P.mP' equals the work of P as its point of application is transferred from P to P'. Thus each term of the second member of equation (42) represents the work of the corresponding pressure, so that if Zu, represent the aggregate work of those pressures whose points of application are transferred towards the directions in which the pressures act, and Zu, the work of those whose points of application are moved opposite to the directions in which they severally act, then the second member of the equation is represented by Eu1Lu2. Moreover the first member of the equation is evidently the work necessary to transfer the resultant pressure P1+ P2+ P &c. through the distance H-h, which is that by which the centre of pressure is removed from or towards the given plane, so that if U represent the quantity of work necessary to make this transfer of the centre of pressure, 2 3 3 1 59. If the sum of those parallel pressures whose tendency is in one direction equal the sum of those whose tendency is in the opposite direction, then P1+ P2+ P2+ ..... =0. In this case, therefore, U=0, therefore Zu,-Eu2=0, therefore Zu1 =Σu; so that when in any system of parallel pressures the sum of those whose tendency is in one direction equals the sum of those whose tendency is in the opposite direction, then the aggregate work of those whose points of application are moved in the directions of the pressures severally applied to them is equal to the aggregate work of those whose points of application are moved in the opposite directions. This case manifestly obtains when the parallel pressures are in EQUILIBRIUM, the sum of those whose tendency is in one direction then equalling the sum of those whose tendency is in the opposite direction, since otherwise, when applied to a point, these pressures could not be in equilibrium about that point (Art. 8.). 60. The preceding proposition is manifestly true in respect to a system of weights, these being pressures whose directions are always parallel, wherever their points of application may be moved. Now the centre of pressure of a system of weights is its centre of gravity (Art. 19.). Thus then it follows, that if the weights composing such a system be separately moved in any directions whatever, and through any distances whatever, then the difference between the aggregate work done upwards in making this change of relative position and that done downwards is equal to the work necessary to raise the sum of all the weights through a height equal to that through which their centre of gravity is raised or depressed. Moreover that if such a system * * This proposition has numerous applications. If for instance it be required to determine the aggregate expenditure of work in raising the different elements of a structure, its stone, cement, &c. to the different positions they occupy in it, we make this calculation by determining the work requisite to raise the whole weight of material at once to the height of the centre of gravity of the structure. If these materials have been carried up by labourers, and we are desirous to include the whole of their labour in the calculation, we ascertain the probable amount of each load, and conceive the weight of a labourer to be added to each load, and then all these at once to be raised to the height of the centre of gravity. Again, if it be required to determine the expenditure of work made in raising the material excavated from a well, or in pumping the water out of it, we know that (neglecting the effect of friction, and the weight and rigidity of the cord) this expenditure of work is the same as though the whole material had been raised at one lift from the centre of gravity of the shaft to the surface. Let us take another application of this principle which offers so many practical results. The material of a railway excavation of considerable length is to be removed so as to form an embankment across a valley at some distance, and it is required to determine the expenditure of work made in this transfer of the material. Here each load of material is made to traverse a different distance, a resistance from the friction, &c. of the road being continually opposed to its motion. These resistances on the different loads constitute a system of parallel pressures, each of whose points of application is separately transferred from one given point to another given point, the directions of transfer being also parallel. Now by the preceding proposition the expenditure of work in all these separate transfers is the same as it would have been had a pressure equal to the sum of all these pressures been at once transferred from the centre of resistance of the excavation to the centre of resistance of the embankment. Now the resistances of the parts of the mass moved of weights be supported in equilibrium by the resistance of any fixed point or points, and be put in motion, then (since the work of the resistance of each such point is nothing) the aggregate work of those weights which are made to descend, is equal to that of those which are made to ascend. 