GRAPHICAL DIFFERENTIATION AND INTEGRATION. 119 



When we compare with the formula (h) of the preceding section, we see that 

 we can write the equation 



0*) divA=^ + div 2 A 



Sz 



where div 2 A represents the divergence of that two-dimensional vector which has 

 the components A x and A y . Now let the three-dimensional vector A be solenoidal, 

 div A = o. Then equation (i) gives 



(7) ^-'=-div,A 



Sz 



This is a differential equation by which we may determine the third component A, 

 of a solenoidal vector, of which we know the two components A x and A y . This will 

 be our most important diagnostic formula. We shall use it to derive the vertical 

 motion from the observed horizontal motion in the atmosphere. 



172. Curl of a Two-Dimensional Vector. Instead of the integral of the normal 

 component A n we shall now consider that of the tangential component A, taken 

 along a curve s. 



(a) fA s ds 



In the special case of a closed curve we shall call this integral the circulation of the 

 vector A around the curve s. Lord Kelvin has introduced this name for cases where 

 the vector A represents velocity. We shall use it, precisely as the expressions 

 transport and outflow, even for cases of abstract vectors, which have nothing to do 

 with motion. Circulation is a quantity which has a definite sign depending upon 

 the direction which we have chosen as positive for rotating motion around a point 

 or circulating motion around a closed curve (section 155). 



Circulations have an additive property similar to outflows. We can join two 

 points of the circuit originally given by a curve. The area limited by the first 

 circuit will then be divided into two areas. We can form the sum of the circulations 

 around the contours of each of them, using in both cases the same direction of circula- 

 tion. In this sum the line-integral taken along the dividing curve will appear twice 

 with opposite signs in the two cases, and will therefore drop out (fig. 98). Thus 

 the sum of the circulations around the contours of the two parts of an area will be 

 equal to the circulation around the contour of this total area. As we can continue 

 the subdivision, we arrive at the result that the circulation around the contour of 

 any area is equal to the sum of the circulations around the contours of all the areas 

 into which it can be subdivided. We can express this result by the equation 



(b) JA s ds = i: JA s ds 



extending the integral of the first member to the contour of the primary area and 

 the integrals of the second to the contours of the areas produced by the division. 



Now let the primary area be subdivided into elementary areas by two systems 

 of curves, namely, the vector-lines and their positive normal curves n. The elements 

 dn of the contour of these areas will then give no addition to the line-integral. 

 The circulation in positive direction around the contour (fig. 99) will be represented 

 by the difference 



(c) -(A'ds'-Ads) 



