KINEMATIC DETERMINATION OF ACCELERATION. 1 67 



It will easily be seen that this field represents the tangential component of the 

 acceleration in the stationary motion. 



(2) We form the field of curvature 7 of the lines of flow, and perform the multi- 

 plication of this field by that of the square of the velocity v 2 . The field 



(e) yv 2 



which we get in this way evidently represents the normal component of acceleration 



in the stationary motion of the particles along the lines of flow. 



(3) We perform the vector-addition of the vector (d) which is directed along, 

 and the vector (e) which is directed normally to the lines of flow. 



Another method of constructing the field of stationary acceleration in which 

 the single operations will not have the same simple physical significance, but which 

 may still under special circumstances be advantageous, will be this (see formula (d') 

 of section 1 74) : 



(V) We construct the ascendant of the half square of the velocity 



(<*') V(^) 



(2') We construct the two-dimensional curl of the velocity (section 172) and 

 form the vector-product of this vector and the velocity. This vector 

 (e') (curl 2 v)Xv 



will be directed along the positive normal to the lines of flow. 



(3') We perform the vector-addition of the two vectors (d') and (e'). 



(B) Local acceleration. While stationary acceleration is derived from one 

 chart which represents the given field of velocities at the given time, local accelera- 

 tion must be derived from two charts which represent velocity at the two different 

 epochs. The method will be that of the regular vector-subtraction and subsequent 

 division by the interval of time as we have developed for the case of pure time- 

 derivations of vector-fields (section 176). 



198. Special Remarks. The chart of local acceleration which we derive 

 from the charts of velocity for the epochs t and /, will correspond to the epoch 

 '0+2 (A - O- O n the other hand the chart of stationary acceleration, which is 

 derived from one of the given charts of velocity, will correspond either to the epoch 

 t a or to the epoch /,. If the interval between these epochs is sufficiently short, the 

 circumstance that the charts of local and of stationary acceleration correspond to 

 slightly different epochs will cause no trouble. But in order to get a satisfactory 

 construction of the local acceleration, we are obliged to select the interval of time 

 ti to with as great a length as possible. For this reason it will be rational to derive 

 the stationary acceleration from both given charts of velocity. The best method 

 will then be this: 



By vector-addition and division by 2 we form the chart of the average velocity 



|(v,+v ) 

 during the time from t Q to t t . From this chart of average velocity we derive the 

 chart of stationary acceleration, which will then correspond to the epoch /o+K^ - O- 

 Then we form the vector-difference of the same two fields of velocity 



v,-v 



