MATHEMATICAL INTRODUCTION. 18 expression in brackets forms a perfect differential,and we may write Xda+Ydy+Zdz=-do (p)............(16). The element of work is thus given by the negative differen- tial of (p);whence it follows that (p)is in this case the Ergal. Again,instead of a single fixed point,we may have any number of fixed points TT,T &c.,the distances of which from p are PaPa &c.,and which exert on it forces (),)()&c.Then if,as in equation (14),we as- aume中，(p,中，(p,中，(p,&c.to be the integrals of the above functions,we obtain,exactly as in equation(15), r=、 dp_d吨，(p_dφp da do de 一… =-)+4++… or x-- (17). Similarly Σ40Z=-是240(7网 Y=一d whence Xdc+Ydy+2aa=-dΣφp)…(18). Thus the sum (p)is here the Ergal. 7.General Eatension of the foregoing. Hitherto we have only considered a single moving point; we will now extend the method to comprise a system com- posed of any number of moving points,which are in part acted on by external forces,and in part act mutually on each other. If this whole system makes an indefinitely small move- ment,the forces acting on any one point,which forces we may conceive as combined iuto a single resultant,will per- form a quantity of work which may be represented by the expression (Xdc+Ydy +ede).Hence the sum of all the Google
.. JUTHEll'ATIC.u. IliTRODUCTION. expression in brackets forms a perfect differential, and we may write: X!k+ YdV+Zdr--d~ (p) ............ (16). The element of work is thus given by the negative differential of ~ (p); whence it follows that ~ (P) is in this ca.se the ErgaAl .• • t d f . I .t!_ d' h gaID, IDS ea 0 a sIDg e lU.e POlDt. we may ave any number of fixed points 'lT1 • 'IT., 'IT., &c., the distances of which from pare Pl' PI' P" &c., and which exert on it forces ;'(Pl)' 4>'(p'), 4>'(PJ, &c. Then if, as in equation (14), we assume ~l(P), ~I(P)' 4>.(P), &c. to be the integrals of the above functions, we obtain, exactly as in equation (15), X - _ d~l(Pl) _ d4>.{pJ _ d~9(P,) _ dJc dJc dJc ••• d --d:r:[~I(PJ +~.(PJ +~.(P.)+ •.. ], or X=- !X~(p) ................................. (l7). Similarly Y==- ~X ~(P), z= -! X ~ (P) .•••••••• (17a), whence Xtk + YdV + Zd" .. - dI, ~ (P) ••••..••• (18), Thus the sum X ~(P) is here the Ergal. § 7. General Eo:tentJion of the foregoing. Hitherto we have only considered a 8in~le moving point; we will now extend the method to compnse a system composed of any number of moving points, which are in part acted on by external forces, and in part act mutually on each other. If this whole system makes an indefinitely small movement, the forces acting on anyone point, which forces we may conceive as combined into a single resultant, will perform a quantity of work which may be represented by the expression (Xda: + Ydy + Zdz). Hence the sum of all the Digitized by Coogle
14 ON THE MECHANICAL THEORY OF HEAT. work done by all the forces acting in the system may be represented by an expression of the form 三(xde+ay+2da), in which the summation extends to all the moving points. This complex expression,like the simpler one treated above,may have under certain circumstances the important peculiarity that it is the complete differential of some func- tion of the co-ordinates of all the moving points;in which case we call this function,taken negatively,the Ergal of the whole system.It follows from this that in a finite move- ment of the system the total work done is simply equal to the difference between the initial and final values of the Ergal;and therefore (assuming that the function which represents the Ergal is such as to have only one value for one position of the points)the work done is completely deter- mined by the initial and final positions of the points,without its being needful to know the paths,by which these have noved from one position to the other. This state of things,which,it is obvious,simplifies greatly the determination of the work done,occurs when all the forces acting in the system are central forces,which either act upon the moving points from fixed points,or are actions between the moving points themselves. First,as regards central forces acting from fixed points, we have already discussed their effect for a single moving point;and this discussion will extend,also to the motion of the whole system of points,since the quantity of work done in the motion of a number of points is simply equal to the sum of the quantities of work done in the motion of each several point.We can therefore express the part of the Ergal relating to the action of the fixed points,as before,by (p),if we only give such an extension to the summation, that it shall comprise not only as many terms as there are fixed points,but as many terms as there are combinations of one fixed and one moving point. Next as regards the forces acting between the moving points themselves,we will for the present consider only two points p and p,with co-ordinates y,s,and ,y,, Google
