relating to Rotating Fluid in the Atmosphere. 667 



(d) The boundary equations; — which we may take to be 

 those represented by the conditions, 



w = Q, — — = 0, at z = and at z = h, 



A being the upper limit of the troposphere which we assume 

 to be a fluid or! finite depth. 



For convenience we have taken p — kp throughout; the 

 results obtained in the paper can, however, be easily modified 

 to suit other physical conditions than that of an isothermal 

 atmosphere, 



These equations are simplified to some extent by the 

 assumption which we now make that the motion of: the fluid 

 in each layer parallel to the ground at point is entirely 



horizontal. This makes w and - 1 — each zero. Moreover, in 



at 



the special cases o£ motion to be considered in the present 

 paper the terms w x and w y are also zero. The values to be 

 assigned to the terms o) x , a) y , a> z depend in part on H the 

 rotational velocity o£ the earth about its axis, on <£, the 

 latitude of the point 0, and on the motion of the point 

 relative to the earth. If we denote by (U, V) the com- 

 ponents of the velocity of point in the directions duo East 

 and due North respectively, we have 



V u 



» x = — cj>= — p-; u>y = H cos (/> + ^- ; ft> z = 12sin </>. 



V . • tr 



03 x =—^; Wy= — VL sin <j) . (f) -f p-; Q) z = nC0S(j> . (j). 



To obtain the values of (co x 2 + co/)z, (co/ + &>/]#, 

 (oo z 2 + ft) x 2 )y to be used in our equations we omit all terms 

 containing I2 2 which appear in co/ and co z 2 . These terms are 

 already allowed for in treating gravity as a force uniform in 

 direction over the whole field around point 0. That is, the 

 terms referred to are compensated for in the variation in 

 the direction of gravity around point 0, and do not affect 

 the motions now being considered. 



Case I. — The first motion of the atmosphere to which we 

 shall apply the above equations is that corresponding to a 

 uniform rotation of the atmospheric layer in contact with 

 the earth about a vertical axis through a point which is 

 at rest relative to the earth's surface. We assume that each 

 particle of fluid in any of the upper layers describes a circle 



