KINEMATIC DETERMINATION OF ACCELERATION. 



165 



to the determination of a definite point B. This point will then represent the situa- 

 tion of the particle at the epoch t, with the highest degree of approximation which 

 can be attained when the determination is to be made by the use of two charts of 

 velocity instead of by the use of only one. Or in other words: AB will represent 

 the displacement in the second approximation. 



(2 ) The distances between the points in the series B' , B" , B'" . . . may 

 remain finite. The process of constructions will then be divergent and will lose every 

 physical significance. Examples of this divergence can easily be given. We should 

 meet with it, for instance, in the case of atmospheric wave-motions (see fig. 51) if 

 the interval of time t, 1 was selected of such a length that the displacement (a) 

 obtained in the first approximation was of the same order of magnitude as the 

 wave-length. This case of divergence must be avoided, and can always be avoided 

 if the selected interval of time /, 1 be sufficiently short. The periods which from 

 this point of view may be used must be found gradually by experience. 



We shall consider henceforth exclusively the case of convergence, and of con- 

 vergence so rapid that already the point B" or the point B'", will define with 

 sufficient approximation the situation of the required point B. According to our 

 experience the interval of three hours which we have used seems always to give 

 convergence, and as a rule of a satisfactory rapidity. 



196. Discontinuous Method for Constructing Charts of Acceleration. Let A 



be the situation of the considered particle at the epoch t ot and B its situation at the 

 time /, as we find it in the second approximation by the construction of the preceding 

 section. v Q being the velocity at the point A at the epoch t Q , and v, the velocity at 

 the point B at the epoch t u we shall then have (fig. 113) 



(a) AB' = v o (/,-0 B'C= v, (/,-/) 



the half vector-sum of these displacements vl.B'and.B'Cdefines the displacement A B. 

 The vector-difference of the same two displacements (a) will be represented 

 by the line B' C, for which we shall thus have 



(b) B'C'= (v,-v ) (t,-t ) 



Now let us divide this equation by (/, Q 2 . We shall then get 

 (A v.-Vq = B'C 



K) t-t (t-t y 



But the first member in this equation represents the average acceleration of the 

 particle during the time t l 1 . The equation therefore expresses the fact that the 

 vector B'C which we have constructed will, after division by the square of the 

 interval of time (/, O 2 , represent the acceleration required. If this acceleration 

 should be attributed to a definite place in the field, it would of course be to the 

 central point between the points A and B. 



We have thus arrived at a discontinuous method of constructing charts of 

 acceleration: For a sufficient number of particles we perform the construction 

 giving the displacement of the particles in the second approximation. This con- 

 struction at the same time gives the vector B' C , which gives the direction and 





