164 



DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



Now let a second chart be given, which represents the state of motion at this 

 second epoch /,. According to this chart we shall find not the velocity v but a 

 certain velocity v,' at the point B' . We may therefore expect to get a better determi- 

 nation of the displacement if we suppose that the particle has moved not with the 

 velocity v 0) but with the velocity (v/+v ). During the time t x t a this velocity 

 would give the displacement 



(b) 4B"-i(T.+T,0(*.-O 



The point B" would then more correctly than the point B' give the situation of 



of the particle at the epoch t t . 



The point B" and the corresponding displacement AB" can be found by a 

 direct continuation of the construction which led to the point B' ; i. e., we measure 

 off from this point the displacement 

 (&') 3' C" = v/ (/,-/) 



The point B" will then be the central point of the line AC" which represents the 

 vector-sum of the two displacements AB' and B'C" '. 



-Construction of displacement in second 

 approximation. 



Fig. 1 13. Displacement and acceleration. 



But according to the second chart the velocity at the point B" will not have 

 the value v/ but a certain other value v,". Therefore we may expect to get a still 

 better determination of the displacement required if we suppose that the particle 

 has moved during the time /, t Q , not with the velocity \ (v +v/), but with the 

 velocity \ (v +v,")> which would have given the displacement 



(c) 



AB'"=\{v +v l ") (*,-0 



The point B'" at which the particle would then arrive can be found by a con- 

 struction similar to that which led to the point B" . From the point B' we set off 

 the displacement 



W) 



B'C'"= v," (/,-*) 



The point B'" will then be the central point of the line A C 

 vector-sum of the two displacements AB' and B'C" . 



Evidently these constructions can be continued indefinitely, 

 then present themselves : 



(i) The distances between the points B', B", B'" . . . may converge toward 

 zero. The process of constructions will then be convergent and will lead ultimately 



which represents the 

 Two cases may 



