340 MISCELLANEOUS STUDIES 



Since the solar radiation consists of a series of waves of varying 

 length, each having a different coefficient of transmission (Murray, 

 1912, p. 248, Kriimmel, 1907, p. 263), the use of a single average value 

 is only a simple approximation to the true relation. 



No analysis will be attempted of the complex way in which the 

 heat in an element of volume, specified by given values of y and L, 

 is lost. This loss depends upon evaporation at the surface and on 

 the mixing process at all depths, and the rate of evaporation increases 

 as the temperature increases. Also heat tends to flow from regions 



Fig. 1. 



of high temperature to those of low temperature (Gehrke, 1910, p. 68). 

 It seems reasonable to suppose, therefore, that the rate of loss would be 

 greater, the greater the temperature. 



Although the precise manner in which the rate of loss of heat 

 depends upon the temperature is not known, some definite form of 

 relation must be assumed in order to formulate th'e temperature 

 problem mathematically. For simplicity assume the rate of loss at 

 any depth to be proportional to (6 0J at that depth, where t is a 

 function of the depth and latitude only. Consider now the time rate 

 at which heat is gained and lost in a given rectangular element of 

 volume of unit cross section and thickness dy whose upper surface is 

 at the depth y (fig. 1). 



The rate of change of heat in this volume element is evidently 



t (1) 



since the volume specific heat multiplied by the volume of the element 

 equals the change in the amount of heat per degree change of tem- 

 perature. 



