8 ON THE MECHANICAL THEORY OF HEAT. 4.Geometrical interpretation of the foregoing results, and observations on partial diferential coefficients. The important difference between the results in the two cases mentioned above is rendered more clear by treating them geometrically.In so doing we shall for the sake of simplicity assume that the function(x,y)in equation (A) is such that it has only a single value for any one point in the plane of co-ordinates.We shall also assume that in the movement of the point p its original and final positions are known,and given by the co-ordinates o and respec- tively.Then in the first case we can find an expression for the work done by the effective force during the motion, without needing to know the actual path traversed.For it is clear,that this work will be expressed,according to con- dition (A),by the difference F()-F(o).Thus,while the moving point may pass from one position to the other by very different paths,the amount of work done by the force is wholly independent of these,and is completely known as soon as the original and final positions are given. In the second case it is otherwise.In the system of equations (B),which belongs to this case,the first equation must be treated as the equation to a curve;and (since the form of the second depends upon it)the relation between them may be geometrically expressed by saying that the work done by the effective force during the motion of the point p can only be determined,when the whole of the curve,on which the point moves,is known.If the original and final positions are given,the first equation must indeed be so chosen,that the curve which corresponds to it may pass through those two points;but the number of such possible curves is infinite,and accordingly,in spite of their coinci- dence at their extremities,they will give an infinite number of possible quantities of work done during the motion. If we assume that the point p describes a closed curve,so that the final and initial positions coincide,and thus the co- ordinates have the same value as o,3,then in the first case the total work done is equal to zero:in the second case,on the other hand,it need not equal zero,but may have any value positive or negative. The case here examined also illustrates the fact that a Google
8 ON THE MECHANICAL THEORY OF HEAT. § 4. Geometrical interpretatiO'1l. of the foregoing Tesulf.8, and observatioll8 on partial differential coejJicients. The important difference between the results in the two cases mentioned above is rendered more clear by treating them geometrically. In so doing we shall for the sake of simplicity assume that the function F (z, '!I) in equation {A) is such that it has only a single value for anyone point in the plane of co-ordinates. We shall also assume that in the moveme.nt of the point p its original and final positions are known, and given by the co-ordinates ~O' '!Io' and ~l' '!Il respec- tively. Then in the first case we can find an expression for the work done by the effective force during the motion, without needing to know the actual path traversed. For it is clear, that this work will be expressed, according to condition (A), by the difference F(Z,'!l,) -F(zo'!lJ. Thus, while the moving point may pass from one position to the other by very different paths, the amount of work done by the force is wholly independent of these, and is completely known as soon as the original and final positions are g'lVen. In the second case it is otherwise. In the system of equations (B), which belongs to this case, the first equation must be treated as the equation to a curve; and (since the form of the second depends upon it) the relation between them may be geometncally expressed by saying that the work done by the effective force during the motion of the point p can only be determined, when the whole of the curve, on which the point moves, is known. If the original and final positions are given, the first equation must indeed be so chosen, that the curve which corresponds to it may pass through those two points; but the number of such possible curves is infinite, and accordingly, in spite of their coinci .. dence at their extremities, they will give an infinite number of possible quantities of work done during the motion. If we assume that the point p describes a closed curve, 80 that the final and initial positions coincide, and thus the coordinates ZI' '!Il have the same value as ZO' '!I., then in the first case the total work done is equal to zero: in the second case, on the other hand, it need not equal zero, but may have any value positive or negative. The case here examined also illustrates the fact that a Digitized by Coogle
MATHEMATICAL INTRODUCTION. 9 quantity,which cannot be expressed as a function of x and y (so long as these are taken as independent variables),may yet have partial differential coefficients according to x and y, which are expressed by determinate functions of those vari- ables.For it.is manifest that,in the strict sense of the words,the components.X and Y must be termed the partial differential coefficients of the work W according to x and y: since,when x increases by dr,y remaining constant,the work increases by Xda;and when y increases by dy,x re- maining constant,the work increases by Ydy.Now whether W be a quantity generally expressible as a function of and y,or one which can only be determined on knowing the path described by the moving point,we may always employ the ordinary notation for the partial differential coefficients of W, and write =X, (8) d W =Y. Using this notation we may also write the condition(4),the fulfilment or non-fulfilment of which causes the distinction between the two modes of treating the differential equation, in the following form: d d da dy …(⑨)。 Thus we may say that the distinction which has to be drawn in reference to the quantity W depends on whether the difference d /aw 西 dx dy is equal to zero,or has a finite value. 5.Extension of the above to three dimensions. If the point p be not restricted in its movement to one plane,but left free in space,we then obtain for the element of work an expression very similar to that given in equation (3).Let a,6,c be the cosines of the angles which the direc- tion of the force p,acting on the point,makes with three Google
