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



indeterminate to the extent of an additive function of t. It 

 becomes determinate when the value of p at some point of the 

 fluid is given for all values of t. 



Suppose, for example, that we have a solid or solids moving through a 

 liquid completely enclosed by fixed boundaries, and that it is possible (e.g. by 

 means of a piston) to apply an arbitrary pressure at some point of the 

 boundary. Whatever variations are made in the magnitude of the force ap 

 plied to the piston, the motion of the fluid and of the solids will be absolutely 

 unaffected, the pressure at all points instantaneously rising or falling by 

 equal amounts. Physically, the origin of the paradox (such as it is) is that 

 the fluid is treated as absolutely incompressible. In actual liquids changes 

 of pressure are propagated with very great, but not infinite, velocity. 



Steady Motion. 



22. When at every point the velocity is constant in magnitude 

 and direction, i.e. when 



dt = ^ = ^~ = (1) 



everywhere, the motion is said to be steady. 



In steady motion the lines of motion coincide with the paths 

 of the particles. For if P, Q be two consecutive points on a line 

 of motion, a particle which is at any instant at P is moving in the 

 direction of the tangent at P, and will, therefore, after an infinitely 

 short time arrive at Q. The motion being steady, the lines of 

 motion remain the same. Hence the direction of motion at Q 

 is along the tangent to the same line of motion, i.e. the particle 

 continues to describe the line. 



In steady motion the equation (3) of the last Art. becomes 



= II J&amp;lt;? 2 4- constant (2). 



P 



The equations may however in this case be integrated to a 

 certain extent without assuming the existence of a velocity- 

 potential. For if 8s denote an element of a stream-line, we have 

 u = q dx/ds, &c. Substituting in the equations of motion and 

 remembering (1), we have 



du _ cZO 1 dp 

 ^ ds dx p dx 



