OcTOBER 16, 1902] 
NATURE 
615 
other at a short interval of time, does this give us any reason to 
suppose that these two events are connected with each other, 
both being due to the same cause, or one being the cause of the 
other? Everyone admits that the simple concurrence of events 
proves nothing, but if the same combination recurs sufficiently 
often we may reasonably conclude that there is a real connection. 
The question to be decided in each case is what is ‘‘ sufficient” 
and what is ‘‘ reasonable.” Tere we must draw a distinction 
between experiment and observation. We often think it suffi- 
cient to repeat an experiment three or four times to establish a 
certain fact, but with meteorological observations the case is 
different, and it would, e.¢., prove very little if on four successive 
full moons the rainfall had been exceptionally high or excep- 
tionally low. The cause of the difference lies in the fact that in 
an experiment we can control to a great extent all the circum- 
stances on which the result depends, and we are generally right 
in assuming that an experiment which gives a certain result on 
three successive days will do so always. But even this some- 
times depends on the fact that the apparatus is not disturbed, 
and that the housemaid has not come in to dust the room. 
Here lies the difference. What is possible ina laboratory, though 
perhaps difficult, is not possible in the upper regions of the 
atmosphere, where some unseen hand has not made a clean sweep 
of some important condition. 
When we cannot control accessory circumstances we must 
eliminate them by properly combining the observations and in- 
creasing their number. The advantage does not lie altogether on 
the side of experiment, because the very identity of condition 
ander which the experiment is performed gives rise to systematic 
errors, which Nature eliminates for us in the observational 
sciences. In the latter also the great variety in the combinations 
which offer themselves allow us to apply the calculus of proba- 
dility, so that in any conclusion we draw we can form an idea of 
the chance that we are wrong. Astronomers are in-the habit of 
giving the value the ‘‘ probable error” in the publication of 
their observations. Meteorologists have not adopted this 
‘custom, and yet their science lends itself more readily than 
any other to the evaluation of the deviations from the mean 
result, on which the determination of the probable error 
depends. Welook forward to the time when weather forecasts 
svill be accompanied by a statement of the odds that the pre- 
diction will be fulfilled. 
The calculation of the probability that any relationship we 
may trace in different phenomena indicates a real connection 
seems to me to be vital to the true progress of Meteorology, and 
although I have on previous occasions (Cambridge Phz/. 7rans., 
vol. xviii. p. 107) already drawnattention to this matter I should 
like once more to lay stress on it. 
The particular case I wish to discuss (though the methods are 
not restricted to this case) is that in which one of the two series 
of events between which relationship is to be established has a 
definite period, and it is desired to investigate the evidence of 
an equal period in the other series. 
Connections between the moon and earthquakes, or between 
sunspots and rainfall if proved to exist, would form examples of 
such relationships. The question to be decided in these cases 
would be, is there a Junar period of earthquakes, or an 
eleven years’ sunspot period of rainfall. 
Everyone familiar with Fourier’s analysis knows that there is a 
lunar or sunspot, or any other period in any set of events 
from volcanic eruptions down to the birth-rate of mice ; what 
we want to find out is whether the periodicity indicates a 
realconnection ornot. Let us put the problem into its simplest 
form. Take x balls, and by some mechanism allow them to 
drop so that each falls into one of m compartments. If finally 
they are equally distributed each compartment would hold 7/7 
balls. If this is not the case we may wish to find out whether 
the observed inequality is sufficient to indicate any preference 
for one compartment or how far it is compatible with equality of 
chance for each. If we were able to repeat the experiment as 
often as we like we should have no difficulty in deciding between 
the two cases, because in the long run the average number 
received by each compartment would indicate more and more 
closely the extent of bias which the dropping mechanism 
might possess. But we are supposed to be confined to a single 
trial, and draw our conclusions as far as we can from it. 
It would be easy to calculate the probability that the number 
of balls in any one compartment should exceed a given number, 
but in order to make this investigation applicable to the general 
problem of periodicities we must proceed in a different manner. 
