DIAGKAMS. 651 



we are required to draw a line equal and parallel to AA', which we cannot 

 do unless we know the absolute final position of A, with respect to its initial 

 position. In this diagram of displacement there is therefore, besides the points 

 a, I), c, &c., 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 ak parallel and equal to B l A l 

 in the initial configuration, and from If draw kb parallel and equal to A n B,, 

 in the final configuration. It is easy to see that the position of the point 6 

 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 A^,, A. 2 B.,, which represent the relative position of points both of 

 which exist simultaneously, instead of vectors such as A^A t , B^B^, 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 simultaneously. 



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 displacements ivithout 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 continue constant for a unit of tune. The diagram of dis- 



822 



