Travelling Atmospheric Disturbances. 5 



this order it also may be neglected in the equations of 

 motion. Then 



~ ~dy 2(o\dy'dx 'bx'dyj I 



•and 



2 ,_ Bp 1_ /B^ B" _ ^^ B«\ | 



d^ 2&> \dy "da ~dx "dy) J 



where for w and u we must substitute their values from (4). 



So far these results are general, subject to the validity of 

 the approximations made, which seems satisfactory in ordjnary 

 cases. Further progress, however, requires some knowledge 

 of the relations connecting pressure and density with 

 position ; and when the actual laws that hold in the atmo- 

 sphere are substituted the formulae soon become unmanage- 

 able. An approximation can nevertheless be employed that 

 enables the actual conditions to be imitated without making 

 the algebra quite intractable. In the ordinary cyclone 

 below the stratosphere the difference of pressure from normal 

 does not vary greatly with the height, and it appears as if 

 the disturbance arose from a change within the stratosphere : 

 thus the conditions within the troposphere could be repre- 

 sented by a variation in the height of the free surface of an 

 incompressible fluid. At the same time there is a well 

 marked temperature gradient, the temperature usually 

 increasing to wards the south in the troposphere ; the opposite 

 seems to hold in the stratosphere*. Thus the troposphere 

 can be represented by an incompressible fluid of finite depth 

 whose temperature and therefore whose density are mono- 

 tonic functions of one of the horizontal coordinates. Let 

 the mean height of the free surface be H, and the excess of 

 the actual height of any point of the free surface above this 

 be f. Also let the density be given by p=po + p\, where p 1 

 is small and a function of x and z only. 



It is further supposed that J and p 1 are small enough for 



* W. J. Humphreys, Bull. Mt. Weather Obs. vol. ii. pp. 292-297 

 (1910). 



