148-149] VORTEX-SHEETS. 235 



it was found that the circumstances could be represented by 

 imagining a distribution of simple-sources, with surface density 



{I (u f u) + m (v v) + n (w f w)}, 

 47T 



where I, m, n denote the direction-cosines of the normal drawn 

 towards the side to which the accents refer. 



Let us next consider the case where the tangential velocity 

 (only) is discontinuous, so that 



I (u f u) + m (v v) + w (w w) = Q (1). 



We will suppose that the lines of relative motion, which are 

 defined by the differential equations 



dx _ _dy_ = dz . 



u u v v w w 



are traced on the surface, and that the system of orthogonal 

 trajectories to these lines is also drawn. Let PQ, P Q be linear 

 elements drawn close to the surface, on the two sides, parallel to 

 a line of the system (2), and let PP and QQ be normal to the 

 surface and infinitely small in comparison with PQ or P Q, . 

 The circulation in the circuit P Q QP will then be equal to 

 (q q) PQ, where q, q denote the absolute velocities on the two 

 sides. This is the same as if the position of the surface were 

 occupied by an infinitely thin stratum of vortices, the orthogonal 

 trajectories above-mentioned being the vortex-lines, and the 

 angular velocity w and the (variable) thickness &n of the stratum 

 being connected by the relation 2o&amp;gt; . PQ . Sn = (q q) PQ, or 



co$n = %(q -q) (3). 



The same result follows from a consideration of the discontinuities which 

 occur in the values of u, v, w as determined by the formulae (3) and (6) of 

 Art. 145, when we apply these to the case of a stratum of thickness dn 

 within which , 77, are infinite, but so that w, r)8n, 8n are finite*. 



It was shewn in Arts. 144, 145 that any continuous motion of 

 a fluid filling infinite space, and at rest at infinity, may be 

 regarded as due to a proper arrangement of sources and vortices 

 distributed with finite density. We have now seen how by 

 considerations of continuity we can pass to the case where the 

 sources and vortices are distributed with infinite volume-density, 



* Helmholtz, 1. c. ante p. 222. 



