METEOROLOGY. 339 



surfaces of discontinuity. (2) To examine the stability or instability of this 

 equilibrium and the laws of the disturbances originating in and propagating 

 along these surfaces. 



Problem (1) has already been solved to a certain extent by Helmholtz 

 and Margules. 



Problem (2) demands the development of a general view, of which the 

 theory of hquid boundary waves is the simplest example. This problem may 

 therefore be solved by successive generalizations of this simplest wave prob- 

 lem, which is too narrow, especially in the following three respects : that the 

 fluid strata above and below the boundary surface have generally been sup- 

 posed to be incompressible; secondly, that these strata have generally been 

 supposed to have no motion in horizontal directions; and thirdly, that the 

 period of the waves has been supposed to be so short that the influence of the 

 earth's rotation becomes insignificant. Three corresponding generalizations 

 of the elementary wave theory will therefore be of fundamental importance, 

 namely : 



(1) To take into consideration the compressibiUty of the fluid strata. 



(2) Retaining the compressibility, to take into consideration the motion 

 of the strata in horizontal directions. 



(3) Retaining the compressibility and the horizontal motion of the strata, 

 to take into consideration the influence of the earth's rotation upon the waves. 



The solution of problems (1) and (2) has been found in satisfactory 

 generaUty, i. e., for waves of the "long" type (wave-lengths great compared 

 with depth of the strata in which they propagate) . The results obtained are of 

 great simplicity. 



A paper concerning problem (1) is in print and another paper dealing with 

 problem (2) is being prepared. 



Certain integrals of the hydrodynamic equations showing the influence 

 of the earth's rotation upon wave-motion have also been found. These 

 solutions make evident a marked difference in the character of the waves 

 according to whether their period is smaller or greater than that of a revolution 

 in the circle of inertia, i. e., smaller or greater than 15 hours at the latitude 

 of 60°. Below this limit these solutions give waves of the permanent type 

 propagating with invariable amplitudes, while above the limit these waves 

 propagate with exponentially increasing amplitudes, indicating instability of 

 the surface of discontinuity for disturbances of this period and a corresponding 

 spontaneous formation of the waves. These solutions arc, however, not of 

 sufficient generality to be immediately applied to the cyclonic waves, but 

 they seem to indicate that the formation of cyclones may be a spontaneous 

 process. The entire clearing up of this question can be gained, however, 

 only when more general integrals of the hydrodynamic equations have been 

 found. 



