392 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 51 



Theorems similar to those just expressed for the whole year can 

 also be established for shorter intervals of time, some of which may 

 here be given. 



IV. "The quantities of heat that are within special portions of 

 a year given to and lost by special portions of the earth's surface or 

 the atmosphere are in general not equal." 



The proof of this theorem consists in the simple fact that the 

 thermal condition of the earth's surface and the atmosphere is sub- 

 ject to periodical variations, that is to say, this is simply an expres- 

 sion of the fact that there occur times of excess of insolation and 

 other times of excess of radiation. 



By use of the notation, introduced above, this theorem takes the 

 following form ■ 



%,r =<Ct (13) 



or also 



> 



; 



q T do = q 0/r = q aT - q 0iT == 



where the integral is to be taken over the whole closed area a and 

 where the omission of the accents over q z and q aT indicates that the 

 quantities are to be considered as algebraic and can therefore have 

 correspondingly different signs. 



If we consider not a closed surface but only a definite portion of 

 the boundary surface of the atmosphere then we have the expression 



Qa.T = <\a,T (14) 



In this expression we consider the case in which the inequality 

 > or< becomes an equality =, as an exceptional case and in general 



we have to use the sign ^. 



If q aT >o then this expresses the fact that there is an excess of 

 heat added over that abstracted; if q 0>T <o then this expresses 

 the deficiency that has been experienced by the masses contained 

 within the volume a during the interval x from t x to t 2 . 



We can also write 



iV-iV = v,-V < 15 ) 



When this difference has a positive sign it indicates an increase 



