616 
Percentage of thunder- 
storms during the fortnight 
including 
Place of observation and Time of A 
author. observations. = 
New moon | Full moon 
and * and 
| first quarter. last quarter. 
Karlsruhe (Eisenlohr) ... 1801-31 | 50°8 49°2 
Gotha (Luedicke) woe 1867-75 | 72°5 27°5 
Vigevano (Schiaparelli) ... | 1827-64 46 54 
Germany (K6ppen) : 1879-83 | 56 44 
Glatz (Richter)... ... | 1877-84 | 62 | 38 
United States (Hazen) ... 1884 1, 56:5) le aaeas 
Prag (Griiss) 1840-59 51 49 
2. $0 220 ... | 1860-79 52°5 47°5, 
Gottingen (Meyer) aleanss7=Soule as | 46 
Kremsmunster (Wagner).. | 1862-87 538 46°2 
Aix la Chapelle (Polis) ... 1833-92 | 54°4 45°6 
Sweden (Eckholm) | 1880-95 | 533 46°2 
Batavia (v.d. Stock) ... | 1887-95 | 51°9 48°1 
Greenwich (McDowall) ... | 1888-91 |} 54 46 
Average a a, a | 549 451 
It will be seen that out of fourteen comparisons, thirteen show 
higher numbers in the first column, there being also, except in 
two cases, a general agreement as regards the magnitude of the 
effect. Two of the stations given in the table, Gottingen and 
Gotha, are perhaps geographically too near together to be treated 
as independent stations, and we may, therefore, say that there are 
thirteen cases of agreement, against which there is only one 
published investigation (Schiaparelli) in which the maximum 
effect is near full moon. 
The probability that out of thirteen cases in which there are 
two alternatives, selected at random, twelve should agree and 
one disagree is one in twelve hundred. If the details of the 
investigations summarised in the above table are examined, 
considerable differences are found, the maximum taking place 
sometimes before new moon and sometimes a week later. There 
is, however, evidently sufficient fvza facze evidence to render an_ 
exhaustive investigation desirable. The most remarkable of all 
coincidences between thunderstorms and the position of the moon 
remains to be quoted. A. Richter has arranged the thunder- 
storms observed at Glatz, in Silesia, according to lunar hours, 
and finds that in each of seven successive years the maximum 
takes place within the four hours beginning with upper 
culmination. If this coincidence is a freak of chance, the 
probability of its recurrence is only one in three hundred 
thousand. The seven years which were subjected to calculation 
ended in 1884. What has happened since? Eighteen years 
have now elapsed, and a further discussion with increased 
material would have definitely settled the question, but nothing 
has been done, or, at any rate, published. To me it seems 
quite unintelligible how a matter of this kind can be left in this 
unsatisfactory state. Meteorological observations have been 
allowed to accumulate for years, one might be tempted to say 
for centuries, yet when a question of extraordinary interest arises 
we are obliged to remain satisfied with partial discussion of 
insufficient data. 
The cases I have so far discussed were confined to periodical 
recurrences of single detached and independent events, the 
condition, under which the mathematical results hold true, 
being that every event is entirely independent of every other 
one. But many phenomena, which it is desirable to examine 
for periodic regularities, are not of this nature. The barometric 
pressure, for instance, varies from day to day in such a manner 
that the deviations feom the mean on successive days are not 
independent. If the barometer on any particular day stands 
half an inch above its average it is much more likely that on 
the following day it should deviate from the mean by the same 
amount in the same direction than that it should stand half an 
inch below its mean value. This renders it necessary to 
modify the method of reduction, but the theory of probability is 
still capable of supplying a safe and certain test of the reality of 
any supposed periodic influence. I can only briefly indicate the 
mathematical theorem on which the test is founded. The 
calculation of Fourier’s coefficients depends on the calculation 
NO. 1720, VOL. 66] 
NATURE 
[OcToBER 16, 1902 
of a certain time integral. This time integral will for truly 
homogeneous periodicities oscillate about a mean value, which 
increases proportionately to the interval, while for variations 
showing no preference for any given period, the increase is only 
proportional to the square root of the time. 
