be given by the equation, T Y dx 4. V dy 4- 7 dz Li JL -7- + J: + A . dq> d$ d$ Multiply the equation (24) by a; or -^-, which, being the same for every element, may be introduced indifferently without or within the sign of integration ; and it is easily seen that the equation may be written u> 2 \p*dm = 2L, d(p which is identical with (23). PROP. II. 6. Let the movements of all the points of a solid body be sup- posed to be parallel to the same fixed plane. Let G be its centre of gravity, and let a plane be drawn through it parallel to the Fig. 19. fixed plane. Let the plane of the paper represent the plane so drawn, and let A be the projection of the instantaneous axis, s 130 MOTION OF A SOLID BODY ON A ROUGH PLANE. which is, of course, perpendicular to this plane, and let a circle be described on the diameter, GA. Draw in the plane of the paper a line GP parallel to a fixed lirle, and through P a line perpendicular to the plane of the paper. Then, if /be the moment of inertia, and L the statical mo- ment of the acting forces with regard to this line, w the angular velocity, and the actual angle of rotation, the equation ob- tained in the foregoing proposition for the instantaneous axis namely, ~ Iw* = iL, afy is true in this case also. Since the movements of all sections parallel to the fixed plane are the same, it will be sufficient to consider motion in two dimensions. Let the axes of x and y be taken parallel, re- spectively, to the fixed directions, PA, GP. Assume/) = AP q = GP, and let k be the radius of gyration round an axis through G perpendicular to the plane of the paper. Then, since the motion is a pure rotation round the axis, whose projection is A, we have from the foregoing proposition, taking the mass of the body = i, where L' is the statical moment with regard to the instantaneous axis. But if a?, y, be the co-ordinates of any point in the body referred to axes through P, we have L = \{yX- (x-p) Y) dm = L+p$Ydm. But from the motion of the centre of gravity, whose co- ordinates with regard to the fixed axis are x l y l9 we have Hence MOTION OF A SOLID BODY ON A ROUGH PLANE. 131 Now, since the motion of the centre of gravity is a pure rota- tion round an axis through J., we have -j- = 2i, a^> a

2 = 2g (p sin a + q cos a). (33) This equation admits of being integrated at once when the equa- tion of the section is known, giving to 2 in terms of 9, and, as be- fore, 9 in terms of t. If, now, in the equation (30) we replace fj. by an unknown function, 0, and substitute the value of w 2 , obtained from (33), we shall have an equation to determine in terms of 9 ; and, therefore, in terms of t. If this value (positive or negative) do not exceed ^ the motion is one of pure rolling. If the value of lie outside these limits, the motion is slipping as well as rolling, and we must recur to the former solution. In the preceding discussion the centre of gravity is supposed to be in front of the perpendicular to the inclined plane through T. All the equations are made applicable to the case in which the centre of gravity lies behind this line by simply changing the sign of q. They are made applicable to all cases if we replace q by its value -^. For it is easily seen that this ex- pression changes sign as the centre of gravity passes through the perpendicular to the inclined plane. We now proceed to consider generally the case of a rolling body. PROP. III. 9. If a solid body of any form roll without sliding upon a rough plane, the absolute accelerating force at any point of the body is equal to the relative force with regard to the point of the body which is in contact with the plane. Taking the fixed plane as the plane of xy, let x y z be the co-ordinates of a point in the body with regard to a fixed origin, and x y z the co-ordinates of the same with regard to the point in the body which is in contact with the plane. Let, also, ?, rj be MOTION OF A SOLID BODY ON A ROUGH PLANE. 135 the co-ordinates of this latter point with regard to the fixed origin. We have then x = x 1 + , y = y + TJ, z = z 1 -f ?, .where is evanescent. c^aj _ cJV d*Z d*y _ d*y' d*y d z z _ d?z d-% di? = 1& + ~dt*' ~dt 2 = ~di? + di*' di?~'d? + di?' But, since the point ? rj is, for the moment, at rest, and since, moreover, its first motion is perpendicular to the given plane, and changes sign at the instant of contact, we must have Whence These equations evidently contain the proposition, which may be otherwise enunciated thus : The absolute accelerating force, at any point of a body which rolls without sliding, is equal and opposite to the relative accelera- ting force at the point of contact, estimated with regard to the first point as origin. PROP. IV. 10. If a solid body roll without sliding upon a fixed rough plane, to determine the force of resistance of the plane at the point of contact. Let it be supposed that in Prop. III. the first point is the centre of gravity of the rolling body. Then we know that the relative motion of the body with regard to this point is the same as if it were fixed all the forces remaining unchanged ;* and, * It must be carefully observed that among tbese forces, whose values are sup- posed to remain unchanged, the resistance of the rough surface is included. It is not meant that the motion of a rolling body round its centre of gravity is actually the same, whether the centre of gravity be or be not fixed. This would evidently be untrue. If the geometrical condition of the system namely, that the rolling 136 MOTION OF A SOLID BODY ON A ROUGH PLANE. therefore, that the relative accelerating force at any point of the body, 'with regard to the centre of gravity, is not altered by conceiving this latter point to be fixed, and thereby converting relative into absolute accelerating force. Hence, and from the principle just stated, it is evident that the absolute accelerating force at the centre of gravity is equal and opposite to the absolute accelerating force which would exist at the point of contact, if the centre of gravity were considered, for the instant, as a fixed point. Let X YZ be the components of the acting force at any point of the body, referred to the system of principal axes pass- ing through the centre of gravity. Assume X = jXdm, Y= \Ydrn, Z=\Zdm. Let, also, X', Y', Z' be the components of the resistance of the rough surface referred to the same axes. Then, the mass of body shall remain continually in contact with a fixed surface, be supposed to con- tinue, the motion of this body round its centre of gravity would, in general, be changed by fixing this centre. So also would the resistance of the supporting sur- face, and, therefore, the condition given in the text would not be fulfilled. Fully stated, the assertion of the unchangeability of the motion round the centre of gravity is as follows : Conceive a second body identical with the rolling body in all respects but one namely, that the supporting surface is replaced by a force equal in every respect to the resistance which the surface actually exerts upon the rolling body. Conceive, also, a third body identical with the second body in all respects but one namely, that its centre of gravity is fixed. Then we know that the motion of this third body round its centre of gravity is identical with that of the second ; and, therefore, that all the equations and formulae connected ^with the motion of a solid body round a fixed point are applicable to the motion of the second body round its centre of gravity. But the equations of motion of the second body are, as is evident from the ordinary methods of mechanics, absolutely identical with those of the first. Hence the equations and formulae alluded to are applicable to the motion of the first body. All difficulty of conception will be removed, if we bear in mind that the third is an imaginary body subjected to one geometrical condition namely, the fixity of the centre of gravity, and acted on by a force equal in every respect to the resist- ance which the supporting surface exerts upon a body not subject to this con- dition. MOTION OF A SOLID BODY ON A ROUGH PLANE. 137 the body being unity, the components of the absolute accele- rating force at the centre of gravity are X+X', F + Y', Z+ Z. Again, if #', y\ z 1 be the co-ordinates of the point of contact with regard to the same axes, and p, g, r the components of the angular velocity, it is known that the components of the accelerating force which would exist at the point of contact, if the centre of gravity were fixed, are , do . dr dt " y dt ' + w - py')p (35) t ' dt ~ pz ^ P + ^ ~ qz ^ 9 ' These three components are, by the principle stated above severally equal to - (X + X')> - (F+ Y), - Z + Z'). But if .4, B, C be the principal moments of inertia, and if we put L = \(yZ-zY}dm, M = \(zX - xZ)dm, N = J (x Y - yX] dm, we have Adp + (C- B)qrdt = (L + y'Z' - 'F) dt Bdq + (A- C)prdt = (M+ z'X' - x'Z')dt (36) Cdr +(B- A)pqdt = (N + x'Y- y'X') dt. Substituting then the values of the differential coefficients in the expressions (35), and equating them severally to - ( X + X') 9 - (Y + Y), - (Z + Z'), we have three linear equations for the determination of the three unknown quantities, X, Y', Z'. The general result is, however, too complicated to be of much interest. 138 MOTION OF A SOLID BODY ON A ROUGH PLANE. ii. As an example, let us consider the case of a sphere which rolls on a rough plane, the acting force passing through its centre: We have then A = = C, Z = o,J7= o,N=o. Since, moreover, all axes through the centre of gravity are principal axes, we may take at the giveru instant the axes x, y', z' 9 parallel and perpendicular to the supporting plane. We have then x - o, y = o, z' =a . We have also, since the motion is at the instant a rotation round an axis parallel to the plane of x y, r = o. The first two expressions (35) become, therefore, do dp a dt>- a ft' and the first two of the equations (36) become dp aT' dq _ aX' ~dt = " T"' ~dt = ~A' Substituting these values, and putting A = k 2 (where k is the radius of gyration), we have Hence (k> + a*)X' = - k*X, (k* + a 2 ) F' = - k*Y. These equations denote that the effective force of friction is parallel and contrary to the component of the acting force parallel to the plane. The intensities of these two forces are to each other in the ratio P : k z + a 2 ; or, in the present case, 2:7. Hence it is evident that the motion of the centre of the sphere is the same as if the plane were smooth, and the acting force were diminished in the proportion 7:5. Thus, for example, if gravity be the sole acting force, the centre of a sphere which moves on an inclined plane sufficiently rough to prevent sliding, will describe a parabola similar to that described by a single particle on a smooth inclined plane if 7 sin (incl. of smooth plane) = 5 sin (incl. of rough plane). INITIAL MOTION OF A SOLID BODY. 139 II. Initial Motion of a Solid Body resting upon one or more Fixed Rough Surfaces, PROP. V. 12. If an indefinitely small movement be communicated to a solid body resting in a given position upon one or more fixed rough surfaces, the resistances of the supporting surfaces will be, in general, determinate. Let #!, y lt z it be the co-ordinates, referred to axes fixed in space, of a certain determinate point in the solid body. Let, also, 0, <, \fj be the polar angles determining the position, with regard to the same axes, of a system of rectangular axes fixed in the body. We may conveniently take the centre of gravity for the determinate point, and the principal axes which inter- sect there for the system of moveable axes. Let a?, y, z be the co-ordinates of the point of contact with one of the supporting surfaces, referred to the fixed axes, and a/, y', z' the co-ordinates of the same point referred to the move- able axes. We have, then, from the equation of the supporting surface, /(*,y,*)-o; (38) and from the equation of the surface of the body itself, F(x, y, z'} = o. (39) Let, also, a, j3, 7 be the direction cosines of the common nor- mal at the point of contact referred to the fixed axes, and a', j3', y the direction cosines of the same line referred to the moveable axes. We have then, df df df dx dy ' dz' (40) dx* dy n dz 1 where tlF* v = l~^~ 1 40 INITIAL MOTION OF A SOLID BODY. But we know from the ordinary equations of transformation that x 1 ) y\ z' may be expressed as functions of the nine quantities, x, y, z, #j, y,, Sj, 6, 0, i//. Similarly, a', ]3', y may be expressed as functions of the six quantities, a, )3, 7, 0, 0, 4/. Substituting these values in the equations (40), we have six equations, five of which are independent, between the quanti- ties, #, y, s, #1, y 1} g lf a, )3, 7, 0, 0, i/>. Eliminating between these and the equations (38), (39), the quantities x y s, a |3 7, we have a single equation of the form F(a l9 y lt z l9 0,0,i/0 = o. (41) Differentiating this equation twice, and making the first differential coefficients vanish, according to the principle stated in p. 1 06, we have \AJJL, I* U/ l*A M. MT s x id dP ' <%i 'de + ~dfdP ~ A similar equation will be found for each supporting surface. Let /be the effective force of friction at the point of contact, and , b, c its direction cosines. Let also JV" be the normal re- action of the supporting surface. Then, if m be the mass of the body, we have, from the motion of the centre of gravity, m -~ = mX + aN + af + aN' + a'f + &c., d% ^ f tlT* 'X> ff m -r ~ m Z + 7-^V + cf -i- 7 Jv f / H- &c. JV", &c., /', &c., being the forces of reaction and friction at the other points of contact. Again, differentiating the well-known equations, INITIAL MOTION OF A SOLID BODY. 141 O d$ dO p - sin d> sm a -77 cos 9 -7-, at at n dif, dO a = cos (b sm y -7- 4 sin rf> -, a^ a C?0 xj (^^ and making the first differential coefficients vanish, as before we have dp a dty d*0 do M . d 2 -* = cos sm -^ + sin , (44) But from the equations of rotation round the centre of gravity applied to the commencement of the motion, \ve have if-^M'S-* (4S) L, M, N being the components of the statical moment of the act- ing forces, including the forces of resistance, and ^4, j8, (7, the principal moments of inertia. Substituting these values in the equations (44), we have three equations to determine the diffe- rential coefficients d*0 &<> d Substituting the values of the several differential coefficients of the second order in the equation (42), we have one equation between the quantities -ZV,/, a, b, c, -ZV',/', &c. A similar equa- tion will be found for each point of contact. If now the motion given to the body be a motion of slipping at each point of contact, a, 6, c, a', &c., are known, and / = /n N 9 f = fi'N\ &c. The only unknown quantities, therefore, are iV, JV', &c., which are fully determined by the equations which have been found. 142 INITIAL MOTION OF A SOLID BODY. But if the motion of the body be, at some one cf the points of contact, a motion of pure rolling, f may not have its maximum value, and its direction in the tangent plane will not be given. This case requires, therefore, to be treated separately. PROP. VI. 13. A solid body rolls without slipping on a rough surface, the velocity being indefinitely small. To determine the direc- tion and magnitude of the effective force of friction at the point of contact. We have given in Prop. IV. the method of determining the resistance which a rough surface exerts upon a body which rolls without slipping upon the surface of another body. The results there obtained may be adapted to the case of initial motion simply by making p = o, or = fjiE be fulfilled, where F and E are to be replaced by the values found above. If q be positive that is to say, if the centre of gravity be, as in the figure, in front of the perpendicular TP 9 the greatest inclination for which an initial movement of pure rolling is possible, will be found by equating to zero the value of It. This will give tanz = - -. (co) pq If q be negative that is to say, if the centre of gravity be behind the perpendicular TP, there is no limit to the inclination for which an initial movement of pure rolling is possible, if the plane be sufficiently rough. Thus, if the coefficient of friction be equal to, or greater than & + q* a motion of pure rolling is possible for all inclinations. 