PAPER BY PROF. OBERBECK. 157 



and furthermore put 



f^P+kcp-XW; 

 f 2 =lcW+\<p 



(12) 



we thus obtain 



?A 4. Ill = _c ( A^-iiH 

 Dx dy ' ' \ dy dx J 



dy dx~ V *p Jy y 



According to the equatious (la) (1ft) and (1) the functions cp aud W 

 must for the outer region satisfy the partial differential equations 



Jcp=0 . . (14a) 



but for the inner region the equation 



- A(p——c (14ft) 



aud for both regions 



JW=C (15) 



where we have used the abbreviation A for 



• dx 2 t ty 2 



For regions of pure horizontal motions solutions of these equations can 

 be given of great generality, which will now be separately treated of. 



IV. ATMOSPHERIC CURRENTS IN REGIONS OF PURE HORIZONTAL 



MOTION. 



When in accord with the assumption of purely horizontal motions we 

 have J(p=Q throughout the whole region under consideration, theu we 

 can also put Z=0. In this case we can satisfy equation (13) if we put 



/,== constant, / 2 == constant. 



The second of these equations gives 



W=-\cp (16) 



in which an arbitrary constant can be omitted. Then from the first of 

 these equations, namely, for/j, there results 



P= constant -Tc<pfl+ 1 ^\ (17) 



