126 DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



graphical addition of the fields a and (p{t, t ). If we have sufficient knowledge 

 of the derivative <p at different times t, we can repeat this operation and thus find 

 at any time t the field a which is expressed analytically by the integral 



(g) a = a + I <p dt 



*s to 



The graphical addition (f) will cause no such difficulty as that of the graphical 

 subtraction (d) . The only difficulty connected with the integration will arise from 

 the gradual summing up of small errors from the one partial operation to the other. 



176. Pure Time-Differentiations and Time-Integrations of Vector-Fields. 

 The principles for the pure time-differentiations will be precisely the same for a 

 vector-field as for the scalar field. 



Let A be a vector which depends upon both coordinates and time. Let it have 

 the value A at a certain point at the time t , and the value A, at this same point at 

 the time /. The vector 



m r- ^ 



will then represent the average during the interval of time /, 1 of the vector 



w - 



which is the pure time-derivative of the vector A at the considered point. If we 

 use sufficiently small intervals of time we can identify the vector F with the value 

 of F at the time 



(c) / = /<,+ 



2 



By these formulae we see at once that if we know the field of the given vector A 

 at two moments t and /,, which are separated by a sufficiently small interval of 

 time t x to, we can form the field of the derivative at the time (c) in this manner: 



We form the field of the vector-difference 



(d) A,-A 



and afterwards perform the division of this field with the constant factor 



0) h-to 



We have thus reduced the pure time differentiation of a vector-field to algebraic 

 problems already treated. The only difficulty connected with this differentiation 

 will consist in the formation of the vector-difference between two vector-fields which 

 are very like each other. For this reason we must not choose too short an interval 

 of time (e), just as we must not choose it too long if we are to be able to identify 

 the two vectors (a) and (b). 



For the formation of the vector-difference (d) we can use any of the methods 

 which we have developed in vector-algebra. We can use the method of section 

 158 or the graphical tables (section 160), or finally the complete resultantometer 

 (section 161). If we wish to use either of the first two methods, the field represent- 

 ing the difference of angle is first drawn as accurately as possible. The curves, as 



