40 42.] SIMPLY-CONNECTED SPACES. 37 



matics, and is true whatever functions u, v, iv may be of x, y, z, 

 provided only they be continuous over the surface*. 



Irrotational Motion. 



41. The rest of this chapter is devoted to the study of irrota- 

 tional motion, as defined by the equations 



f = 0, 77 = 0, =0 (8). 



The existence and the properties of the velocity-potential in the 

 various cases that may arise will appear as consequences of these 

 equations. 



Considering any region occupied by irrotationally-moving fluid, 

 we see from (6) that the circulation is zero in every circuit which 

 can be filled up by a continuous surface lying wholly in the 

 region, or in other words capable of being contracted to a point 

 without passing out of the region. Such a circuit is said to be 

 evanescible. 



Again, let us consider two paths ACS, ADB, connecting two 

 points A, B of the region, and such that either may by con 

 tinuous variation be made to coincide with the other, without ever 

 passing out of the region. Such paths are called mutually 

 reconcileable/ Since the circuit A CBDA is evanescible, we have 



I(ACBDA) = 0, or since I(BDA) = - / (ADB), 



I(ACB)=I(ADB)-, 

 i.e. the flow is the same along any two reconcileable paths. 



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. 53, we con 

 template only simply-connected regions. 



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

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



* It is not necessary that their differential coefficients should be continuous. 



The theorem (7) is attributed by Maxwell to Stokes, Smith s Prize Examination 

 Papers for 1854 The proof given above is due to Thomson, I.e. ante. For other 

 proofs, see Thomson and Tait, Natural Philosophy, Art. 190 (j), and Maxwell, 

 Electricity and Magnetism, Art. 24. 



