46 48.] STREAM-LINES. 43 



(/3) Again, if -~ be zero at every point of the boundary of 

 such a region as is above described, (f&amp;gt; will be constant throughout 

 the interior. For the condition -~ = 0, expresses that no stream 

 lines enter or leave the region, but that they are all contained 

 within it. This is however, as we have seen, inconsistent with 

 the other conditions which the stream-lines must conform to. 

 Hence, as before, there can be no motion, and &amp;lt;/&amp;gt; is constant. 



This theorem may be otherwise stated as follows : no irrota- 

 tional motion of a liquid can take place throughout a simply- 

 connected region bounded entirely by fixed rigid walls. 



(7) Again, let the boundary of the region considered consist 

 partly of surfaces S over which (f&amp;gt; has a given constant value, and 



partly of other surfaces X over which -^- = 0. By the previous 



argument, no stream -lines can pass from one point to another of &, 

 and none can cross 2. Hence no stream -lines exist; &amp;lt; is there 

 fore constant as before, and equal to its value at S. 



48. Kecalling the dynamical interpretation of &amp;lt; given in Art. 

 26, we see that the first theorem of Art. 47 asserts that a uniform 

 impulsive pressure applied to the boundary of a liquid mass pro 

 duces no motion. This is otherwise obvious. 



A non-uniform impulsive pressure applied to the boundary of 

 the mass will of course generate some definite motion. Further, 

 it appears highly probable that this motion will be everywhere 

 finite and continuous throughout the mass; although if this be 

 the case right up to the boundary, it is necessary that the 

 given surface-value of the impulsive pressure should be continuous, 

 and should also have its rate of variation from point to point of 

 the boundary everywhere finite and continuous. We are thus led 

 to the following analytical theorem : 



(a) There exists a single-valued function &amp;lt;f&amp;gt; which satisfies the 

 equation \7 2 &amp;lt;/&amp;gt; = at every point of a finite region of space, which 

 is with its first derivatives finite and continuous throughout that 

 region, and which has a given arbitrary value at every point of 

 the boundary. If the finiteness and continuity of &amp;lt;f&amp;gt; hold up to the 



