544 Dr. Hans Geiger on the Irregularities in 
with the period of the substance) is given by XtN, the 
number of atoms still unchanged after the time r is given by 
(1 — \t)N. From this it follows that the probability of a 
single atom breaking up during the time t is \r, while the 
probability that the same atom will exist after that time is 
1 — \t. Hence the absolute average error is 
e=+ v /N\T(l-\T) J 
or, neglecting the square of Xr compared with Xt itself, the 
error is 
e=+ VNXt or ± VZ, 
where Z is the number of atoms disintegrating during the 
time t. This result was first deduced by E. v. Schweidler 
(loc. cit.) in a similar manner. 
According to the simple radioactive theory, the average 
number of atoms breaking up during the time r is given by 
Z = NXt. 
r 
The actual number observed may show a deviation from 
this, or an average error equal to the square root of the 
number of atoms breaking up during the time t. The 
absolute average error increases therefore with the number 
of atoms breaking up, i. e. with the intensity of the radiation ; 
while the relative error ^4- decreases. The movement of 
the needle of the electrometer registers the absolute error. 
The correctness of this theoretical conclusion may be 
tested as follows : — Rutherford has shown that one a particle 
from radium itself produces 80,000 ions in its path of 
3*5 cms. in air at atmospheric pressure ; while Durack has 
found that each of the swifter /3 particles from radium 
expelled at a speed approaching that of light makes a new 
pair of ions in every 6 cms. of air at 1 mm. pressure. Conse- 
quently, in the cylinders A and B which were about 12 cms. 
in length one /3 particle will produce at atmospheric pressure 
4 x 760 or about 3000 ions. Therefore, in order to produce 
the same ionization current with /9 particles, about 25 times 
as many /3 particles as a particles are necessary. Hence the 
average error in the number of /3 particles shot out is 
6=±x/25Z. 
The average error measured by the electrometer is 
e = ± 5 Vv / 25Z, 
as one ft particle produces only an effect ^ of that produced 
