420 ANNALS NEW YORK ACADEMY OF SCIENCES 



sums will be positive or negative and the whole series of sums will form a 

 rhythmical curve. 



Even if the purely accidental errors of the observations be so large as to 

 mask completely the periodic character of the residuals, yet the effects of 

 these errors will be largely eliminated in forming the products and taking 

 the sums, and the final sums will form a rhythmical curve. Instead of 

 using these sums directly, Newcomb finds the ratio of each sum to the first, 

 [ao aj, and calls these successive ratios Xj, Xj, X3, etc., so that 



[aoaj 

 x-= 



' [Soao] 



where i = 1, 2, 3, etc. 



Now if the observations be periodic, or if there be a tendency toward a 

 rh}^hmical deviation whose period is approximately a multiple of t, then 

 such period or tendency will be shown by an increasing value of x at the 

 time corresponding most nearly to the completion of the period. If there 

 be no tendency toward any period between 2t and nt, then the series of x's 

 should converge toward zero. 



This method was used in an investigation of the observations of the 

 difference between the polar and equatorial diameters of the sun as made 

 by Schur and Ambronn during the years 1892-1902. As has been noted, 

 the general mean of all of Schur's observations made during the period was 

 + 0".018, while that of Ambronn was only + 0".002. In the case of Schur, 

 therefore, the residuals found by subtracting this mean (+ 0".02) from 

 each observation were used instead of the observations themselves. In 

 the case of Ambronn, the mean being so nearly zero, the observations were 

 used directly. The whole series of observations was then divided into 

 consecutive periods of seven days, and the mean residual for each period 

 found. In all there were 654 such seven-day periods, out of which number 

 seventy-one periods only contained observations by both observers. In 

 eighteen periods, Schur had two or more observations, and in six periods 

 Ambronn had two observations. 



The series is disconnected; there are many periods in which no observa- 

 tions were made, and these periods are scattered irregularly throughout 

 the series. The longest period in which consecutive observations were 

 made was begun in May, 1899, when observations were made in eleven 

 successive seven-day periods. In the entire series there are found only nine 

 cases in which observations were made on six or more consecutive seven-day 

 periods and which therefore could be used in the present investigation. 

 In addition to these nine, two other sets were utilized, in one of which 



