18 ON THE MECHANICAL THEORY OF HEAT. point a similar equation to the above;and by summation we shall obtain the following: Now the quantitytheof the whole syatem of points.If we take a simple expression for the vis viva, and put T=2… …(26), tl:en the equation becomes d-(+y+2）k②m But the right-hand side of this equation is the expression for the work done during the time dt.Integrate the equa- tion from an initial time t to a time t,and call To the vis viva at time t:then the resulting equation is the meaning of which may be expressed as follows: The Work done during any time by the forces acting upon a system is equal to the increase of the Vis Viva of the syetem during the same time. In this expression a diminution of Vis Viva is of course treated as a negative increase. It was assumed at the commencement that all the points were moving freely.It may,however,happen that the points are subjected to certain constraints in reference to their motion.They may be so connected with each other that the motion of one point shall in part determine the motion of others;or there may be external constraints,as for in- stance,if one of the points is compelled to move in a given fixed plane,or on a given fixed curve,whence it will natur- Translator's Note.The vis riva of a particle is here defined as balf the mass multiplied by the square of the velocity,and not the whole mass, as was formerly the custom. Google
18 ON THE MECHANICAL THEORY OF HEAT. point a similar, equation to the above; and by summation we shall obtain the following: , ~ m I":"~ (X dx ydy. Zdz)d (9 W ) '" "2 v - '" dt + dt + dt t .............. o. Now the quantity I ~", is· the vis viva of the whole sy~tem of points. If we take a simple expression for the tis viva, and put T= I ~~ v' ..............•••.••......... (26), then the equation becomes 'dT~I (X~~ + y ~~ +Z~;) dt ..•......•.. (27). But the right-band side of this equation is the expression for the work done during the time dt. Integrate the equation from an initial time to to. a time t, and call To the vis viva at time to: then the resulting equation is _ (t, (' d:JJ d.1J dz) '} T- To- Jt I X dt + Y,it + Z dt dt ............ {_8}, , 0 . the meaning of which ma.y be expressed as follows: The Work done during any time by the forces acting upon a system is equal to the increase of the Vis Viva of the sYBtem during the 8a1~e time. ' In this expression a diminution of Vis Viva is of course treated as a negative increase. It was assumed at the commencement that all the points were moving freely. It may, however, happen that the points are subject~ to certain constraints in reference to their motion. They may be so connected with each other that the motion of one point shall in part determine the motion of others; or there may be external constraints, as for instance, if one of the points is compelled to move in a given fixed plane, or on a given fixed curve, whence it will natur- , * Translator's Note. The 'Vi, {'iva of a particle is here defined as half the mass multiplied by the square of the velocity, anll not the Whole mass, as was fprmerly the custom. Digitized by Coogle
MATHEMATICAL INTRODUCTION. 19 ally follow that all those points,which are in any connection with it,will also be to some extent constrained in their motion. If these couditions of constraint can be expressed by equations which contain only the co-ordinates of the points, it may be proved,by methods which we will not here con- sider more closely,that the reactions,which are implicitly comprised in these conditions,perform no work whatever during the motion of the points;and therefore the principle given above,as expressing the relation between Vis Viva and Work done,is true for constrained,as well as for free motion. It is called the Principle of the Equivalence of Work and Vis Viva. 9.On Energy. In equation (28),the work done in the time from to to t is expressed by (密++z in which t is considered as the only independent variable, and the co-ordinates of the points and the components of the forces are taken as functions of time only.If these functions are known(for which it is requisite that we should know the whole course of the motion of all the points),then the inte- gration is always possible,and the work done'can also be determined as a function of the time. Cases however occur,as we have seen above,in which it is not necessary to express all the quantities as functions of one variable,but where the integration may still be effected, by writing the differential in the form (Xdc+Ydy+ed), and considering the co-ordinates therein as independent vari- ables.For this it is necessary that this expression should be a perfect differential of some function of the co-ordinates, or in other words the forces acting on the system must have an Ergal.This Ergal,which is the negative value of the above function,we will denote by a single letter.The letter U is generally chosen for this purpose in works on Me- cbanics:but in the Mechanical Theory of Heat that letter is needed to express another quantity,which will enter as 2
