EECENT PKOGRESS m THEORJiTICAL PHYSICS. 209 



The components of the vector-potential are related to those of 

 the scalar-potential as follows: 



a, b, c, being as before the components of magnetic force, derived 

 from the scalar-potential V by differentiation along .r, y, z. 



The components of the electric current, u, v, u\ are known to 

 satisfy the condition 



du dv dw 



dx ' dy dz 



The components of magnetic induction a, 6, c, also satisfy a sim- 

 ilar equation 



da db dc 



'd^ ^ ~d^ '^ ~d^ ^ ^• 



On a careful study of these formulse. which have been deduced 

 for the potentials and forces of ordinary magnets and electro- 

 magnets, we are impressed with their similarity to the formulee that 

 express the ordinary motions of an incompressible frictionless fluid. 

 For example, in fluid motion, where u, v, and 'C, represent the 

 component velocities of an element of the fluid in three rectangu- 

 lar directions, and D represents the density of the fluid, we have 

 the following so-called "Equation of Continuity" of the fluid: 



-P^ fdw dv <]w] dD dD„ , dD , dD J 



I^ Idx + d^+d^ j +-d7 + ^-"+7i7" +^-= 0. 



This equation is proven for ordinary motions of fluids, in all 

 works upon the djnamics of fluids. It is merely an analytical 

 statement that in all motions of fluids, however they may expand 

 or contract, and move about in currents or otherwise, there is no 

 change in the was."? of the fluid caused by such motions. If the 

 fluid be incompressible, there can also be no variation in its 

 14 w A B 



