342 MISCELLANEOUS STUDIES 



where the constant A' g A',, since the amount of energy used cannot 

 exceed the amount available. Equation (4) then becomes 



..) (7) 

 Let 



where B is a constant and 



& = -* (9) 



where ft, is the absorption coefficient (Kriimmel, 1907, p. 263). 

 Equation (7) then reduces to the ordinary linear differential equation 



- + kO=B[(a l +o 2 x) cosaf + asZ + lJe-^ + fc^ (10) 

 Therefore 



(11) 



where F(x,y) is an arbitrary function of x and y. Integrating 

 equation (11) gives 



. asm at -\-kcos at . Be~^ v (l-fa,a-) 

 -a 2 x) - --- \- - --- \-0 l 



(12) 

 which can be readily transformed into 



v \suty yi**' / i^ w I I 



Va 2 -f fc 2 k } 



where 



tan = - (14) 



k 



and only the periodic part of the integral is retained. 



If X is assumed to be independent of JT the latitude gradient g 

 of the mean annual temperature is, from equation (13), 



. (15) 



Therefore, since g is independent of y (sixth statement, p. 339) 



k = k l e~^ (16) 



where fc, is a constant. That is, 



g=2 



