352 MISCELLANEOUS STUDIES 



From equations (41) and (42) 



e>i = |4?r =1-168 or 6, = .039 



fi* I U 



and 



flje- 36 ' e- 71 *]=R 1 (.1285) = A5 or #, = 3.50. 



From equation (43) and the value, 3.5, already found for R 1 



e-** er' r * = = = .1142 and & 1 = .034 

 o.D 



which agrees approximately with the value .039 obtained from the 

 first equation. The average value .0365 should be used instead of the 

 large value of b l exceeding .12 (p. 351), in order to obtain the actual 

 rate at which the water gains heat as a result of both absorption and 

 mixture of water from other layers. That is, the rate of gain of 

 heat in the actual system takes place as if there were no such mixture 

 of the water and the coefficient of absorption were less than the true 

 value. Hence, as far as the rate of gain and loss of heat is concerned 

 we can substitute this more simple ideal system for the actual one. 

 We have now determined all of the constants of the original differential 

 equation (4), on page 341, which expresses the time rate of change of 

 heat in an element of volume, on the assumption that the average 

 flow, either vertically or horizontally, is zero. The modified tempera- 

 ture resulting from any additional factor, for example a current, can 

 be deduced by solving the above differential equation, to which has 

 been added the rate of change of heat due to this factor. 



DEDUCTION OF THE CHANGE IN SURFACE TEMPERATURE PRODUCED 

 BY A HORIZONTAL FLOW OF WATER 



Preliminary discussion. Statement of assumptions and mathematical 

 formulation of the problem. 



It is well known, as stated by a prominent British hydrographer, 

 Wharton (1894, pp. 699-712), that "the most obvious phenomenon of 

 the ocean is the constant horizontal movement of its surface water, 

 which in many parts takes well defined directions. ' ' 



The work of both practical seamen and scientists has after many 

 years revealed the essential features of the main ocean currents, and 

 in a few limited regions a fairly detailed knowledge of the currents 

 has been obtained. However, all investigators agree that the esti- 



