26 Proceedings of Royal Society of Edinburgh. [sess. 
e being the base of the natural logarithms. Suppose that in the 
graded action the total quantity of material in the given volume is, 
in chemical units, A, and that at the beginning of the reaction no 
B or C is present. Let there be at the time t } x untransformed, y 
in the intermediate state, and z in the final state ; then 
x+y+z= A. 
Let further the velocity constant of A->B be m, and that of B->C 
be n. At the time t , then, we have for the rate of diminution of x, 
dx 
and for the rate of increase of z 
dz 
dt 
— ny . 
Eliminating x and y from these equations by means of the relations 
dy 
d 2 z dy 
/Yi 
dt =mx- ny and^-n^. 
we obtain 
y72» Jy 
jp + (m + n) a dt + mn(z-A)*= 0, 
whence, by treating 2-A as the variable, we have 
z - A = C 1 e~ na + C. 2 e ~ nt . 
To determine the constants C x and C 2 we have z — 0 when t = 0, 
1 dz 
and also y = - • -r, = 0 when t = 0. We accordingly obtain 
n at 
nA mA 
Cj = — — - and Co = 
m-n 
so that 
2 n-m' 
m 
z — A( 1 + -e~ mt + e~ nt ) . 
' m-n n — nm. ' 
n — m 
We have here, then, an expression for z in terms of the time and 
the velocity constants of the separate actions. Suppose one of 
