40 IRROTATIONAL MOTION. [CHAP. Ill 



A region such that all paths joining any two points of it are 

 mutually reconcileable is said to be 'simply-connected.' Such a 

 region is that enclosed within a sphere, or that included between 

 two concentric spheres. In what follows, as far as Art. 46, we con- 

 template only simply-connected regions. 



36. The irrotational motion of a fluid within a simply-con- 

 nected region is characterized by the existence of a single-valued 

 velocity-potential. Let us denote by < the flow to a variable 

 point P from some fixed point A, viz. 



rP 



d> = (udx + vdy + wdz) (1). 



J A 



The value of < has been shewn to be independent of the path 

 along which the integration is effected, provided it lie wholly 

 within the region. Hence < 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 



u=- d ,v = -f,w = - d . ..(2), 



dx dy dz 



i.e. </> is a velocity-potential, according to the definition of Art. 18. 



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

 in (1), simply adds an arbitrary constant to the value of <f>, viz. the 

 flow from A to B. The original definition of <f> in Art. 18, and the 

 physical interpretation in Art. 19, alike leave the function indeter- 

 minate to the extent of an additive constant. 



As we follow the course of any line of motion the value of </> 

 continually decreases ; hence in a simply-connected region the 

 lines of motion cannot form closed curves. 



37. The function < 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, <j> must also satisfy the 

 equation of continuity, (5) of Art. 21, or as we shall in future 

 write it, for shortness, 



(1), 