1 5. It appears a paradoxical result that a motion of pure rolling should be possible if the plane be vertical. In such a case it might be supposed that no pressure could exist between the cylinder and the plane, and, therefore, no friction. If this be INITIAL MOTION OF A SOLID BODY. 145 so, it is plain that the cylinder should fall freely without any rotation. To explain this seeming paradox we must remark, in the first place, that, if the body begin to revolve, there will be de- veloped instantaneously, before any finite velocity has been attained, a finite pressure at the point of contact. This finite pressure will, of course, be accompanied by a finite force of friction, and there will no longer be any difficulty as to the motion supposed. Now, if the mechanical arrangement, by which the cylinder was supported before the motion commenced, was such as to cause no pressure at the point of contact, and if none be de- veloped by the particular mode in which the body is set free ; or, in other words, if the initial value of the pressure be zero, the cylinder will fall freely without rotation. But, let it be supposed that the initial value of the pressure, and, therefore, of the possible force of friction, is finite. Then, how short soever be the time during which this force is sup- posed to act, a rotatory motion will commence, which will, as we have seen, instantaneously develop a finite pressure at the point of contact, and, therefore, a finite force of friction ; thus rendering the supposed motion possible. Even if the cessation of the initial pressure be instantaneous, so also is the com- mencement of the rotatory motion, and, therefore, the develop- ment of a new pressure. In this case it does not seem possible to decide what is the nature of the initial motion. The ambiguity which meets us here is one of those which are due to the abstractions of Kational Mechanics, and can be removed only by an examination of the problem as it really ex- ists in nature. Before considering it in this point of view, however, we shall illustrate the same principle by another example Example 2- 1 6. A rigid, weightless, rod carries at one extremity a heavy particle, the other extremity resting against a rough vertical wall. If the system be left to itself without initial velocity, to u 146 INITIAL MOTION OF A SOLID BODY. determine the motion, the loaded extremity being initially higher than the other. Let AB (Fig. 21) be the wall, EP the rod, and EH a hori- zontal line. Let it be supposed that HEP is not greater than the angle of friction. Then, if the rod begin to revolve round E, there will be a finite pressure against the wall in the direction PE ; and, as HEP is less than the angle of friction, there will be no motion at E, and the rod will continue to revolve round this point. Then, inasmuch as the pressure against the wall jnust always lie in the line PE, and as the angle HEP is con- Fig. 21. stantly diminishing, there can be no slipping at E, and the motion will be a pure rotation round E until the pressure vanishes. To find in what position of the rod this happens, put HEP = 0, EP = a. Then, from the ordinary formulae for the pressure on the point of support in the case of a simple pendulum, we find, Pressure (directed in PE) = g sin 9 - ato 2 , where w is the angular velocity. But if be the initial value of0, tfw 2 = 2g (sin - sin 0). Hence, Pressure = g (3 sin - 2 sin ). Equating this expression to zero, we have the value of 0, for which the motion ceases to be a rotation round E. Thus, for example, if the angle of friction = 60, and if the rod be origi- nally inclined to the horizon at this angle, the pressure will not vanish until the rod has revolved through an angle of nearly 25. 17. We see from this example that, if the body begin to re- volve, it will revolve through a finite angle before the nature of the motion changes. But so long as we adhere to the abstractions of Rational Mechanics, we have no means of deciding whether or not it will begin to revolve. For, according to these ab- INITIAL MOTION OF A SOLID BODY. 147 stractions, the rod being regarded as absolutely incompressible, and, therefore, inelastic, the liberation of the body from the constraint by which it was prevented from moving, the cessation of the pressure corresponding to the state of rest, and the de- velopment of a new pressure corresponding to the state of motion, are absolutely simultaneous. It is, therefore, impossible to decide, so long as we preserve these abstractions, whether the initial motion be or be not a motion of rotation. But this ambiguity is at once removed if we consider the rod as it really is a body gifted with a certain amount of elasticity. If this be taken into account, it will be seen that the liberation of the body from restraint, and the cessation of the pressure corresponding to the state of rest, are not simultaneous. For this pressure, resulting from the compression of the rod, will not cease until the rod resume its natural state. If, then, the mechanical arrangements be so contrived that the rod shall be set 'completely free before it has resumed its natural state, the pressure will, at the commencement of the motion, still continue to exist, and the motion will be, as we have seen, a rotation round E. If, on the other hand, the mechanical arrange- ment be such that the rod shall have resumed its natural state before it is free to move, the pressure will, at the commence- ment of the motion, have ceased to exist, and the rod, with the attached weight, will fall freely without revolving. Example 3. 1 8. Two equal cylinders are placed, with their axes hori- zontal, on a rough inclined plane, and are connected by a cord passing over the lower cylinder and under the higher cylinder, this cord being perpendicular to the axes. If the cylinders be now left to themselves, determine the initial motion. Let the figure (Fig. 22) represent a section of the inclined plane and of the cylinders perpendicular to their axes, and let this section contain the cord. Let be a fixed point, and assume x = OA , x = OA e . Let also 0, 0' be the angles, estimated in the 148 INITIAL MOTION OF A SOLID BODY. direction of the arrows through which the cylinders have revol ved since any fixed epoch. Then supposing the plane of the paper to bisect both the cylinders, and taking the common mass to be unity,| we have from the motion of the centres of gravity the equa- tions, Fig. 22 . -Ji;2-0Bin- Tcosft-F', (51) where T is the tension of the cord, and F, P are the effective forces of friction. Also, from the motion of the cylinders round their centres of gravity, we have 41 S = (F ~ T} a> ki S' = (F - T) a ' (52) k being the radius of gyration. Moreover, the amount of cord coiled round the cylinders at any time together with the length of the cord between them is constant. Now the amount of cord coiled round the two cylinders from the time of the commence- ment of the motion is a (0 + & + 2)3 - 2j3 ), where j3 is the initial value of |3. Also, the length of the cord between the cylinders is (x 1 - x) cos j3, we have, therefore, a (8 + 0' + 2/3) + (x - x) cos /3 = const. (53) whence, remembering that (x' - x) sin /3 = 2fl> dO dO' dx 1 dx INITIAL MOTION OF A SOLID BODY. 149 For the initial motion, differentiating a second time and re- jecting all the first differential coefficients, we have Substituting the values of these differential coefficients taken from the equations of motion, and putting a 2 = w 2 & 2 , we find (n 2 - cos ]3) F f + (n z + cos ]3) F = 2 (n 2 + cos 2 j3) T. (56) We now proceed to consider the various kinds of motion of which the system is geometrically capable. i. Both cylinders may roll without slipping. 2. The upper cylinder may roll without slipping, the lower cylinder rolling and slipping. 3. The lower cylinder may roll without slipping, the upper cylinder rolling and slipping. 4. Both cylinders may roll and slip. 5. Both cylinders may slip without rolling. The last case is plainly impossible dynamically. For if there be no rotation we have, from the equations of motion, F- T= F'. But if P, P be the pressures at the points of contact, we have, since both cylinders slip, F = f iP = fi (g cos a- Tsm j3) + TsinjS). If, then, F= F', we must have T = o ; and, therefore, F= o, F' = o, which is impossible if the plane be rough. With respect to the other four cases, we know from the general principle (p. 123) that the first movement, if it be pos- sible, will actually take place, and that the fourth will take place only when the other three are impossible. But the gene- ral principle does not give us any mode of deciding between the second and third. We shall see, however, that the second kind of movement is impossible. (i). Both cylinders may roll without slipping. In this case dx = adO, dx' = ddO' ; we have, therefore, from equation (54), 15 INITIAL MOTION OF A SOLID BODY. (I + COS 0) ~ + (I- COS /3) g = 0. (58) It is plain from this equation that if both cylinders roll without slipping they must roll in opposite directions ; also from the equations of motion, remembering that and putting a 2 = n*& 2 , ( i + n z ) F = g sin a + (n z + cos ]3) T (59) (i + n 2 ) F' = g sin a + (n 2 - cos |3) T. Hence, as w 2 > 1, we must have F>F', F- T>F' - T. But, inasmuch as the cylinders roll in opposite directions, one of the quantities, F - T, or F' - T, must be negative. The fore- going condition shows that F' T is the negative quantity. We see then that if both cylinders roll without slipping, the upper cylinder must roll down, and the lower cylinder up. To determine the smallest coefficient of friction which will make this motion possible, it will be sufficient to determine this coefficient, so that there shall be no slipping at the point A. For we have already seen that F > F' ; and it is plain that pres- sure at A < pressure at A'. Hence, coefficient of effective friction at A > coefficient of effective friction at A'. If, there- fore, there be no slipping at A, there is, a fortiori, none at A. We have also, from equation (55), when both cylinders roll, Qvd*0' ^#0 (i +cos/3)-^ +(i -cos/3) = o; whence, from equations (52), (i + cosj3)(^- T) + (i - cosj3)CF- T) = o. (60) Substituting from (59) for F, F', we have (i + cos 2 /3) T= #sina ; (61) INITIAL MOTION OF A SOLID BODY. whence i -f n 2 \ i + cos 2 /3/ Also, if P be the pressure at A 9 P= #cos a - T'sin j3- Substituting in this expression the value of T 9 and putting F = /zP, we obtain a value for n which is the smallest coefficient of friction which will cause both cylinders to roll. In order that this motion may be possible, it is necessary that the value for p should be positive and finite. The foregoing value of F being always finite and positive, this condition will be fulfilled if P > o. We have then g cos a- T sin )3 > o, or (i + cos 2 /3) cos a > sin |3 sin a. If /3 satisfy the equation i + cos 2 /3! = sin j3i tan a, all values of j3 lying between /3i and o will satisfy the condition. Hence, whatever be the inclination of the plane, we can always find a coefficient of friction and a series of positions of the cylinders for which there will be perfect rolling. If a = or < 45, it is evident that for all finite values of ]3, ( i + cos 2 j3) cos a > sin ]3 sin a. In this case, therefore, we can always find a coefficient of fric- tion such that for all positions of the cylinder there will be per- fect rolling. (2). The upper cylinder may roll without slipping, the lower cylinder slipping and rolling. It is plain that the slipping of the lower cylinder must be downwards. For, omitting friction, we have as the value of the accelerating force at A' 9 This quantity being essentially positive, it is evident that the accelerating force at A' must be downwards. Hence, in con- 152 INITIAL MOTION OF A SOLID BODY. formity with the principle stated (p. 7), the force of friction must be upwards. The slipping, if it exist, must, therefore, be downwards. Hence the complete accelerating force at A' (in- cluding friction) must be positive, or dt VW ~ a dt] = 9 Sin + ^ ~ C S T ~ ( n * + ') F > ' Hence (n z + i).P<#sina + (n a -cosj3) T.' But if the upper cylinder roll, we have (n* + i) F = g sin a + (n z + cos ]3) T. From these two conditions it appears that F>F'. But if P, P be the pressures at A, A\ we have, since the motion at A' is slipping, F' - juP'. Hence, as F cannot exceed juP, we must have P> P. But this is impossible, inasmuch as P = g cos a - T sin ]3, P = g cos a + T sin ]3, showing that necessarily P < P. Hence it is impossible that the motion should be such as is here supposed. (3). The lower cylinder may roll without slipping, the upper cylinder slipping and rolling. Since the motion of the lower cylinder is pure rolling, we have from the second equation (59), (i +n).F = 0sma + (n 2 -cos/3) T; and, since the upper cylinder slips, we must have F= nP = fi (g cos a - Tsin/3). These two equations, together with the equation (56), are suffi- cient to determine F, F', T. To determine the least value of ju which will make this move- ment possible, we have, in addition to the two preceding equa- tions, and the general equation (56), F = pp = p ( g cos a + Tain |3). (4). If the value of ^ be less than that derived from these equations, both the cylinders will slip and roll. ( 153 ) CHAPTER VI. NECESSARY AND POSSIBLE EQUILIBRIUM. i. WE now proceed to consider the distinction, peculiar to our present subject, between positions in which equilibrium may exist, and positions in which it must exist. The former of these classes may be denominated positions of possible equilibrium, and the latter, positions of necessary equilibrium. A position of necessary equilibrium, then, may be defined to be a position in which, if a system be placed in it without velocity, it will continue indefinitely. A position of possible equilibrium is one in which a system may be so placed, without velocity, as to continue indefinitely, and yet may be so placed, still without velocity, as to move. 2. In order to understand the nature of this difference, it will be necessary to recur to a distinction already noticed that, namely, between statical and dynamical friction. It has long been known that the coefficient of maximum friction is not the same for the two different cases of rest and motion. Where one body rests upon another, the coefficient of maximum friction has a greater value than it would have if the two bodies were in a state of relative motion. But there is a more important difference. When a material particle moves upon a rough surface, the direction of the force of friction is always directly opposed to that of the motion of the particle. If, therefore, this motion be governed by any geo- metrical condition which determines its line of direction on the supporting surface, the line of direction of the force of dynami- cal friction is also determined. Thus, for example, if the sup- porting surface be a plane, and if the particle be attached by a rigid rod to a fixed point, the position, and, therefore, the motion, of the particle, will necessarily be in a determinate circle. It is plain, therefore, that the line of direction of the frictional force x I 54 NECESSARY AND POSSIBLE EQUILIBRIUM. is necessarily determinate, being for every position of the particle, while in motion, a tangent to this circle. 3. More generally, if a system of particles whose positions are connected by certain geometrical relations, and each of which rests upon a rough surface, be in motion, we are not at liberty to assume for the direction of the force of friction at each point any line in the tangent plane at that point. For it is plain from what has been said, that a system of possible lines of direction for the forces of friction must be coincident with a system of possible lines of motion of the several particles. We shall now proceed to consider how far the directions of the forces of fric- tion are restricted by this consideration. Let ,ri ?/i Si, x z y z z^ # y n %n be the co-ordinates of the several particles, and let it be supposed that these co-ordi- nates are connected by the equations, L\ = o, LZ - o, &c., the number of these equations being m. Let !, Ss 2 , $s n be a system of possible displace- ment of the particles ; and let a^ b Ci, 2 b z c^ . . . . a n b n c n be the cosines of the angles which these displacements make with the axes. Then it is plain that we shall have the following equations of condition : f ! -~ + bi - + Ci -~ J Si + ( a z -j-1- -i- &c. J Ss 2 + &c. = o (0 -r-^ + &C. ) 8! + &C. = O, &c., &c. Now, if the number of these equations is sufficient to enable us to eliminate the n - i ratios, we shall have m - n + i equations of condition between a, 6 2 c 2 , &c., and x l y l 2, x^y* x z , &c. NECESSARY AND POSSIBLE EQUILIBRIUM. 