104 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



depth, provided that the depth of the lower stratum in the image varies 

 in the same ratio as the remaining linear dimensions of the waves. 



The velocity of propagation of such waves in shallow water depends 

 on the depth of the water. For water waves of slight height and with- 

 out wind it can be computed by well-known formula 1 . When we indi- 



cate the depth of the water by // and put «=""-, then is 



A 



which for h=& becomes 



f2 _g e^—e"" 

 ~n ' e n7l +e "" 



2 _ ff _ U X 



b* = • / = 



H l7t 



and for small values of /* becomes 



b 2 =c/h 



When however the depth of the water is not small relatively to the 

 wave length, then the retardation is unimportant, thus for 



= the speed of propagation diminishes as 1:0.95768 



A 2 



1 

 = 4 



=- the speed of propagation diminishes as 1:0.80978 



= the speed of propagation diminishes as 1:0.39427 



When it is calm at the earth's surface the wind beneath the trough of 

 the aerial billow is opposed to the directiou of propagation, but un- 

 der the summit of the billow it has the same direction as that. Since 

 the amplitudes at the earth's surface are diminished in the proportion 

 e- nh : 1 with respect to the amplitudes at the upper surface, therefore 

 these latter variations can only make themselves felt below when the 

 depth is notably smaller than the wave-length. Variations of baro- 

 metric pressure are only to be expected when decided changes in the 

 wind are noticed during the transit of the wave. 



VII. FUNDAMENTAL FORMULAE FOR THE COMPUTATION. 



1 will here give the theory of the calculation only so far as is neces- 

 sary, so that any investigator familiar with analytical methods can 

 verify my results. I introduce two new variables, ?/ and 0, which are so 

 connected with rectangular coordinates x and y that 



e»(»+*')=a[co8 (0+»t»)-cos e] (1) 



wherein n, a, and e are constants. The boundary line between the two 

 fluids corresponds to a constant positive value of rf, namely : 



