OCEAN TEMPERATURES 361 



and t 1 is the average of the values of t for the beginning and end of 

 the time interval. Let 6 = 6' -f- 6" where 6' is the solution when 

 H l = 0. Then 



>.) (69) 



i 



Substituting (6'-\-6") inequation (68) gives 



de " * d0 ' 



(70) 



For & use the normal value determined from the expression j 



of equation (59) for t = t 1 then for surface temperatures using the 

 value 6 for y (see page 343) 



60' 



where tan c =7- 



k 



This is consistent with equation (69) since X may be any function of z. 

 Equation (70) then becomes 



Integrating equation (71) gives 



_ fc(z z n ) a TJ 

 0" = Oe-^^- 1 -^ (72) 



where is arbitrary. Adding the two solutions ff and 6" gives 



_ / ^ , , ^z ) Ba*m n , B 

 cos ( a # e) + - -- h' i -- rr 



(73) 



where the expression j Ms the normal temperature. To deter- 



mine the temperature at a given position, distant (z z ) from the 

 initial position z , give 0. such a value that the expression for in 

 equation (73) will reduce to the given temperature when z = z . Then 

 substitute this value and the given value of z in equation (73). 