155 Hence we infer that, If the number of equations connecting the positions of a system of material particles, each of which rests on a rough surface, be equal to, or greater than, the number of the particles, the lines of direction of the forces of dynamical friction cannot be assumed arbitrarily in the tangent planes at the several points, but are sub- ject to certain equations of condition, the number of these equations being m - n + i. But even if m < n, in which case the lines of direction are arbitrary, the actual directions themselves are not so. Thus, if AB, CD,EF, &c., be a system of possible directions for the forces of dynamical friction, we cannot take arbitrarily any combination of the signs in the series, ABCD EF, &c. For, so long as there are any equations of the form (i), we can- not assign arbitrarily any combination of signs to the displace- ments Ss l} Ssjj, &c. In fact, every equation of the form Alfa + A 2 $s 2 -f &c. = o, excludes two such combinations those, namely, which make every term positive, and those which make every term negative. The same limitation, of course, holds for the signs of AB, CD, &c., which are opposed in direction to fa, Ss 2 , &c* 4. As an example of this principle, let us consider the case of a system of three material points, each resting on a rough surface, and connected with each other by three rigid inextensi- ble rods. If the lengths of these rods be a, b 9 c, we have the equations of condition, (a?i - # 2 ) 2 + (3/1 - 2/ 2 ) 2 + (*i - s 3 ) 2 = 4 2 (*3 - *l) 2 + G/3 - */l) 2 + (*8 - *l) 2 = 4 2 (2) (03 ~ ^s) 2 + (y a - y 3 ) 2 + (*a - Zs) 2 = li\ whence, by differentiation, &* 2 ) + ( yi - y a ) (%A - Sy 2 ) + (z, - s 2 ) (fa - fa) = o - fa) + (3/3 - yO (&/ 3 - Syi) + (z 3 - z,) (fa - fa) = o (3) a.) (fa - fa) + (y a - y 3 ) (^2 - 8y) + ( 8 - ar 8 ) (& a - fe) - o. Let, as before, $!, s 2 , 8* 3 be the absolute displacements, and 156 NECESSARY AND POSSIBLE EQUILIBRICPM. let us represent by / 3 Si, 4Ss 2 , &c., the angles between the lines 4 4 4, and the displacements $i, Ss 2 &s 3 . Then it is easy to see that the foregoing equations may be written, S! COS &?i = Ss COS ZSs Ss 3 cos / 2 Ss 3 = &i cos 4&i (4) & 2 cos 4^2 = Ss 3 cos i, Ss 2 , S*3> being the absolute displacements taken positively. Hence, by eliminating Ss 1? Ss 2 , s 3 , we have cos 4^s 2 cos 4^3 cos 4$ s i = cos 4$ s s cos 4$ s i cos 4$ S 2' (5) This equation of condition must be satisfied by the directions of every system of possible movements &i, Ss 2 , Ss 3 ; and since every system of possible directions, which can be assigned to the forces of dynamical friction, must coincide with a system, - &i, - Ss 3 , - Ss 3 , it is plain that the same equation of condition must be satisfied by the lines of direction of these forces. Moreover, it is plain from the equations (4), that if the direc- tion of $81 change sign, those of Ss 2 & 3 must change sign also. Hence, if + Ss 2 , + Ss 2 , + Ss 3 be a possible system, the only other possible combination of signs will be - Ss^ - 5s 2 , - s 3 . 5. As another example, ifPQ (Fig. 23) be an inextensible rod laid upon the two rough rods, PA, QA, each of the extremities, P, Q, has two possible directions of movement, PA, AP, and Q A ,AQ; or, as it may be other- wise expressed, + PA, QA. \ ^^ But of the four combinations in- cluded in PA, QA, two namely, -f PA, + QA, and - PA, A - QA, are rendered impossible Fig. 23. by the inextensibility of the rod. The remaining combinations, + PA, - QA, and - PA, 4 QA, are, therefore, the only possible directions of movement; and, consequently, - PA, 4- QA and + P A , - QA the only possible directions for the forces of dyna- mical friction. It appears from the foregoing discussion, that in every system connected by equations of condition, the existence of these NECESSARY AND POSSIBLE EQUILIBRIUM. 157 equations imposes certain restrictions upon the direction of the forces of dynamical friction. We shall now proceed to consider whether any such limitations exist in the case of statical friction. 6. The direction of the force of statical friction at each point of a system is governed by a rule analogous to, though not identical with, the rule which has been already stated for the case of dynamical friction. If, indeed, as we shall hereafter see, we discard the abstractions of .Rational Mechanics, and state the conditions of each problem as they really exist in nature, these rules admit of a common statement. But, for the present, it is desirable to keep them distinct. The force of statical friction in other words, of that friction which is developed between two bodies which are in contact, the points of contact not having any relative motion is, for each particle of a system which rests upon a rough surface, directly opposed to the tangential component of the resultant of all the other forces, external and geometrical, which act upon that par- ticle. In order, therefore, to determine what limitations upon the directions of the forces of statical friction exist, we must in- quire what limitation upon the direction of this resultant exists at each point. (i.) If there be no equations of condition, and, therefore, no geometrical forces, the resultant of all the forces acting at each point (excluding friction) is perfectly definite both in direction and in magnitude. The direction of the force of statical friction is, therefore, also perfectly definite. (2.) If there be m equations of condition (the number of particles being ?z), each of these equations will introduce a geo- metrical force, determinate in direction, but indeterminate in magnitude. The expressions for the directions of the resultant forces acting at each particle will, therefore, contain m indeter- minate quantities. Since, then, the number of equations required for the determination of the directions of these forces is 2^, the elimination of the m indeterminate quantities will leave in - m equations among these directions. The same number of equa- tions will evidently exist between the directions of the tangential components of these resultant forces. Deducting from these the n equations which express the condition that each of these 158 NECESSARY AND POSSIBLE EQUILIBRIUM. components is situated in the tangent plane to the supporting surface, there will remain n- m equations between their directions, and, therefore, between the directions of the forces of statical friction. 7. But these limitations upon the directions of the forces of statical friction do not in anywise interfere with the equilibrium of the system ; in fact, they are themselves conditions of equili- brium. If the system can be kept in equilibrium at all, it will be so kept by forces whose directions satisfy these conditions. For it is evident that, when a system is in equilibrium, any one of the forces which act at each point must be directly opposed to the resultant of all the rest. The force of friction, therefore, which is one of these forces, and which necessarily acts in the tangent plane, must be directly opposed to the tangential com- ponent of the other forces. We have seen that this is precisely the direction which it does really assume. The force of friction, therefore, takes of itself the proper direction for equilibrium. So far, then, as the directions of these forces are concerned, the conditions of equilibrium are necessarily fulfilled. But for actual equilibrium it is further necessary, as we have before seen, that, at each particle, the resultant of all the forces, external and geometrical, shall lie within or upon the cone of resistance. If this condition be fulfilled, equilibrium will exist. Now, it must be observed that the equations which express this condition are functions of the geometrical forces of the system, constituting a certain number of relations between these forces. Unless, therefore, the geometrical forces be such as to satisfy these equations, we cannot say that equilibrium actually exists. But the geometrical forces do not necessarily take such values as to satisfy these equations. In fact, as we shall see more clearly when we come to state the question without the abstractions of Rational Mechanics, the initial values of these forces are properly among the data of the problem. Their direc- tions are, indeed, as we have before seen, determinate ; but their magnitudes may be assigned arbitrarily. Unless, therefore, these magnitudes have been so assigned as to satisfy the con- ditions of equilibrium, the initial state of the system will be a state of motion. NECESSARY AND POSSIBLE EQUILIBRIUM. 159. 8. We are now able to understand the meaning and reality of the kind of equilibrium spoken of at the commencement of this chapter namely, possible equilibrium. The equilibrium of a system is possible, if we can assign to the geometrical forces such values as will keep the system at rest ; impossible, if no such values, consistent with the conditions to which they are subject, can be assigned to these forces. It remains that we inquire when the equilibrium of a system is necessary, and what are the conditions requisite to the actual existence of such a state of equilibrium. We shall, for the pre- sent, omit to notice the difference between the coefficients of friction for the state of rest and the state of motion, and shall conduct the investigation as if these coefficients were identical, reserving for future consideration the question, how far the re- sults are modified by the difference between these coefficients. 9. The forces of statical friction take of themselves, as we have seen, the best directions for securing equilibrium those, namely, which are, at each point, immediately opposed to the tangential component of the resultant of all the other forces. But these directions do not necessarily coincide with any system of directions which can be assigned to the forces of dynamical friction. These latter are not, therefore, necessarily the best directions for securing equilibrium, to which they may even be inclined at finite angles. It is, then, possible that a system may be in equilibrium under the influence of the forces of statical friction, and yet be removed by finite distances from the position in which it would be in equilibrium, if the friction became dynamical. In such a case, if the system be disturbed from its position of equilibrium by the communication of indefi- nitely small velocities to its several points, when the friction at each point will, of course, become dynamical, a finite force tend- ing to augment the displacement may at once be developed at some or all of these points.* * We have here a case differing in an important respect from ordinarily unstahle equilibrium, where friction is not taken account of. In this latter case, if a num- ber of indefinitely small velocities be communicated to the several points of the system, no finite force is called into play by the movement so given, and, therefore, the movement will not in any finite time become finite. Thus, for example, if a 160 NECESSARY AND POSSIBLE EQUILIBRIUM. Let it be supposed, then, that the initial values of the geometri- cal forces are inconsistent with the conditions of equilibrium. The system will then commence to move. Immediately on the com- mencement of this motion the force of friction at each moving point will become dynamical ; that is to say, opposed in direc- tion to the motion of the particle, and taking its maximum value. We have already shown (Prop. II., p. 105) that, in such a case, the geometrical forces acting upon the moving particles become determinate functions of the directions of.the initial movements. 10. "We are now enabled to determine the conditions of neces- sary (as distinguished from possible) equilibrium. For this purpose it will be necessary to change slightly the ordinary enunciation of the principle of virtual velocities. This principle may be enunciated as follows : A number of material particles, each of which rests upon a fixed surface, rough or smooth, are connected by given geo- metrical relations, and acted on by given forces, and are in equilibrium. Let #, y, z be the co-ordinates of one of these particles, and let X, Y 9 Z be the components of the resultant of all the forces, external, geometrical, and frictional, which act upon it. Conceive the system to be set in motion by the com- munication of an indefinitely small velocity to each particle. Let v be this velocity for the particle #, y, s, and a, b, c, the cosines of the angles which it makes with the axis. Then if the force* X, Y 9 Z, do not change sensibly as the system passes from a state of rest to a state of motion, the condition of equilibrium, ne- cessary and sufficient, is 2mv (aX + bY+ cZ) = or < o, in which we may understand X, Y 9 Z to refer either to the state of rest or to the state of motion, inasmuch as the forces are supposed to be the same for both states. material particle, placed on the highest point of a smooth surface, receive an indefi- nitely small velocity, this velocity, although it continually increases, will not in any finite time hecome finite. In the case of friction, on the contrary, where a finite force, tending to augment the displacement, may he called into play by an in- definitely small velocity, this velocity may in a finite time hecome finite. NECESSARY AND POSSIBLE EQUILIBRIUM. l6l If, on the other hand, any of the forces do change percep- tibly in passing from a state of rest to a state of motion, we must understand by X, Y, Z in the foregoing condition the forces which act upon the particle in the state of rest. Now, in such a case, it is quite possible that, although these forces may satisfy the foregoing condition, the forces which take their places, as the system passes from rest to motion, may not satisfy it. In such a case, then, if the system while at rest were acted on by the forces which do act upon it when in motion, the equilibrium could not exist. Thus, for example, if the forces acting upon the particles when at rest be susceptible of a number of different systems of values, the forces which act upon the particles in motion being, as we have seen, absolutely determinate, it is possible that among the former systems o.f values might be found one or more to satisfy this condition, while, on the other hand, the single system corresponding to the state of motion might not satisfy it. In such a case equilibrium is possible, but not necessary. If, however, the condition be true for all values of v, , &, c, the forces X, F, Z, having the values corresponding to the motion represented by these letters, equilibrium is necessary. For even if values be given to the forces inconsistent with this condition, these values, when the system begins to move, will at once be replaced by others which satisfy the condition. We shall then have a system acted on by forces which are in equili- brium, and whose several points are endued with indefinitely small velocities. Now, if we adopt the abstractions of Eational Mechanics, the vis viva of such a system is plainly less than any assignable quantity. For, inasmuch as the forces corresponding to a state of rest are instantaneously changed into forces corresponding to a state of motion, which latter forces satisfy the condition, 2mv (aX + bY + cZ) < or = o, (6) and are, therefore, incapable of producing vis viva, the time during which the forces corresponding to the state of rest, which are capable of producing vis viva, act, is absolutely evanescent. Hence, whatever values be assigned to the forces of rest, if the Y 1 62 NECESSARY AND POSSIBLE EQUILIBRIUM. condition (6) be satisfied, the vis viva of the system can never become equal to any assignable magnitude. The system is, therefore, necessarily in equilibrium. We infer, therefore i. If it be possible to assign such values to the geometrical forces of the system, that the condition 2m (XSx + Yy + Zlz) = o or < o be satisfied, the position is one of possible equilibrium. 2- If this condition be satisfied by every system of values of the forces which corresponds to a system of possible movements, the equilibrium is necessary. ii. As an example of the principle here laid down, let us consider the case of a material particle, R, (Fig. 24) connected by a rigid, weightless rod, with a fixed point, 0, and resting against a rough vertical plane. We have then, independently of fric- tion, three forces acting upon the particle, ' namely i. The weight g, whose direction is vertical. 2. The normal reaction, P, of the vertical plane. 