58 IRROTATIONAL MOTION. [CHAP. Ill 



definition of an n + I -ply-connected region, indicated in Art. 48, we remark 

 that in a simply-connected region every equipotential surface must either be a 

 closed surface, or else form a barrier dividing the region into two separate 

 parts. Hence, supposing the whole system of such surfaces drawn, we see 

 that if a closed curve cross any given equipotential surface once it must cross 

 it again, and in the opposite direction. Hence, corresponding to any element 

 of the curve, included between two consecutive equipotential surfaces, we have 

 a second element such that the flow along it, being equal to the difference 

 between the corresponding values of $, is equal and opposite to that along 

 the former ; so that the circulation in the whole circuit is zero. 



If however the region be multiply-connected, an equipotential surface may 

 form a barrier without dividing it into two separate parts. Let as many 

 such surfaces be drawn as it is possible to draw without destroying the 

 continuity of the region. The number of these cannot, by definition, be 

 greater than n. Every other equipotential surface which is not closed will 

 be reconcileable (in an obvious sense) with one or more of these barriers. A 

 curve drawn from one side of a barrier round to the other, without meeting any 

 of the remaining barriers, will cross every equipotential surface reconcileable 

 with the first barrier an odd number of times, and every other equipotential 

 surface an even number of times. Hence the circulation in the circuit thus 

 formed will not vanish, and &amp;lt; will be a cyclic function. 



In the method adopted above we have based the whole theory on the 

 equations 



dw dv _ du dw _ dv du 



dy dz dz dx dx dy &quot; 



and have deduced the existence and properties of the velocity -potential in the 

 various cases as necessary consequences of these. In fact, Arts. 35, 36, and 

 49, 50 may be regarded as a treatise on the integration of this system of 

 differential equations. 



The integration of (i), when we have, on the right-hand side, instead of 

 zero, known functions of ^, y, 2, will be treated in Chapter vn. 



52. Proceeding now, as in Art. 37, to the particular case of 

 an incompressible fluid, we remark that whether &amp;lt;/&amp;gt; be cyclic or not, 

 its first derivatives d&amp;lt;f&amp;gt;jdx, dfyjdy, d^/dz, and therefore all the 

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

 c/&amp;gt; will still satisfy the equation of continuity 



v^ = o (i), 



or the equivalent form 



where the surface-integration extends over the whole boundary of 

 any portion of the fluid. 



