MATHEMATICAL INTRODUCTION. 3 the one hand the force need not itself be the same at diffe- rent points of space;and on the other,although the force may remain constant throughout,yet,if the path be not straight but curved,'the component of force in the direction of motion will still vary.For this reason it is allowable to ex- press work done by a simple product,only when the distance traversed is indefinitelyment of space. Let ds be an element of space,and S the .component in the direction of ds'of the force acting on the point p.We have then the following equation to obtain dw,the work done during the movement through the indefinitely small space ds dW=Sds…(1). If p be the total resultant force acting on the point p,and the angle which the direction of this resultant makes with the direction of motion at the point under consideration, then 8=Pcos中， whence we have,by (1), dw=P cos ods..(2). It is convenient for calculation to employ a system of rectangular co-ordinates,and to consider the projections of the element of space upon the axes of co-ordinates,and the components of force as resolved parallel to those axes. For the sake of simplicity we will assume that the motion takes place in a plane in which both the initial direction of motion and the line of force are situated.We will employ rectangular axes of co-ordinates lying in this plane,and will call and y the co-ordinates of the moving point p at a given time.If the point moves from this position in the plane of co-ordinates through an indefinitely small space ds, the projections of this motion on the axes will be called dz and dy,and will be positive or negative,according as the co-ordinates x and y are increased or diminished by the motion.The components of the force p,resolved in the directions of the axes,will be called X and Y.Then,if a and b are the cosines of the angles which the line of force makes with the axes of a and y respectively,we have X=aP;Y=bP
'lIA.THEMATICAL . INTRODUCTION. 3' the one'hand the force 'need' not itself be the same' at·diffe- , rent points of space; and on the other, although the force may remain constant througho,ut, yet, if the path be not straight but curved/the component of force in the direction of motion will still vary. For this reason it is allowable to express work done by a simple product, only when ihe distance' traversed is indefinitely small.. ie. fO!.,An element of space. Let ds be an element of y:p~~/a.na 8 the ,component in the direction of dR'of the force acting on the point p. We have then the following ~quation to ,obtain d W, the work done during the movement through the indefini~ly small ,~aceds: ' dW=8ds ......................... (I). If P be the total resultant force acti~g on the point p, and ~ the angle which the direction of this resultant makes, with the direction of motion at the point under consideration, then. 8=Pcosq" ' whence we have, by (It, dW = P cos ~ds ..................... (2). It is convenient for calculation to employ a system of rectangular co-ordinates, and to consider the projections of the element of space upon the axes of co-ordinates, and the components of force as resolved parallel to those axes. For the sake of simplicity we will assume that the motion takes place in a plane in which both the initial direction of motion and the line of force are situated. We will employ rectangular axes of co-ordinates lying in this plane, and will call IlJ and '!J the co-ordinates of the moving point p at a given time. If the point moves from this position in the plane of co-ordinates through an indefinitely small space ds, the projections of this motion on the axes will be called d:c and dV' and will be positive or negative, according as the co-ordmates fD and '!J are increased or diminished by the motion. The components of the force P, resolved in the directions of the axes, will be called X and Y. Then, if a and b are the cosines of the angles which the line of force makes with the axes of fC and '!J respectively, we have X=aPj Y=bP. --1-2- - Digitized by L.oogle
4 ON THE MECHANICAL THEORY OF HEAT. Again,if a and B are the cosines of the angles which the element of space da makes with the axes,we have da=ads;dy=Bds. From these equations we obtain Xda+Ydy =(aa+68)Pds. But by Analytical Geometry we know that aa+bB=co8中， where is the angle between the line of force and the element of space:hence Xde+Ydy cos中Ps, and therefore by equation(②)， dW=Xdc+Ydy.…(③) This being the equation for the work done during an indef- nitely small motion,we must integrate it to determine the work done during a motion of finite extent. S3.Integration of the Differential Eguation for Work done. In the integration of a differential equation of the form given in equation(3),in which X and Y are functions of x and y,and which may therefore be written in the form dW=φ()dx+(x划）d…(③a, 'a distinction has to be drawn,which is of great importance, not only for this particular case,but also for the equations which occur later on in the Mechanical Theory of Heat;and which will therefore be examined.here at some length,so that in future it will be sufficient simply to refer back to the present passage. According to the nature of the functions (zy)and (zy),differential equations of the form (3)fall into two classes,which differ widely both as to the treatment which they require,and the results to which they lead.To the Google
