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CAJJ.J.U.    J.VJJ.      IUJLO     JJJ. J.J.J.UJ-|JCUI.      v j.nji.i^uJ.v^J.J.      ui.\j    tuigji^iii^iiu     ^J.     uiivy    VJVUJ.J.J--/    j.u      u-J      r^v^wv^..ii~     >^
when y = \l>.    Hence
If cos #*                                    /t)QN
'        ...................... (28)
Also           <f> =  wdz = Wz + /3~l If sin $z . cos [j3y \/( W*/a2 - 1)},
1   |YcZ</>\
= -.    -
dz - --------
-.     -r          - --------  -
W) \dyjy,                   /3W
so that the retardation is greatest at the places where v\ is least, that is where the jet is narrowest. This is in agreement with observation, since the places of maximum retardation act after the manner of a convex lens. Although a complete theory of the optical effects in the case of a symmetrical jet is lacking, there seems no reason to question Emden's opinion that they are natural consequences of the constitution of the jet.
But although many features are more or less perfectly explained, we are far from anything like a complete mathematical theory of the jet escaping from high pressure, even in the simplest case. A preliminary question is  are we justified at all in assuming the adiabatic law as approximately governing the expansions throughout ? Is there anything like the " bore " which forms in front of a bullet advancing with a velocity exceeding that of sound ? * It seems that the latter question may be answered in the negative, since here the passage of air is always from a greater to a less pressure, so that the application of the adiabatic law is justified. The conditions appear to be simplest if we suppose the nozzle to end in a parallel part within which the motion may be uniform and the velocity that of sound. But even then there seems to be no reason to suppose that this state of things terminates exactly at the plane of the mouth. As the issuing gas becomes free from the constraining influence of the nozzle walls, it must begin to expand, the pressure at the boundary suddenly falling to that of the environment. Subsequently vibrations must set in; but the circumstances are not precisely those of Prandtl's calculation, inasmuch as the variable part of the velocity is not small in comparison with the difference between the mean velocity and that of sound. It is scarcely necessary to call attention to the violence of the assumption that viscosity may be neglected when a jet moves with high velocity through quiescent air.
* Proc. Roy. Soc. A, Yol. LXXXIV. p. 247 (1910); Scientific Papers, Vol. v. Art. 346, p. 608.tionary in space only when the stream moves with' the velocity of sound; but here the motion is not limited to one dimension, as is shown by the swellings between the disks. Indeed the propagation of any wave at all is inconsistent with uniformity of pressure within the jet.l as of the particles of fluid with each other, is more directly shewn by an experiment on the continuance of a column of mercury, in the tube of a barometer, at a height considerably greater than that at which it usually stands, on account oi the pressure of the atmosphere. If the mercury has been well boiled in the tube, it may be made to remain in contact with the closed end, at the height of 70 inches or more " (Young's Lectures, p. 626,1807). If the mercury be wet, boiling may be dispensed with and negative pressures of two atmospheres are easily demonstrated.