51-52] MULTIPLE CONNECTIVITY. 59 



The theorem (a) of Art. 40, viz. that $ must be constant 

 throughout the interior of any region at every point of which (1) 

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

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

 boundary, is necessarily single-valued. 



The remaining theorems of Art. 40, being based on the assump 

 tion that the stream-lines cannot form closed curves, will require 

 modification. We must introduce the additional condition that 

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



Removing this restriction, we have the theorem that the 

 irrotational motion of a liquid occupying an n-ply-connected region 

 is determinate when the normal velocity at every point of the 

 boundary is prescribed, as well as the values of the circulations in 

 each of the n independent and irreducible circuits which can be 

 drawn in the region. For if $ lt &amp;lt;p. 2 be the (cyclic) velocity-poten 

 tials of two motions satisfying the above conditions, then &amp;lt;f&amp;gt; = fa &amp;lt;/&amp;gt; 2 

 is a single-valued function which satisfies (1) at every point of 

 the region, and makes d(f&amp;gt;/d)i = at every point of the boundary. 

 Hence by Art. 40, (f&amp;gt; is constant, and the motions determined by 

 &amp;lt;/&amp;gt;! and &amp;lt;p. 2 are identical. 



The theory of multiple connectivity seems to have been first developed by 

 Riemarm* for spaces of two dimensions, a propos of his researches on the 

 theory of functions of a complex variable, in which connection also cyclic 

 functions, satisfying the equation 



through multiply-connected regions, present themselves. 



The bearing of the theory on Hydrodynamics, and the existence in certain 

 cases of many-valued velocity-potentials were first pointed out by von Helm- 

 holtzf. The subject of cyclic irrotational motion in multiply-connected regions 

 was afterwards taken up and fully investigated by Lord Kelvin in the paper 

 on vortex- motion already referred to J. 



* Grundlagen flir eine allgemeine Theorie der Functional einer veranderlichen 

 complexen Grosse, Gottingen, 1851; Mathematische WerJce, Leipzig, 1876, p. 3; 

 &quot;Lehrsatze aus der Analysis Situs,&quot; Grelle, t. liv. (1857) ; Werke, p. 84. 



t Crelle, t. lv., 1858. 



J See also Kirchhoff, &quot;Ueber die Krafte welche zwei unendlich diinne starre 

 Einge in einer Fliissigkeit scheinbar auf einander ausiiben konnen,&quot; Crelle, t. Ixxi. 

 (1869); Ges.AWi.,^. 404. 



