54 IRROTATIONAL MOTION. [CHAP. III. 



This is otherwise evident from Art. 25 (3),. which shews that 

 S? is single-valued, and that therefore the cyclic constants of &amp;lt;/&amp;gt; 

 cannot alter. 



In Examples 5 and 6, Arts. 3 6, 37, we had instances to which 

 the above result is not applicable ; the reason being that in Ex. 5 

 the external forces have not a potential, whilst in Ex. 6 their 

 potential is itself a cyclic function. 



CO. Proceeding now, as in Art. 43, to the particular case of 

 an incompressible fluid, we remark that whether &amp;lt;p be many-valued 



or not, its first derivatives -~ , ~ , -? , and therefore all the 



ax dy dz 



higher derivatives, are essentially single-valued functions, so that 

 (j&amp;gt; will still satisfy the equation of continuity 



and the equivalent form 



where the surface-integration extends over the whole boundary of 

 any portion of the fluid. 



In the theorems of Arts. 45 and 46 the spaces to which (11) 

 is applied are simply-connected, so that it is allowable to suppose 

 &amp;lt;/&amp;gt; single-valued throughout them even when the region of which 

 they form a -part is multiply-connected. On this understanding 

 the theorems, in question still hold when is a cyclic function. 



The theorem (a) of Art. 47, viz. that &amp;lt; must be constant 

 throughout the interior of any region at every point of which (10) 

 is satisfied if it be constant over the boundary, still holds when the 

 region is multiply-connected. For &amp;lt;/&amp;gt;, being constant over the 

 boundary, is necessarily single-valued. 



The remaining theorems of Art.. 47, being based on the assump 

 tion that the stream-lines cannot form closed curves, are however 

 no longer exact. We must introduce the additional condition that 

 the circulation is to be zero in each circuit of the region. 



