360 MISCELLANEOUS STUDIES 



which reduces to equation (58) when a 4 and a, equal zero. The 

 temperature at any time and place down stream from the position 

 where z equals z can be found if the relation of the temperature to 

 the time is known where z equals 2 , by giving the arbitrary function 

 /[ ] values such that for z equals z , e = 0'+6" + 6'" will equal 

 the observed temperature which is a known function of the time at 

 that position. Thus the arbitrary function is determined since its 

 value is known for a series of values of the independent variable 



(f --- cos at' -| -- - sin at' ) , using i' for the time where z equals Z . 

 a a / 



For any other value of the time V and for a distance (z 2 ) down 

 stream 



sin af- 



where 



f[(t ^cos * + -?*- si 



L \ a a 



=/[V -^-cosaf + -^- sinaf 1 (66) 



a a. 



F/^ 4 , -, A z z n ~] 



\ (t- -cos at +- sinafl -- =~ 



L \ a a //! I 



= (f ^-cosaf + -^-sina*'). (67) 



\ a a / 



Solution for the particular case in which the time interval is so small 

 that the solar radiation may be assumed to depend 



only on the latitude. 



In certain cases it will be convenient to take a time interval so 

 short that the insolation may be regarded as independent of the time 

 and the current may be assumed to have a constant velocity from a 

 position z equals z where the temperature may be assumed constant. 

 Under these conditions the temperature at any point distant (z z ) 

 down stream will be independent of the time if sufficient time has 

 elapsed for an element of volume of the water passing through the 

 position z and having the given constant temperature to move through 

 a distance equal to or greater than (z z ). For this steady state, 

 equation (47) becomes 



O l }Hi = (68) 



where 6 2 = 1 -j- a L cos at 1 



& 3 = a 2 cos at : -}- a 3 