3. The force, Q, arising from the attachment of the particle to the fixed point, acting in the direction OR. Then, if these forces be resolved parallel and perpendicular to the rough vertical plane, it is easily seen that the square of the former component will be, putting ROC = j3, RCY = 0, Q? sin 2 j3 + g 2 - 2gQ sin j3 cos 6. This component must, for equilibrium, be destroyed by the force of friction. Now, if the particle be in an extreme position, the force of friction will be /uP. But since the forces, resolved along the normal, must equilibrate P = Qcos/3. Since, then, the force of friction will, from its own nature, take a direction opposed to the above-mentioned component, the single condition of equilibrium will be Q 2 sin 2 j3 + f - igQ sin ]3 cos 9 = or < ^u 2 Q 3 cos 2 j3. (7) NECESSARY AND POSSIBLE EQUILIBRIUM. 163 This condition is impossible if sin 2 /3 cos 2 < sin 2 j3 - ju 2 cos 2 j3. We have then, for the limiting position, sin 9 = jit cot ]3 ; an equation which gives the extreme obliquity of the radius RC, drawn to the particle. If the obliquity be greater than this, equilibrium is im- possible ; but equilibrium does not necessarily exist, even when the obliquity lies within this limit, unless the geometrical force Q satisfy the condition (7). We must, therefore, in placing the system in its position of equilibrium, provide that the force, Q, shall satisfy this condition. Let us now consider what are the conditions of necessary equilibrium. In accordance with the principle stated (p. 160), let it be supposed that the particle receives a small velocity, whose direc- tion will, of course, be a tangent to the vertical circle which is the locus of the point R. Of the two possible directions of dis- placement, RX, RY, it will evidently be sufficient to consider the former. For, if the equilibrium can be broken by an inde- finitely small displacement, this will be effected rather by a displacement downwards than by a displacement upwards. Then, taking CX, (7 Fas axes, and denoting, as before, by Q the reaction of the rod, we have the equations of motion of the particle d*x = Q (sin /3 sin 9 - JUL cos |3 cos 9) at () d II ^ = Q (sin ]3 cos 9 + ju cos ]3 sin 9) - g. ctt But by the geometry, x = a sin 9, y = a cos 9. Differentiating these equations twice, and rejecting, in accord- 164 NECESSARY AND POSSIBLE EQUILIBRIUM. ance with the general principle of p. 106, the square of the first differential coefficient, we have d-x d z 9 d* d*0 Substituting these values in the equations of motion, and divid- ing these equations one by the other, we find Q sin ]3 = g cos 0. Now, the accelerating force resolved in the direction of the displacement will be xJ^2 sp /-j% yt cos ; -- sin 9 | - g sin - fj.0, cos ]3 = g (sin - p cot )3 cos 0). o.t (it If this force be positive, the velocity will increase, and, in a finite time, become finite. This will happen if tan 9 > ju cot /3. If the force be negative, the velocity will diminish, and, in an indefinitely short time, be destroyed. If the force be zero, the velocity will at first' continue the same, but as its position changes, the force will increase, and the equilibrium will be broken. This, however, will not happen until after an indefinitely great length of time. These conditions are expressed by tan 9 > ju cot ]3, tan 9 < /j. cot ]3, tan = ju cot j3- Now, we have before seen that unless sin > /u cot ]3, the equilibrium is possible, if the value of Q be rightly assigned. It is plain, therefore, that between the positions X and 3 defined respectively by the equations tan Oi - fj. cot ]3, and sin 9 2 = /* cot ]3, equilibrium is possible, but not necessary. The same discussion is made applicable to the case in which the original displacement is in the direction R Y, by simply sub- stituting - JUL for ^, and resolving the accelerating force in the direction RY instead of the direction RX. The value of the force so resolved will be - g (fj. cot |3 cos 9 + sin 9), NECESSARY AND POSSIBLE EQUILIBRIUM. 1 65 which, being essentially negative, will, in an indefinitely short time, reduce the particle to rest. But it is easily seen that be- tween Oi and 2 ) the particle will not remain at rest. For it has been shown that there will not be equilibrium, unless the internal force, Q, satisfy the condition Q 2 sin 2 /3 + g* - 2gQ sin ]3 cos < or = ju 2 cos 2 /3. But it will be found that the value of Q is (as in the former part of the discussion) determined by the equation Q sin |3 = g cos 9. Substituting this value in the preceding condition, we find tan < or = JJL cot [3. Hence it is evident that between the positions Oi and 2 > if the particle receive a velocity in either direction the equilibrium will be broken. With regard to the stability or instability of the equilibrium, it is necessary to make a distinction. Equilibrium may be called unstable when it is capable of being broken, either by the communication of an indefinitely small velocity, or by an inde- finitely small change in the value of the coefficient of friction. These species of instability are not necessarily identical. Thus, in the present case, between the positions Oi and 2 ? the equili- brium is, as we have seen, necessarily unstable in the first sense. But whether or not it be unstable in the second sense, will de- pend upon the value which has been given to Q. For if the value of Q be such as to make Q 2 + g z - igQ, sin j3 cos sensibly less than ju 2 Q 2 cos 2 /3, it is plain that ^ may receive a sensible diminution without a rupture of the equilibrium. In this case the equilibrium is stable in the second sense. If, on the other hand, the value of Q be such as to make these expressions equal, the equilibrium is unstable in the second as well as in the first sense. The whole discussion may be summed up as fol- lows : 1. 9 < Oi. Equilibrium necessary and stable. 2. = Oi. Equilibrium necessary and unstable in the first 1 66 NECESSARY AND POSSIBLE EQUILIBRIUM. sense, and may also be unstable in the second sense, if Q have the proper value. 3. 9 > Oi < 6 2 . Equilibrium possible but not necessary. Unstable in the first sense, and may also be unstable in the second sense, if Q have the proper value. 4. 6 = 2 . Equilibrium possible and unstable in both senses. 5. > O z . Equilibrium impossible. PROP. VII. 12. A solid body acted on by given forces rests upon two rough surfaces. Given the position of the body, to determine whether the equilibrium be necessary or only possible. The principle of this investigation is identical with that which has been already applied to the case of a system of ma- terial particles. Here, as in the former case, we must inquire whether it be possible to give to the body such an indefinitely small movement as will give rise to a continually increasing vis viva. If this cannot be done, the equilibrium is necessary ; if it can be done, the equilibrium is only possible. Now, we have seen (pp. 139-42), that if a given indefinitely small motion be communicated to a solid body resting upon one or more rough surfaces, the ixsistances at the points of support are determinate. We have then a body acted on by given forces, and can calculate the increment of vis viva which these forces would produce. If this increment be negative, or indefinitely small, as compared with the small motion communicated to the body, the equilibrium is necessary. If neither of these suppo- sitions be true, the equilibrium is only possible. Let X, F, JZ, be the components of the resultant of all the forces applied to the body, and L, M, N, the components of the resultant moment. The parts of these forces and moments arising from the resistances of the supporting surfaces will, in general, as we have seen, depend upon the nature of the motion communicated to the body. Hence, supposing, as in p. 122, that the position of the body is determined by the variables # lf 2A, *!, 0, ^, 0, and putting dx! = avdt, dy v = bvdt, dsi = cvdt, (9) d(j> = awdt, d^ = fiwdt, dO NECESSARY AND POSSIBLE EQUILIBRIUM. 167 the forces and moments are of the form f(a, b, c, a, /3, y, v, a>), the form of the function being, in general, different for rolling and for slipping motion. Now we know that the vis viva of a solid body = mass ot body x (vel. of centre of gravity) 2 + vis viva corresponding to the rotation of the body round the centre of gravity. But the vis viva corresponding to this latter motion is (Poisson Mecanique, Tom. II., p. 150), Ap* + B(f + Cr 2 . Hence ^ inc. vis viva of solid body <= d V dp -n dq ~ dr dt dt dt But, from the equations of motion of a solid body, and Ap -- + Bq ~ + Cr = Lp + Mq -I- Nr Ti-rdd) ,,-r Ttr \ n *T /ii d\L , . = N -j- + {(L sm0 -f M cos 0) sin - JV cos v] (i i) + (M sin $ - L cos 0) , dt substituting for p, ^, r, from the general equations (3), p. 122. Hence, and from equations (9) we find dV= (aX + bY+ cZ) vdt + {Na + (L sin sin + M sin 9 cos - Ncoscf)) |3 + (M sin - L cos 0) y) wdt. (12) If this value be zero or negative for all values of a, b, c, a, )3, y, v, w, which are consistent with the conditions of the question, the equilibrium is necessary. Now, we have seen (p. 140), that, for each supporting surface, we have an equation o.f the form 1 68 NECESSARY AND POSSIBLE EQUILIBRIUM. Differentiating this equation, and substituting from (9), we have / dF .dF dF\ f dF -dF dF\ via-: + o-j + c- r -}+(t)[a-r^p- rr + y-^} = o. \ dx l dy l dzj \ d ^ d^ * dO j We have also a 2 + i, a 2 -f /3 2 + = i. If by means of these equations as many as possible of the quan- tities a, 6, c, &c., be eliminated from the value of dV> the con- dition of necessary equilibrium requires that, independently of the values of those which remain, we should have dV ' = or < o. If the position of the solid body depend upon a single variable, u, two movements only are possible, represented by the two values +-T-- In this case the condition of necessary equilibrium ut requires that the sign of the second differential coefficient -^-, as given by the equations of motion, should be (unless this co- efficient vanish) different from the sign of. ut Example i. 13. Let ABCD (Fig. 25) be a rectangular drawer, and let a force be applied to one of its handles, H, in the direction EH. Determine the con- dition and nature of equilibrium, if it exist. It is evident that the drawer may be considered to be acted upon by three forces, namely, the moving force at H, and the two resistances of the rough sides at A and C. If equilibrium exist these three forces must meet in the same point. Draw the lines Aa, Ad, Cc, Cc', each making, with the tran verse sides AD, BC, Fig. 25. an angle equal to the angle of friction. Then we know that the NECESSARY AND POSSIBLE EQUILIBRIUM. 1 69 directions of resistance at the points A, C, must be respectively within the angles aAa, cCc. If, then, as in fig. 25, the lines Aa, Aa, and the lines Cc, 'Cc' 9 do not intercept any common portion of the line HE, it will be impossible to assign directions to the forces of resistance meeting at the same point of this line. In this case equilibrium is impossible. It is plain that this will happen, if the angle of friction be less than CAD. If, as in Fig. 26, the angle of friction be greater than CAD, it will always be possible to assign such directions and mag- nitudes to the forces of resis- tance at A and C, that their resultant shall be equal and opposite to the acting force. For this purpose it is only A| necessary to take any point p, Fi S- 26 - in the line ca', and join it with the points A, C. These joining lines Ap, Cp, are evidently possible directions of the forces of resistance. The magnitudes of these forces are found by taking a line pq to represent the acting force, and drawing the pa- rallels qr, qs. The lines rp, sp, will represent the forces of resistance in magnitude and direction. It is plain then that the condition of possible equilibrium is tan CAD < ju. To determine the condition of necessary equilibrium let it be supposed that the drawer receives a small movement. The forces of friction will now assume their maximum values. Denoting then the applied force by F, the normal reactions at A and C by P, P, and taking the axes of co-ordinates parallel to AB, BC, the equations of motion of the centre of gravity of the drawer will be (13) But since there is no rotation, the sum of the moments of the acting forces with regard to the centre of gravity will be zero. Hence, putting (Fig. 25) AB - za, Iff = b, we have 2Pa = Fb. 170 NECESSARY AND POSSIBLE EQUILIBRIUM. The accelerating force in. the direction JSfl"will, therefore, be If this force be positive, the small velocity communicated to the drawer will become finite, and the equilibrium will be perma- nently broken. If the force be negative, the drawer will return to rest in an indefinitely short time. If the force be zero, the drawer will simply retain the indefinitely small velocity which has been communicated to it. In neither of these two latter cases will the velocity ever become finite. The condition of necessary equilibrium is, therefore, fj.b,not < a. The whole result may, therefore, be stated as follows : If the coefficient of friction have any value from o to tan CAD, equilibrium is impossible. If it have a value greater than tan CAD but less than a -r- b, equilibrium is possible but not necessary. If the coefficient of friction be greater than a ~- 6, equilibrium is necessary. Example 2. 14. Two rods AB 9 CD, firmly connected at B, rest, as in fig. 27, upon a rough vertical board. Determine the limits of the inclination of AB to the vertical, and the nature of the equili- brium. It is evident that the system of rods is kept in equilibrium by three forces, namely, i. The weight act- ing in the line G V, (rbeing the centre of gravity of the system. 2. The reaction of the rough board acting at the point E. 3. The reaction of the rough board acting at A. The only restriction by which these lat- ter are bound is that their lines of direction shall not make with the perpendiculars to EB, EA, respec- tively, angles greater than the Fig. 27. NECESSARY AND POSSIBLE EQUILIBRIUM. 17 I angle of friction. Let VE, A F, be these lines of direction respectively. Then it is necessary and sufficient for possible equilibrium, that these lines shall intersect upon the vertical G V. The extreme position consistent with possible equilibrium is, therefore, attained when two lines making the angles VAF, VEB, each equal to the complement of the angle of friction, intersect upon the line G V. Let e be the angle of friction, and assume AB = a, AG = 6, BAE = 0. Then evidently AEV=z- 0, A VE= 90- (it - 9). Also AE = a sec 0, A F= b sin sec *. But AEsin (e - 9) = A Fcos (as - 0), or by substitution a sec 9 sin (g - 9) = b sin 9 sec c cos (ae - 0), (14) an equation by which 9 is determined. Now it is plain, that of the two lines AV,A V, which can be drawn, each making with the horizontal line the angle of fric- tion, the line A V } inclining downwards, corresponds to a more oblique position of the system than the line A V, inclining up- wards. For, since VE makes with the perpendicular to the rod at E a smaller angle than V'E, it is evident that if VE be in an ex- treme position, VE will be beyond the extreme position. Hence it appears that the extreme position of possible equilibrium is found by taking the direction of the force of friction at A down- wards. But now let it be supposed that the system receives a small velocity, so as to cause the point A to descend. The force of friction at A will at once commence to act upwards, and conse- quently the direction of the force of resistance at A will be AW. Taking now EX, EY, as axes of co-ordinates, we have as the equations of motion of the centre of gravity m -j-j = - R sin (e - 9) + R cos e, (-5) d i/ m ^ = R cos (c - 9) + R sin c - mg. 172 NECESSARY AND POSSIBLE EQUILIBRIUM. JR, R' t being the complete resistances at E and A. Again, the equation of rotation round the centre of gravity will be mfc ^?jR(atan cos e -( a -6) sin ) - R'b cos (e-f 0). (16) But by geometry, x - b sin 0, y = - (a sec -b cos 6). Differentiating these equations according to the principle stated, we have But in order that the velocity may never become finite, it is d~0 necessary and sufficient that the value of -=-^ should never be- come positive. The limiting position of necessary equilibrium is, therefore, found by making in (15), (16), d*Q d\x ^ = o, and therefore -JT = o. at at From these two equations we find, after reducing b sin cos 2 = a cos e sin (e - 0), (17) an equation which denotes that the lines AV', EV, meeton^the vertical through the centre of gravity. As this is the condition of equilibrium when the resistance at A takes the direction A V this result might have been foreseen. It is easy to show that the equations (14) and (17) have each one, and but one, real positive root. Putting tan = x, b = na, we may write the first equation sc* + (in - i) x* tan c +{n (i - tan a c) + i }x - tan e = o. It is easily seen by the method of successive substitution that this equation has one real positive root between o and tan e. But since b is always greater than -Ja, we have in - i > o. Hence, whatever be the sign of the third term, the equation will always have one variation and two permanences. There is, therefore, one and only one, real positive root. If the equation (17) be similarly treated, it may be written x* - #' tan + (n sec 2 + i) x - tan = o. NECESSARY AND POSSIBLE EQUILIBRIUM. 