20 THE UNIVERSITY SCIENCE BULLETIN. 



11.13 years, has varied from 7.3 to 17.1 years during the last 115 

 years. 



When any of these four difficulties exists it is almost impossible 

 successfully to treat the problem unless the investigator stumbles 

 upon the true period, either by a fortunate suggestion or by some 

 reason extraneous to the problem, or by the patient trial-and-error 

 method by which Kepler found his three laws of planetary motion. 

 Schuster (6) has developed a method designated as the periodogram, 

 which will avail in some cases. 



METHOD USED BY TURNER IN EXAMINING THE EARTHQUAKE 



DATA. 



The exact form of this method seems to be due to Schuster (6), 

 and is a slight modification of the one astronomers have used for 

 generations. Suppose that we have a mass of material — for ex- 

 ample, the number of earthquakes recorded per month, or the rain- 

 fall per month — through many years. Plotting shows no perio- 

 dicity, or at the most only a faint hint of such. Chance or Schuster's 

 periodogram leads us to suspect a period of, for example, 15 months. 

 We can write the first 15 months' data in a row as the heads of as 

 many columns. The sixteenth month, the thirty-first, etc., will fol- 

 low successively in the first column, the seventeenth, thirty-second, 

 etc., in the second column, and so on, the thirtieth, forty-fifth, etc., 

 in the fifteenth column. Each column will then contain only months 

 which are in the same phase of the suspected period, if it actually 

 exists. 



We will refer to one such row as a cycle, and to the columns as 

 phases. Suppose the period to exist. It may not show in a single 

 cycle, probably will not, because of large accidental errors or incom- 

 mensurable periods, either or both of which may be present. But 

 the months of any phase of an incommensurable period will, in the 

 long run, be almost evenly distributed through all the phases of our 

 assumed period, and will, therefore, be subject to the same laws as 

 accidental errors, namely, their influence will be inversely propor- 

 tional to the square root of the number of cycles. In the course of 

 four cycles (five years in our present example) their importance 

 will be only half as great as for any one cycle; after sixteen cycles 

 one-quarter as great, etc. However, the effect of our assumed 

 fifteen-month period will be equal in each, and therefore as prom- 

 inent in the average as in any one cycle. Thus, no matter how 

 large the accidental errors, or the variation due to incommensurable 

 periods, the true variation from phase to phase will begin to appear. 



