39-40] CONDITIONS OF DETERMINATENESS. 45 



Otherwise : we have seen in Arts. 36, 37 that the lines of 

 motion cannot begin or end at any point of the region, and that they 

 cannot form closed curves lying wholly within it. They must 

 therefore traverse the region, beginning and ending on its bound- 

 ary. In our case however this is impossible, for such a line always 

 proceeds from places where </> is greater to places where it is less. 

 Hence there can be no motion, i.e. 



_ - - 



7 - "J ~7 - V) ~~7 - ^J 



ax dy dz 



and therefore < is constant and equal to its value at the boundary. 



(13) Again, if d<j)/dn be zero at every point of the boundary of 

 such a region as is above described, </> will be constant throughout 

 the interior. For the condition d(f)/dn = expresses that no lines 

 of motion 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 lines must conform to. Hence, as 

 before, there can be no motion, and </> is constant. 



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

 tinuous irrotational motion of a liquid can take place in a 

 simply-connected region bounded entirely by fixed rigid walls. 



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

 partly of surfaces 8 over which $ has a given constant value, and 

 partly of other surfaces 5} over which dcf>/dn = 0. By the previous 

 argument, no lines of motion can pass from one point to another 

 of S, and none can cross 2. Hence no such lines exist; </> is 

 therefore constant as before, and equal to its value at 8. 



It follows from these theorems that the irrotational motion of a 

 liquid in a simply-connected region is determinate when either the 

 value of </>, or the value of the inward normal velocity d<j>/dn, is 

 prescribed at all points of the boundary, or (again) when the value 

 of $ is given over part of the boundary, and the value of d(j>/dn 

 over the remainder. For if fa, fa be the velocity-potentials of 

 two motions each of which satisfies the prescribed boundary- 

 conditions, in any one of these cases, the function fa fa 

 satisfies the condition (a) or (/3) or (7) of the present Article, 

 and must therefore be constant throughout the region. 



