flows up from the water below. Hence the first boundary condition is 



The other boundary conditions are: 



Ti = T s = C x at x * 0, (7) 



T x = T 2 « at x - £ , and (8) 



T 2 - C 2 at x * «. (9) 



There are also three other boundary conditions derived from the fact 

 that when t = 0, f is fixed, while T^ and T2 must be given as functions 

 of x. Ti lies between and £ while T2 lies between <f and « . As 

 equation (6), containing the unknown function ^ , is not linear and 

 homogeneous, a solution cannot be reached by the combination of special 

 solutions. The method of solution then will be to find particular inte- 

 grals of equations (l) and (2) and after modifying them to fit boundary 

 conditions equations (7), (8), and (9) to find under what conditions the 

 solution will satisfy equation (6). This will also determine the initial 

 values of f , T]_ and T 2 . 



Now the function <£>(x,i7)(the probability integral) is a solution of 

 such differential equations as (l) and (2). Consequently if B]^, D]_, B 2 and 

 D2, are constants and 77 = J- /VT and y = k V a z t , 



T| = B, + D, $ ( x , ??, ) (10) 



and 



T 2 =B 2 +D 2 ^ (x,%) (ID 



are also solutions. Boundary condition (8) means that ^v?* 7 ?!' and 

 <!>(£, »7 2 ) must each be constant, which will be true if £=0j> £ ««, 

 or if £ is proportional to v ftT . The first two of these assumptions 

 are evidently inconsistent with (8). Thus, there remains only the last 

 which may be put into the form 



C = b^ (12) 



