38 IRROTATIONAL MOTION. [CHAP. III. 



velocity-potential. Let &amp;lt; denote the flow from some fixed point 

 A to a variable point P, viz. 



f 

 = 



J 



(9). 



The value of &amp;lt;/&amp;gt; has been shewn to be independent of the path 

 along which the integration is effected, provided it lie wholly 

 within the region. Hence (f&amp;gt; is a single-valued function of the 

 position of P\ let us suppose it expressed in terms of the co 

 ordinates (x, y, z) of that point. By displacing P through an 

 infinitely short space parallel to each of the axes of co-ordinates 

 in succession, we find 



dd&amp;gt; dd&amp;gt; dd&amp;gt; 



-f u &amp;gt; ~f = v &amp;gt; -f = w &amp;gt; 

 dx dy dz 



i.e. (f&amp;gt; is a velocity- potential, according to the definition of Art. 22. 



The substitution of any other point B for A, as the lower limit 

 in (9), simply adds an arbitrary constant to the value of &amp;lt;, viz. the 

 flow from B to A. The original definition of &amp;lt;p in Art. 22, and its 

 physical interpretation in Art. 26. leave the function indeterminate 

 to the extent of an additive constant. 



As we follow the course of any stream-line the value of &amp;lt; con 

 tinually increases ; hence in a simply-connected region the stream 

 lines cannot form closed curves. 



43. The function &amp;lt; with which we have here to do is, together 

 with its first differential coefficients, by the nature of the case, 

 finite, continuous, and single-valued at all points of the region 

 considered. In the case of incompressible fluids, which we now 

 proceed to consider more particularly, &amp;lt;j&amp;gt; must also satisfy the 

 equation of continuity, (5) of Art. 25, or as we shall write it, for 

 shortness, 



at every point of the region. Hence &amp;lt; is now subject to mathe 

 matical conditions identical with those satisfied by the potential of 

 masses attracting or repelling according to the law of the inverse 

 square of the distance, at all points external to such masses ; so 

 that many of the results proved in the theories of Attractions, 

 Statical Electricity, &c., have also a hydrodynamical application. 



