28 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 7I 



than the true period, the vakies fell along a line rising above the 

 horizontal as AB. To compute the length of the true period it was 

 first necessary to project the line until it had cross 360° and count the 

 number of periods and fractions needed. 



Thus, supposing AB to have been calculated for a period of 26 

 hours when the true period was 24 hours, it would cross the plot from 

 0° to 360° in 12 periods. Since this crossing was equivalent to a 

 loss of one i:)eriod, there must have been 13 true periods ; hence 

 26x12 



13 



24, or expressed mathematically, 



P=^ ; (14) 



but when the trial period is shorter than the true period, the formula 

 becomes 



in which n is counted in periods and fractions of a period, p is the 

 true period and p' is the trial period. In such a case a plot of the 

 amplitude a is nearly a horizontal line. Next a test was made by 

 combining a number of periods. Assuming certain phase angles and 

 amplitudes for various periods of different lengths, values were com- 

 puted for half-day intervals by the formula 



lr=A + a sin cj>r, (16) 



in which /,- are successive computed values corresponding to suc- 

 cessive values of </> at half-day intervals. The terms Ir and <f)r remain 

 as previously defined. 



Figure 9 shows a plot from the sum of the computed values of the 

 periods selected. The computations were extended to an interval of 

 80 days. The plot covers only a portion of this interval. Without 

 any clue to the periods used in forming them the sums of the periods 

 were given to my associate, Mr. Angus Rankin, for analysis by the 

 harmonic formulas. Computations were made for regularly increas- 

 ing periods, first for a period of three half-day intervals, then for 

 periods of four half-days, five half-days, etc., successively to 20 half- 

 days, and afterwards for periods of successive whole days to 20. 



Table IX shows examples of the computations where r equalled 

 eight half-days and nine half-days. The plot of the computed values 

 of and a for the period of four days is given in figure 10. The 

 ordinates are in degrees and are repeated three times from o to 360 

 while the abscissas are successive periods. The values of are shown 



