128 DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



When the field of the vector-derivative F is given at a series of epochs, and the 

 field of the vector A at the initial epoch t , we can perform the pure time-integration, 

 which is the inverse operation to the pure time-differentiation considered. We have 

 then to identify the value of the derivative F at a moment / with the average 

 derivative F during a finite but short interval of time /, 1 when t </<^ 1 . We 

 then perform the multiplication of the average derivative F with the constant factor 

 t t to, and afterwards perform the addition of the two vector-fields according to 

 the formula 



This operation may be repeated any number of times, and will lead to the field of 

 the vector A at the time /, which is expressed analytically by the integral 



(g) a=a +j;;f^ 



The delicate point in this process of integration will be the addition of the 

 generally very small vector F(/, / ) to the finite vector A. But as the isogons 

 and the intensity-curves of the two fields will usually cut each other under finite 

 angles, we shall not meet with the same difficulties as those connected with the 

 differentiation. The only difficulty will be the gradual summing up of the small 

 errors which enter at each partial operation. 



177. Complex Time and Space Differentiation. Besides the pure space-differ- 

 entiations and the pure time-differentiations we shall also meet with complex space- 

 time-differentiations. They will be seen to occur in all investigations concerning 

 moving continuous media. 



Let / be any function of coordinates and time, 



(a) f(x, y, z, t) 



It has four partial derivatives 



{ ' 5x ?y 9z 3t 



The last is what we have called above the pure time-derivative. In order to form 

 it we have to consider x, y, z as constant, and let only time vary; i. e., we compare 

 the values of / in the same locality at two different epochs. We shall therefore also 

 call it the local time-derivative. 



But on other occasions we shall have to compare the values which the function 

 / has at two epochs at one and the same physical particle. What we keep constant 

 in this comparison will then be not the locality x, y, z, in which the values of / are 

 observed, but the individuality of the particle at which the values of / are observed. 

 Now let v x , v y , v z be the velocity-components of the particle. If at the time / it has 

 the coordinates x, y, z, it will at the time t+dt have the coordinates x+v x dt, 

 y+v y dt, z+vdt. We have then to compare 



(c) f(x+v x dt, y+v y dt, z+v z dt, t+dt) 



