Avcust 16, 1918] 
portional to the times required to bring the 
reacton to the same stage. 
This holds not only for reactions of the first 
order (where a single substance decomposes) 
but for reactions of higher orders (where two 
or more substances combine) as well as for 
consecutive reactions? and autocatalysis.* 
It follows that when a chemical process 
proceeds at different rates under different 
conditions, we can compare the velocity con- 
stants by simply taking the reciprocals of the 
times required to bring the reaction to the 
same stage. If we merely wish to know the 
relative rates (as is usually the case in biol- 
ogy) it is not necessary to determine the 
velocity constants at all. 
Whenever the initial conditions are the 
same with respect to concentration we need 
only compare the times required for equal 
amounts of work, since these bring the re- 
action to the same stage. 
Tf on the other hand one attempts to arrive 
at the relative rate by comparing the amounts 
of work performed in equal times (as is fre- 
quently done in biological research) he can 
easily fall into serious error. This is evi- 
dent from Fig. 1, which shows the curves of a 
reaction proceeding at two different rates, the 
velocity constant of B being twice as great 
as that of A. It is evident that the abscissa 
of A at any point is just twice that of B while 
no such relation obtains among the ordinates.* 
For example at the point C the ordinate of 
B is twice as great as that of A, while at the 
point D it is only 1.1 times that of B. Hence 
it is evident that we should compare abscisse 
rather than ordinates (i. e., times required to 
2 The principle holds for consecutive reactions in 
ease all the constants are multiplied by the same 
factor, otherwise not. Cf. Osterhout, W. J. V., 
Jour, Biol. Chem., 32: 23, 1917 
3 Cf. Mellor, J. W., ‘‘Chemical Statics and Dy- 
namies,’’ p, 291, 1909. 
4 We can not avoid the difficulty by comparing 
the rates of the two processes at a given time; for 
the rates so obtained will bear no constant Yatio to 
each other. Only when they are compared at the 
same stage of the reaction will they show a con- 
stant relation; this gives the relation between the 
velocity constants. 
SCIENCE 
173 
do equal amounts of work rather than amounts 
of work performed in equal times). 
This principle will also be found to apply 
to a variety of physical processes. 
The principle is sufficiently obvious where 
successive determinations are made and curves 
are drawn. But there is a common type of 
AMT. 
fe i re 
c D 
Fie. 1. Curves showing the same process pro- 
ceeding at different rates one of which, B, is twice 
as rapid as the other, A. 
experimentation in which, for various reasons, 
a single observation at one rate is compared 
with a single observation at another rate. 
The principle in question is then easily over- 
looked. In some eases this leads to serious 
errors. 
If we wish to compare the normal rate of 
a biological process with an abnormal rate 
(e. g., under the influence of a reagent) it is 
evident that we can use this principle, but 
the method of application will depend on cir- 
cumstances. The normal rate may be con- 
stant and its graph a straight line. If this 
is also true of the abnormal rate it will make 
no difference whether we compare times or 
amounts of work. 
When the abnormal rate is variable we may 
have the condition shown in Fig. 2. The 
normal rate FE is constant: the variable ab- 
normal rate F at any point such as H may be 
determined by drawing the tangent at that 
point and taking the ratio J ~K. 
In many eases it is not possible to secure 
data for drawing directly such a curve as that 
shown in Fig. 2. We may, however, deter- 
