Page:Encyclopædia Britannica, Ninth Edition, v. 7.djvu/168

150 ir.o DIAGRAMS system and on the other a set of points, each point corre sponding to a particle of the system, and the whole repre senting the configuration of the system at a given instant. This is called a diagram of configuration. Diagram, of Displacement. Let us next consider two diagrams of configuration of the same system, corresponding to two different instants. We call the first the initial configuration and the second the final configuration, and the passage from the one con figuration to the other we call the displacement of the system. We do not at present consider the length of time during which the displacement was effected, nor the inter mediate stages through which it passed, but only the final result a change of configuration. To study this change we construct a diagram of displacement. Let A, B, C be the points in the initial diagram of con figuration, and A, B , C be the corresponding points in the final diagram of configuration. From o, the origin of the diagram of displacement, draw a vector oa equal and parallel to AA, ob equal and parallel to BB, oc to CC , and so on. The points, a, b, c, Jzc., will be such that the vector al&amp;gt; indicates the displacement of 6 relative to a, and so on. The diagram containing the points a, b, c, &c., is therefore called the diagram of displacement. In constructing the diagram of displacement we have hitherto assumed that we know the absolute displacements of the points of the systsm. For we are required to draw a line equal and parallel to AjAg, which we cannot do unless we know the absolute final position of A, with respect to its initial position. In this diagram of displace ment there is therefore, besides the points a, 6, c, &amp;lt;fec., an origin, o, which represents a point absolutely fixed in space. This is necessary because the two configurations do not exist at the same time ; and therefore to express their relative position we require to know a point which remains the same at the beginning and end of the time. But we may construct the diagram in another way which does not assume a knowledge of absolute displacement or of a point fixed in space. Assuming any point and calling it a, draw ok parallel and equal to B t A x in the initial configuration, and from k draw kb parallel and equal to A 2 B. 2 in the final configura tion. It is easy to see that the position of the point b relative to a will be the same by this construction as by the former construction, only we must observe that in this second construction we use only vectors such as AjBp A 2 B 2, which represent the relative position of points both of which exist simultaneously, instead of vectors such as A 1 A 2, B 1 B 2 , which express the position of a point at one instant relative to its position at a former instant, and which therefore cannot be determined by observation, because the two ends of the vector do not exist simul taneously It appears therefore that the diagram of displacements, when drawn by the first construction includes an origin o, which indicates that we have assumed a knowledge of absolute displacements. But no such point occurs in the second construction, because we use such vectors only as we can actually observe. Hence the diagram of displace ments without an origin represents neither more nor less than all we can ever know about the displacement of the material system. Diagram of Velocity. If the relative velocities of the points of the system are constant, then the diagram of displacement corresponding to an interval of a unit of time between the initial and the final configuration is called a diagram of relative velocity. If the relative velocities are not constant, we suppose another system in which the velocities are equal to the velocities of the given system at the given instant and con tinue constant for a unit of time. The diagram of dis placements for this imaginary system is the required diagram of relative velocities of the actual system at the given instant. It is easy to see that the diagram gives the velocity of any one point relative to any other, but cannot give the absolute velocity of any of them. Diagram of Acceleration. By the same process by which we formed the diagram of displacements from the two diagrams of initial and final configuration, we may form a diagram of changes of rela tive velocity from the two diagrams of initial and final velocities. This diagram may be called that of total accelerations in a finite interval of time. By the same process by which we deduced the diagram of velocities from that of displacements we may deduce the diagram of rates of acceleration from that of total accelera tion. We have mentioned this system of diagrams in elementary kinematics because they are found to be of use especially when we have to deal with material systems containing a great number of parts, as in the kinetic theory of gases. Ths diagram of configuration then appears as a region of space swarming with points representing molecules, and the only way in which we can investigate it is by consider ing the number of such points in unit of volume in different parts of that region, and calling this the density of the gas. In like manner the diagram of velocities appears as a region containing points equal in number but distributed in a different manner, and the number of points in any given portion of the region expresses the number of molecules whose velocities lie within given limits. We may speak of this as the velocity-density. Path and Ilodograph. When the number of bodies in the system is not so great, we may construct diagrams each of which represents some property of the whole course of the motion. Thus if we are considering the motion of one particle relative to another, the point on the diagram of configura tion which corresponds to the moving particle will trace out a continuous line called the path of the particle. On the diagram of velocity the point corresponding to the moving particle will trace another continuous line called the hodograph of the particle. The hodograph was invented and used with great success by Sir W. R. Hamilton as a method of studying the motions of bodies. DIAGRAMS OF STRESS. Graphical methods are peculiarly applicable to statical questions, because the state of the system is constant, so that we do not need to construct a series* of diagrams corresponding to the successive states of the system. The most useful of these applications relates to the equilibrium of plane framed structures. Two diagrams are used, one called the diagram of the frame and the other called the diagram of stress. The structure itself consists of a number cf separable pieces or links jointed together at their extremities. In practice these joints have friction, or may be made pur posely stiff, so that the force acting at the extremity of a piece may not pass exactly through the axis of the joint ; but as it is unsafe to make the stability of the structure depend in any degree upon the stiffness of joints, we assume in our calculations that all the joints are perfectly smooth, and therefore that the force acting on the end of any link passes through the axis of the joint.