24 27.] EQUIPOTEXTIAL SURFACES. 21 



26. A comparison of equations (2) with the equations of Art. 15 

 gives a simple physical interpretation of the velocity-potential. 



Any actual state of motion of a liquid, for which a velocity- 

 potential exists, could be produced instantaneously from rest by the 

 application of a properly chosen system of impulsive pressures. This 

 is evident from equations (21) Art. 15, which shew, moreover, that 



&amp;lt;j&amp;gt; + const. ; so that CT = G p(f) gives the requisite system. In 



the same way OT = p&amp;lt;f&amp;gt; + C gives the system of impulsive pressures 

 which would completely stop the motion. The occurrence of an 

 arbitrary constant in these expressions shews, what is otherwise 

 evident, that a pressure uniform throughout a liquid mass produces 

 no effect on its motion. 



In the case of a gas, &amp;lt;/&amp;gt; is the potential of the external im 

 pulsive forces by which the actual motion at any instant could 

 be produced instantaneously from rest. 



A state of motion for which a velocity-potential does not exist 

 cannot be generated or destroyed by the action of impulsive pres 

 sures, or of external impulsive forces having a potential. 



27. The existence of a velocity-potential indicates, besides, 

 certain kinematical properties of the motion. 



A line of motion is denned to be a line drawn from point 

 to point, so that its direction is everywhere that of the motion of 

 the fluid. The differential equations of the system of such lines 

 are 



dx _ dy _ dz ,^ 



u v w&quot; 



The relations (2) shew that when a velocity-potential exists the 

 lines of motion are everywhere perpendicular to a series of surfaces, 

 viz. the surfaces &amp;lt;f&amp;gt; = const. These are called the surfaces of equal 

 velocity-potential, or more shortly, the equipotential surfaces. 



Again, if from the point ( t r, y, z] we draw a linear element ds 

 in the direction (I, m, w), the velocity resolved in this direction is 



7 dd&amp;gt; dx deb dii dd&amp;gt; dz , . , d&amp;lt;h ml t 



III + mv + nw, or 7 ~ -j- + -y- -f- + -y- -=- , which = ~ . The veloc- 

 dx as dy ds dz ds as 



ity in any direction is therefore equal to the rate of increase of &amp;lt;f&amp;gt; 

 in that direction. 



