GRAPHICAL DIFFERENTIATION AND INTEGRATION. 1 23 



We will denote the vector which has the components (a) by the sign BVA, thus 

 (a') F = BVA 



The vector-equation (a') may be considered as a shortened symbolic expression of 

 the three scalar equations (a). 



We shall consider especially the two-dimensional vector F in the case when 

 B = A. This vector will have the two components 



?A ?A 2>A %A 



K ' Sx y Sy y x Sx y 2y 



and will in accordance with (a') be represented by the vector-formula 



(b') F = AVA 



If the fields of the two components A x and A y are given separately, we can form the 

 fields of F x and F y in accordance with these formulae, performing for each of them 

 two graphical differentiations, two graphical y 

 multiplications, and one graphical addition. 

 In order to examine more closely the rela- 

 tion of the derived vector F to the given vector 

 A, we can make a special choice of the system 

 of coordinates (fig. 100). At the considered 

 point the axis of X shall be tangential to the 

 vector-line s of the given vector A. F x will A, 

 then be the same as the component F s tan- 

 gential to the line s. As at the considered 



a t F IG - IO - 



point A x = A and A y = o, and as ultimately 



dx will be identical with ds, we get for the tangential component 



to *>^7-S^') 



cS cS 



As the curve 5 near the point of tangency forms the infinitely small angle a with the 

 axis of x, we can write here A y = A a. Derivation then gives c * a.^ \-A 



cX cX cX 



As at the point of tangency a is zero, we get here s - = A = A. But-- represents 



cX cX ?S cS 



3 A 



the curvature y of the curve s. Instead of - y - in the second equation (6) we can 



2x 



thus write Ay. When we introduce this, and remember that at the considered 

 point A x = A and A y o, we get this expression of F y or F 

 (C) F n = A*y 



Thus the derived vector F will have two rectangular components, one which has 

 the direction of the given vector and is equal to the derivative of the half square 

 of the intensity of this vector with respect to its vector-lines, while the other is 

 normal to the given vector and equal to the square of the intensity of this given 

 vector multiplied by the curvature of its vector-lines. Hence we can form the field 

 of this derived vector F by the following construction : 



