118 Mr. H. Jeffreys on Periodic 



If, then, V 2 ' be the value of V 2 over the sea, we shall get 

 the values of F', <!>', and V 2 ' simply by putting the suffix 

 1 for throughout in the above formulae. All of the 

 exponents are unaltered. We therefore see that V 2 can be 

 put in the form 



Y 2 = A.e~ KZ + 



B,-- + C.exp(-^< V /^)' • ( 16 > 



where B and C have different values according as the point 

 considered is over land or sea, but A is the same in both 

 cases. 



Now it is obvious that if the temperature of the air were 

 a function of the height and time alone, so that all the sur- 

 faces of equal temperature were horizontal, there could be 

 no horizontal variation of pressure and consequently no hori- 

 zontal movement of air. Without altering the effect of 

 temperature on horizontal movement we can therefore sub- 

 tract from each of A, B, and C their mean values over the 

 surface of the earth. Then, except near the coast, we can 

 treat A as zero, while B and C have equal and opposite 

 values over land and over sea. 



Even if we include the effect of the coast, it is evident 

 that both B and C can be expanded over the whole plane in 

 sines and cosines of multiples of nrx/l. 



As henceforth we shall be considering only the -atmo- 

 sphere, the suffix 2 can now be omitted. 



O V IT OC 



The solution of -*—- = k\7 2 Y that contains sin — r— as a 

 factor and vanishes when z is + co is 



<?~ TOZ sin ttxJI, 

 where m is the solution of 



2 



?n 2 = -~ + -p- that has a positive real part. . (17) 



We shall consider the motion of the air that is produced 

 when we have 



Ye'^ = be~ vz sin wa/l + ce' 1 ^ sin iraf/l, . . (18) 



where b and c are constants. 



The solution for all other terms can be derived from those 

 for these two. 



