61 (54.] THEOREMS MODIFIED BY MULTIPLICITY. 57 



(a) The normal velocity has a prescribed value at every point 

 of the internal boundary ; 



(b) The circulations in the n independent circuits of the 

 region have prescribed values ; and 



(c) The velocity vanishes at an infinite distance from the 

 internal boundary. 



If, for instance, we have an anchor-ring moving in an infinite 

 mass of liquid which is at rest at infinity, the irrotational motion 

 of the fluid at any instant is determinate when we know the motion 

 of the ring (and therefore the velocity of every element of its 

 surface normal to itself), and also the value of the circulation in 

 any circuit embracing it. 



63. The theory of multiple continuity seems to have been 

 first developed by Riemann*, for spaces of two dimensions, d propos 

 of his researches on the theory of functions of a complex variable, 

 in which connection also cyclic functions satisfying the equation 



^4-^ = 



do? &quot;*&quot; df 

 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 Helmholtzt. The subject of cyclic irrotational motion in 

 multiply-connected regions was afterwards taken up and fully 

 investigated by Sir W. Thomson in his paper on vortex-motion 

 already referred to. 



Greens Theorem. 



64, In treatises on Electrostatics, &c., many important pro 

 perties of the potential are usually proved by means of a certain 

 theorem due to Green J. Of these the most interesting from our 

 present point of view have been already given ; but as the theorem 



* Lehrsatze aus der Analysis Situs. Crelle, t. 54. 1857. 



t Crelle, t. 55. 1858. 



I An essay on Electricity and Magnetism, 3. 



