210 WISCONSIN ACxiDEMT SCIENCES, ARTS, AND LETTERS. 



density, caused either by its own motions or by the lapse of time. 

 Then, the total differential of D in the preceding equation, will be 

 zero, and there will be left 



<f« ^r rft. ^ K. 



dx^ d2/^ dz ^' 



which is the simplified form of the ''equation of continuity" for in- 

 incompressible fluids. Equation (K), is an equation of condition 

 which all incompressible fluids are required to fulfill. 



Km, V, w, be made to depend upon the variations of a quantity 

 Q, such that 



dO dO dO T 



" dz dn ' dz' 



we may call Q the velocitij potential of the velocities u, i\ and iv; be- 

 cause it is a quantity the first differential co-efiicient of which along 

 a line, x, y, or z, gives the velocity of the fluid along that line. This 

 quantity Q, must, in incompressible fluids, which, since w, v, w, as 

 has been said, satisfy Equation (K), therefore also give 



d^Q d^Q d^Q ^ ,, 



dj2 ^ dy2^ dz2 ^' 



or, in other words, like the scalar magnetic and electric potential 

 F, or the potential of magnetization I, it must satisfy Laplace's 

 equation. 



Returning to magnets, and electro-magnets, we have seen that 

 the general expression for the value of the potential of any magnet 

 of finite dimensions, at any point in space whose co-ordinates are 

 x', y', z\ is, designating the potential by V, 



V-=//'dS+///^d.d,dz 



where the surface part of the integral extends over the whole sur- 

 face of the magnet, designated by S, and the solid part of it (every 

 element of which = dx, dy, dz) extends to all portions within the 

 surface. 



r = the distance from the magnet to the point, x^, y^, z^, where 



the value of V is taken; 

 J— I A + mB + nC i. e., the intensity of magnetization normal to 

 the surface of the magnet; because 