,MATHEMATICAL INTROpUCTION. 19 ally follow that all those points, which are in any connection with it, will also be to some extent constrained in their motion. If these 'conditions of constraint C'U1 be expressed by equations which contain only the co-ordinates of the points, it may be proved, by methods which we will not here consider more closely, that the reactions, which are implicitly comprised in these conditions, perform no work whatever during the motion of the points; and therefore the principle given above, as expressing the relation between Vis Viva. and Work done, is true for constrained, as well as for free motion. It is called the' Principle of tM Equivalence of Work and Vis Viva. ' § 9. On Energy. In equation (28), the work done in the time from to to t is expressed by ft (' d:c 'dy nz) t.?" X dt + Y dt + Z dt dt, in which t is considered as the only independent variable, and the co-ordinates of the points and the components of the forces are taken as functions of time only. If these functions are known (for which it is requisite that we should know the whole course of the motion of all the points), then the integration is always possible, and the work done 'can also be determined as a function of the time. Cases however occur, as we have seen above, in which it is not necessary to express all the quantities as functions of one variable, but where the integration may still be effected, by writing the differential in the form I (Xdx + Ydy+Zdz), and considering the co-ordinates therein as independent variables. For this it is necessary that this expression should be a perfectdifi'erential of some function of the co-ordinates, or in other words the forces acting on the system must have an Ergal. This Ergal, which is the negative value of the above function, we will denote by a single letter. The letter U is generally chosen for this purpose in works on Mechanics: but in the Mecha.nical Theory of Heat that letter is needed to express another quantity, which will enter as 9-2 Digitized by Coogle
20 ON THE MECHANICAL THEORY OF HEAT. largely into the discussion;we will therefore denote the Ergal by J.Hence we put: (Xdx+Ydy+Zde)=-dJ...............(29), whence ifJo be the value of the Ergal at time to,we have: (Xda+dy+zda)..) which expresses that the work done in any time is equal to the decrease in the Ergal. If we substitute Jo-J for the integral in equation(28), we have: T-T。=J。-JorT+J=T。+J。(31) whence we have the following principle:The sum of the Vis Viva and of the Ergal remains constant during the motion. This sum,which we will denote by the letter U,so that U=T+J.(32), is called the Energy of the system;so that the above prin- ciple may be more shortly expressed by saying:The Energ3 remains constant during the motion.This principle,which in recent times has received a much more extended application than formerly,and now forms one of the chief foundations of the whole structure of physical philosopby,is known by the name of The Principle of the Conservation of Energy. Google
20 ON TilE MECHANICAL THEORY OF HEAT. largely into the discussion; we will therefore denote the Ergal by J. Hence we put: I(Xtk + Ydy + Zrls) = - 0. .............. (29), whence if Jo be the value of the Ergal at time to, we have: J~I(Xtk+ Ydy +Zdz) =t1o-J ............ (30), which expresses that the work done in a.ny time is equa.l to the decrease in the Ergal. If we substitute Jo-J for the integral in equation (28), we have: T- To=t1o-J or T+ J = To+t1o ......... (31); whence we have the following principle: The sum of the Vis Viva and of the Ergal remains constant during the motion. This sum, which we will denote by the letter U, so that U = T + J. ............................. (32), is called the Energy of the system; so that the. above principle may be more shortly expressed by saying: The Energy remains constant during the motion. This principle, which in recent times has received a much more extended application than formerly, and now forms one of the chief foundations of the whole structure of physical philosophy, is known by the name of The Principle of the Oonservation of Energy. Digitized by Coogle
(21) CHAPTER I. FIRST MAIN PRINCIPLE OF THE MECHANICAL THEORY OF HEAT,OR PRINCIPLE OF THE EQUIVALENCE OF HEAT AND WORK. §1.Nature of Heat. Until recently it was the generally accepted view that Heat was a special substance,which was present in bodies in greater or less quantity,and which produced thereby their higher or lower temperature;which was also sent forth from bodies,and in that case passed with immense speed through empty space and through such cavities as ponder- able bodies contain,in the form of what is called radiant heat.In later days has arisen the other view that Heat is in reality a mode of motion.According to this view,the heat found in bodies and determining their temperature is treated as being a motion of their ponderable atoms,in which motion the ether existing within the bodies may also participate;and radiant heat is looked upon as an undulatory motion propagated in that ether. It is not proposed here to set forth the facts,experiments, and inferences,through which men have been brought to this altered view on the subject;this would entail a refer- ence here to much which may be better described in its own place during the course of the book.The conformity with experience of the results deduced from this new theory will probably serve better than anything else to establish the foundations of the theory itself. We will therefore start with the assumption that Heat consists in a motion of the ultimate particles of bodies and of ether,and that the quantity of heat is a measure of the Vis Viva of this motion.The nature of this motion we 时GOO8le
