l8 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 128 



means — A of 8 repetitions of consecutive values for 68^ months 

 prior to 1900, and B of 7 repetitions of it after 1900. They show con- 

 siderable similarity, but the phase of A is, on the whole, 3 months 

 farther advanced than B. If we displace A backward 3 months, and 

 take the mean of the two determinations, curve C results. Curve C 

 gives an indication of three maxima. Taking the average of these and 

 subtracting, we get curve D. Curve D obviously presents seven humps 

 somewhat similar and nearly equally spaced. Averaging these features 

 of curve D and subtracting, we get curve E. Now the halves stand 

 out very plainly. Averaging the halves of curve E and subtracting, 

 we get curve F. It obviously contains five similar parts. Having de- 

 termined their average, and subtracting, we have curve G. It pos- 

 sibly has II maxima, but is so nearly smooth that I merely drew a 

 smooth line and used values from that in my synthesis of St. Louis 

 precipitation. The range of this 68^-month period, curve G, is 13 

 percent. Thus this single diagram presents for us five submultiples 

 of 22| years, viz, ^, ^, M2, %o, and V28, none of which would be ap- 

 parent to one who was unacquainted with the family of periods related 

 to 22| years. 



I now take the 30^-month period in Peoria ^° precipitation for illus- 

 tration. To somewhat abbreviate the presentation, I start with de- 

 partures from mean percentages of normal precipitation. That is, 

 using the symbols above explained and used, I start with ai, 02, ai-, a-^. 

 In figure 10, A, B (left) and C, D (right) are curves corresponding 

 to these four symbols. I shift curve A backward 5 months to com- 

 bine with curve B. This reduction gives us curve E. Proceeding with 

 curve E, I detect in it the half of 303^ months. Removing this, we 

 get curve F. In curve F a period of 303- -r- 3 months is seen. Remov- 

 ing this period, we get curve G. Its range is 7 percent. Proceeding 

 with curves C and D, they seem best adapted to be treated separately. 

 From curve C I remove its half-period, yielding curve H, and from 

 curve H its third-period, yielding curve I. Treating curve D in the 

 same way, curves J and L result. I now combine results I and L, by 

 moving I 6 months forward and taking the mean. It yields curve M. 

 And now a period of 303^-^-7 months is seen. Removing it, I derive 

 curve N. It has a range of 10 percent. Its phase is about 8 months 

 in advance over curve G, but the two forms are similar. 



As another example I take the period of 45^ months, which is ^ of 



10 Having had much experience since my Peoria publication (Smithsonian 

 Misc. Coll., vol. 117, No. 16, 1952) I have revised that synthesis. This present 

 illustration is from the revised tabulations, and so are all Peoria data in this 

 paper. 



