236 VORTEX MOTION. [CHAP. VII 



but finite surface-density, over surfaces. In particular, we may 

 take the case where the infinite fluid in question is incompressible, 

 and is divided into two portions by a closed surface over which the 

 normal velocity is continuous, but the tangential velocity dis- 

 continuous, as in Art. 58 (12). This is equivalent to a vortex- 

 sheet; and we infer that every continuous irrotational motion, 

 whether cyclic or not, of an incompressible substance occupying 

 any region whatever, may be regarded as due to a certain distri- 

 bution of vortices over the boundaries which separate it from the 

 rest of infinite space. In the case of a region extending to 

 infinity, the distribution is confined to the finite portion of the 

 boundary, provided the fluid be at rest at infinity. 



This theorem is complementary to the results obtained in 

 Art. 58. 



The foregoing conclusions may be illustrated by means of the results of 

 Art. 90. Thus when a normal velocity S n was prescribed over the sphere 

 r = a, the values of the velocity-potential for the internal and external space 

 were found to be 



/A n CY i AY 



<f> = - ( - ) S and <f> = ( - ) 



* n\aj n + l \rj 



respectively. Hence if dc be the angle which any linear element drawn on 

 the surface subtends at the centre, the relative velocity estimated in the 

 direction of this element will be 



dS n 



The resultant relative velocity is therefore tangential to the surface, and 

 perpendicular to the contour lines (S n = const.) of the surface-harmonic $ n , 

 which are therefore the vortex-lines. 



For example, if we have a thin spherical shell filled with and surrounded 

 by liquid, moving as in Art. 91 parallel to the axis of #, the motion of the 

 fluid, whether internal or external, will be that due to a system of vortices 

 arranged in parallel circles on the sphere ; the strength of an elementary 

 vortex being proportional to the projection, on the axis of #, of the breadth 

 of the corresponding strip of the surface*. 



Impulse and Energy of a Vortex-System. 



150. The following investigations relate to the case of a 

 vortex-system of finite dimensions in an incompressible fluid 

 which fills infinite space and is at rest at infinity. 



* The same statements hold also for an ellipsoidal shell moving parallel to one of 

 its principal axes See Art. 111. 



