32 



CLIMATIC CYCLES AND TKEE GROWTH 



without having to rearrange or interpolate the numbers. This very need 

 caused in 1913 the construction and use of the cyclograph to be described 

 shortly. 



Integration has its dangers and may be deceptive. Just as above in averag- 

 ing curves together into a group, the curves must be equalized or standardized 

 or one may dominate to the exclusion of the others, so here we are in fact 

 averaging curves together into a group, and a strong configuration in one 

 block of data may produce a cycle that exists nowhere but in that one block. 



Since this type of integration is the basis of all mathematical forms of 

 cycle analysis, its dangers have been well seen and various protections have 

 been devised; for example, a common safeguard is to separate the data into 

 two halves and see if the cycle comes out equally well in each half. 1 Such 

 a process applied by Schuster in 1906, in his classical work on the sunspot 



Table 2 



1 If the extra half-year is placed at some other point in the cycle, the result is practi- 

 cally the same. 



numbers, made him see at once that there are other periods in those numbers 

 than the well-known one near 11 years. 



Thus integrations of selected values fall far short of giving satisfactory- 

 cycle analysis. Not only should each proposed period be tried separately in 

 all parts of the data, but all nearby period values should each be tried through- 

 out the data. We believe that our method described below is the only safe 

 one yet suggested that fills these exacting requirements. 



Successive Integration — In harmonic analysis a definite interval of time, 

 usually the full length of the data, is taken as a fundamental, and integral 

 parts of this, |, f, J, etc., perhaps to 30 different fractions, are taken as suc- 

 cessive periods on which the data are summated, and the qualities of the 

 resulting means ascertained. This is multiple integration applied in an effort 

 to reach all possible periods that may exist in the data. But while this proc- 

 ess covers many possibilities in short period lengths, it leaves large gaps in 



1 For a cycle that is permanent, this test is very helpful, but for discontinuous 

 periods it is obviously of little value. 



