62 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



Motions. The mathematical difficulties of this problem however can 

 be overcome only in a few of the simpler cases. In many other cases, 

 however, one can from the above given method of consideration of this 

 matter at least draw conclusions as to the general nature of the varia- 

 tions that occur. 



Especially is it to be mentioned that in accordance wifth the laws 

 established for vortex motions, the vortex filaments and with them the 

 vortex sheets in the interior of a frictionless liquid can neither originate 

 nor disappear, but rather each vortex filament must retain perma- 

 nently the same constant moment of rotation ; furthermore that the 

 vortex filaments themselves advance along the vortex sheet with a 

 velocity that is the mean of the two velocities existing on the two sides 

 of the discontinuous surface. Thence it follows that a surface of dis- 

 continuity can only elongate in the direction toivards which the stronger of 

 the two currents that meet in it is directed. 



I have first sought to find examples of permanent discontinuous sur- 

 faces in steady currents, for which the integration can be executed, in 

 order thereby to prove whether the theory gives forms of currents that 

 correspond to experience better than when we disregard the discon- 

 tinuity of motion. If a surface of discontinuity that separates quiet 

 and moving water from each other is to remain stationary, then along 

 this surface the pressure within the moving layer must be the same as 

 in the quiet layer, whence it follows that the tangential velocity of the 

 particles of liquid must be constant throughout the whole extent of the 

 surface; equally so must the deusity of the fictitious vortex filament 

 be constant. The beginning and end of such a surface can only lie on 

 the boundary of the inclosure or at infinity. Where the former alter- 

 native is the case they must be tangent to the wall of the inclosure, 

 assuming that the latter is continuously curved, because the compo- 

 nent-velocity normal to the wall of the inclosure must be zero. 



Moreover the stationary forms of the discontinuous surface are dis- 

 tinguished, as experiment and theory agree in showing, by a remarkably 

 high degree of variability under the slightest perturbations, so that to 

 a certain extent they behave similarly to bodies in unstable equili- 

 brium. The astonishing sensitiveness to sound waves of a cylindrical 

 jet of air impregnated with smoke has already been described by Tyn- 

 dall ; I have confirmed this observation. This is evidently a peculiarity 

 of surfaces of discontinuity that is of the greatest importance in oper- 

 ating sonorous pipes. 



Theory allows us to recognize that in general wherever an irregularity 

 is formed on the surface of an otherwise stationary jet, this must lead to 

 a progressive spiral unrolling of the corresponding portion of the sur- 

 face, which portion, moreover, slides along the jet. This tendency to- 

 wards spiral unrolling at every disturbance is moreover easy to see in 

 the observed jets. According to the theory a prismatic or cylindrical 

 jet can be indefinitely long. In fact however such an one can not be 



