632 PROCEEDINGS OF THE AMERICAN ACADEMY. 



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

 subject to the condition u = Uq when a; = 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 = Vq for all positive values and v — for all negative 

 values of a; 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 : 



e^=^r%-^vA* (7) 



v = ^re-^'d(3A (8) 



A/TTt/ — J^ 



2*xA^ 



in which /8 is merely a variable of integration. 



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

 differentiate them accordins 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 ^ = and x is positive, the 

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

 limit ; the integral is therefore and «e = 0. If ic = the lower limit 



is and the value of the integral is J_ 



2 



.-. u = Uo 



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



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

 should be given by the following equation : 





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

 initial concentrations of Uq and V^^. 



If we knew the diffusion constants a'" and IP' 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, § 306 (1822). 



t Fourier, loc. cit. ; Stefan, Wiener Sitzungsber., 79, II. 17G (1879). 



