68 DEFINITE INTEGRALS. [INT. III. 



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

 of which it is the vector parameter is harmonic, for 



dx dy 



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 fluid. If the velocity is R, 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 R, and base dS, whose volume is 



R cos (Rn) dS. 



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

 surface S, is the surface integral 



I JB cos (Bn) dS = ff(X cos (nx) + T cos (ny) + Z cos (m)) dS. 



Such a surface integral may accordingly be called the flux of 

 the vector R through 8. 



A tube of vector R is a tube such that no fluid flows across its 

 sides, such as a material tube through which 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 solenoidal vectors we have any vector 

 which is the curl of another vector, for 



d_ (d_z _ 



identically. 



The equation 



3 2 3 2 3 2 

 is called Laplace's equation, and the operator A = ^- + ^-, + ^-. 



dx? dy 2 dz z 



Laplace's operator. 



_a_ ,az _dz\ + d_ rar _ az| = Q 



dy\dz dx] dz\dx dy } 