4 ON THE MECHANICAL THEORY OF HEAT. Again, it a and fJ are the cosines of the angles which the element of space ds makes with the axes, we have d:x: - ads ; dy = fJaB.· From these equations we obtain Xda;+ Ydy= (aa+ bfJ) Pds~ But by Analytical Geometry we know that aa + bfJ = cos q" where q, is the angle between the line of force and the element of space: henc~ Xdz+ Ydy-cosq,Pds, and therefore by equation (2), dW =Xtk+ Ydy •.••..••.••.••.••.•••••• (3). This being the equation for the work done during an indefinitely small motion, we must integrate it to determine the . work done during a motion of finite extent. § 3. IntegralMm of the Differentt'al EtJUatilm for Work dooe. In the integration of a differential equation of the form given in equation (3), in which X and Y are functions of tJ: and y, and which may therefore be written in the form dW = q, (zy) da; +",. (xy) ay •....•.......•. (3a), :a distinction has to be drawn, which is of great importance, not only for this partiCUlar case, but also for the equations which occur later on in the Mechanical Theory of Heat; and which will therefore be examined . here at some length, so that in future it will be sufficient simply to refer back to the present passage. According to the nature of the functions tf, (a:y) and +' (xy), differential equations of the form (3) fall into two classes, which differ widely both as to the treatment which they require, and the results to which they lead. To the Digitized by Coogle
MATHEMATICAL INTRODUCTION. 5 first class belong the cases,in which the functions x and Y fulfil the following condition: dx dy dy dae () The second class comprises all cases,in which this condition is not fulfilled. If the condition (4)is fulfilled,the expression on the right-hand side of equation(3)or(3a)becomes immediately integrable;for it is the complete differential of some func- tion of x and y,in which these may be treated as indepen- dent variables,and which is formed from the equations dr (u)-x, dF (c)=Y. dx dy Thus we obtain at once an equation of the form 0=F()+const...(). If condition(4)is not fulfilled,the right-hand side of the equation is not integrable;and it follows that w cannot be expressed as a function of x and y,considered as independent variables.For,if we could put W=F(cy),we should have I= dw dF(ay) da dx Y dw_dF(x划 dy dy whence it follows that dx dF(ay) dy dxdy dy dF(ay) dx dyd.c But since with a function of two independent variables the order of differentiation is immaterial,we may put FFC)rm dxdy dydx Google
MATHEMATICAL INTRODUCTION. class belong the which the functions the following ,",VUCUULVU dX dY (iii = dx ••.•.................... (4). The second class comprises all cases, in which this condition is not fulfilled. If the condition fulfilled, the expression right-hand side of or (3a) becomes integrable; for it is differential of of x and '9, in may be treated variables, and formed from the dF (x!!l = X ilF (x.1f) = y. rk ' dy • Thus we obtain at once an equation of the form w = F (X'!}) + const. .................... (5). If condition (4) is not fulfilled, the right-hand side of the ,orn'''T.',nTl is not integrable it follows that OVT',"OO,Q.,n as a function '9, considered as 'llu'o",'"a"'"'' For, if W = F (wy), we ilF(xy) dx Y _ dW _ dF(xy) - dy ---;ty-' , whence it follows that dX d'F(xy) ilxdy , since with a. two independent order of differentiation is immaterial, we may put cPF(xy) _ d'F(rcy). , . dxd!J - ~ydx J
G ON THE MECHANICAL THEORY OF HEAT. whence it follows that dx.dy =在，ie.condition(倒is fulfilled for the functions X and Y;which is contrary to the assump- tion In this case then the integration is impossible,so long as x and y are considered as independent variables.If however we assume any fixed relation to hold between these two quantities,so that one may be expressed as a function of the other,the integration again becomes possible.For if we put f(xy）=0..(6) in which fexpresses any function whatever,then by means of this equation we can eliminate one of the variables and its differential from the differential equation.(The general form in which equation(6)is given of course comprises the special case in which one of the.variables is taken as constant;its differential then becomes zero,and the variable itself only appears as part of the constant coefficient).Sup- posing y to be the variable eliminated,the equation (3) takes the form dw=()da,which is a simple differential equation,and gives on integration an equation of the form w=F(a)+const................... The two equations(6)and(7)may thus be treated as form- ing together a solution of the differential equation.As the form of the function f(xy)may be anything whatever,it is clear that the number of solutions thus to be obtained is infinite. The form of equation (7)may of course be modified. Thus if we had expressed x in terms of y by means of equa- tion(6),and then eliminated x and da from the differential equation,this latter would then have taken the form dW=中，()dy, and on integrating we should have had an equation W=F()）+const..(7a). This same equation can be obtained from equation(7)by substituting y for a in that equation by means of equation(6). Or,instead of completely eliminating a from (7),we may
