93, 94.] DISCONTINUOUS MOTIONS. 99 



where else exceedingly slow, be very great, and so will still 

 transgress the limit pointed out in Art. 30. 



It is a matter of ordinary observation that under such circum 

 stances a surface of discontinuity, beginning at the sharp edge, is 

 formed. Thus the current issuing from an orifice in the thin wall 

 of a vessel does not, as would appear from Example 7, spread out 

 and follow the walls, but forms a compact stream bounded on all 

 sides by fluid sensibly at rest. Similarly, the current issuing from 

 a straight pipe or canal does not spread out in all directions in 

 the manner indicated in Example 9, but forms, at all events for 

 a short distance, a uniform stream whose boundaries are pro 

 longations of the sides of the canal. As practical exemplifications 

 of these statements we may point to the smoke issuing from a 

 chimney, and to the motion of a rapid torrent through a bridge 

 whose span is considerably less than the breadth of the channel 

 below. 



It is not very easy to form, from theory, a precise idea of 

 the manner in which the existence of these surfaces of discon 

 tinuity is brought about. If the motion in any of the cases above 

 referred to be generated gradually from rest, as for instance in the 

 case of Example 9 by the motion of a piston fitting the canal, 

 then if the edges be slightly rounded the continuous motion 

 already discussed will in the first instance be possible. As how 

 ever the motion of the piston is accelerated, a time arrives when 

 the pressure at the edge sinks to zero. The boundary-conditions 

 are then altered and the nature of the analytical problem is 

 entirely changed. Helmholtz supposes that at the points of 



zero pressure the values of the derivatives -f- . , -J- , be- 



dx cty dz 



come discontinuous, and that it is to these discontinuous com 

 ponents of the total force acting on a fluid element that the 

 generation of the discontinuous motion which is actually produced 

 is to be ascribed. 



The conditions to be satisfied at a surface of discontinuity are 

 easily found. We have of course the kinematical relation (13) 

 of Art. 10 ; and the dynamical condition is that the pressure at 

 every point of the surface must be the same on both sides. If 

 the motion be steady, this requires that the values of the squares 



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