ALTER: RAINFALL AND SUN-SPOT PERIODS. 27 



a large number of sections and an- extrapolation made for a cycle in 

 advance without serious error. Indeed, in such a case it might be 

 possible to predict the time of the next sun-spot maximum or mini- 

 mum quite accurately from the rainfall data. 



Effect of Annual Cycle. In many cases the residual left from 

 the seasonal variation is large enough to distort the curves ma- 

 terially. I have, therefore, always carefully eliminated it, no matter 

 how large or how small. To do this I have, wherever it is very pro- 

 nounced, prepared two tables for each section according to the plan 

 previously outlined, repeating and averaging in each one the months 

 determined by table 3. In the first of these tables I have used the 

 actual values of the rainfall. In the second I have used instead of 

 each January the mean of all the Januaries, and so on for each 

 month of the year. In this second table the mean monthly values 

 were repeated or averaged exactly as in the first one, to give a table 

 entirely similar to the first table. The variation from phase to phase 

 in this second table is, therefore, entirely the seasonal residual and 

 contains all of it. For the average state in the United States it is 

 approximately four per cent each side of the normal, the rest of the 

 seasonal variation having been damped out by the process of tabu- 

 lating the incommensurable period which is being investigated. The 

 quotients of the sums of each phase of the first table by the second 

 give us the percentage of normal rainfall of that phase for the section 

 concerned throughout all the years of the data. Each month is in 

 this way weighted in accordance with its normal rainfall. In no 

 case has there been any smoothing of results other than that marked 

 in the tables where the 77iean has sometimes been smoothed by aver- 

 aging each phase with the ones immediately adjoining for better ex- 

 amination. 



In the eastern United States and northern Europe the yearly 

 variation of rainfall is small enough that each month may be 

 weighted the same without serious error. I have, therefore, in these 

 two cases divided the actual rainfall of each month by its normal 

 and thus obtained the percentage of normal to plot. This has the 

 advantage for the reader that he need look at but one table instead 

 of two to see how the period has been followed from cycle to cycle. 



It may occur to some that possibly there is in some manner a 

 residual of the seasonal effect left in this period, despite the elimina- 

 tion explained above. There are three answers that may be givn to 

 this objection, all of which are merely the same one in different 

 forms. 



