1877.] 



the Rainfall with the Sun-spot Period. 



253 



to the corresponding non-periodical elements, so the cyclical differences 

 will be inversely less or more different from the differences between the 

 individual observations and the mean of the whole of them ; and if there 

 be no periodicity, the cyclical means will tend to disappear, and the two 

 sets of differences would, in a sufficiently long series, be identical. 



Hence it may be inferred that when the cyclical differences closely ap- 

 proximate in magnitude to the mean difference of the original observa- 

 tions from the arithmetical mean of all of them, the periodical elements in 

 those observations must be correspondingly small ; and this applies mani- 

 festly to the whole of the cycles for which the differences are shown in 

 Table III. 



Further to test the reality of the supposed periodicity shown in Table I., 

 I have rearranged the series of 64 years' observations, in a purely arbitrary 

 manner, in cycles of eleven years, by drawing the actual observations at 

 random one after another, and setting them down in succession till the 

 whole were exhausted. From three arbitrary sets of six cycles thus 

 prepared the mean cyclical difference averaged 10*9, 11*2, and 11-6 — 

 results which again indicate that, by adopting the actual sequence of the 

 observed quantities of rain instead of taking them at random, we pro- 

 duce no material effect on the mean cyclical differences, nor any such 

 tendency to a diminution in their numerical value as necessarily accom- 

 panies a true periodical element. 



It is, moreover, important to bear in mind that the mere circumstance 

 of: any series of cyclical means showing a single maximum and single 

 minimum gives no more real indication that such a result is a truly 

 periodical feature than would be supplied by the appearance of two or 

 more maxima and minima. The law of periodicity, if it exist at all, can 

 only be inferred by the facts indicated by observation ; and it is obviously 

 to argue in a circle, first to assume a cycle on which to work, which 

 shall give a single maximum and minimum, and then to infer that there 

 is true periodicity because of the single maximum and minimum. The 

 test of the periodicity is in truth to be sought altogether outside of the 

 particular values of the successive cyclical means. 



It is, of course, manifest that a complication of periodical elements 

 may so mask one another as to prevent positive results being obtained 

 by such an examination of the cyclical means and differences as I have 

 made in the case before us. But the whole scope of my present argu- 

 ment is negative, and it necessarily leads, I think, to the conclusion that 

 the cyclical variations shown in Table II. from the mean values in 

 Table I. are so great as to show that any apparent regularity or ten- 

 dency to a maximum in one part of the 11-year cycle, and a minimum 

 in another, has no real weight, and that there is no proof of greater 

 tendency to periodicity in the 11-year means than in the original isolated 

 observations. 



This might perhaps be considered all that need be said on this subject 



