532 Mr. G. H. Bryan on a Simple Graphic Illustration 



under any law of force. Let M be the position of the particle at 

 any instant, and let the velocity of the particle at this instant be 

 represented by the ordinate MP drawn at right angles to OX. 

 Then, since the momentum is proportional to the velocity, the 

 coordinates OM, MP represent the coordinate and momentum 

 of the particle at the given instant, and we may call P the 

 representative point. 



Now let four such particles of equal mass be projected 

 simultaneously, having the initial coordinates x and x + Bx 

 and the initial velocities v and v -f Bv. The representative 

 points will form a small rectangle P Q R S of area Bx . Bv. 



Let P' Q' E/ S' be the corresponding representative points 

 at any subsequent instant t. 



Then the determinantal relation asserts that the area of the 

 small parallelogram P' Q ; B/ S' is equal to that of the rect- 

 angle P Q R S. 



[Instead of taking four particles we might suppose the 

 points P, Q, R, S to refer to the same particle projected with 

 different initial conditions and allowed to move for a fixed 

 time-interval t.~\ 



This property may be verified from first principles in the 

 following simple cases : — 



Case I. Let the motion be uniformly accelerated. Then from 

 the equations 



v'=v+ft, x f =x + vt + ift 2 , 



and the corresponding equations obtained by substituting 

 x + Bx for x and v + Bv for v, it is easy to see (fig. 1) that the 



Fig. 1. 



p' 



FT s 



a 



M 



M' K' N' V X 



parallelogram P / Q / R / S / has its base P / Q / parallel to OX and 

 equal to PQ or Bx, and its altitude equal to PQ or Bv. 

 Therefore 



area FQ'R'S^area PQRS. 



The parallelogram will, however, have undergone a shear, 



