149-150] IMPULSE OF A VORTEX-SYSTEM. 237 



If X', Y, Z' be components of a distribution of impulsive 

 force which would generate the actual motion (u, v, w) instan- 

 taneously from rest, we have by Art. 12 (1) 



v , 1 der T7 , 1 dix , 1 d'SF . . 



X --- -j- = u, 7 --- T- = V > * j -=w ...... (1), 



p dx p dy pdz 



where OT is the impulsive pressure. The problem of finding X', F', Z', 

 OT in terms of u, v, w, so as to satisfy these three equations, is clearly 

 indeterminate; but a sufficient solution for our purpose may be 

 obtained as follows. 



Let us imagine a simply-connected surface 8 to be drawn 

 enclosing all the vortices. Over this surface, and through 

 the external space, let us put 



^ = />< .............................. (2), 



where </> is the velocity-potential of the vortex-system, determined 

 as in Art. 148. Inside 8 let us take as the value of w any 

 single-valued function which is finite and continuous, is equal to 

 (2) at $, and also satisfies the equation 



at 8, where &n denotes as usual an element of the normal. It 

 follows from these conditions, which can evidently be satisfied in an 

 infinite number of ways, that the space-derivatives d^/da), div/dy, 

 dtvjdz will be continuous at the surface 8. The values of X , F, Z' 

 are now given by the formulae (1); they vanish at the surface 8, 

 and at all external points. 



The force- and couple-equivalents of the distribution X' t F', Z' 

 constitute the ' impulse ' of the vortex-system. We are at present 

 concerned only with the instantaneous state of the system, but it 

 is of interest to recall that, when no extraneous forces act, this 

 impulse is, by the argument of Art. 116, constant in every respect. 



Now, considering the matter inclosed within the surface $, we 

 find, resolving parallel to oc, 



fffpX'dxdydz = pfffudxdydz - pfJl<f>dS ............... (4), 



if I, ra, n be the direction-cosines of the inwardly-directed normal 

 to any element 88 of the surface. Let us first take the case of a 

 single vortex-filament of infinitely small section. The fluid 



