120 



DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



They can therefore be 



A' and ds' will vary as we proceed along the curve n 

 developed as functions of the length of arc n 



A' 



A+^dn 

 en 



ds' = ds+ hrdn 



Introducing this and leaving the term of second order out of consideration, we get 

 for (c) 



(^dnds +A hrdn) 

 \cn Sn J 



or introducing the area da = dnds of the element 

 ( i\ fSA , ,, i Sds^ 



-m +A m da 



\Sn ds Sn) 



The factor of da then represents the circulation per unit area, or the curl of the 

 two-dimensional vector A. We shall introduce the notation 



fa --. . rsA , , i aft 



2 \?n ds SnJ 



Fig. 98. Fig. 99. 



the suffix 2 denoting that the operation curl is performed only in two dimensions. 

 We shall see presently that the curl of the three-dimensional vector is a vector. 

 But, precisely as in the case of the vector-product, the vector-nature of the curl does 

 not appear if we confine ourselves to the consideration of two-dimensional fields. 



We can now write every term in the sum appearing as second member of 

 equation (b) in the form curh A da-. This sum then takes the form of an integral 

 extended to the area formed by all the elements da. Thus we get the formula 



(e) j A s ds = |curl 2 Ada 



that is, the line-integral of the tangential component of a two-dimensional vector 

 taken around a closed curve is equal to the integral of the curl of the vector taken 

 over the area bounded by the closed curve. 



As the expression 



1 cdn 

 dn ?s 



represented the divergence of the vector-lines, section 



ds 



170 (/), i. e., the curvature of their positive normal curves, the expression 



ds Sn 



will represent the divergence of the positive normal curves, i. e., the negative curvature 



( 7) of the vector-lines which are the negative normal curves to the curves n (section 



168). That is, we can write the expression of curl 2 A 



(/) 



curl 2 A = ^ --M7 

 Sn 



