ALTER: RAINFALL AND SUN-SPOT PERIODS. 19 



from each to complete the tables. To complicate the task, these 

 data were given for fifty-five districts during the early years and for 

 thirty-three dm-ing the later. From some countries averages made 

 correctly were sent in form to use, but in the main the data, as se- 

 cured, required much work to put it in a form to begin the investiga- 

 tion. Such tables are added to this paper in order that other in- 

 vestigators may be saved the preliminary computations. All long 

 records have been studied, with the exception of Canada, which is so 

 close to the United States that it was felt the results secured would 

 not be worth the work of averaging many stations together to get 

 district values in usable form. In the proper places comments will 

 be made on the methods of securing district averages in the United 

 States and other countries. It is believed that many of these should 

 be remade. 



MATERIAL SUITABLE FOR HARMONIC ANALYSIS. 



A mass of observational material, when plotted with time as ab- 

 scissae and observed values as ordinates, may show no repetition of 

 the same curve, even though such a curve might exist. There may 

 be nothing definite about it to indicate a period. In such cases or- 

 dinary methods of harmonic analysis become useless. This failure 

 to repeat values, when a period exists, may be due to any one or 

 more of the four following causes : 



(a) Incommensurable periods may coexist. In this case the curve 

 will never repeat itself, although for short periods of time there may 

 be a fairly close approximation to such repetition. If there are three 

 or more incommensurable periods the curve obtained for the data is 

 very complex. For example, the seasonal variation of the rainfall 

 would be incommensurable with a possible one equaling the sun-spot 

 period. Of course, if one of such periods is known, as in the case of 

 the seasonal variation in the example above, it may be eliminated. 



(b) There may be large accidental errors. Such errors mask a 

 periodicity almost completely in any one cycle and disappear only 

 when the data values in each of a number of well-distributed phases 

 are added through many cycles. From the theory of errors, their 

 influence will be inversely proportional to the square root of the 

 number of cycles added. 



(c) Long-period variations may exist. If there are periods longer 

 than the interval of the data they will produce much the same effect 

 as accidental errors or incommensurable periods. 



(d) There may be periods which vary in length. An example of 

 such a period is the sun-spot period, which, although averaging 