( 21 ) CHAPTER I. FIRST MAIN PRINCIPLE OF THE MECHANICAL THEORY OF HEAT, OR PRINCIPLE OF THE EQUIVALENCE OF HEAT AND WORK. § 1. N atuTe of Heat. Until recently it was the generally accepted view that Beat was a special substance, which was present in bodies in greater or less quantity, and which produced thereby their higher or lower temperature; which was also sent forth from bodies, and in that case passed with immense speed through empty space and through such cavities as ponderable bodies contain, in the form of what is called radiant heat. In later days has arisen the other view that Heat is in reality a mode of motion. According to this view, the heat found in bodies and determining their temperature is treated as being a motion of their ponderable atoms, in which motion the ether existing within the bodies may also participate; and radiant heat is looked upon as an undulatory motion propagated in that ether. It is not proposed here to set forth the facts, experiments, and inferences, through which men have been brought to this altered view on the subject; this would entail a reference here to much which may be better described in its own place during the course of the book. The conformity with experience of the results deduced from this new theory will probably serve better than anything else to establish the foundations of the theory itself. We will therefore start with the assumption that Heat consists in a motion of the ultimate particles of bodies and of ether, and that the quantity of heat is a measure of the Vis Viva of this motion. The nature of this motion we Digitized by Coogle
22 ON THE MECHANICAL THEORY OF HEAT. shall not attempt to determine,but shall merely apply to Heat the principle of the equivalence of Vis Viva and Work, which applies to motion of every kind;and thus establish a principle which may be called the first main Principle of the Mechanical Theory of Heat. 2.Positive and negative valwes of Mechanical Work. In $1 of the Introduction the mechanical work done in the movement of a point under the action of a force was defined to be The product of the distance moved through and of the component of the force resolved in the direction of motion.The work is thus positive if the component of force in the line of motion lies on the same side of the initial point as the element of motion,and negative if it falls on the opposite side.From this definition of the positive sign of mechanical work follows the principle of the equiva- lence of Vis Viva and Work,viz.The increase in the Vis Viva is equal to the work done,or equal to the increase in total work. The question may also be looked at from another point of view.If a material point has once been set in motion,it can continue.this movement,on account of its momentum, even if the force acting on it tends in a direction opposite to that of the motion;though its velocity,and therewith its Vis Viva,will of course be diminishing all the time.A material point acted on by gravity for example,if it has received an upward impulse,can continue to move against the force of gravity,although the latter is continually diminishing the velocity given by the impulse.In such a case the work,if considered as work done by the force,is negative.Conversely however we may reckon work as positive in cases where a force is overcome by the momen- tum of a previously acquired motion,as negative in cases where the point follows the direction of the force.Applying the form of expression introduced in 1 of the Introduction, in which the distinction between the two opposite directions of the component of force is indicated by different words,we may express the foregoing more simply as follows:we may determine that not the work done,but the work destroyed, by a force shall be reckoned as positive. On this method of denoting work done,the principle of Google
22 ON '.rIlE MECHANICAL THEORY OF HEAT. shall not attempt to determine, but shall merely apply to Heat the ptinciple of the equivalence of Vis Viva and Work, which applies to motion of every kind; and thus est~blish a principle which may be called the first main Principle of the M.echanical Theory of Heat. § 2. Positive and negative values of Mechanical Work. In § 1 of the Introduction the mechanical work done in the movement of a point under the action of a force was defined to be The product of the distance moved through and of the component of the force resolved in the direction of motion. The work is thus positive if the component of force in the line of motion lies on the same side of the initial point as the element of motion, and negative if it falls on the opposite side. From this definition of the positive sign of mechanical work follows the . principle of the equivalence of Vis Viva and Work, viz. The increase in the Vis' Viva is equal to the work done, or equal to the increase in total work. The question may also be looked at from another point of view. If a material point has once .been set in motion, it can continue. this movement, on account of its momentum, even if the force acting on it tends in a direction opposite to that of the motion; though its velocity, and therewith its Vis Viva, will of course be diminishing all the time. A material point acted on by gravity for tlxample, if it has received an upward impulse, can continue to move against the force of gravity, although the latter is continually diminishing the velocity given by the impulse. In such a case the work, if considered as work done by the force, is negative. Conversely however we may reckon work as· positive in cases where a force is overcome by the momentum of a previously acquired motion, as negative in cases where the point follows the direction of the force. Applying the form of expression introduced in § 1 of the Introduction, in which the distinction between the two opposite directions of the component of force is indicated by different words, we may express the foregoing more simply as follows: we may determine that not the work done, but the work destroyed, by a force shall be reckoned as positive. On this method of denoting work done, the principle of Digitized by Coogle