TitANgACJttOJfS OT^ SfiCTIO^T A. 607 



noticed loug ago by Franklin. On the otlier hand, such waves are easily produced, 

 owinw to the smallnees of the potential energy. An interesting application of the 

 theory has in recent times been made by VV, V. Ekman, to explain the abnormal 

 resistance which ia occasionally experienced by ships off the mouths of various 

 Norwegian fiords, where a layer of fresh water rests on one of salt water. This 

 is interpreted as a wave-resistance due to waves generated on the common 

 boundary. If p' is greater than p, the wave-velocity is imaginary: there is then of 

 course instability, and a corrugation of wave-length A, once produced, will tend to 

 increase in amplitude, and the more rapidly the smaller the value of X. This has 

 a bearing on the theory of the formation of cirrus clouds, which was pr'oposed and 

 experimentally illustrated by Jevons in 1857. An upper layer of air is supposed 

 to cool by radiation until it becomes heavier than the air below it ; a sort of 

 interpenetration then takes place by means of narrow filaments which are made 

 manifest by the condensation of vapour when the warmer air comes in contact 

 with the cooler. 



In practice a discontinuity of density in the atmosphere is likely to be accom- 

 panied by a discontinuity of velocity. The corresponding circumstances in the 

 case of incompressible fluids were investigated by Lord Kelvin; if the velocities of 

 the two currents are v, v' , the formula for the wave- velocity may be written : — ■ 



p+p' ^V i^TT p+p ip+p'T^ ' )' 



or, in case the difference of density is relatively small, 



v=i(^'+^')±Ay{-£ • p^-\{v-v'f\. 



The fiist term represents the mean velocity of the two currents; the second 

 represents a wave-velocity which may be superposed on this in either direction. 

 This second term is imaginary, indicating instability, for values of \ less than 



2np {v-v'Y 



p-p'' 2ff ' 



Under atmospheric conditions very great wave-lengths will be required for 

 stability, unless v — v' is very small. The importance of the fact that with an 

 absolule discontinuity of density the slightest difference of velocity. would imply 

 partial instability has been strongly emphasised by Helmholtz. He pointed out 

 that in the case of air a rapid mixture of the two strata must take place, and 

 that this cause is far more powerful than friction in establishing a transition 

 stratum within which the change of density from p to p' takes place gradually. 

 The presence of such a layer would modify the theory to some extent, but this has 

 hardly yet been investigated. 



The theoretical results so f\ir quoted relate to disturbances of infinitely small 

 amplitude ; in cases of instability they represent therefore only the initial 

 tendency. Helmholtz has shown that although a plane surface of separation 

 between two currents is unstable, there are other forms of relative equilibrium 

 which are possibly stable. Since any common velocity may be superposed on 

 the two currents without affecting the dynamics of the question, this amounts 

 to a theory of waves of finite (as distinguished from infinitely small) amplitude. 

 The calculations, which are somewhat complicated, have been revised by W. Wien ; 

 but it does not appear yet to have been decisively settled whether, or in what 

 Sense, such waves may be stable. It is very desirable that this very interesting 

 and definite problem should be attacked de novo by some mathematician. 

 Helmholtz argues that the theory has a real application to the atmosphere, and 

 that the crests of these waves are under favourable conditions made visible by 

 the condensation of vapour cooled by expansion. 



It will be evident from this review that the theory of wave-motion is, from 

 the point of view of atmospheric phenomena, on many points uncertain. It 

 may perhaps be urged that it would be unfair to expect any great improvement 



