124 DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



(i) We form the half square of the intensity-field of the given vector (section 

 147) and then the derivative (c) with respect to the vector-lines. 



(2) We form the field of curvature of the given vector-lines (section 168) and 

 perform the graphical multiplication of this field by that of the square of the 

 intensity {c'). 



(3) We perform the graphical addition of two mutually normal vectors (section 

 157) : the vector F s which has the same direction as the given vector A and the 

 intensity determined by the operation (1); and of the vector F which is normal 

 to the given vector and has the intensity determined by the operation (2). 



We can also give another method for determining the vector F. We can change 



the second member of equations (b) : in the first of these equations by adding and 



S A ?A 



subtracting the term A y ?; in the second by adding and subtracting A x -. This 



Sx 3y 



gives 



yd) 



These equations represent the vector F as the vector-sum of two vectors. The first 

 is the ascendant of the scalar \(Al + A 2 y ) = \ A 2 . The second is the vector- 

 product of the vectors curl 2 A and A. When we remember that the vector curl 2 A is 

 normal to the surface which contains A, we see by the properties of the vector- 

 product that this second vector will be directed along the positive normal to A. 

 Thus we can represent the scalar equation (d) by the vector-equation 



(<*') F= V(i^ 2 ) + (curl,A)XA 



Thus we can also use the following method for constructing the field of the vector F. 



(1) We construct the scalar field of the half square of the intensity of the given 

 vector (sec. 147), and then the field of the ascendant of this scalar (sec. 169). 



(2) We construct the field of the curl of the given vector (section 172) and 

 perform the graphical multiplication of this field by that of the intensity of the given 

 vector. This field is considered as the intensity-field of a vector which has the 

 direction of the positive normal to the given vector. 



(3) We form the field of the sum of the two vectors, the fields of which we have 

 found by the first two operations. 



In most cases the first method will be preferable, as the two fields the vector- 

 sum of which we shall form are then normal to each other. But still in special cases 

 the second may be the shorter, for instance if some of the partial fields upon which 

 the construction depends are already constructed for other purposes. 



175. Pure Time-Differentiations and Time-Integrations of Scalar Fields. 

 While a pure space-differentiation is performed upon one chart, representing the field 

 of a scalar or a vector at a given moment, the pure time-differentiations will consist 

 in the comparison of two charts, which represent the field at two different moments. 



