810 
regularity measured by the probable error 
P and at the interval ¢ a probable error p, 
it seems justifiable to say that the interval 
tis 5 times as difficult as 7. If 7 is the 
natural interval selected by the subject, 
then the artificial interval ¢ would be more 
difficult than 7, and we should measure the 
difficulty by comparing probable errors. 
Tt is now possible to state with some 
definiteness the law of difficulty for free 
rhythmic action. Let 7 be the natural 
period and let its probable error—that is, 
its difficulty—be P. It has already been 
observed (Science, 1896, N.5S., IV., 535), 
that any other larger or smaller period 
(slower or faster beating) will be more diffi- 
cult than the natural one and will have a 
larger probable error. Thus any interval ¢ 
will have a probable error p which is greater 
than P, regardless of whether ¢ is larger or 
smaller than 7. 
Three years ago (SciENcE, as above) I 
promised a complete expression for this 
law. Continued observations during this 
time enable me to give an idea of its general 
form. The results observed can be fairly 
well expressed by the law 
p=P(i+eE5")) 
in which 7 is the natural period, P the 
probable error for 7, ¢ any arbitrary period, 
p the probable error for ¢ and ¢ a personal 
constant. 
This may be called the law of difficulty in 
free rhythmic action. A curve expressing 
the equation for 7’'=1.0°, P=0.02’ and 
¢ = 1 is given in the figure. 
It will be noticed that periods differing 
but little from the natural one are not 
much more difficult and that the difficulty 
increases more rapidly for smaller than for 
larger periods. 
In plotting this curve I have assumed 
unity as the value for all personal con- 
SCIENCE. 
[N. 8. Von. X. No. 257. 
stants. The personal constants will un- 
doubtedly vary for different pérsons, for 
different occasions and for different forms 
08 
06 
OF 
02 
Si 10 1S 20 
of action; an investigation is now in prog- 
ress with the object of determining some 
of them. 
In case it is desired to know what peri- 
ods are of a difficulty 2, 3, ---, n times that 
of 7, a table of values for p may be drawn 
up in the usual way and that value for ¢ 
sought for (with interpolation) which gives 
for p a value 2, 3, ---, n times as great. 
Thus, in a table for the above example it is 
found that the periods 0.38° and 2.6° are 
twice as difficult. 
This law can be stated in another form 
which is of special interest to the psychol- 
ogist. To the person beating-time a period 
of 0 is just as far removed from his natural 
period as one of o ; both are infinitely im- 
possible. The objective scale does not ex- 
press this fact; objectively a period of 0 is 
as different from a period of 1° as a period 
of 2° would be. Similar considerations hold 
good for the lesser periods; the scale by 
which the mind estimates periods is differ- 
ent from their objective scale. This differ- 
ence may be expressed by asserting that the 
following relations exist between the two: 
LG = T)? 
