228 



NA TURE 



{July 7, 1887 



the probable character of this periodical fluctuation in an 

 eleven-year cycle, the coefficients of the first two periodical 

 terms of the harmonic formula have been computed, 

 taking 1864 as the initial epoch. These coefficients 



are — 



?/ = 5-340 inches. ?/' = 2 '873 inches. 



U' = 206° 29' U" = 247° 15' 



and the values of the eleven annual phases of the cycle 

 thus found are — 



Inches. 



1864 and 1875 - 5-1 



1865 ,, 1876 - 67 



1866 „ 1877 - 4-4 



1867 „ 1878 - 1-5 



1868 ,, 1879 - 06 



1869 ,, 1880 - 07 



1870 ,, 1881 + 0-8 



1871 ,, 1882 + 4-4 



1872 „ 1883 + 7-3 



1873 „ 1884 + 5-9 



1874 ,, 1885 + 0-5 



Taking the differences of these values from the recorded 

 rainfall of each of the twenty-two years, the mean devia- 

 tion of the latter in any one year from its periodical value 

 is found to be — 



± 3"5 inches, 



which is only one-fourth of the range of the periodical 

 variation as above determined ; and the probable error e, 

 of the periodical value, as found by the formula— 



e = 0-6745 \/ , \ ' 



A' n{n — I) 



is ± 070 inch. 



On the other hand, the mean deviation of a single year 

 from the general average is 



± 5'2 inches, 



and the probable error of that average ± 0*94 inch. 



What, then, is the numerical probability of the cyclical 

 variation, thus determined, being a true periodical 

 fluctuation, representing a regularly recurrent pheno- 

 menon ? As a general problem this cannot be solved, 

 because we do not know all the variations to which the 

 rainfall may conceivably be subject. But we can com- 

 pare the relative probability of this particular variation 

 being the result of a periodic law, and of its being a mere 

 fortuitous series of variations from a constant average. 

 That it is the most probable variation, having the assumed 

 period of eleven years (with the exception of such as 

 might be computed from a larger number of periodic 

 terms), is assured by the method of its computation, 

 which is based on that of least squares ; and one may 

 assume that this relative probability for a single year is 

 represented by the inverse ratio of the probable errors of 

 the two means above determined, viz. — 



o'94 

 070' 



This ratio of probability increases in geometrical pro- 

 gression, as the number of years during which it is found 

 to hold good increases in arithmetical progression ^ ; and, 

 for twenty-two years, becomes — 



(H|4)-= 655 : I. 



This ratio, although by no means amounting to demon- 

 stration of the exact validity of this particular cycle, 



' The probability of throwing any given series of numbers of a single die, in 

 any prescribed order, repeatedly for n throws, is obviously the same as that 



of throwing a single given number n times in succession, viz. (— ) ; and 



the probability of throwing, in like manner, one out of a given series of dyads 



or triads, the dyads or triads varying in any prescribed order is ( — I or 



\~) • The relative probability of the dyad to the triad series is I— )" ; and 

 generally the relative probability of a phenomenon, the law of variation of 



affords at least a very high probability that the apparent 

 undecennial fluctuation is no chance phenomenon. 



Apart from the approximate identity of its period, the 

 oscillation of the rainfall, thus disclosed, is very different 

 in character from that of the sunspot curve. The periodical 

 minima of both rainfall cycles preceded those of the corre- 

 sponding sunspot cycles by two years ; the actual year of 

 minimum rainfall coincided with that of sunspot mini- 

 mum in the first cycle, and preceded it by two years in 

 the second. The periodical maximum of the first cycle 

 followed the sunspot maximum by two years, that of the 

 second cycle coincided with the corresponding phase of 

 sunspots, which, in this case, was retarded by two years. 

 The actual rainfall maximum occurred two years later 

 than the sunspot maximum in the first cycle, and one 

 year later in the second. 



Hence, as far as the evidence of two cycles goes, the 

 minimum of the rainfall tends to precede the minimum 

 of the sunspots, the maximum of the former to follow 

 that of the latter ; and it is noteworthy, as I shall after- 

 wards show, that the droughts which, during the last 

 century, have visited with more or less intensity certain 

 portions of the Indian peninsula, have, on an average, 

 preceded years of sunspot minimum by about one year. 



In the other provinces of tropical India, an eleven-year 

 cycle is hardly, if at all, to be detected ; a conclusion fully 

 in accord with that which I drew, in 1877, from an ex- 

 amination of the rainfall registers of Bangalore, Mysore, 

 Bombay, Nagpur, &c. The more pronounced phases of 

 the Carnatic cycle are indeed reproduced as a rule, more 

 or less distinctly, as seasons of high or low rainfall re- 

 spectively, in most parts of the peninsula ; but some of 

 the intermediate years are characterized by vicissitudes 

 as great, and even greater than these, destroying the 

 appearance of anything like regular fluctuation. 



The Carnatic minimum of 1867, which was the cul- 

 mination of five years' (1864-68) deficient rainfall, was 

 represented also in Mysore and Bellary, in Malabar and 

 the Deccan ; but, in the last two of these provinces, 1866 

 had a still lower rainfall : and in Berar and Khandesh, 

 while the deficiency of 1866 was (relatively to the average) 

 greater than in any of the more southern provinces, that 

 of 1867 was above the average. In the Konkan again, 

 there was no very great deficiency before 1871, and this 

 was shared more or less by the whole of the peninsula, 

 excepting only the Carnatic and Malabar, which had an 

 excess of 16 and 13 per cent, respectively. 



The Carnatic maximum of 1872 was reproduced in 

 Orissa and the Northern Circars — that is to say, in all the 

 eastern provinces of the peninsula — and also in Berar and 

 Khandesh ; but in other parts of the peninsula the rain- 

 fall of this year differed but little from the average. 

 1874, however, was a year of excessive rainfall in all 

 the western and southern provinces of the peninsula. 



The great drought of 1876 (the second Carnatic mini- 

 mum) extended with even greater intensity to Mysore, 

 Bellary, Hyderabad, and the Deccan districts of Bombay, 

 and aifected more or less the whole of the peninsula, and, 

 in addition, a great part of extra-tropical India. But in 

 the Konkan and Malabar the deficiency was only 18 per 

 cent, of an average fall. In the Konkan the deficiency of 

 the following year was much greater ; and in the northern 

 provinces of Bombay, as well as in the greater part of 

 North- Western India, the summer rainfall of 1877 failed 

 almost completely ; whereas in the Carnatic the rainfall of 

 that year was remarkably copious. 



which is unknown, varying n times in succession, between limits i / and 

 i (/ + 2) respectively, is — 



V + ■^■' 



Similar reasoning holds good when / and ^ + jk are the measures of the 

 jiiean variation ; and also when, as in the case before us, they represent th- 

 probable errors of alternative averages. Finally, the relative improbabi'^y 

 of the more limited range, as a chance n^sult — in other words, the P''*'^ 

 bility of the limitation being the result of a regulating cause — is exp"^*^^° 

 by the inverse ratio. 



