ALTER: RAINFALL AND SUN-SPOT PERIODS. 21 



If the assumed period does not exist, the mean values of the phases 

 Avill approach each other as we increase the number of cycles. 



This last point gives us two very powerful criteria for the verity 

 of our assumed period: 



(a) Having given a large numher of cycles, we may compare the 

 phase values of the first half of the cycles with those of the latter 

 half. If the variation be real the curves from the two halves of the 

 data should agree fairly well. If the variation be accidental there 

 can be only chance resemblance. Unless the assumed period exists, 

 the two halves of the data are entirely independent, when there are 

 enough cycles to eliminate residuals of other periods that might 

 exist. A very simple test for a real relationship between the two 

 curves may be made as follows : There is an even chance that if the 

 results are purely accidental, any pair of values from the same 

 phase in the two curves will lie on the same side of the normal. If 

 there are three curves, one-fourth of them should show all three 

 curves on the same side. Much departure from this accidental 

 grouping indicates strongly a correlation. 



(6) Having obtained the phase values, as above, for each half of 

 the data, we may consider half the difference of identical phases in 

 the first and last halves of our data as a measure of the deviation of 

 the two curves from each other and of the amount of chance error 

 left in each phase. Call this half difference d. We will have in this 

 example r/^, d.,, . . . d^j. The probable error of any point on the 

 curve which is formed from the whole of the data will be given by 

 the formula, 



e = 0. 6745 ^f-(^') . 



If this probable error is as large as half the variation from maxi- 

 mum to minimum phase there is approximately an even chance that 

 the variation is accidental. If the ratio of e to the variation is 

 smaller than about one-eighth, the chances are less than one in a 

 thousand that it is accidental. These ratios are tabulated in the 

 general discussion of results for each set of data. Both these criteria 

 must be applied in any case under discussion. 



Let us suppose that the assumed period is not an exact number of 

 months; for example, 14% months. In this case 7 cycles will equal 

 104 instead of 105 months. We must spread our 104 months over 7 

 cycles of 15 phases each; that is, over 105 phases. To do this we 

 will fill each of the first 6 cycles and the first 14 phases of the 

 seventh cycle just as formerly, using all the data that we have for 7 