173 It is evident that this equation has, like the former, one real positive root between o and tan e. But if we denote the left hand side of this equation by w, we shall have du ~ - 3# 2 - 2x tan g + u sec 2 f + i . U/iC It is easily seen that this quantity is always positive, hence u can only vanish once. There is, therefore, only one real root. ( '74 ) CHAPTER VII. ON THE DETERMINATION OF THE ACTUAL VALUE OF THE ACTING FORCE OF FRICTION. i. WE have seen that in systems of bodies kept in equilibrium by friction, the actual amount of this force which is developed at each point of contact, is in certain cases indeterminate. Thus, in the case of a beam resting against two rough surfaces we'may assume, consistently with the preservation of equilibrium, for the directions of the reaction of the supporting surfaces, any two lines situated on or within the cones of resistance, and intersect- ing on the vertical through the centre of gravity of the beam. It becomes, therefore, an interesting question to determine the actual amount of friction which is developed in any such case of equilibrium. In the phenomenon itself there can be nothing uncertain or indefinite, and if any such ambiguity appear in the solution which we have obtained, it must arise solely from some abstraction or simplification which we have introduced into the investigation, and which has no real existence in nature. There is no difficulty in detecting the abstraction which ra- tional mechanics introduces into questions such as that of the equilibrium of a beam resting upon two rough surfaces ; it is the supposition of the perfect rigidity of the beam. In the solution of all such problems, it is supposed that the beam preserves its original shape, and that there is no yielding on the part of the support which sustains it. If then the posi- tion be one of equilibrium, that is to say, a position in which no finite movement of the system ensues, it is supposed that the system, when placed in such a position, is at once reduced to a state of perfect rest. This, however, is not a true account of the phenomenon. The force which in bodies resists a change of shape, being a function of that change vanishing with it, it is plain that, under VALUE OF THE ACTING FORCE OF FRICTION. 175 the influence of any distorting force, some change of shape will always ensue. Even then, when in one of those positions usually styled positions of equilibrium, the system will always begin to move, the resisting force being, at the commencement of the motion, absolutely zero. 2. Thus, when a heavy body is laid on a horizontal plane, the reaction of this plane is not an instantaneous, but a progres- sive force, commencing from zero. When the distance be- tween the superficies of the body and the supporting plane has diminished to a certain value, both the body and the plane will begin to yield and change their shape. From this change of shape there arises a force opposing the removal of the particles of each body from their original relative position, and increasing with the change of that position. Under the influence of these two opposing forces the body will continue to descend until they become equal, and, in virtue of its acquired velocity, for some time longer, and will finally be brought to rest when the velocity is distroyed by the preponderating force of resistance. If the body and the plane be perfectly inelastic, that is to say, if the force developed by distortion be merely a force tending to prevent further distortion, and not to restore the molecules to their original position, the phenomenon ends here ; but if they be in any degree elastic, the developed force which has now increased above the force of gravity, will cause the body to rise, and then will ensue a series of small oscillations depending upon the elasticity of each substance. 3. Similarly, in the case of a beam resting on two rough sur- faces, as soon as the beam comes in contact with the surfaces it commences to bend and contract under the influence of gravity, its own elasticity, and the resistance and friction of the sur- faces, the last force always opposing itself to the motion of the extremities of the beam. Under the influence of these forces the beam will execute a series of small oscillations, the law of which will depend on that of the elastic force. If we know this latter force we can form the equations of the small oscillations of the beam, observing to change the sign of the force of fric- tion whenever the motion of the point at which it is exerted changes its direction. If these equations can be integrated, Ij6 VALUE OF THE ACTING FORCE OF FRICTION. we shall know the position of the beam at the end of each oscil- lation, and therefore the value of the elastic force which depends on that position. Knowing this force in magnitude and direc- tion at the two extremities of the beam, we know the pressure against the supports, and therefore (in magnitude and direction) the reaction of the surfaces. Changing now the sign of the force of friction at the extremity, which has for the moment ceased to move, we can determine whether the maximum value of the force of friction (with its sign thus changed), or any less value, would satisfy the condition of equilibrium at that point. If so, the point will cease to move. If not, a fresh oscillation will com- mence, and the investigation must be continued until we find the permanent position assumed by the beam. Knowing this position, and, therefore, the values of the elastic force or pressure at the two extremities, we know of course, both in magnitude and direction, the reactions of the fixed surfaces which are the objects of our inquiry. The difficulty which attends the integration of the equations of motion of an elastic body, renders the complete solution of questions of this kind in general impossible, but the following case in which the solution is easily obtained will serve to exemplify the general method. Example. 4. A heavy body is placed on a rough inclined plane, whose inclination is greater thari tan -1 ^u, and is supported by a string parallel to the plane, and attached to a fixed point. To determine the actual force of friction developed, and the tension of the string. The only equation which statical science furnishes for the determination of these forces is T+F=Mgsma (i.) leaving them both of course indeterminate, unless the value of the geometrical force T be given ; or, in other words, unless the system be, not simply laid on the inclined plane, but strained by a given force, at the time when it is placed in its position of equilibrium. If this be so, and if the given force be within the VALUE OF THE ACTING FORCE OF FRICTION. 177 limits Mg (sin a /j, cos a) equilibrium exists, and the actual value of Fis given by eq. (i). But if the system be simply laid on the plane, or, more ge- nerally, if the straining force lie without these limits, the prob- lem ceases to be statical. The condition of equilibrium is not fulfilled, and the body begins to move. We shall suppose that there is no straining force, and that the original distance of the body from the fixed point is precisely the unstretched length of the string. We must now, in conformity with the principle stated in p. 175, replace the geometrical force, T, by the elasticity of the string, a force initially zero. The maximum force of friction being less than the resolved force of gravity, the body will begin to descend. Neglecting the elasticity of the body and of the plane, and assuming the elasticity of the cord to be pro- portional to its extension, divided by its original length, we have as the equation of motion of the body down the plane KT d ' X KT f ' ^ && aP = & ' Sm a ~ /" cos a ) ~ -p (2) where / is the original length of the cord, and x is measured from the point at which the body would rest if the cord were drawn tight, but not extended. Assuming for the sake of brevity k* 2 p = g (sin a - /u cos a), 3 we may write the equation Integrating this we have x - p = A cos Ot + B sin Ot. Suppose that at the beginning of the motion 1 dx * = o,- = , 2 A 178 VALUE OF THE ACTING FORCE OF FRICTION. we have Hence x = p (i - cos 0) -I- TT- sin 0. The moment at which this motion ceases will be of course given by the equation dx or tan Ot = - 7r . 9p Substituting this value in (3) we have, recollecting that Ot is in the second quadrant, The corresponding value of the elastic force of the string will be 0V 2 ). Now, if this value be less than the sum of the resolved weight of the body, and the maximum force of friction estimated in the same direction, the body will remain at rest, and we have the equations T = H(frp + v / 5 i p 5 T0V 5 ) (4) F= Mg sin a - T, to determine the actual values of the tension and the friction. Suppose, for the sake of simplicity, that v = o, or that the body starts without initial velocity from the point at which the string is at its natural length. Then the foregoing equations give T= zMQ*p = iMg(sm a - ju cos a), (5) jP= Mg sin a - T= Mg(2fji cos a - sin a). Now, in order that the body may remain permanently at rest in this position, we must have T < or - Mg (sin a + /j. cos a), VALUE OF THE ACTING FORCE OF FRICTION. 179 or tan a < or = 3^. If this condition be satisfied, the equations (5) will give the permanent values of the tension and the friction. / With respect to this latter force, it appears from the value which we have obtained 1. That if tan a < 2ju, the value of F will be positive, or the friction will act upwards. 2. That if tan a = 2jit, the value of Fis zero, or there is no friction developed. 3. That if tan a > 2ju and not > 3^, the value of F will be ne- gative, or the friction will act downwards. If tan a > 3ju, the body will begin to ascend, and we shall have for its upward motion the equation -,. CL 00 -. . . . x K X ^ COS a ) --/- where 2 <7 = g (sin a + ju cos a). Integrating this as before, and determining the constants by the condition that at the commencement of the motion dx_ we find x = q + (zp - q) cos Ot. (6) From this equation, by the same reasoning as before, we find, 1 . That if tan a be not greater than 5^, the body will come to a state of permanent rest at the end of this oscillation, when Ot = TT. 2. That if this condition be fulfilled, the permanent values of the tension and friction will be given by the equations, T = 4fj. Mg cos a, (y) F '= Mg (sin a - 4fj, cos a). If tan a be greater than 5^, the body will descend, and the I 80 VALUE OF THE ACTING FORCE OF FEICTION. same method must oe pursued until the body come permanently to rest. The result of the whole discussion of the case of a heavy body placed on a rough inclined plane, at a distance from the fixed point equal to the natural length of the supporting string may be summed up as follows : i. If tan a be greater than (471.+ i)/*, and not greater than (4H + 3)^1, the body will come permanently to rest at the end of the (aw + i) th semi-oscillation, and the values of the tension and friction will be T= ^Mg (sin a - (an + i)/* cos a). (8) F = Mg {(47* + 2)fj. cos a - sin a). 2. If tan a be greater than (471 + 3) /i, and not greater than (471 + 5)^, the body will come to rest at the end of the (an + 2) th semi-oscillation and the values of Tand F will be T= 4(n+ I)IUL Mg cos a, F= Mg{sin a ~4(n + i)/u cos a). (9) 3. If tan a be any even multiple of /u, F= o, and the tension is the same as if the plane were smooth. 4. If the value of tan a lie between 471/1 and (471 + a)ju, the value of F is positive, and the tension is less than if the plane were smooth. 5. If the value of tan a lie between (411 4- 2}/m and (471 + 4}^ the value of ^is negative, and the tension is greater than if the plane were smooth. CHAPTER VIII. MISCELLANEOUS PROBLEMS. I. Problem of the Top. i. THE problem of the top maybe generally stated as follows : A solid of revolution, to which has been communicated a very rapid motion round its axis of figure, is placed upon a rough horizontal plane. Required to determine its motion. Let Fig. 28 represent the top, the plane of the paper being the vertical plane passing through the axis, and, therefore, through the point of contact. Let G be the centre of gravity of the top, and GZ\ the axis of figure. Let also - QNQ'N', QZiQ' be the equatorial and meridional sections, re- spectively ; the line QQ' being in the plane of the paper. Then the equatorial plane, QNQ'N' is evidently d perpendicular to the plane of the paper. The same is true of the horizontal plane Fig. 28. on which the top rests. Hence the line NX', which is parallel to the intersection of these planes, is perpendicular to the plane of the paper. Let the origin of the fixed axes be taken in the horizontal plane, the axis of s being vertical ; and let x, y, s be the co- ordinates of G. Let OX, OF, OZ be parallel to these axes. 1 82 MISCELLANEOUS PROBLEMS. Then if The the point of contact of the top with the horizontal plane, it is evident that TR is perpendicular to the nodal line NN'. Let GXt, GY GZ l be the movable axes, and GR the perpendicular on the horizontal plane ; also, let TP, TP f be perpendicular to GZ, QQ', respectively. Then it is plain that the polar angles which determine the position of the body are TO'T' = *, NGX, = 0, RGZ, = 0. We shall suppose that the lower extremity of the top is a spheri- cal surface, of which C is the centre. If, then, we assume Radius of sphere = TC = a, GC = b TP = 5, TP = n, TR = *', we have from the figure the following geometrical equations : = a sin 0, i) = b + a cos 9, z = a + b cos 0, z' = b sin 0. ( i ) Let R be the complete reaction of the horizontal plane, and c the angle of friction. We shall assume, for the purpose of simplifying the question, that the motion of rotation round the axis of figure is so rapid, as compared with either of the other angular motions of the body, that in determining the direction of the force of friction the motion of T, considered as a point in the body, may be taken to be perpendicular to the plane of the paper. If, therefore, the rotation be in the direction of the arrow at N 9 the force of friction is parallel to the line NN', the velocity of rotation being so great that there is necessarily slipping at T. The complete reaction, R, may, therefore, be resolved into R cos c parallel to OZ, and R sin e parallel to NN 1 . The second of these components, which is the force of friction, may be further resolved into R sin c cos i//, R sin c sin ^, parallel to the axis of x and y, respectively. To determine the components, with respect to the movable axes GXi, GYi, GZ\ of the moment of R round the centre of gravity, we shall suppose this force to be replaced by its com- MISCELLANEOUS PROBLEMS. 183 ponents R cos , R sin e. Now, the moment of R cos with re- gard to G is R cos s X TR, and the axis of the moment is NN'. This moment may be resolved into R cos e x TR cos , 72 cos x 7$ sin , o round GX 6rF 15 6rZj, respectively. Similarly, the moment of R sin E may be resolved into R sin E x TP' sin 0, JS sin x TP f cos 0, ^ sin e x TP round the same axis. To fix the signs of these moments, it is to be remarked that if the angles 0, be measured as in the figure, the positive directions of the movable axes will be 6rX 1} 6rPi, GZ\. Now, since the components p, q, r of the angular velocity are proportional to the indices of position of the instantaneous axis with regard to these axes, it is evident that the positive directions of the rota- tions indicated by p, q, r must correspond to a position of the instantaneous axis lying within the solid angle made by the positive directions GX l9 GY\, GZ V Hence it is easy to see that if the direction of the impressed rotation round GZ l be that indicated by the arrow ss, the positive directions of p and q will be indicated by the arrows xx, yy, respectively. The moment of the normal resistance R cos will, therefore, be negative with respect to GX ly and positive with regard to G PI. The moment of the force of friction R sin E will be negative with regard to all three axes. If, then, LMN be the moments of the total forces with respect to the movable axes, we have L = - R (s' cos cos + r? sin sin 0) M =- R(z' cos sin - rj sin cos 0) (2) 2. We now proceed to form the differential equations of motion. We have, in the first place, for the motion of the centre of gravity, I 84 MISCELLANEOUS PROBLEMS. d*x . = /csme cosif; etc ^-.flsinesin^ (3) d*z ^ = ^ cos - ' sin 9 sin < - 0' cos < q = i// sin cos ;// + 0'sin (5) r = 7 - i// cos 0; putting in like manner, ety = tfdt, d$ = tfdt, de = O'dt. 3. Assuming A - C = nA, udt = sin Odfa-u'dt = ^/M, we find from the first two equations (4), A { q' sin ^ - ;/ cos + nr (p sin + q cos 0) j = Rz' cos . (6 > ^t { g 1 ' cos -f p' sin ^ + nr (p sin - q cos 0) j = - Rr] sin . Also, from the first two equations (5) we have p sin + q cos fy = u, p cos $ - q sin <{> = - 0'. Whence, by differentiation, MISCELLANEOUS PROBLEMS. 