Investigations of periodicities are much facilitated by a certain 
preliminary treatment of the observations suggested by an 
optical analogy. The curve, which marks the changes of such 
variables as the barometric pressure, presents characteristics 
similar to those marking the curve of disturbance along a ray of 
white light. The exact outline of the Juminous disturbance is 
unknown to us, but we obtain valuable information from its 
prismatic analysis, which enables us to draw curves connecting 
the period and intensity of vibration. For luminous solids we 
thus get a curve of zero intensity for infinitely short or infinitely 
long radiations, but having a maximum for a period depending 
on temperature. Gases, which show preference for more or less 
homogeneous vibrations, will give a serrated outline of the 
intensity curve. 
I believe meteorologists would find it useful to draw similar 
curves connecting intensity and period for all variations which 
vary round a mean value such as barometric, thermometric 
or magnetic variations. These curves will, I believe, in abl 
cases add much to our knowledge; but they are absolutely 
essential if systematic searches are to be made for homogeneous 
periods. The absence of any knowledge of the intensity of 
periodic variation renders it, e.g., impossible to judge of the 
reality of the lunar effect which Eckholm and Arrhenius believe 
to have traced in the variations of electric potential on the 
surface of the earth. The problem of separating any homo- 
geneous variation, such as might be due to lunar or sunspot 
effects, is identical with the problem of separating the bright 
lines of the chromosphere from the continuous overlapping 
spectrum of the sun. This separation is accomplished by 
applying spectroscopes of great resolving powers. In the 
Fourier analysis, resolving power corresponds to the interval of 
time which is taken into account, hence to discover period- 
icities of small amplitude we must extend the time interval of 
the observations. 
I believe that the curve which connects the intensity with 
the period will play an important 7d/e in meteorology. It is a 
curve which ought to have a name, and for want of a better 
one I have suggested that of periodograph. To take once more 
barometric variations as an example, it is easy to see that just 
as in the case of white light the periodograph would be zero for 
very short, and probably also for very long, periods. There 
must be some period for which intensity of variation is a 
maximum. Where is that maximum? And does. it vary 
according to locality? The answer to these questions might 
give us valuable information on the difference of climate. 
Once the periodograph has been obtained, the question of testing 
the reality of any special periodicity is an extremely simple 
one. If be the height of the periodograph, the probability 
that, during the time interval chosen, the square of the Fourier 
coefficient should exceed 4% is e~*. If we wish this quantity 
to be less than a million, £ must be about 11; so that in order 
to be reasonably certain that any periodicity indicates the 
existence of a truly homogeneous variation, the square of the 
Fourier coefficient found should not be less than 11 times the 
corresponding ordinate of a periodograph. 
I have calculated in detail the periodograph of the changes 
of magnetic declination at Greenwich, taking as basis the 
observations published for the 25 years 1871-95. It was not, 
perhaps, a very good example to choose, on account of the 
complications introduced by the secular variation, but my object 
was to test the very persistent assertions that have been made as 
to the reality of periodic changes of 26 days or thereabouts. 
The first suggestion of such a period came from Hornstein, of 
Prague, who ascribed the cause of the period to the time of 
revolution of the sun round its axis. He only diseussed the 
records for one year’s observations, but the evidence he offered 
was sufficient to impress Clerk Maxwell with its genuineness. 
Since Hornstein’s first attempts, a great many rough and some 
very elaborate efforts have been made by himself and others to 
rove a similar period in various meteorological variations. The 
period found by different computors differed, but there is a 
good deal of latitude allowed if the rotation of the sun really 
has an effect on terrestrial phenomena, because the angular 
velocity of the visible solar surface varies with the latitude. 
Hornstein himself and some of his followers deduced a period 
