131, 132.] VORTEX-SHEETS. 155 



velocity, and e the thickness, at any point of such a film. Let us 

 examine the form which our previous results assume when o&amp;gt; is 

 increased, and e diminished, without limit, yet in such a way that 

 the product coe, ( == w, say,) remains finite. The infinitely thin 

 film is then called a vortex-sheet. 



The functions L, M t N will now consist in part of potentials 

 of matter distributed with surface densities ~ , ( - , over S. 



LIT Z.7T Z.7T 



We know from the theory of Attractions that L, M, N are con 

 tinuous even when the point to which they refer crosses S, but 

 that their derivatives are discontinuous ; viz. the derivative taken 

 in the direction of the normal (drawn in the direction of crossing) 

 experiences an abrupt decrease of amount 4?r x surface-density. 



Hence the changes (diminutions) in the values of -j- , -, , -7- 



will be 2/fe, Zmge, 2tte, if I, m, n be the direction-cosines of 

 the normal drawn as just explained. The values of u, v, w ob 

 tained from (6) will therefore be discontinuous at S, the components 

 of the relative velocity of the portions of fluid on opposite sides 

 of S being 



2(.?-i,H 2(-7f), 2(7,-iii) e (19), 



respectively. These are the amounts by w r hich the components 

 on the side towards which the normal (I, m, n) is drawn fall short 

 of those on the other. This relative velocity is tangential to S, 

 and perpendicular to the vortex-lines. Its amount is 2&&amp;gt;e, or 2o&amp;gt; , 

 and its direction is that due to a rotation of the same sign as a/ 

 about the vortex-lines in the adjacent part of S. 



Hence a surface of discontinuity at which the relation 



h/j-f mi\ + nn\ = lu z + mv 2 +mu 2 (20), 



[(13) of Art. 10] is satisfied may be treated as a vortex-sheet, in 

 which the vortex-lines are everywhere perpendicular to the direc 

 tion of relative motion of the fluid on the two sides of the surface, 

 and the product o&amp;gt; of the (infinite) angular velocity into the 

 (infinitely small) thickness is equal to half the amount of this 

 relative velocity. 



In the same way, a discontinuity of normal velocity is obtained 

 by supposing 6 to be infinite throughout a thin film, but in such 



