60 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



bring about discontinuity of motion— namely, the pressure, which can 

 assume any positive value whatever while the density of the liquid will 

 continuously vary therewith ; but as soon as the pressure passes the 

 zero value and becomes negative, a discontinuous variation of the 

 density occurs; the liquid is torn asunder. 



Now, the magnitude of the pressure (at any point) in a moving fluid 

 depends on the velocity (at that point), and in incompressible fluids 

 the diminution of pressure under otherwise similar circumstances is 

 directly proportional to the living force of the moving particles of 

 liquid. Therefore if the latter exceeds a certain limit the pressure 

 must, in fact, become negative, and the liquid tears asunder. At such 

 a place the accelerating force, which is proportional to the differential 

 quotient of the pressure, is evidently discontinuous, and thus the con- 

 dition is fulfilled which is necessary in order to bring about a discon- 

 tinuous motion of the liquid. The movement of the liquid past auy 

 such place can now take place only by the formation from that point 

 onward of a surface of discontinuity. 



The velocity that will cause the tearing asunder of the liquid is that 

 which the liquid would assume when it flows into empty space under 

 the pressure that the liquid would have at rest at the point iu ques- 

 tion. This is indeed a relatively considerable velocity; but it is to be 

 remarked that if liquids flow continuously like electricity the velocity 

 at every sharp edge around which the current bends must be infinitely 

 great.* Thence it follows that at every geometrically perfect sharp edge 

 past which liquids flow, even for the most moderate velocity of the rest of 

 the liquid, it must be torn asunder and form a surface of discontinuity. 

 On the other hand, for imperfectly somewhat rounded edges such phe- 

 nomena first occur for certain larger velocities. Pointed protuber- 

 ances on the surface of a canal through which a current flows will have 

 similar effects. 



As concerns gases, the same circumstance occurs as with liquids, only 

 with this difference, — that the living force of the motion of a particle is 

 not directly proportional to the diminution of the pressure (p); but 

 taking into consideration the cooling of the air by its expansion the 



living force is proportional to the diminution of p m , where m=l— — 



and y is the ratio of the specific heat at constant pressure to that for 

 constant volume. For atmospheric air the exponent m has the value 

 0.291. Since this is positive and real, therefore p m , like p, for high 

 values of the velocity can only diminish to zero and not become negative. 

 It would be otherwise if gases simply followed the law of Mariotte and 

 experienced no change of temperature. Then instead of p m the quan- 

 tity log p would occur, which can become negative and infinite without 



*At the very small distance p from a sharp edge -whose surfaces meet each other 



7T — (X 



at the angle a the velocities will be infinite, or as p — m , where m- 



2n-a 