HA.THElUTICAL INTRODUCTION'. 9 quantity, which cannot be expressed as a function of m and '!I (so long as these are taken as independent variables), may yet have partial differential coefficients according to a: and '!I. which are expressed by determinate functions of those variables. For it. is ma.nifest that, in the strict sense of the words. the components.X and Y must be termed the partial differential coefficients of the work Waccording to m and '!I: since, when m increases by th. 11 remaining constant, the work increases by X d:r;; and when '!I increases by dy, m remaining constant, the work increases 'by Ydy. Now whether W be a quantity generally expressible as a function of a: 'and '!I, or one which can only be determined on knowing the path described by the moving point, we may always employ the ordinary notation for the partial differential coefficients of W. and write . (~) ==x·l ........................... .... (8}. (d'!l == Y. J Using this notation we may also write the condition (4), the fulfilment or non-fulfilment of which causes the distinction between the two modes of treating the differential equation. in the following form: .. ~ (~) == ! (~:) .......................... (9). i Thus we may say that the distinction which has to be drawn in reference to the duan~ W depends on whether the difference ;y (d!) -th (ddV ) is equal to zero, or has a finite value . . § 5. E:dension of the MO'IJe to three dimensiona. H the point p be not restricted in its movement to one plane, but left free in space, we then obtain for the element of work an expression very similar to that given in equation (3). Let a, h, c be the cosines of the angles which the direction of the force P, acting on the point. ·makes with three Digitized by Coogle
10 ON THE MECHANICAL THEORY OF HEAT. rectangular axes of co-ordinates;then the three components X,Y,2 of this force will be given by the equations X=aP,Y=bP,Z=cP. Again,let a,8,y be the cosines of the angles,which the element of space ds makes with the axes;then the three projections da,dy,de of this element on those axes are given by the equations dx=ads,dy=Bd8,dz=yds. Hence we have Xdx+Ydy Zds=(aa+68+cry)Pds. But if be the angle between the direction of P and ds, then az+b3+y=co8φ： hence xdx+Ydy+Zaz=cos中×Pds. Comparing this with equation(②)，we obtain dw=Xda+Ydy+zdz.............(10). This is the differential equation for determining the work done.The quantities x,Y,2 may be any functions what ever of the co-ordinates a,y,;since whatever may be the values of these three components at different points in space, a resultant force P may always be derived from them. In treating this equation,we must first consider the fol- lowing three conditions: dx dydy dz dz dx dyde’ d返d西’da=de…(11) and must enquire whether or not the functions x,Y, satisfy them. If these three conditions are satisfied,then the expression on the right-hand side of (11)is the complete differential of a function of a,y,in which these may all be treated as independent variables.The integration may therefore be at once effected,and we obtain an equation of the form W=F(y习+c0nst…(12)
10' ON THE MECHANICAL THEORY OF BEAT. rectangular axes of co-ordinates; then the three components :X, Y, Z of this force will be given by the equations X =aP, Y=~P, Z=CP. Again, let a; fJ, "I be the cosines of the angles, which the element of space a8 makes with the axes; then the three projections dte, dy, dz of this element on those axes are given by the equations ' d:x: = ads, ay = fJ ds, az = "Ids. Hence we have Xa:x:+ Ydy +Zdz= (aa+bfJ+ cry)Piis. But if t/> be the angle between the direction of P and a8, then a:r + bfJ + cry = cos t/>: hence X d:c + Y dll * Zdz = cos t/> x Pds. Comparing this with equation (2). we obtain' dW = Xd:c + Ydy+Zdz .................. (10). Thill is the differential equation for determining the work done. The quantities X. Y, Z may be any functions whatever of the co-ordinates te. y. Z; since whatever may be the values of these three components at different points in space, a resultant force P may always be derived from them. In treating this equation. we must first consider the following three conditions: • dX dY dY dZ dZ dX dy = d:x:' dz = dy' d:x: = dz ............ ~11). and must enquire whether or not the functions X. Y, Z satisfy them. If these three conditions al·~satisfied. then the expression on the right-hand side of (11) is the complete differential of a function of 11:, '!I. z. in which these may all be treated as independent vanables. The integration may therefore be at once effected. and we. obtain an equation of the form W = F (:x:yz) + COnst. ........................ (12). Digitized by Coogle
MATHEMATICAL INTRODUCTION. 11 If we now conceive the point p to move from a given initial position()to a given final position the work done by the force during the motion will be repre- sented by F(（,1,2)-F(eo,o,2 If then we suppose F(,y,z)to be such that it has only a single value for any one point in space,the work will be completely determined by the original and final positions; and it follows that the work done by the force is always the same,whatever path may have been followed by the point in passing from one position to the other. If the three conditions (1)are not satisfied,the integra- tion cannot be effected in the same general manner.If, however,the path be known in which the motion takes place, the integration becomes thereby possible.If in this case two points are given as the original and final positions,and various curves are conceived as drawn between these points, along any of which the point p may move,then for each of these paths we may obtain a determinate value