I 



A STUDY OF CORRKLATIONS AMONG TERRE8TRIAI. TP^.MPERATTTRES. 349 



It is to the summation found ut the bottom of this table that our attention will 

 be especially directed. It must be admitted that the periodicity among the numbers 

 seems to be very well marked, the apparent period being about six years. This is so 

 nearly one-half that of the sun-spot period that, if the result is not purely fortuitous, 

 we may well regard this as an actual period. 



Assuming the correlation to be real, the fact brought out may be found by 

 dividing the first sum [ag"] into each of the sums following. This is done in Table 

 IX. The second column of the table gives the values of [aoao], [aoaj], •••, [aoaj, 

 which are the sums S just found. The third column gives the quotients [ai,a,]^[aoao]. 

 Accepting them as real, the result may be expressed as follows : Whatever the mean 

 annual world departure in any one year, we have had since 1871, as a mean rule, a 

 departure in the same direction of 0.23 of its amount the year following. In the third 

 year following we have had a departure in the opjwsite direction of 0.3U, of its initial 

 amount ; in the fourth year of 0.21 ; in the sixth year a departure, now in the original 

 direction, of 0.16 ; and in the ninth a departure in the opposite direction,^of 0.40 of 

 the initial departure. 



To estimate the probability that this periodicity is real we must estimate the 



probable accumulated amount of the purely fortuitous deviations. We have for this 



purpose 



Standard annual deviation = ±0.14 



The probable mean value of a product of two such deviations will depend upon 



the law of statistical distribution. Our best result will be derived not by assuming 



the normal law of distribution, which may not be strictly applicable, but by taking 



the indiscriminate average, without regard to sign, of the entire 261 products. We 



thus find 



Genei'al average aa = .0155 



The average expected accumulations of 30 such sums, if fortuitous, will be about ±..08. 

 This, then, is the expected average value of a non-systematic [aoa,] (/ = 1, 2, 3, etc.), 

 for the period 1871-1904. The actual average we see to be 0.13. The excess is no 

 greater than might well be the result of chance deviations. But the inference of its 

 reality is strengthened by the evident 6-year pei-iodicity of the sums. On the other 

 hand, the existence of this period as an unbroken one is negatived by the fact that 

 during the last ten years of the series the epoch is practically reversed. The proof of 

 a permanent period half that of the sun-spots therefore falls to the ground. If there 

 is any real periodicity the case is similar to that of the waves of the ocean when, after 

 a series of definite period, a new series sets in with the same period, but not a con- 

 tinuation of the first. 



A. p. S.-XXI. VV. 14, 1, '08. 



