632 PROCEEDINGS OF THE AMERICAN ACADEMY. 



stirring, at the concentration U , so that equation (5) must be solved 

 subject to the condition u = U when x = for all values of t, and u = 

 for all positive values of x at the time t = 0. Equation (6) has for 

 initial values v ■= V for all positive values and v = for all negative 

 values of x when t = 0. The theory of differential equations of this type 

 teaches us that if we can obtain any one solution for each of these equa- 

 tions that satisfies its boundary conditions it is the only solution. The 

 following are unique solutions of (5) and (6) subject to these conditions : 



u = — = I 



-/3 3 





dfl* (7) 



«4fWt (8) 



Siy/t 



in which /? is merely a variable of integration. 



These may be seen to be solutions of (5) and (6) respectively if we 

 differentiate them according to the rules for the differentiation of a defi- 

 nite integral and substitute the results in (5) and (6). Equation (7) is 

 also seen to satisfy the initial conditions, for if t = and x is positive, the 

 lower limit of integration becomes + oo , which is the same as the upper 

 limit ; the integral is therefore and u =■ 0. If x = the lower limit 



is and the value of the integral is 'XJL 



2 



.-. u = U 



Similar reasoning shows equation (8) to satisfy initial conditions for v. 



The metastable product H, as we have defined it on page 8 (Eq. 2), 

 should be given by the following equation : 



It is our problem to ascertain whether II is a constant for several 

 initial concentrations of U and V . 



If we knew the diffusion constants a 2 and b~ we could expand the in- 

 tegrals in (7) and (8), integrate term by term for a given value of x and 

 t, and thus obtain u and v and from them H. This would necessitate a 

 previous independent determination of the diffusion constants, and would 



* Fourier, Th. an. de Chaleur, § 366 (1822). 



t Fourier, loc. cit. j Stefan, Wiener Sitzungsber., 79, II. 176 (1879). 