185 p' sin (j> + q cos ^ - Q'fy' = u' p' cos (p - q' sin - w0' = - 0" ; or, replacing $' by its value given by the third equation (5) namely, 0' = r + u cot 0, p' sin + cf cos = i/ + O'(r + u cot 0) p' cos - o. Hence, inasmuch as, initially, 9' = o, 9 f will at first become positive ; or, in other words, the axis will commence by descending. But it appears from the second equation (7), that this downward velocity cannot attain any sensible value. For if any sensible positive value were assigned to 0', it is plain that, on account of the magnitude ofr, the left side of the equation would neces- sarily become positive, and could not, therefore, vanish. 5. If, in the third equation (3), we substitute for z its value a + b cos 0, and eliminate R by means of the first equation (7), we find (A + 6 2 sin 2 0) - + 6 2 sin 0cos 9 -bg$'m9 = u(Cr + Attcot9) etc c(t This equation may be written, -jj. (A + b 2 sin 2 0)jr z + ify cos = 2(Cr \ Ait cot 0). ( i 2 ) Eliminating r from this equation by means of (9), we have an equation only involving 0, u, and t. A second equation be- tween the same variables is obtained by eliminating R between the equations (7). But these equations do not seem to admit of further inte- gration. i88 MISCELLANEOUS PROBLEMS. 1 1 . Friction Wheels . 6. A wheel is attached firmly to an axle so as to turn with it. The system is then placed, as in Fig. 29, so that the axle rests Fig. 29. upon four vertical wheels, whose axes rest upon fixed bearings. If the first wheel have a given velocity of rotation, to compare the length of time which will elapse before it come to rest, with the duration of the movement which would have taken place if the wheel itself had rested on the fixed bearings. Let the fixed bearings be four cylindrical holes, and let the supporting wheels be equal in every respect. Let Fig. 29 repre- sent a section of the entire system by a plane perpendicular to the axles, and cutting two of the supporting wheels the directions of the rotation being indicated by the arrows. Then it is plain that the directions of the forces of friction acting upon the axle, #, at T lt T z , will be in both cases from left to right. Hence, if Ti/, T 2 I be the directions of the complete reactions of the sup- porting wheels on the axle (7, these lines must be, as in the figure, to the right of the corresponding radii C Ji, C T 2 . Put C.T, = C 2 T 2 = a, CT, = CT 2 = b, C.R, *> C 2 E Z = r ; and let 2TFbe the weight of the supported wheel and axle. Put also VCC, = VCC, = a. Let /?,, R 2 be the normal reactions at T i9 T z , and jP,, F z the MISCELLANEOUS PROBLEMS. I 89 forces of friction at these points. Then, since C remains at rest, and since, moreover, 2 W is equally supported by each pair of wheels, we have (E l + E 2 ) cos a- (Fi- F z ) sin a = W (13) (R v - R z ) sm a + (F l + F z ) cos a = o. We assume that each supporting wheel bears upon each of its cylindrical supports in one line only. This assumption is neces- sarily true, if the cylindrical hole be in the smallest degree larger than the axle which turns in it. Let n\ 9 r? 2 be the angles which planes passing through these lines and the axes of the wheels, respectively, make with the vertical. Let also Pi,P 2 be the normal reactions of two of the cylindrical bearings. Then, since the axes of the wheels remain at rest, and since the forces of friction ex- erted by these bearings have evidently their maximum values, we have jRi cos a - F l sin a + W' = 2 PI (cos ^ 4 fj. sin rji) = aPi cos( qi - e) sec c jR : sin a + FI cos a = aPi (sin ^ - fj. cos rji) = 2/\ sin (i^ - c) sec e (14) .R 2 cos a 4 F z sin a 4 W'= 2 P 2 (cos i] Z - fj. sin r/ 2 ) = aP 2 cos (173 4 t ) sec P 2 sin a - F 2 cos a= 2P 2 (sin r/ 2 + ju cos r; 2 ) = 2P 2 sin(sin rj, + f)sec, where W is the weight of each of the supporting wheels. Assume, for the sake of brevity, N. B. It is plain that 0, 0, are the angles which the com- plete reactions of the fixed bearings make with the vertical. Solving these equations (14) for R l9 R^ F^ F z , we find RI = 2Pi sec e cos (9 - a) - W cos a FI = 2 PI sec c sin (8 - a) 4 W sin a (15) R z = aP 2 sec e cos (a - 0) - JF' cos a jF 2 = aP 2 sec e sin (a - 0) - W sin a*. Substituting these values in (13), we have 2 (P l cos 6> + P 2 cos 0) = (TT 4 2TF 7 ) cos e , P : sin = P 2 sin ; 190 MISCELLANEOUS PROBLEMS. whence _ ( W + 2 W) sin cos E p _ (W+ 2 FT) sin cos 2 sin (0 + 0) ~~I^nT(0T0)~ Substituting these values in (15), we have, putting W+ ^W I -nW, ^TFsin (b cos (0 - a) , rr , /?, = -- r-t L - W cos a sm ( + t/) w^Fsin0sin(0- a) .__, . T! - '+ IF sin a (17) sm + ) in 0cos (a cos a sm (0 + whence we find easily, in sin (a - 0) T^, . ^- - ^T sin a ; in 0sin 0cos a l - z - - tf 7 sm a. sin (0 + 8) We must now distinguish three periods, each characterized by a different kind of motion. (i). When the upper wheel, to which has been communi- cated a velocity of rotation, is laid upon the supporting wheels, which are initially at rest, the motion at each of the four points of contact is combined slipping and rolling. This motion v con- tinues for a certain time, which may be called the first period, terminating when perfect rolling has been established at two of the four points of contact. (2). The second period commences with the termination of the first, and ends when perfect rolling has been [established at the four points of contact. (3). The third period embraces all the rest of the motion. It is evident that the equations (13) and (14), depending merely on the immobility of the axes, hold during"all these periods. MISCELLANEOUS PROBLEMS. 1 9 I (i) During the first period, the forces of friction at the several points of contact have their greatest values. We have, therefore, F l = R, tan f ', F z = R 2 tan e' where t' is the angle of friction at T l9 T 2 . Substituting for R l R 2 , jR, F z from (17), we find easily n W sin sin (9 - a - e) + W sin (0 + 0} sin (a + e] - o, n W sin sin ( - - E') - JP sin (0 + 0) sin (a - g') = o. Dividing these equations respectively by sin sin 9 sin (a 4 e') and sin sin sin (a - t')> we have W (Sot - ( W ^ W'} cot e + nWcot (a 4 e') = o, (18) W cot - ( W + W) cot + ft Wcot (a - e') = o. Hence, ^cot = JF cot (a + O + ( ^ + ^') cot ( ~ / ) (19) TTcot = W cot (a -Y) + ( W+ W) cot (a + e'). Now, if ?72jfc 3 be the moment of inertia of the upper wheel, and tt) the angle through which it has revolved, the equation of its rotation will be Substituting for F {1 F 2 from (17), we have . d\o TT7 , . / cot - cot 9 \ mk -j-^ = nWbsina * n , dt 2 \cot + cot 9J or, replacing cot and cot 9 by their values (19), we have - 7Q d 2 b) rrrl . (cot (a- c')- cot (a + x )) mk* -r = - 2 7r 6 sin a J 7 7 [ dt 2 (cot (a -E') + cot (a + e)( = - Wb sin 2c' sec a. (20) If then, 12 be the given initial velocity of the upper wheel, we have, by integration, da Wb sin 2e' sec a , 192 MISCELLANEOUS PROBLEMS. Again, substituting for 0, 0, in the values of F^ F 2 (17), we find _ tFsin 6 ' sin (q - e') _ W sin t' sin (q + x ) , sin 2a sin 2 a It is plain from these equations that F 2 > F^. Now, the equations of rotation of the friction wheels being respectively, and **'V* = aF t - 2/irP 2 , (23) it is easy to see that, on account of the smallness of r, the velocity of rotation of the wheel C 2 T 2 will increase more rapidly than that of C^. Hence perfect rolling will be established at T z sooner than at T^ Integrating the second of the foregoing equations, we have, putting r = ia t Now, when perfect rolling is established at Hence the time at which this "takes place is given by the equation, t = Wb sn 2e' sec a a 2 in which the values of F*P% are to be substituted from (22) and (16). Or, putting b = i'a, m'k'* = emk*, ~ Wi'*e sin 2e' sec a + F 2 - 2iP* tan i The velocity of the upper 'wheel at the moment when per- fect rolling is established at T 2 is found by substituting this value of t in (21). This velocity is MISCELLANEOUS PROBLEMS. 193 tan E Q l< ' dt Wi z e sin 2e' sec a + F z - 21*2, tan Hence we have Wi' z e sin 2e' sec a O Qo - &1 = -rrr^ : ; ^ ^FZ r nrr*6 sm 26 sec a + -r 2 - 2'^jr^ tan e Now, inasmuch as the diameter of the wheel is in general very great as compared with that of the axle, i i f are small frac- tions. Hence, unless e be large, which is never the case in friction wheels, Q - &i is very small, and may in practice be neglected. The velocity of the wheel C\T\ at the same moment is given by the equation m W __ = p l _ 2 iP l tan ) at tan Q Wi' z e sin ^^ sec a + F. 2 - 2iP 2 tan Hence, i' (F l - 2iP l tan C JQ O . ~dt ~ Wi' z e sin 2e 7 sec a + jP 2 - 2iP z tan e* Let this velocity be called Q'I. It is evidently a small quan- tity of the same order of magnitude as i'. If quantities of the order i'* be neglected, the value of Q\ is ., .sn a - a i = ' l sin (a + e ) It is evident from this equation that a cannot be less than E'. If a = e', the velocity of the wheel C l T l at the end of the first period is a small quantity of the second order. We have assumed, in this investigation, that the axis of the upper wheel remains at rest, or, in other words, that the axle does not roll up the circumference of either of the supporting wheels. Now, it is plain from the figure that this rolling up 2 C 1 94 MISCELLANEOUS PROBLEMS. could only take place at T 2 . In order, then, that the axis of the upper wheel may remain at rest, it is necessary and sufficient that the normal pressure Ri should not vanish. We must have, then, RI > o ; or, which is evidently the same, F l >o. Substituting the value of F l from the first of equations (22), we find, sin (a - t') > o. In order to the immobility of the axis, it is therefore neces- sary and sufficient that a should exceed the angle of friction at the circumference of the supporting wheel. As the smallest excess is sufficient, we may take a' as the limiting value of a. (2). During the second period of the motion there is perfect rolling at T 2 , and combined rolling and slipping at 7\. The equations applicable to this period are therefore (i). The equa- tions (16) and (17), which depend solely upon the immobility of the axes. (2). The equation F l = ^ tan s', denoting that the force of friction at T\ has its greatest value. (3). The equa- tion dtt> f j da) a Tt =b dt> denoting that perfect rolling has been established at T z . Dif- ferentiating this equation, and substituting for the second diffe- rential coefficients, we have 2iP* tan - F 2 = iei' 2 (F l + F 2 ), or substituting the values of FI, F 2 , P 2 from (17) and (16), n W sin 9 {i sin e - sin (a - $) \ + W sin a sin (0 + ) a sin 0-. Dividing by sin 0, and substituting for cot from the first of equations (18), we find (n + i) i sin i - A cos + B sin o (30) MISCELLANEOUS PROBLEMS. 195 where A = 2 (i - 2ei' 2 ) sin a, B = (n + i) cos a + (ft - i + 4nei' 2 ) sin a cot (a + e'). This equation determines 0. Substituting the value so found in the first equation (18), we obtain the value of 0, and thence those of F 19 jP 2 , &c. To find at what instant perfect rolling is established at T\, we have the equation of rotation of the lower wheel TCi, ' k( *& = (^1-2^1 tans). Integrating this equation, and observing that at the com- mencement of the second period, the velocity of rotation of the wheel CiT l is 12', we have m'V* ( -^ - Q'i J = at (Fi - 2iPi tan ), (31) t being the time which has elapsed since the commencement of the second period. Similarly, integrating the equation of rotation of the upper wheel, we have -afe(*', + .Fl) (32) But when perfect rolling is established at T^ Hence, tan.) a5'(J-, + - - -- < (33) where t is the duration of the second period. The velocity of the upper wheel at the termination of this period is found by eliminating t between (32) and (33). This value, which we shall call Q 2 , is ~dt = jR - 2iPi tan + 2ei'* (F l + F 2 ) ' ^ 196 MISCELLANEOUS PROBLEMS. in which F l9 F^ P l are to be replaced by their values as found above. Hence, i tan Now, we have seen (p. 193) that Q\ is a small quantity of the order i'. Hence it is evident that Q ll - Q. 2 is (unless e be large) a small quantity of the order z" 3 , and may therefore be neglected. We may, therefore, without sensible error, assume that, at the commencement of the third period, the upper wheel preserves the velocity originally communicated to it. (3). During the third period, perfect rolling being established both at T! and 7^, the velocity of rotation must be the same for all the supporting wheels. Hence, evidently, . D . -- - = F l - 2iPi tan = F 2 - 2iP 2 tan . (35) Substituting for F lt F*, P lt P 2 , from (17) and (16), we have 2W sin 9 sin cos a - sin a sin (0 + 0) = ni sin e (sin - sin 8). (36) Again, if mk~ be the moment of inertia of the upper wheel, and w the angle through which it has revolved, we have 7 c? 2 w , /r T,, TT7 .sin (0-A) . , x rf _ . _ 2 j (Fl t j.) = - a TFS g^Z-j .m . (37) Now, since it is supposed that there is no slipping at T l or T t9 we have o> = aw', whence, and from equations (35) and (37), sin 0- S sin (0 + 0) m'kP m'k'* or from (16), (17), ar (sin + sin 0) sin c - a 2 sin (0 - 0) sin a 'k"* sin (0 + 0) or sin + sin =^> sin (0 - 0), (38) where ..,. sin a p = (i -f 4ei 2 ) ^ s . ' i sin MISCELLANEOUS PROBLEMS. 197 The equations (36) and (38) are sufficient to determine 0, <. To solve these .equations, we may put the second under the form cos J (0-0) {sini(0 + 0) -psini(0-0)J = o, giving either - = TT or sin J (0 + 0) = p sin i (0 - 0). The first of these is inadmissible, inasmuch as we should then have from equations (16), an impossible equation, as P 1? P 3 are essentially positive quanti- ties. We must, therefore, take the second factor, and put sin i(0 + 0)=p sin i(0-0). (39) Eliminating, now, sinj (0 - 0) from equation (36), which may be put under the form cos J (0 + ) { sin a sin i (0 + $) - MI sin c sin 1 (0 - < } = n cos a sin sin 0, we have sin (0 + 0) = 2 sin a - in sin * Assume sin (0 - 0) = w sin (0 + 0). (41) Then, eliminating 0, between the equations (39), (40), (41), we find (42) Now, recurring to equation (37), we see that the moment of the force retarding the movement of the upper wheel is i + F t = 2bnv W sin a. If this wheel rested in fixed bearings, the retarding moment would be, evidently, 198 MISCELLANEOUS PROBLEMS. Hence, inasmuch as the time required to bring the wheel to rest is inversely proportional to the retarding force, we have, Time of rotation with friction wheels tane Time of rotation with fixed bearings ' nv sin a If, as before, we neglect quantities of the order ^ 3 , i' 2 , we have, approximately, sin a i sin Hence, and from (43), we find, Time of rotation with friction wheels i Time of rotation with fixed bearings " i cos c ^/^ + tan z a ' It is evident from this expression that friction wheels are used to most advantage when a is made as small as possible. Now, we have seen (p. 194), that the smallest admissible value of a is e', or c, if we suppose the coefficient of friction to be the same everywhere. If, then, in the foregoing expression (44) we make a = , we have, finally, Time of rotation with friction wheels i Time of rotation with fixed bearings i *fn? C os *e + sin * ' LOCOMOTIVES. A. Locomotive with a Single Pair of Driving Wheels. 7. A locomotive rests upon a number of wheels, of which two only are connected with the engine, these two being firmly at- tached to the same axle, so as to turn with it. If there be a slight difference between the diameters of these wheels, deter- mine whether either wheel will roll without slipping, and, if so, which. Let M be the mass of the entire train, F lt F 2 the effective forces of friction developed at the points of contact of the driving wheels (these forces being reckoned in the direction of the train's motion), and R the entire retarding force, consisting of the re- sistance of the air, the friction acting on all the other wheels, MISCELLANEOUS PROBLEMS. 