244 SIR WILLIAM THOMSON ON VORTEX MOTION. 



bounding solid detached, as in that case in which one at least of the two rings is 

 loose, we have varieties of multiple continuity curiously different from that illus- 

 trated by a single ordinary straight or bent tunnel, illustrated sufficiently by the 

 simplest types, which are obtained by boring a tunnel along a line agreeing in 

 form with the axis of a cord or wire on which a simple knot is tied ; and by fixing 

 the two ends of wire with a knot on it to the bounding solid, so that the surface 

 of the wire shall become part of the bounding surface of the space considered, the 

 knot not being pulled tight, and the wire being arranged not to touch itself in 

 any point ; or by placing a knotted wire, with its ends united, in the interior of 

 the space. No amount of knotting or knitting, however complex, in the cord 

 whose axis indicates the line of tunnel, complicates in any way the continuity of 

 the space considered, or alters the simplicity of the barrier surface required to 

 stop the circulation. But it is otherwise when a knotted or knitted wire forms 

 part of the bounding solid. A single simple knot, though giving only double con- 

 tinuity, requires a curiously self-cutting surface for stopping barrier : which, in 

 its form of minimum area, is beautifully shown by the liquid film adhering to an 

 endless wire, like the first figure, dipped in a soap solution and removed. But no 

 complication of these types, or of combinations of them with one another, eludes 

 the statements and formulae of § 57. 



59. I shall now give a dynamical lemma, for the immediate object of preparing 

 to apply Green's corrected . theorem (§ 57) to the motion of a liquid through a 

 multiply continuous space. But later we shall be led by it to very simple 

 demonstrations of Helmholtz's fundamental theorems of vortex motion ; and 

 shall see that it may be used as a substitute for the common equations of 

 hydrokinetics. 



(Lemma). An endless finite tube* of infinitesimal normal section, being given 

 full of liquid (whether circulating round through it, or at rest) is altered in shape, 



* A finite length of tube with its ends done away by uniting them together. 



