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



is an exact differential. Hence, integrating, we see that 



udx + vdy + wdz 



consists of two parts, one of which is an exact differential, whilst 

 the other does not contain t. In some cases, for example, when 

 the motion is wholly periodic, we can assert that the latter part is 

 zero, and therefore, that a velocity-potential exists. 



25. Under the circumstances stated in Art. 23, the equations 

 of Art. 6 are at once integrable throughout that portion of the mass 

 for which a velocity-potential exists. For, in virtue of the rela- 



dv dw dw du du dv , . , . ,. , . /n . ,, 



tions -y- = -j , T~ = j~ &amp;gt; j- = ~r &amp;gt; which are implied in (2), the 

 dz dy dx dz dy dx 



equations in question may be written 



d 2 cf) du dv dw _ d V 1 dp 

 dxdt dx dx dx~ dx pdx 



These have the common integral 



Here q denotes the resultant velocity ^(u^ + v 2 + w 2 ), and F(t) is 

 an arbitrary function of t, which may however be supposed in 



cluded in -j- , since, by (2), the values of u } v, w are not thereby 

 affected. 



For incompressible fluids the equation (3) becomes 



whilst the equation of continuity ((9) of Art. 8) assumes the form 



# #4 w 



dx* + dy z + dz* 



In any problem to which these equations apply, and where the 

 boundary-conditions are purely kinematical, the process of solution 

 is as follows. We must first find a function &amp;lt;f&amp;gt; satisfying (5) and 

 the given boundary-conditions ; then substituting in (4) we get the 

 value of p. Since the latter equation contains an arbitrary func 

 tion of t, the complete determination of p requires a knowledge of 

 its value at some point of the fluid for all values of t. 



