234 VORTEX MOTION. [CHAP. VII 



The formula (3) may be otherwise written 



dS'.. ..(4), 



where ^ denotes the angle between r and the normal (I, ra, n). 

 Since cos OSS'/'r 2 measures the elementary solid angle subtended 

 by SS' at (#, y, z\ we see that the velocity-potential at any point, 

 due to a single re-entrant vortex, is equal to the product of ra'/27r 

 into the solid angle which any surface bounded by the vortex 

 subtends at that point. 



Since this solid angle changes by 4?r when the point in 

 question describes a circuit embracing the vortex, we verify that 

 the value of <f> given by (4) is cyclic, the cyclic constant being 

 twice the strength of the vortex. Cf. Art. 142. 



Comparing (4) with Art. 56 (4) we see that a vortex is, in 

 a sense, equivalent to a uniform distribution of double sources over 

 any surface bounded by it. The axes of the double sources must be 

 supposed to be everywhere normal to the surface, and the density 

 of the distribution to be equal to the strength of the vortex 

 divided by 2?r. It is here assumed that the relation between 

 the positive direction of the normal and the positive direction 

 of the axis of the vortex-filament is of the 'right-handed' type. 

 See Art. 32. 



Conversely, it may be shewn that any distribution of double sources over 

 a closed surface, the axes being directed along the normals, may be replaced 

 by a system of closed vortex-filaments lying in the surface*. The same thing 

 will appear independently from the investigation of the next Art. 



Vortex-Sheets. 



149. We have so far assumed ?/, v, w to be continuous. We 

 will now shew how cases where surfaces present themselves at 

 which these quantities are discontinuous may be brought within 

 the scope of our theorems. 



The case of a surface where the normal velocity is discon- 

 tinuous has already been treated in Art. 58. If u, v, w denote the 

 component velocities on one side, and u', v', w' those on the other, 



* Cf. Maxwell, Electricity and Magnetism, Arts. 485, 652. 