14 ON THE MECHANICAL THEORY OFlIEAT. work done by all the forces acting in the s,stem may be. represented by an expression of the form I (X~ + Y'lly + Zdz), in which the summation extends to all the moving points. This complex expression, like the simpler one treated above, may have under certain circumstances the important . peculiarity that it is the complete differential of some function of the co-ordinates of all the moving points; in which case we call this function, taken ne~atively, the Ergal of the whole system. It follows from thls that in a finite move-' mElnt of the system the total work done is simply equal to the difference between the initial and final values of the Ergal; and therefore (as.'luming that the function which represents the Ergal is such as to have only one value for one position of the points) the work done is completely determined by the initial and final positions of the points, without its being needful to know the paths, by which these have moved from one position to the other. . . This state of things, which, it is obvious, simplifies greatly the determination of the work done, occurs when all the forces acting in the system are central forces, which either act upon the moving points from fixed points, or are actions between the moving points themselves. First, as regards central forces acting from fixed points, we have already discussed their effect for a single moving point; and this discussion will extend· also to the motion of the whole system of points, since the qua.ntity of work done in the motion of a number of points is simply equal to the sum of the quantities of work done in the motion of each several point. We can therefore express the part of the Ergal relating to the action of the fixed points, as before, by ~ tp (P), if we only give such an extension to the summation, that it shall comprise not only as many terms as there are fixed points, but as many terms as there are combinations of one fixed and one moving point. N ext as regards the forces acting between the moving points themselves, we will for the present consider only two points p and p', with co-ordinates /If, y, z, and :t, y', e', Digitized by Coogle
MATHEMATICAL INTRODUCTION. 15 respectively.If r be the distance between these points,we have r=2-+-)'+(2-.…(19) We may denote the force which the points exert on each other byf(r),a positive value being used for attraction,and a negative for repulsion. Then the components of the force which the point p exerts in this mutual action are f,,f,,f,; and the components of the opposite force exerted by p'are f,,f,兰，f号. But by (19),differentiating so that the components of force in the direction of x may also be written -寸2：-时密； and if f(r)be a function such that f)=f”6dg0, the foregoing may also be written -世；-4(m dx da Similarly the components in the direction of y may be written -时0，-f; dy dy and those in the direction of =r;-r0. Google
JUTHEJrU.TICAL INTRODUCTION. . 15 respectively. If r be the distance between these points, we have r::J(z'-a;)1 +C!; - y)'+ (z' -Z)I ......... (19). We may denote the force which the points exert on each other by I'(r). a positive value being used for attraction, and a negative for repulsion. Then the components of the force which the point p exerts in this mutual action are , , . f'(r) a; ~ a;, I'(r) y ~ .v, f(r) z ~ II ; and the components of the opposite force exerted by p' are , , , f'(r) a; - a;, f'(r) y- y, !'(r) z - z • r r r But by (19), differentiating dr :c' - :c ar a; - a;' da;=--r-; da;'==- -r-; 80 that the components of force in the direction of a; may also be written . - I' (r) ~:; - f' (r) :. i . and if f (r) be a. function such that I(r) == J f' (r) dr ... ............... (20). the foregoing may also be written -dl(r) -df(r) --;tZ ; da;' • Similarly the components in the direction of y may be written • - a/(r}. - df(r). dy '-d"il-' and those in the direction of • -tll(r). -dl(r) dz'da" Digitized by Coogle
