117] SOLENOIDAL VECTORS. 349 



Such a vector will be termed solenoidal, or tubular, and the 



O ~y Q -T7" O f7 



condition - - + - + = will be termed the solenoidal condition 



ex cy oz 



(Maxwell). We may abbreviate it, div. E = 0. If a vector point- 

 function JR is lamellar as well as solenoidal, the scalar function V 

 of which it is the vector parameter is harmonic, for 



0X . dY . dZ 



A solenoidal vector may be represented by its tubes, its direction 

 being given by the tangent to an infinitesimal tube, and its magnitude 

 being inversely proportional to its cross -section. As an example of 

 a solenoidal vector we may take the velocity of particles of a moving 

 liquid. If the velocity is B, with components X, Y, Z, the amount 

 of liquid flowing through an element of surface dS in unit time is 

 that contained in a prism of slant height E, and base dS, whose 

 volume is 



E cos (En) dS. 



The total flux, or quantity flowing in unit time through a sur- 

 face $, is the surface integral 



I I EGOS (En) dS = I I [X cos (nx) -f Fcos (ny) -f Zcos (w*)] dS. 



Such a surface integral may accordingly be called the flux of 

 the vector E through S. 



A tube of the vector E is a tube through whose sides no fluid 

 flows, such as a material rigid tube through which a liquid flows, 

 and the divergence theorem shows that as much liquid flows in 

 through one cross -section as out through another, if the solenoidal 

 condition holds. If the liquid is incompressible, this must of course 

 be true. 



As a second example of a solenoidal vector we have any vector 

 which is the curl of another vector, for 



d (dZ 2Y\ _d_l^_<^2\ , JMU_^| = 

 J^\dy~ dz\ + dy\ dz dx\^~dz\dx dy 



identically. 



The equation 



is called Laplace's equation, and the operator 



_a* d^ d^ 



~ dx* + dy*~^~ cz^ 

 Laplace's operator. 