61. If a plane be taken perpendicular to the directions of any number of parallel pressures and there be two different positions of the points of application of certain of these pressures in which they are at different distances from the plane, whilst the points of application of the rest of these pressures remain at the same distance from that plane, and if in both positions the system be in equilibrium, then the centre of pressure of the first mentioned pressures will be at the same distance from the plane in both positions. 2 ... =0, For since in both positions the system is in equilibrium, therefore in both positions P1+ P2+ P3+ .•.(Y1—y1)P1+(Y2—Y2)P2+(Y3—Y3)P3+... +Pn(Y2—Yn)=0. Now let P, be any one of the pressures whose point of application is at the same distance from the given plane in both positions, .. Y=y, and Y1-y=0, .'.(Y1—y1)P ̧+(Y2−Y2) P2+...+(Yn-1—Yn−1)Pn_1=0, ·.Y1P1+Y2P2+...+Y-1P-1=y1P1 + y2P2+...+Y-1P-19 2 2 are the frictions of its elements upon the road, and these frictions are proportional to the weights of the elements; their centre of resistance coincides therefore with the centre of gravity of the mass, and it follows that the expenditure of work is the same as though all the material had been moved at once from the centre of gravity of the excavation to that of the embankment. To allow for the weight of the carriages, as many times the weight of a carriage must be added to the weight of the material as there are journeys made. where H represents the distance of the centre of pressure of P1, P, . . . P-1, from the given plane in the first position, and hits distance in the second position. Its distance in the first position is therefore the same as in the second. Therefore, &c. From this proposition it follows, that if a system of weights be supported by the resistances of one or more fixed points, and if there be any two positions whatever of the weights in both of which they are in equilibrium with the resistances of those points, then the height of the common centre of gravity of the weights is the same in both positions. And that if there be a series of positions in all of which the weights are in equilibrium about such a resisting point or points, then the centre of gravity remains continually at the same height as the system passes through this series of positions. If all these positions of equilibrium be infinitely near to one another, then it is only during an infinitely small motion of the points of application that the centre of gravity ceases to ascend or descend; and, conversely, if for an infinitely small motion of the points of application the centre of gravity ceases to ascend or descend, then in two or more positions of the points of application of the system, infinitely near to one another, it is in equilibrium. WORK OF PRESSURES APPLIed in differENT DIRECTIONS TO A BODY MOVEABLE ABOUT A FIXED AXIS. 62. The work of a pressure applied to a body moveable about a fixed axis is the same at whatever point in its proper direction that pressure may be applied. For let AB represent the direction of a pressure applied to a body moveable about a fixed axis O; the work done by this pressure will be the same whether it be applied at A or B. For conceive the body to revolve about O, through an exceed ingly small angle AOC, or BOD, so that the points A and D may describe circular arcs AC and BD. Draw Cm, Dn, and OE, perpendiculars to AB, then if P represent the pressure F applied to AB, P. Am will represent the work done by P when applied at A (Art. 52.), and P. Bn will represent the work done by P when applied at B; therefore the work done. by P at A is the same as that done by P at B if Am is equal to Bn. Now AC and BD being exceedingly small, they may be conceived to be straight lines. Since BD and BE are respectively perpendicular to OB and OE, therefore DBE = L BOE; and because AC and AE are perpendicular to OA and OE, therefore CAE = ▲ AOE. Now Am=CA. cos. BD OE. Similarly Bn=DB cos. DBE=DB. cos. BOE= OB BD OB cos. BOE= x OE, i. e. Am=OE. 63. If any number of pressures be in equilibrium about a fixed axis, then the whole work of those which tend to move the system in one direction about that axis is equal to the whole work of those which tend to move it in the opposite direction about the same axis. For let P be any one of such a system of pressures, and O a fixed axis, and OM perpendicular to the direction of P, then whatever may be the point of application of P, the work of that pressure is the same as though it were applied at M. Suppose the whole system to be moved through an exceeding small angle about the point O, and let OM be represented by p, then will pê represent the space described by the point M, which will be actually in the direction of the force P, therefore L * It is a well known principle of Geometry, that if two lines be inclined at any angle, and any two others be drawn perpendicular to these, then the inclination of the last two to one another shall equal that of the first two. |