(j ON THE MECHANICAL THEORY OF HEAT. whence it follows that ~X~ ~; i.e. condition (4)-is fulfilled for the functions X and)r; which is contrary to the assnmption. . ... . In this case then the integration is impossible, so long as a: and !I are considered as independent variables. If however we assume any fixed relation to hold between these two quantities, so that one may be expressed as a function of the other, the integration again becomes possible. For if we put f (xy) = 0 ........................ (6), in which f expresses any function whatever, then by means of this equation we can eliminate one of the variables and its differential from the differential equation. (The general form in which equation (6) is given of conrsj:l comprises the special case in which one of the. variables is taken as cOIlRtant; its differential then becomes zero, and the variable itself only appears as part of the COIlRtant coefficient). Supposing '!I to be the variable eliminated, the equation (3) takes the form dW = 4> (x) dz, which is a simple differential equation, and gives on integration an equation of the form to = F (a:) + const ..................... (7). The two equations (6) and (7) may thus be treated as forming together a solution of the differential equation. As the form of the function f(a:y) may be anything whatever, it is clear that the number of solutions thus to be obtained is infinite. The form of equation (7) may of course be modified. Thus if we had expressed a: in terms of !I by means of equation (6), and then eliminated a: and cIa: from the differential eqnation, this latter would then have taken the form dW = 4>1 (y) dy, and on integrating we should have had an equation W =~(!I) +const ...................... (7a). This same equation can be obtained from equation (7) by substituting '!I for a: in that equation by means of equation (6). Or, instead of completely eliminating a: {rom (7), we may Digitized by Coogle
MATHEMATICAL INTRODUCTION. 7 prefer a partial elimination.Forif the function F()con- tains x several times over in different terms,(and if this does not hold in the original form of the equation,it can be easily introduced into it by writing instead of x an expression such C为 as (1-)&)then it is possible to substitute y for a in some of these expressions,and to let a remain in others. The equation then takes the form W=F(3,）+const...(7b, which is a more general form,embracing the other two as special cases.It is of course understood that the three equa- tions (7),(7a),(76),each of which has no meaning except when combined with equation(6),are not different solutions, but different expressions for one and the same solution of the differential equation. Instead of equation (6),we may also employ,to integrate the differential equation (3),another equation of less simple form,which in addition to the two variables a and y also contains W,and which may itself be a differential equation; the simpler form however suffices for our present purpose, and with this restriction we may sum up the results of this section as follows. When the condition of immediate integrability,expressed by equation (4),is fulfilled,then we can obtain directly an integral equation of the form: W=F(,y)+const...................(A). When this condition is not fulfilled,we must first assume some relation between the variables,in order to make inte- gration possible;and we shall thereby obtain a system of two equations of the following form: f(,)=0, W=F(z,y)+const. (B) in which the form of the function r depends not only on that of the original differential equation,but also on that of the function f,which may be assumed at pleasure. Oigzdby Google
MATHEMATICAL INTRODUCTION, 7 prefer F (x) contains x over in different (and if this does not hold in the original form of the equation, it can be easily introduced into it by writing instead of x an expression such 11+1 as (1- a) x + ax, ~, &c.) then it is possible to substitute x '!I for :l< in some of these expressions, and to let x remain in others. The equation then takes the form W=F.(x,y) + " .. ".",,(7b), which other two as special three equations (7a), (7b), each of which meaning except when combined with equation (6), are not different solutions, but different expressions for one and the same solution of the differentiar equati<?n. Instead of equation (6), we may also employ, to integrate the differential equation (3), another equation of less simple fi)rID, which in addition to the two variables x and y also contains which may itself eq uation ; the however suffices purpo!;e, and restriction we may sum results of this section When of immediate integrability, expressed by (4), fulfilled, then obtain directly an integral equation of the form: W = F(x, y) + const. .. ".""." .. " ..... (A). When this condition is not fulfilled, we must first assume some relation between the variables, in order to make integration and we shall thereby a system of two following form =0, W=F(x,y) + const. " in which of the function not only on that of the original differential equation, also on that of the functionf, which may be assumed at pleasure