

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 </> = w/p + const. ; so that CT = p<f> 4- C gives the requisite sys- 

 tem. In the same way ^ = p<j> + G 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, <f> 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 Jdnematical 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 



dv == dy = dz 



U V W " " \ /' 



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 <f> = const. 



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

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

 lu 4- mv + nw, or 



_ d(j> dx d(j) dy _ d<f> dz , . , _ _ d<f> 

 dx ds dy ds dz ds' ds ' 



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

 decrease of </> in that direction. 



Taking 8s in the direction of the normal to the surface </> = 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. 



