42 45.] TUBES OF FLOW. 39 



We proceed to develope those which are most important from this 

 point of view. 



44. The proof of (10) given in Art. 12 is based essentially on 

 the consideration that since the fluid is incompressible the total 

 volume which enters any element dx dy dz in unit time is zero. To 

 apply the same principle to a finite region occupied entirely by 

 liquid, let dS be an element of the surface of the region, dn an 



element of the normal to it drawn inwards. By Art. 27 -? is the 



dn 



inward velocity of the fluid normal to the surface, and therefore 



T- dS is the volume whi&amp;lt; 

 dn 



the element dS. Hence 



T- dS is the volume which in unit time enters the region across 

 dn 



c? *j? n n-n 



-y- CtO = U (HJJ 



the integration extending over the whole boundary of the region. 

 Equations (10) and (11), expressing the same fact, must be mathe 

 matically equivalent ; see Art. 64. 



The stream-lines drawn through the various points of an 

 infinitesimal circuit constitute a tube, which may be called a tube 

 of flow. The product of the velocity (j) into the cross-section (cr) 

 is the same at all points of such a tube. 



We may, if we choose, regard the whole space occupied by the 

 fluid as made up of tubes of flow, and suppose the size of the 

 tubes so chosen that the product qcr is the same for each. The 



f[d&amp;lt;b 



value of the integral l\ -f- dS taken over any surface is then 

 Jj dn 



proportional to the number of tubes which cross that surface. If 

 the surface be closed, the equation (11) expresses the fact that as 

 many tubes cross the surface inwards as outwards. Hence a 

 stream-line cannot begin or end at a point of the fluid. 



45. The function &amp;lt;f&amp;gt; cannot be a maximum or minimum at a 

 point in the interior of the fluid ; for, if it were, we should have 



V- everywhere positive, or everywhere negative, over a small 



closed surface surrounding the point in question. Each of these 

 suppositions is inconsistent with (11). 