16 ON THE MECHANICAL THEORY OF HEAT. That part of the total work done in the indefinitely small motion of the two points,which is due to the two opposite forces arising from their mutual action,may therefore be expressed as follows: [2a+留+2+盟业+智y da dy f()ai But as r depends only on the six quantities ,ys,,y,, and f(r)can therefore be a function of these six quantities only,the expression in brackets is a perfect differential,and the work done,as far as concerns the mutual action between the two points,may be simply expressed by the function -df(r). In the same way may be expressed the work due to the mutual action of every other pair of points;and the total work done by all the forces which the points exert among themselves is expressed by the algebraical sum -f(r)-时(r)-d时(r-… or as it may be otherwise written, -d[f(r)+f(r)+f(r")+...]or -dEf(r); in which the summation must comprise as many terms as there are combinations of moving points,two and two.This sum f(r)is then the part of the Ergal relating to the mutual and opposite actions of all the moving points. If we now finally add the two kinds of forces together, we obtain,for the total work done in the indefinitely small motion of the system of points,the equation (xdx+Ydy+Za)=-dΣφ(p)-f(r) =-d[区φ(p）+f(r】.(21), whence it follows that the quantity (p)+f(r)is the Ergal of the whole of the forces acting together in the system. The assumption lying at the root of the foregoing analy- sis,viz.that central forces are the only ones acting,is of course only one among all the assumptions mathematically Google
16 ON THE lIEClIA.NICAL THEORY OF HEAT. '!'hat part of the total work done in the indefinitely small motion of the two points, which is due to the two opposite forces arising from their mutual action, may therefore be expressed as follows: _ [df(r)rk+4f(r)d +df(r)d.+~f(r)d.1{+df(r)d' do: dU '!I d. dJ! fIT Y . +d~;)dZJ. But as r depends only on the six q~antities :1:, !I. II, al, g', II', and fer) can therefore be a function of th'ese six quantities only. the expression in brackets is a perfect differential, and the work done. as far as concerns the mutual action between the two points. may be simplyexpreued. by the function -df(r). In the same :way~ay be. expressed the work due to the mutual action of every .other pair of points; and the total work done by all the forces which the points exert among themselves is expressed by the algebraicalsum - df(r) - df(r') - df (r") - ••• ; or as it may be otherwise written, - d[f(r) + fer') + fer') + ... ] or - dIf(r); in which the summation must comprise as many terms as there are combinations of moving points, two and two. This sum If(r) is then the part of the Ergal relating to the mutual and opposite actions of all the moving points. If we now finally add the two kinds of forces together, we obtain, for the totpl work done in the indefinitely small motion of the system of points, the equation I(X!k+ Ydg+Zd/l)=-dI~(p)-dI.f(r) =-d[I~(P) + If(r)] ........ (21), whence it follows that the quantity I~(P) + If(r) is the Ergal of the whole of the forces acting together in the system. The assumption lying at the root of the foregoing analysis. viz. that central forces are the only ones acting, is of course only one among all. the assumptions mathematically Digitized by Coogle
MATHEMATICAL INTRODUCTION. 17 possible as to the forces;but it forms a case of peculiar importance,inasmuch as all the forces which occur in nature may apparently be classed as central forces. 8.Relation between Work and Vis Viva. Hitherto we have only considered the forces which act on the points,and the change in position of the points them- selves;their masses and their velocities have been left out of account.We will now take these also into consideration. The equations of motion for a freely moving point of mass m are well known to be as follows: m票-x器=y-么…( If we multiply these equations respectively by and then add,we obtain 能+皇腺+密器-（能+2+. The left-hand side of this equation may be transformed into [+留+（钔 or,if v be the velocity of the point, md()d= 2 di dt and the equation becomes 帽=(x+r皇+ d玩.(2). If,instead of a single freely moving point.a whole system of freely moving points is considered,we shall have for every C. 2
HATHEMATICAL INTRODUCTION. 17 possible as to the f~rces; but it forms a case of peculiar lmportance, inasmuch as all the forces which occur in nature may apparently be classed as central forces. § 8. Relation between Work and Via Viva. Hitherto we have only considered the forces which act on the points, and the change in position of the points themselves; their masses and their velocities have been left out of account. We will now take these also into consideration. The equations of motion for a freely moving point of mass m are well known to be as follows: d'a: iI1y iI1z m dt' = X, m dt' = Y, m de" = Z ..•••••.••... (22). If we multiply these equations respectively by dr.c dy dll dt dt, dt dt, dt dt, and then add, we obtain ( tk d':r: d.1J iI1y dz dlz) _( dr.c d.1/ dz) m dt dt' + dt dt' + dt dt" dt- X dt + Y dt +Z dt dt ... (23). The left-hand side of this equation may be transformed into m d [(dr.c)1 (dy)' (dz)1 2dt dt + dt + dt Jdt, or, if v be the velocity of the point, d (m v'\ ~ dCt!) dt= \2 J dt=d (m ",). 2 de dt 2' and the equation becomes d (i VI) = (X: + y7t + Z:;) dt ............ (2~). If, instead of a single freely moving point. a whole system of freely moving points is considered, we shall have for every C. 2 Digitized by Coogle