SIR WILLIAM THOMSON ON VORTEX MOTION. 251 



GO. (n.) Draw any surface cutting a vortex tube, and bounded by it. The 

 surface integral of the component rotation round the normal has the same value 

 for all such surfaces ; and this common value is what we now call the rotation of 

 the tube. 



(o). In an unbounded infinite fluid, an axial tube must be either finite and 

 endless or infinitely long in each direction* In an infinite fluid with a boundary 

 (for instance, the surface of an enclosed solid), an axial tube may have two ends, 

 each in the boundary surface ; or it may have one end in the boundary surface, 

 and no other; or it may be infinitely long in each direction, or it may be finite 

 and endless. In a finite fluid mass, an axial tube may be endless, or may have 

 one end, but, if so, must have another, both in the boundary surface. 



(j>). Proposition. Applying the notation of (/), to axes parallel to those of 

 co-ordinates x, y, z, and denoting, as formerly, by u, v, w, the components of the 

 fluid velocity at {x, y, z), Ave have — 



, (Aw dv s 



-\ — i f c hi — llir \ y — i ( cll L _ c Ilf\ 



2 \dy dzj ' v ~ ' \dz ' dx J' * ~ * \dx dy) • 



The proof is obvious, according to the plan of notation, &c, followed in § 13 

 above. 



(q). Hence by (/), (e), and {b)~ 



ff" Kl - 1) + - (£ - §0 + » C| ■ - 1) } = /<«*■ + ■* + -) • 



where ffdS denotes integration over any portion of surface bounded by a closed 

 curve ; f(udx + &c.) integration round the whole of this curve ; and {I, m, n) the 

 direction cosines of any point (x, y, z) in the surface. It is worthy of remark 

 that the equation of continuity for an incompressible fluid does not enter into the 

 demonstration of this proposition, and therefore u, v, w may be any functions 

 whatever of x, y, z. In a purely analytical light, the result has an important 

 bearing on the theory of the integration of complete or incomplete differentials. 

 It was first given, with the indication of a more analytical proof than the pre- 

 ceding, in Thomson and Tait's " Natural Philosophy," § 190 (J). 



{?'). Propositions (h) (J) (n) (o) of the present section (§ 60) are due to Helm- 

 holtz ; and with his integration for associated rotational and cyclic irrotational 

 motion in an unbounded fluid, to be given below, constitute his general theory of 

 vortex motion, (n) and (o) are purely kinematical ; (h) and (j) are dynamical. 



(s). Henceforth I shall call a circuit any closed curve not continuously reducible 

 to a point, in a multiply continuous space. I shall call different circuits, any 



* Vortex tubes apparently ending in the fluid, for instance, a portion of fluid bounded by a 

 figure of revolution, revolving round its axis as a solid, constitute no exception. Each infinitesimal 

 vortex tube in this case is completed by a strip of vortex sheet and so is endless. 



