OCEAN TEMPERATURES 341 



The rate of gain of heat in this element of volume due to the 

 absorption of solar radiation equals the difference between the rate 

 at which the radiant energy passes in through its upper surface and 

 out through its lower surface. At the upper surface the rate is 

 QPS and at the lower surface it is Qfl^* dy . Therefore the rate of 

 gain due to absorbed radiation is 



(2) 

 since 



= (log ft 



The rate of loss of heat will be assumed to be ka(0 QJdy where 

 k is a function of y only. 



Equating the rate of change of heat in the element to the rate of 

 gain from solar radiation less the rate of loss, we have the following 

 differential equation 



(3) 

 which becomes 



_ 



ot 

 after division by ady. 



Let L = L l -f- x where L l is a standard latitude chosen arbitrarily 

 and x is the distance in degrees from this position, x is positive for 

 latitudes higher than L l and negative for lower latitudes. The function 

 f-i(L, t), (p. 339), then becomes f(x,t), which expresses the way in 

 which the radiation varies with respect to latitude and time. The 

 precise form of f(x,t] is unknown; however, estimates of the amount 

 of radiant energy available at the earth's surface made by Angot 

 (Hann, 1915, p. 40) can be closely approximated to within a ten-degree 

 interval of latitude by an expression of the form 



Q 1 = K 1 [ (a x -|- a 2 x) cos at -f- a 3 x -f- 1] (5) 



7T 



where a = , and the coefficient of cos at is negative. 

 6 



Assuming the amount of energy Q that enters the water to be pro- 

 portional to the amount available Q l 



(6) 



