DISCONTINUOUS PERIOD IN CYCLOGRAM ANALYSIS 53 



individual maxima but it gives results only when enough of these maxima 

 unite into a definite pattern. 



The sunspot cycle was recently described to me as a succession of changes 

 in phase. Upon first hearing this, it seemed a complete disagreement with 

 our results with cyclogram methods. But on consideration it grew to be a 

 good partial statement from the viewpoint of the harmonic approach. Since 

 harmonic analysis assumes fixed testing periods, any departure in period 

 length comes into view as a change in phase and is so expressed. Hence, it is 

 possible that this statement of phase changes in solar data is an attempt to 

 express the same thing that I call a sequence of discontinuous periods. The 

 critical observational fact in the harmonic method of expressing solar changes 

 is whether the phase changes occur just one at a time or shows a succession 

 of similar changes. The former is a genuine change in phase; the latter, of 

 course, is a genuine change in period, and that, it seems to me, is more closely 

 describing what is taking place in the sequence of annual sunspot numbers. 

 In the cyclogram this difference is highly prominent. 



The great advantage of harmonic analysis is that its results may be evalu- 

 ated by the laws of probability whose principles distinguish between "real" 

 and "unreal" results. These adjectives practically refer to permanence; a 

 cycle is real that lasts through the data and eventually rises above random 

 effects; even a very weak cycle observed in a long series of data may be 

 strengthened by repetition and may give an average easily recognized as 

 real, while random effects are smoothed out by averaging the repetitions. 



Students of harmonic analysis had long since observed that cycles in 

 natural phenomena often exhibit a relation between several successive values, 

 statistically called conservation. This has an effect on the cyclical characters 

 in the data and affects the reliability of the results. The cause of the con- 

 servation is not involved in its purely statistical study. 



BARTELS' DIALS 



Two German students of cycles have approached somewhat nearer to 

 the idea of discontinuity, Bartels and Stumpff, the former by development 

 of harmonic analysis methods and the latter with theory and a mechanical 

 process. 



Bartels' work appears in Random Fluctuations, Persistence and Quasi- 

 Persistence in Geophysical and Cosmical Periodicities (Terrestrial Magnetism 

 and Atmospheric Electricity, vol. 40, No. 1, pp. 3-60, March 1935), and in 

 other papers. In carrying out a skilful study of probabilities as applied to 

 periodicities in magnetic phenomena, he has presented mathematical processes 

 that take conservation into account and has recognized and named certain 

 characters that identify with the phenomena long observed with the cyclo- 

 gram. These characters are found in his "harmonic dial" and his "summa- 

 tion dial," particularly the latter with its significance and the tests applied to 

 it. Thus I hope cyclogram usage and results will become clear to astronomical 

 and statistical students. 