for the work done;but the values corresponding to these different paths need not be equal,as in the first case,but on the contrary are in general different. §6.On the Ergal. In those cases in which equation (12)holds,or the work done can be simply expressed as a function of the co-ordinates, this function plays a very important part in our calculations. Hamilton gave to it the special name of"force function";a name applicable also to the more general case where,instead of a single moving point,any number of such points are considered,and where the condition is fulfilled that the work done depends only on the position of the points.In the later and more extended investigations with regard to the quantities which are expressed by this function,it has become needful to introduce a special name for the negative value of the function,or in other words for that quantity,the sub- traction of which gives the work performed;and Rankine proposed for this the term potential energy.'This name sets forth very clearly the character of the quantity;but it Google
MATHEMATICAL INTRODUCTION • 11 . If we now conceive the point p to move from a given initial position (3:0, '!Io' zo) to a given final position (3:1, '!I .. ZI) the work done by the force during the motion will be represented by F (3:1, '!II' ZI) - F (3:0, '!Io, Zo)· If then we suppose F(:r:, '!I, Z) to be such that it has only a single value for anyone point in space, the work will be completely determined by the original and final positions; and it follows that the work done by the force is always the same, whatever path may have been followed by the point -in passing from one position to the other. If the three conditions (1) are not satisfied, the integration cannot be effected in the same general manner. If, 'however, the path be known in which the motion takes place, ~the integration becomes thereby possible. If in this case two points are given as the original and final positions, and various curves are conceived as drawn between these points, along any of which the point p may move, then for each of these paths we may obtam a determinate value for the work done; but the values corresponding to these different paths need not be equal, as in the first case, but on the contrary are in general different . . § 6. On the Ergal. In those cases in which equation (12) holds, or the work done can be simply expressed as a function of the co-ordinates, this function plays a very important part in our calculations. Hamilton gave to it the special name of "force function"; a name applicable also to the more general case where, instead of a single moving point, any number of such points are considered, and where the condition is fulfilled that the work done depends only on the position of the points. In the later and more extended investigations with regard to the quantities which are expressed by this function, it has become needful to introduce a special name for the negative value of the function, or in other words for that quantity, the subtraction of which gives the work performed; and Rankine proposed for this the term 'potential energy.' This name seta forth very clearly the character of the quantity; but it Digitized by Coogle
12 ON THE MECHANICAL THEORY OF HEAT. is somewhat long,and the author has ventured to propose in its place the term“ErgaL” Among the cases in which the force acting on a point has an Ergal,the most prominent is that in which the force originates in attractions or repulsions,exerted on the moving point from fixed points,and the value of which depends only on the distance;in other words the case in which the force may be classed as a central force.Let us take as centre of force a fixed point T,with co-ordinates,and let p be its distance from the moving point p,so that p=√（作-+(-+(5-2y.(13. Let us express the force which Tr exerts on p by (p),in which a positive value of the function expresses attraction, and a negative value repulsion;we then have for the com- ponents of the force the expressions x=0号；了=0；石=o月. Bat by()2=-月：hence=-9o2,nd imi. larly for the other two axes.If (p)be a function such that 中(p)=pp)dp .(14) we may write the last equation thus: X=- do(p)dp dφ(p dp do dx (15), and similarly P=- dφ(p ,Z=- do(p) dy da (15a. Hence we have 0m+ xXde+Ydy+ads--d dy dd(p)ds ody+az But,since in the expression for p given in equation(13)the quantities a,y,s are the only variables,and (p)may there- fore be treated as a function of those three quantities,the Google
12 . ON THE MECHANICAL THEORY OF lIEAT. is somewhat long, and the author has ventured to propose in its place the term " ErgaL" . Among the cases in which the force acting on a point has an Ergal, the most prominent is that in which the force originates in attractions or repulsions, exerted on the moving point from fixed points, and the value of which depends only on the distance; in other words the case in which the force may be classed as a. central force. Let us take as centre of force a fixed point 71', with co-ordinates E, '1, ~and let p be its distance from the moving point p, so that p =J(E-a;)v+ ('1-9)"+ ('-zt .... · ...... ·(13). Let us express the force which 71' exerts on p by~' (p), in which a positive value of the function expresses attraction, and a negative value repulsion; we then have for the components of the force the expressions X=~'(P) E-:l:; Y=~'(P) '1-y; Z=~'(P)~-·. p p P But by (13) Z = - E; a;: hence X = -~' (P) :. and similarly for the other two axes. If ~ (P) be a function such that ~ (P) = J~'{p) dp •........•..••••.. (14), we may write the last equation thus: d~ (p) dp d~ (p) X=-liP tk='-?;D ........... (15). d~(P) d~(P) and similarly Y =-a:u' Z=-(k"' .......... (15a). Hence we have Xd.t:+ Yd9+Zdz=- [d~(P) tk+ d~(p) dy+ d~(p) dIJ]. d:D d!l dz But, since in the expression for p given in equation (13) the quantities:l:, y, • are the only variables, and ~ (p) may therefore be treated as a function of those three .quantities, th.e Digitized by Coogle