199 and the resolved force of gravity, if the train be ascending an incline. Then the equation of motion of the centre of gravity is Jtf-^-tf + F.-JZ. . (45) Moreover, if mk* be the moment of inertia of the driving wheels and axle round their axis of rotation, and 9 the angle through which they have revolved during the time t, the equa- tion of rotation is mk* = P - ^ - F^ (46) where P is the moment of the force of steam round the axis of rotation diminished by the moment of the friction of the axle against its supports, and a^ a 2 are the radii of the driving wheels. (i). Let it be supposed that a\ > 2 - Then, if the motion of the wheel a\ be perfect rolling, the motion of the wheel a z is mixed slipping and rolling. Moreover, since a 2 is the lesser wheel, it is evident that for this wheel the motion of rotation is too slow for perfect rolling. Hence F z has its maximum value, and is directed against the motion of the train. If, then, W z be the pressure which this wheel exerts, and e the angle of friction, we have ITT rrr j. J &X dO F t - -W, tana, and ^ - a, ^ since the wheel i rolls perfectly. Differentiating this latter equation, and eliminating the differential coefficients from (45) and (46), we have, putting M = im. Substituting for F z , and neglecting, the small difference be- tween ! and 2 , we have, denoting either indifferently by a, ten iPa Fi = W^ tan E + y^ -f 7- 2 . Now, we know that FI cannot exceed Wi tan f, where JFi is the pressure exerted upon the wheel a l9 We must have, there- fore, _ _ PR+iPa W\ = or > fT 2 + -7-0 TIT- cot e. 200 MISCELLANEOUS PROBLEMS. But the last term being in general considerable, it is plain that this condition is not fulfilled unless the pressure W\ be much greater than W z - If, then, we provide (as we may), that this shall not be the case, the hypothesis of perfect rolling at the cir- cumference of the larger wheel, and, consequently, of a rotation of the smaller wheel too slow for perfect rolling, becomes impos- sible. (2). Let i < 2 - Then the investigation is the same, with this exception, that inasmuch as the smaller wheel rolls per- fectly, the larger wheel must rotate too fast for perfect rolling. In this case F z acts in the same direction as the motion of the train. Hence, F 2 = TF 2 tan e. Substituting this value in (47), we have -or . &R + iPa F l + W z tan e = 7-^ - . Hence, it is evident that the condition requisite for the truth of this hypothesis is ^ -Tr k*R + iPa W\ + W-i. = or > y cot c. If this condition be not fulfilled, both wheels will slip, in consequence of too rapid rotation. For it is plain that, unless a break be applied to one or both of the driving wheels, the ro- tation cannot be too slow for both wheels. In every case, there- fore, the friction developed at the circumferences of both wheels is in the direction of the motion of the train. The accelerating force by which the train is moved is found by substituting the values of F 19 F 2 in (45). The required value is, i Pa-a?R 3FTSF+TF"' so long as there is perfect rolling at the circumference of the smaller wheel. If both wheels be slipping, the accelerating force is JF 2 ) . JB The greatest steam-tension which can be advantageously used is found by equating these values. We have, then, MISCELLANEOUS PROBLEMS. 2OI iPa = ( JT, T JF a ) (a 2 + 2 ) tan - k?R, (48) in which it will be remembered that P denotes the difference between the moment of the steam pressure to turn the driving wheels, and the moment of the friction of the axle in its supports to retard them. Any excess of steam pressure above the value so found is useless.* B. Locomotive ivith Linked Driving Wheels. 8. A locomotive rests on in wheels, which are so attached that they must necessarily revolve with the same angular velocity. If the diameters of these wheels be not mathematically equal, to determine the value of the motion of each, so far as it is deter- minate. Before entering upon the special question of the locomotive, we shall consider the following general problem : A material system consists of a solid body with a number of wheels attached to it, which are capable of turning round axes fixed in the body, and are parallel to each other. These wheels are so connected that they are compelled to revolve with the same angular velocity, and are acted on by given forces. To determine the equation of rotation. Let it it be supposed that the connexion between any two of the wheels is effected by means of a coupling bar, attached by two arms of equal length to the axles of the wheels ; these arms being firmly fixed to the axles so as to turn with them, and being attached to the coupling bar by a joint round which they can turn freely. For the sake of greater generality, let it be sup- posed that the weights of these bars are taken into account. The general dynamical equation for any system is Applying this equation to the case of a number of solid bodies connected in any way, we may break up the sum denoted by S into a number of integrals, each of which is extended through * It i., in fact, mischievous. Vid. Note A at the end of the volume. 2 I) 202 MISCELLANEOUS PROBLEMS. one of the solid bodies. In the present case, we have three classes of bodies included in S ; namely, i. The solid body itself. 2. The attached wheels. 3. The coupling bars. Let, then, M be the mass of the solid body, m that of one of the wheels, and m' that of one of the coupling bars The general dynamical equation becomes, then, We shall suppose that the motion of the solid body is, at all its points, parallel to a fixed line, which we shall take for the axis of x. We shall suppose, further, that the several axes of rota- tion are perpendicular to this line, and that the axis of?/ is pa- rallel to these axes. Let xfl&i be the co-ordinates of the centre of gravity of the solid body, and a^c^ a 2 6^ 2 , &c., the co-ordi- nates of the centres of gravity of the several wheels. Let, also, x'y'z* be the co-ordinates of any point in the'solid body, referred to axes through its centre of gravity, and rj the co-ordinates of a point in one of the wheels referred to axes through its centre of gravity. We have, then, for the solid body, x =x l + x' t y = yi + y, z = z { + z, (5 1 ) and for the wheel a = 0i + , y = bi + q, * = (?! + . (52) We now proceed to consider the values to be assigned to Sx, Sy, s for the several parts of the system. (i). With regard to the solid body itself, as we suppose the motion to be parallel to the axis of x, we may, without limiting the generality of the solution, assume S* = &*, fy = o, $z = o. (2). With respect to the wheels, the rotation being round axes parallel to the axis of y, we may take for the first wheel MISCELLANEOUS PROBLEMS. 203 with similar values for the other wheels. (3). With regard to the coupling bars, it is easy to see that each one of these is at all times perpendicular to the axes of the wheels which it connects, and that each point of the bar describes a circle, whose radius is equal to the common length, 6, of the arms, by which the bar is connected with these axes. We may take, therefore, for this part of the system, x = ^x,-b sin 080, 8y = o, z = b cos 080. Substituting these values in the general equation (50), and equating to zero the coefficients of 8#i, 80, we have from the former (S3) \XdM + S iXdm -f S \Xdrn. If M be the mass of the entire system, and X Q the x co-ordi- nate of its centre of gravity, this equation may be written This is the ordinary equation for the motion of the centre of gravity. The coefficient of 80 being equated to zero gives (54) Substituting in the first of these sums the values of x and z from (52), and recollecting that $%dm = o, l%dm=o, we have 204 MISCELLANEOUS PROBLEMS. Let p be the distance of any point in the wheel from its centre. Then = p cos 0, = p sin 0, whence and, consequently, |Y y d 2 .,* ffif**a where k is the radius of gyration. It is evident, also, that where L is the moment of the forces acting on the wheel, taken with regard to its axis. The first of the sums denoted by S is, therefore, SL-|fsrf (56) With respect to the second of these sums, since the motion of each coupling bar is parallel to itself, we have, evidently, (fcc = C^TO d*(b cos 0) flfts _ d*(b sin 0) dt* ~ W H ~~W l ' dt* ~ ~d& ' whence, v d? "& "~dF " w- Moreover, as we do not suppose any force, except gravity, to act on the coupling bar, we have, putting a = incl. of axis of x to the horizon, If these values be substituted in the second of the above sums, it becomes **<>-, ,, - &0 Sm o sin - Sm'o - g^m b cos 0. d/t (it Substituting these values in (54), we obtain, finally, MISCELLANEOUS PROBLEMS. 205 the values of 0, in the several terms of which 2 is'composed, not being necessarily the same, but differing by constant quantities. If the masses of the coupling bars be so small that they may be neglected, this equation becomes being the same as that of the motion of a single wheel whose moment of inertia is the sum of the moments of the linked wheels, and to which the original forces are applied so that the sum of their statical moments with regard to its axis may be equal to the sum of these moments, taken each with regard to the axis of the wheel to which it is actually applied. The equation d*0 Sum of statical moments dt 2 Sum of moments of inertia is always true, if we suppose the mass of each coupling bar to be concentrated at the extremity of either arm 6, and to be acted on by two accelerating forces ; namely, i. Gravity. 2. The total accelerating force on the system. In the case of a locomotive attached to a train, the second of these forces is, in general, very small as compared with the other, and may in practice be neglected. 9. To apply the equation (59) to the case of a locomotive, let (7, C' (Fig. 30), be the centres of two wheels linked by the bar AA '. Let the steam pressure, $, be applied in the direction B'B. Then, if we put AB = /, AC = AC 1 = b, ABC = 0, ACB = 9, the moment of the steam pressure with regard to C will be 2o6 MISCELLANEOUS PROBLEMS. Sb cos sin (0 + 0), where and are connected by the equa- tion / sin = b sin 6. Similarly, for the wheel immediately con- nected with the engine at the opposite side, the moment will be S'b cos 0' sin (0' + 0'), ^' and 9' being connected by the same equation as and 0, and 0' being a constant quantity, inasmuch as the wheels revolve at the same rate. The other forces acting on the wheels are the forces of friction - F lt - F z , . . . -F 2n acting tangentially at the points T lt T 2 , &c., and the forces of friction fij -ft, . . . -/an? acting tangentially at the common surfaces of the axles and their sockets. Let a lt a z , . . . a 2n be the radii of the wheels, and r x , r z . . . r zn the radii of the axles. Then, the sum of the moment of these forces will be or, neglecting the small differences between the radii, as being here unimportant, - a (F, + F, +, &c., + F M ), -r(f l +f t +,&c.,+ /,). With regard to the terms contained in the sum ^m'b cos 0, it is plain that the values of 6 will be the same for wheels at the same side, differing by a constant from these values for wheels at the opposite side. Hence, putting 2a for the constant difference, and Jffor the sum of the masses of the bars, we have, assuming the bars at the two sides to be q. p. equal S (m'b cos 9) = - Wb { cos 4- cos (0 + 2a) j = M'b cos a cos (9 + a). Substituting these terms in the general equation (59), and ne- glecting the small differences between the wheels, we have the following equation for 6 + M'b 2 ) Sb cos sin ($ + 0) + S'b cos 0' sin (0' 4 0') (it (60) - a (Fi + F 2 -f &c.) - r (/i+/ 2 + &c.) - M'bg cos a cos (0 4 a), where 0, $', 0' are to be eliminated by the equations / sin = b sin 6, I sin ' = b sin 0', 0' - 9 = 2a. MISCELLANEOUS PROBLEMS. 2 07 It must be remarked, with regard to this equation, that the term Sb cos $ sin (0 + 0) represents the moment of the steam tension only for values of contained in the first semicircle. Beyond this it would follow from the trigonometrical formula that the moment should change sign. But the moment of the steam tension never does change sign, inasmuch as the alternating motion of the piston causes the tension itself to change sign at the same time with the perpendicular on its direction. The same is true of the wheel at the opposite side, and the angle 0'. 10. Let, as before, R denote the resultant of the forces resisting the motion of the train, these forces being the resistance of the air and the resolved weight of the train, if ascending an incline. Then the equation of motion of the centre of gravity will be (61) Let it be supposed i . That there is perfect rolling at the cir- cumference of the wheels i, 2, . . . p. 2. That the rotation of the wheels p + i, p + 2, . . . a z . Then it is evident that the velocity of rotation of the second wheel, is too slow for perfect rolling, and therefore that the force of friction, FZ, acts against the motion of the carriage, or in the direction in which we have supposed these forces to be measured. We must, therefore, take F 2 positively. Eliminating now the diffe- rential coefficients as in Art. 7, and neglecting the small diffe- rence between ai and 2 , we have (k* + m) (F, + F z ) = Ek\ whence Let P!, P z be the pressures on the circumferences of the two wheels respectively. Then, since F z has its maximum value, FZ = P 2 tan . Hence Now we know that FI cannot lie without the limits + Pi tan e. Hence it is necessary to the possibility of the hypothesis of per- k*R feet rolling at the circumference of the larger wheel that -75 A 2 + ea* should not lie outside the limits (P 2 + PI) tan c, (P z - PJ tan e. If P! > P 2 , or, in other words, if the greater pressure correspond to the larger wheel, the latter condition is necessarily fulfilled. (2). Let ! < 2 , and let it be supposed, as before, that perfect rolling is established at the circumference of a } . Then it is evident that the rotation of a 2 is too fast for perfect rolling, and therefore that F 2 acts in the direction of the motion of the carriage, or in a direction opposite to that in which we have supposed it to be measured. We have then FZ = - P 3 tan f. Substituting this value in equation (64), which holds equally in this case, k*R F } = 75 - + P 2 tan c. * * MISCELLANEOUS PROBLEMS. 211 7,2 D If, therefore, r r-$ be greater than (P l - P 2 ) tan f , this hypo- /C "r C'Cv thesis is impossible. This is necessarily so if PI < P 2 , or in other words, if the greater pressure correspond to the larger wheel. To recapitulate the results of this discussion (i). Let it be supposed that the greater pressure corresponds to the larger wheel, and let the moving force R be gradually augmented from zero. Then the larger wheel will roll perfectly until R reaches the limit indicated by the equation J^ 2 =(P 1 + P 3 )tan E . (65) When this limit is passed, both wheels will slip in consequence of too slow, rotation. (2). Let it be supposed that the greater pressure corresponds to the smaller wheel, and let R be gradually augmented as before. Then the smaller wheel will roll perfectly until R reaches the limit indicated by the equation =(P '- P ' )tan - (66) When this limit is passed, the larger wheel will begin to roll perfectly, the smaller wheel slipping until R reaches the limit indicated by (65). If R be still further increased, both wheels will slip as before. It appears from (64) that the same result holds here as in the case of driving wheels, namely, that the sum of the effective forces of friction developed at the circumferences of two wheels, constituting, in the case of driving wheels, the moving force, and, in the case of supporting wheels, part of the retarding force, is but slightly affected by a small difference between the diameters of the wheels.* * The conclusions arrived at in discussing the problem of locomotives are subject to certain modifications, in consequence of the difference between the coefficients of statical and dvnamical friction. Vid. Note A. ( 212 ) III. QUESTIONS FOR EXERCISE. 1 . One end of a beam rests upon a rough plane inclined to the horizon, at an angle greater than the angle of friction ; the other end being sustained by a string which, passing over a pulley, supports a weight. If the position of the beam be given a. Determine the position of the pulley when the suspended weight is the least possible. b. Determine the least weight which will support the beam. c. If the inclination of the plane were equal to or less than the angle of friction, how would these results be modified ? 