Curl in a Vector Field. 535 



The curves drawn in these figures represent, therefore, 

 the projections of the line of force AB upon these planes. 

 The field is determined by quantities similar to those men- 

 tioned above for the two-dimensional field, only a greater 

 number is required to account for the changes in three 

 dimensions. 



2 a. Divergence in three dimensions. 



Consider an infinitesimal tube of flux near 0, of a rect- 

 angular cross-section, having a width dn in the direction of 

 the principal normal, and a width db in the direction of the 

 binomial. The flux comprised in the tube is D dn db, and 

 the volume of the tube of a length ds in the direction of D is 

 dndbds. Thus, by definition 



^— (D dndb)ds 



divD= ■ dndUs > ■ ■ ■ ■ < 18 > 

 or 



divD=-^-+D — -j hD — j — . . . (19) 



ds ds ds v ' 



The last two terms on the right-hand side are similar to 

 those in eq. (2), and by a similar reasoning we obtain : 



dlvD= ^ + R + s;.' • • • • (20 > 



where R is the radius of spread in the osculating plane and 

 R r is that in the rectifying plane *. 



It will thus be seen that the divergence does not depend 

 upon the rate of change of flux density in the normal plane. 



For a solenoidal flux, 



!?+£♦£- < M > 



For a solenoidal and lamellar flux, if <p is the potential 

 function, 



SM£*i)2-*;- • -< 22 > 



which is equivalent to Laplace's equation in the usual form. 



* A similar formula mav be deduced by considering directly the radii 

 of curvature of a surface orthogonal to the flux. See W. v. Ignatowskyy 

 Die Vectoranalysis, 1909, vol. i. p. 84. 



