394 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 51 



appropriate method of elimination, and if furthermore u m is the 

 absolute maximum of u, and t m the moment of time at which this 

 maximum is attained, then 



dq ° > when t % ° 

 d t t m 



and 



moreover 



and 



d(]a < when t ^ *" 

 dt * 



^.T, ~ U «.«m ~ U a,c 



Qa, T2 ~~ U a,0 " u a,t 



if by T t we understand the interval of time from o to t m and by 

 r 2 the interval of time from t m to T. 

 Hence also follows 



fla.r, = ~ q a ,r 2 ( 16 ) 



that is to say, the sum total of the heat received by a given portion 

 of the earth or the atmosphere in the half year when the gain 

 exceeds the loss, is exactly equal to that which is given up during 

 the half year when the loss exceeds the gain. 



Moreover equation (16) also holds good if the year is divided into 

 any two perfectly arbitrary portions provided only that z x + r 2 = 

 T; in every case the heat that is gained in one portion must again 

 be lost in the other portion; but in any such arbitrary division 

 c \ a T cannot have a maximum. If , however, this value is a maxi- 

 mum then this must be designated as "the annual heat exchange 

 for the portion of air or earth under consideration." 



Hence it follows that "The annual heat exchange for any portion 

 of the earth or atmosphere or both is equal to the difference between 

 the maximum and the minimum quantities of energy contained 

 in such portion. " 



For shorter periods, such as the diurnal periods, this theorem 

 needs a slight modification, since in general for any such period 

 not so much is taken away during its season of diminishing heat 

 as is gained during its season of increasing heat; but less during 

 one half of the year and more during the other half. 