NO. 1720, VOL. 66] 
If the compartments are numbered, it does not matter in which 
order, and a curve be drawn in the usual manner representing 
the connection between the compartments and the number of 
balls in each, we may, by Fourier’s analysis, express the result 
by means of periodic functions. The amplitude of each period 
I - é 
can be shown on the average to be EF /mz. It is often more 
convenient to take the square of the amplitude—call it the in- 
tensity—as a test, and we may then say that {the ‘‘ expectancy” 
of the intensity is 47//*. The probability that the intensity of 
any period should be & times its average or expectancy is e—*. 
We may apply this result to test the reality of a number of 
coincidences in periods which have been suspected. A lunar 
effect on earthquakes is in itself not improbable, as we may 
imagine the final catastrophe to be started by some tidal de- 
formation of the earth's crust. The occurrence of more than 7000 
earthquakes in Japan has been carefully tabulated by Mr. Knott 
according to lunar hours, who found the Fourier coefficient for 
the lunar day and its three first sub-multiples to be 10°3, 179, 
10°9, 3°97 ; the expectancy on the hypothesis of chance distri- 
bution for these coefficients I find to be 19°3, 15°7, 10°6, 5'02. 
The comparison of their numbers disproves the supposed con- 
nection; on the other hand, the investigations of Mr. Davison 
on solar influence have led to a result much in favour of such 
influence, the amplitude found being in one series of observ- 
ations equal to five times, and in the other to fifteen times the 
expectancy. The probability that so large an amplitude is due 
to accident in the first case is one in 300 millions, and in the 
second the probability of chance coincidence would be repre- 
sented by a fraction, which would contain a number of over 70 
figures in the denominator. We may, therefore, take it to be estab- 
lished that the frequency of earthquakes depends on the time of 
year, being greater in winter than in summer. With not quite 
the same amount of certainty, but still with considerable prob- 
ability, it has also been shown that earthquake shocks show a 
preference for the hours between 9 a.m. and noon. 
A great advantage of the scientific treatment of periodical 
occurrences lies in the fact that we may determine a prior¢ how 
many events it is necessary to take into account in order to prove 
an effect of given magnitude, Let us agree, for instance, that 
we are satisfied with a probability of a million to one as giving 
us reasonable security against a chance coincidence. Let there 
be a periodic effect of such a nature that the ratio of the occur- 
rence at the time of maximum to that at the time of minimum 
shall on the average be as I+A to 1—A, then the number of 
observations necessary to establish such an effect is given by the 
equation 7 = 200/A*. If there are 2 per cent, more occurrences 
at the time of maximum than at the time of minimum A= ‘ol, 
and 7 is equal to two million. If the effect is 5 percent., the 
number of events required to establish it is 80,000. 
To illustrate these results further, I take as a second example 
a suggested connection between the occurrence of thunderstorms 
and the relative position of sun and moon. Among the various 
statistical investigations which. have been made on this point, 
that of Mr. MacDowall lends itself most easily to treatment by 
the theory of probability. One hundred and eighty-two thunder- 
storms observed at Greenwich during a period of fourteen years 
have been plotted by Mr. MacDowall as distributed through the 
different phases of the moon, and seem to show a striking con- 
nection. I have calculated the principal Fourier coefficient 
from the data supplied, and find that it indicates a lunar 
periodicity giving for the ratio of the number of thunderstorms 
near new moon to that near full moon -the fraction 8°17 to 
4°83. 
This apparently indicates a very strong effect, but the 
inequality is only twice as great as that we should expect if 
thunderstorms were distributed quite at random over the month, 
and the probability of a true connection is only about 20 
to 1. No decisive conclusions can be founded on this, the 
number of thunderstorms taken into account being far too small. 
We might dismiss as equally inconclusive most of the other 
researches published on the subject were it not for a remarkable 
agreement among them, that a larger number of storms occur 
near new moon than near full moon. 
I have put together in the following table the results of all 
investigations that are known to me ; following the example of 
Koeppen, I have placed in parallel columns the number of 
thunderstorms which have occurred during the fortnight in- 
cluding new moon, and the first quarter and the fortnight 
including the other two phases. 
