TRANSACTIONS OF SECTION A. 517 



The probability that oat of thirteen cases in which there are two alterna- 

 tives, selected at random, twelve should agree and one disagree, is one in 1,200. 

 When 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, apparently sufficient 

 prima facie 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. .\. Rlchter has arranged the thunderstorms observed 

 at Glatz, in Silesia, according to lunar hours, and finds that in each of seven 

 successive years the maximum takes place within the few hours beginning with 

 upper culmination. If this coincidence is a freak of chance, the probability of 

 its recurrence is only one in 300,000. 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 defi- 

 nitely settled the question, but nothing has been done, or, at any rate, pub- 

 lished. To me it seems quite unintelligible how a matter of this kind can be left 

 in this unsatisfactory state. INIeteorological observations have been allowed to 

 accumulate for years — one might be tempted to say for centuries — yet when a ques- 

 tion 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 mathe- 

 matical 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 from 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 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 pro- 

 portional 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 charac- 

 teristics similar to those marking the curve of disturbance along a ray of white 

 light. The exact outline of the luminous 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 all 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 pro- 

 blem of separating any homogeneous variation, such as might be due to lunar or 

 suDspot effects, is identical with the problem of separating th^ bright lines of the 



