20 INTEGRATION OF THE EQUATIONS IN SPECIAL CASES. [CHAP. II 



by the application of a properly chosen system of impulsive pressures. 

 This is evident from the equations cited, which shew, moreover, 

 that &amp;lt;f&amp;gt; = tr/p + const. ; so that w = p&amp;lt;f&amp;gt; 4- C gives the requisite sys 

 tem. In the same way ^ = p(f&amp;gt; + C gives the system of impulsive 

 pressures which would completely stop the motion. The occur 

 rence 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;f&amp;gt; may be interpreted as the potential 

 of the external impulsive 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 

 pressures, or of extraneous impulsive forces having a potential. 



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

 certain kinematical properties of the motion. 



A line of motion or stream-line * is defined 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; &quot; \ / 



The relations (1) shew that when a velocity-potential exists the 

 lines of motion are everywhere perpendicular to a system of sur 

 faces, viz. the equipotential surfaces &amp;lt; = const. 



Again, if from the point (x, y, z) we draw a linear element Ss 

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

 lu + mv + nw, or 



_ d&amp;lt;t&amp;gt; dx d$dy d(f&amp;gt; dz , . , _ d(/&amp;gt; 

 dx ds dy ds dz ds ds 



The velocity in any direction is therefore equal to the rate of 

 decrease of &amp;lt;/&amp;gt; in that direction. 



Taking 8s in the direction of the normal to the surface &amp;lt;/&amp;gt; = const, 

 we see that if a series of such surfaces be drawn corresponding to 



* Some writers prefer to restrict the use of the term stream-line to the case of 

 steady motion, as defined in Art. 22. 



