26 
Poissoiis Exponential Binomial Limit 
satisfy these conditions, is tlie number of a-particles discharged per i-miuute or 
J-minute interval from a film of polonium*. 
In vital statistics the sample may be an individual or house or community and 
the event an accident or disease and so on. But it must be borne in mind that 
for such series as the above to be applicable the occurrence of one event in the 
sample must not preclude or influence in any way the occurrence of a second. 
The probability of x occurrences, m being the mean number, in a sample, is 
and in the tables which follow this is evaluated for O'l, 0 2 ... to lo'O and for 
A' = 0, 1, 2 ... up to such an integer as gives a figure in the sixth place of decimals, 
the number of places tabulated. 
The terras of the series were calculated, each by a fractional operation upon 
the preceding, beginning with the modal term and going both forward and back. 
Thus if m = 7-6 the term e"' " x (7'6)77 ! was first calculated by tables of logarithms, 
and the succeeding terms were then obtained seriatim by the operations 
H Jl H i- 
8 ' 9 ' 10 ' ' 
and the preceding ones by the operations 
X A A f 
7-6 ' 7-6 ' 7-6 ' ' 
done with a mechanical calculator, first a multiplication and then a division. 
Seven places of decimals were thus calculated and the series is checked by the 
total, which differs from unity by the remainder (a figure in the eighth or later 
place of decimals in all the present cases) and the algebraical sum of the errors of 
seventh figure approximations. 
Poisson's exponential series has been previously calculated to four places of 
decimals by L. von Bortkewitsch-f- for values of m from O'l to lO'O. 
The present tables give the probability of each number of times of occurrence 
of the event. For the sums of these values, that is, the probability of occur- 
rence of the event, a given number of times or greater, or a given number of times 
or less, reference must be made to a second paper in this issue of Biometrikal, 
where such probabilities are calculated for integral values of m from 1 to 30. 
* See Eutherford and Geiger : "The Probability Variations in the Distribution of a-Particles," 
Philosophical Magazine, Vol. xx. p. 700, 1910. See also E. C. Snow, "Note ou the Probability Varia- 
tions, &c.," Vol. XXII. p. 198, 1911, who finds the variance of experiment from theory to be such 
as would occur once in six experiments and once in three experiments respectively of the limited time 
taken, were theory exact. In a note to the first paper H. Bateman gives a proof of the exponential series 
of probabilities arrived at from considerations of this problem. 
t Das Gesetz der kleinen Zahlen, 1898. A comparison of the table printed therein with the present 
table shows agreement except as to the fourth figure ; the nearest fourth figure is not given, in rather 
many instances, in the tables of Bortkewitsch. 
J Lucy Wbitaker, B.Sc. " On the Poissou Law of Small Numbers," Vol. x. p. 37 et seq. 
