Curl in a Vector Field. 537 



where D n and D b are the projections of the vector D upon 

 b and n, at points near 0. 



Consider points 0' and 0" (fig. 4) infinitely close to O, 

 and situated upon the «-axis. At these points the total 

 components of D upon the normal plane of the vector at 

 are directed along the ?z-axis, as is shown by heavy arrow- 

 heads. This is because the osculating planes at 0' and 0" 

 form an infinitesimal angle with that at 0, and D has no 

 projection upon the n—b plane at 0. Thus, the component 

 Db at . 0' or 0" is an infinitesimal of the second order as 

 compared with the component D n which is an infinitesimal 

 of the first order. 



Therefore, in eq. (25) we may put 



"§ = 0, ...... (26) 



and consequently 



(c„rlD>=^. . . . (27) 



Consider now points M / and M" infinitely close to 0, and 

 situated upon the 6-axis. If the component of the curl 

 according to eq. (27) has a finite value, the components D n 

 at these points must be in opposite directions, because 

 D n = at point 0. Thus, the existence of a curl in the 

 direction of the vector itself is due to a lack of symmetry of 

 the field with respect to the osculating plane nt. A two- 

 dimensional flux is always symmetrical with respect to its 

 plane, and consequently there is no component of the curl 

 in the direction of the vector itself *. 



Take a point T in the plane nb, with coordinates dn and db. 

 The component of the vector D at this point, parallel to n. is 



B n =^-db+ ^dn (28) 



db on 



* The foregoing expressions for the components of a curl may be also 

 derived as a specific case from the general expressions for " rotation " in 

 curvilinear orthogonal coordinates. See, for example, A. E. H. Love, 

 * Mathematical Theory of Elasticity,' second edition, pp. 54-56. It is 

 believed, however, that the simple derivation given above is better 

 adapted for the needs of phj^sicists, and perhaps gives more insight 

 into the nature of curl. See also J. Spielrein, " Geometrisches zur 

 elektrischen Festigkeitsrechnung," Archiv fur Elektrotechnik, vol. iv. 

 p. 78(1915). 