2. Two ladders AB, AB', resting on a rough horizontal plane A A', are kept in equilibrio by a cord BE connecting their upper ends. If the friction of the plane be gradually diminished, find which ladder will slip first. 3. A circular cylinder rests upon two planes inclined at the same side of the vertical, having its axes parallel to their inter- section, and is supported by a cord coiled round it, passing over a pulley, and sustaining a weight. Determine the conditions of equilibrium. 4. A cylinder is supported on a rough inclined plane by a string coiled round it, and fixed to a point in the plane. a. If the coefficient of friction be less than unity, determine the point to which the string should be fixed in order that the inclination of the plane may be the greatest possible. b. Find the greatest inclination of the plane consistent with equilibrium. 5 . Determine these positions, if the coefficient of friction be equal to or greater than unity. 6. In the preceding problem, if the string be attached to a weight which hangs freely, find the conditions of equilibrium. a. If these conditions be' not fulfilled, what is requisite in order that the initial motion may be one I. Of pure slipping, II. Of rolling up, in. Of rolling down. QUESTIONS FOR EXERCISE. 2 I 3 7. A stick passes over the lower of two rough pegs,, and under the other, the pegs being equally rough ; find the distance of the centre of gravity of the stick from either peg when equi- librium is on the point of being broken. 8. A rod rests in the limiting position of equilibrium on two equally rough rectangular planes, whose line of intersection is horizontal and at right angles to the rod. If now the position of the planes be changed by causing them (still at right angles to each other) to revolve round their line of intersection, the corresponding positions of the rod will pass through the same point. 9. A cube placed on a smooth inclined plane is kept in equi- librium by a string attached to the middle point of its highest edge, passing over a pulley, and sustaining a weight ; show that this is impossible if the inclination of the plane exceed 45. 10. If the plane be rough and the direction of the string be given, find the least coefficient of friction which will give equili- brium. 1 1 . A material particle is laid on a rough plane inclined to the horizon at the angle of friction ; if it be required to drag it down the plane, determine the limits of the direction of the re- quisite force. 12. A beam resting on two smooth planes is kept in equili- brium by a cord attached to its centre of gravity and to a fixed point ; where should the point be situated in order that this may be effected with the weakest possible cord ? 13. How is this modified if the planes be rough? 14. Three equal rods AB, BC, CD are connected by smooth hinges at B and (7, the extremities AD resting on a rough horizontal plane, and the whole system being situated in a ver- tical plane. Determine the extreme symmetrical positions of equilibrium. 15. A beam rests with one extremity on a rough horizontal plane, the other extremity being supported against a cylinder of any form, which also lies upon the horizontal plane. Show that the actual amount of friction developed at each point of con- tact is indeterminate, unless the cylinder be, separately, in a position of equilibrium. 214 QUESTIONS FOB EXERCISE. 1 6. A ladder is placed against a rough vertical wall, the co- efficients of friction of the wall and the ground being given. Determine the weight of the heaviest man who can ascend to the top of the ladder without causing it to slip, and state under what circumstances this weight is unlimited. 17. A beam is laid horizontally upon two rough planes equally inclined to the horizon, to which their intersection is parallel. If the beam be in an extreme position of equilibrium, show that the angle between the vertical plane passing through it and either of the inclined planes is equal to the complement of the angle of friction, and hence deduce the condition neces sary in order that any horizontal position may be a position of equilibrium. 1 8. It is required to sustain a cubical block on a rough in- clined plane, by a cord attached to its upper edge and to a fixed point. Where should the point be placed in order that this may be done with the weakest possible cord ? 19. A cubical block is laid on a rough horizontal plane ; a string attached to the middle point of one of the upper edges of the cube passes over a pulley and sustains a weight which is gradually increased till equilibrium is broken. Determine the nature of the initial motion. 20. A cubical block, and a cylinder whose diameter is equal to a side of the cube, are laid upon a rough plane, and are attached to each other by a cord coiled round the middle of the cylinder, and fixed to the middle point of one of the edges of the cube which is parallel to the axis of the cylinder. If the plane be then slowly raised (the cube being uppermost) until equilibrium is broken, what will be the nature of the initial motion ? 2 1 . A cylinder is laid on a rough horizontal plane, and is in contact with a rough vertical wall, the coefficients of friction being equal ; a string, coiled round it at right angles to the axis, passes over a fixed pulley and sustains a weight which is gra- dually increased till equilibrium is broken. Determine the nature of the initial motion. QUESTIONS FOR EXERCISE. 215 22. A heavy bead strung on a rough vertical circle is pro- jected from the lowest point with a velocity just sufficient to carry it round ; determine this velocity. 23. A particle resting on a rough inclined plane is attached by a fine string to a fixed point in the plane ; if the particle be disturbed from its position of equilibrium, determine the angle through which it must be moved in order that it may just return to its original position. [N. B. The original position of equilibrium is supposed to be that which it "would be if the plane were smooth.] 24. If a heavy bead, strung on a fixed, rough, circular wire, whose plane is horizontal, receive a given impulse in the direc- tion of the tangent to the circle ; determine the arc described before it is reduced to rest. 25. An elastic ball rolls along a rough horizontal plane with a given velocity, and impinges perpendicularly against a smooth vertical wall. Determine the velocity with which it is reflected, and the point of its path at which a perfect rolling motion is re- established. 26. A material particle is attached by a string to a fixed point in a rough inclined plane on which it moves ; if the particle start without velocity from the position in which the string is horizontal, determine the greatest coefficient of friction which would allow it to reach the lowest point. 27. A cylinder is laid on a rough inclined plane to a point in which, situated above the point of contact, it is attached by a string coiled round it at right angles to the axis, and leaving it at a point which is not one of contact with the plane ; if the cylin- der be allowed to descend, find the accelerating force on its centre of gravity in any given position. 28. A right cone is laid on an inclined plane, so rough as to prevent slipping, and is slightly disturbed from its position of equilibrium. Determine the motion. 29 A weight is laid on an imperfectly rough inclined plane ; a thin cord, attached to the weight, passes under and round a cylinder which is laid on the same plane above the weight; the 2l6 QUESTIONS FOR EXERCISE. coefficients of friction being different for the two bodies. Ex- cluding lateral movements, the simple movements of which each of these bodies is capable arc for the first, slipping up or down ; and for the second i. pure rolling up or down ; 2. rolling and slipping up or down; 3. pure slipping up or down; 4. rotation without motion of the centre of gravity. Examine the various combinations of these movements, and determine which of them are possible. 30. A sphere is moving on a horizontal plate (so rough as to prevent slipping) which has a uniform motion of rotation round a vertical axis. Determine the motion. 31. A sphere descends a rough inclined plane of given height and length, under the influence of gravity. Show that there is necessarily a value of the coefficient of friction for which the vis viva gained in the descent is a minimum. 32. Two spheres are projected with different velocities and in different directions along a rough horizontal plane. Show that the motion has three periods, during the first of which the path of the centre of gravity is an arc of a parabola, during the second an arc of a different parabola, and during the third a right line. 33. A cylinder of any form is laid upon a rough horizontal plane, and allowed to oscillate. Prove that the initial motion will be pure rolling if P* tan a tan c > or = . where is the angle of friction, a the angle which the plane passing through the centre of gravity of the cylinder and the line of contact makes with the horizon, p the vertical height of the centre of gravity, and A; the radius of gyration of the cylinder round an axis parallel to the sides, and passing through the centre of gravity. 34. A carriage resting on four equal wheels rolls down an QUESTIONS FOR EXERCISE. 217 inclined plane ; determine the time in which it will reach the bottom. 35. A circular cylinder is laid upon two others which are placed upon a rough inclined plane, the axes of all three being horizontal. Show that equilibrium is impossible, and determine the nature of the initial movement. 36. It is required to draw a heavy body up a rough inclined plane by the application of a constant force parallel to the plane. Determine the intensity, so that the total expenditure of the force may be the least possible. 2 F 2l8 NOTES. NOTE A. STATICAL AND DYNAMICAL FRICTION. FN all questions which concern the transition of a particle from a state of rest to a state of motion, the results obtained must be modified by the consideration that the coefficient of friction is changed in this passage. Thus, for example, a force applied to a material particle which rests upon a rough surface may be sufficient to keep it in motion, and even to communicate to it a finite velocity when it has once begun to move, and yet may be insufficient to set it in motion. The first object will be attained if the ratio of the tangential to the normal component of the acting force (supposed to be finite) exceed, by any finite quantity, the coefficient of dynamical friction. For the attain- ment of the second object, it is necessary that the same ratio should exceed the coefficient of statical friction, which differs sensibly from the former. It is plain, therefore, that the problem of initial motion differs sensibly according as we suppose the velocities initially communicated to a particle or system to be absolutely zero, or to be only indefinitely smalL This difference is due to two causes : the first (which has been already considered in pp. 104-9) being the fact that the geometrical forces change sensibly in the transition of the system from a state of rest to a state of motion ; the second being the change, also sensible, which, in this transition, affects the coefficient of friction. Both causes tend to produce the same result namely, the possible development of a finite force by the communication of indefinitely small velocities to a system which was previously in equilibrium.-, A result of some importance to Practical Mechanics follows from these considerations. If we wish to utilise fully the force of friction as a resisting power, we must be careful not to augment the ratio of the tangential to the normal component of the force acting at the point of contact sufficiently far to produce relative motion. For, if relative motion be produced, the available force of friction is at once sensibly diminished. The same result will often follow an unsteady application of force. A slight blow, or a sudden muscular contraction, may, by augmenting the tangential component, produce a relative velocity, or slipping, at NOTES. the point of contact. Now, if there were no difference between statical and dynamical friction, this velocity, which is in general small, would be unimportant, as it would be speedily destroyed by the forces which had previously produced equilibrium. But, how small soever the velocity be, it is sufficient to convert statical into dynamical friction, and therefore to diminish sensibly the force by which equilibrium is preserved. It is, therefore, quite possible that the tangential compo- nent of the acting force, which, in the state of rest, was less than the maximum force of friction, may, in the state of motion, exceed it, and thus augment the small velocity originally communicated. This seems to be the explanation of the advantage which belongs to a steady step in walking over slippery ground. The same conclusion is applicable to the transition of a solid body from a motion of pure rolling to a motion of combined rolling and slip- ping. So long as the motion is pure rolling, the point of contact is at each instant at rest, and therefore the friction developed is statical fric- tion. The limit of the force so developed is therefore the maximum value of statical friction, and we may utilize this or any smaller amount. But when the point of contact begins to slip, the friction becomes dyna- mical, and its maximum value is sensibly reduced. If, therefore, we desire that the amount of friction developed at the point of contact should be as great as possible, we must be careful so to arrange the moving force that the motion of the body may be one of pure rolling. Now, this is precisely the case with a locomotive. The force by which the train is drawn forward is necessarily equal to the sum of the forces of friction developed at the points of contact of the driving wheels. In order, therefore, that this sum may be as great as possible, we must endeavour to provide that the motion of these wheels shall be a motion of pure rolling. If, then, the motion of rotation impressed by the engine be too rapid for pure rolling, the force by which the train is drawn for- ward will be augmented by reducing the steam pressure until perfect rolling is established. Practical engineers are perfectly aware of this principle, which they express by saying that, when the wheel begins to slip, its lite is diminished. 220 NOTES, NOTE B, p. 170. THE following difficulty, connected with. Ex. 2, may suggest itself to the student : If the system be in equilibrium, this equilibrium will not be dis- turbed by fixing the lower point, A. If this point be fixed, the only possible movement of the system will be a rotation round A. Such a rotation, which would lower the centre of gravity, does not at first sight appear to be resisted by any force whatever. For at the very commence- ment of the rotation the rod CD would separate from the vertical board at the point E. As this separation would be effected without any motion of the one body along the surface of the other, no friction would be developed at the point E ; and there would be nothing to prevent the rotation from taking place. This reasoning would be conclusive, if we assumed the perfect rigi- dity of the system. If there be absolutely no elasticity in the rods, or in the joint at B, the initial movement of CD would necessarily sepa- rate it from the vertical board, and the force of friction could not, as is evident from its nature, resist this separation. But if there be any elasticity in the system, no matter how small the amount may be, the initial movement of CD will not separate it from the vertical board. When the system is placed as in the figure, the angle ABE is slightly increased by the strains at A and E. As AB revolves round A, the elasticity of the joint will tend to restore ABE to its original value, and thus cause BC to remain in contact with the board for a time, which is not absolutely zero. There will be, thus, a movement of sliding of BC along the board at E, which movement will necessarily develope and be resisted by the force of friction at this point. This result is obviously independent of the amount of elasticity possessed by the rods, provided that it be not absolutely zero. THE END. 14 DAY USE 5 DESK FROM WHICH BORROWED LOAN DEPT. (Pa^lt^n^ iMSSSi/gSK*. Berkeley YC !0950 U.C. BERKELEY LIBRARIES