456 Dr. H. Jeffreys on Periodic 



the neglected terms are small compared with that retained. 

 The largest of them is Rd/dr, and the ratio of this to 'd/'dt 

 is of the same order as that of y\ 2 U./4:co 2 to 7 ; so that the 

 neglect is justified so long as II is small compared with 

 (<w/\) 2 , which is usually true. Thus, for the annual variation 

 in Asia, II is of order 10~(cm./sec.) 2 , and (co/\) 2 about 

 10 12 (cm. /sec.) 2 , and in most cases the neglect is similarly 

 justifiable. 



Interpretation of the Results. 



The first effect of heating is that the air near the ground 

 expands and forces up that above it. On this account alone, 

 the vertical displacement in the upper air would contain 

 terms like abe^jv, and thus the free surface would change 

 in height and the pressure variation in the upper air would 

 contain a term gp^abe^jv ; there would be no pressure varia- 

 tion on the ground. This corresponds exactly to the case 

 when (4&) 2 — y 2 )/\ 2 gh is large ; if <y>2co this motion is 

 mostly radial, but if y<2co the transverse wind increases 

 in magnitude as the pressure in the upper air rises, and 

 remains always near to the geostrophic value. This trans- 

 verse wind thus helps to maintain the pressure differences. 



This ideal case only arises, however, when A, is of the 

 order 10~ 9 /lcm., corresponding to areas as large or larger 

 than Asia. For smaller areas two further factors need to 

 be considered. The uplifting of the upper air by the 

 expansion of the lower is of the nature of a tidal wave, 

 which would in the absence of rotation tend to spread out 

 with velocity \^{gh), comparable with the velocity of sound. 

 If such a wave can spread from the centre to the circum- 

 ference within a single period, the upper air in the heated 

 area has time to flow out over the lower, and the pressure in 

 the upper air may thus be very much reduced, the surface 

 pressure at the same time being caused to vary in the 

 opposite sense to the temperature. This corresponds to 

 the case when gh is large compared with y 2 /\ 2 . If the 

 rotation is rapid compared with the temperature variation, 

 the velocity of such a wave is {gJiy 2 / (4:(o 2 — y 2 )}i, and the 

 outward flow becomes important if gh is comparable with 

 (4»« T7 «)/X». 



If friction is taken into account, the outward flow is 

 much hindered, and if the friction were large enough there 

 would be no outward flow and no variation of pressure 

 on the ground. The ratio of the Motional terms in (35) to 

 the last term is, however, of order (k/icoyir 1 , which has 

 already been regarded as small, being of order 1/50, 



