The Motion and Distribution of tlie Sun-spots 



29 



mathematical statistics to various phenomena, now shows that this accordance with 

 the simple Gaussian law must not be assumed a priori. On the contrary, the 

 statistical analysis of different mass-appearances shows many cases in which the 

 usual law of distribution can only be regarded as the first approximation to the 

 actual distribution. 



A glance at table XI shows that there is no pronounced skewness, wliether 

 positive or negative. In half the cases 8x is positive, in half negative. If we tal^e 

 into consideration the uncertainty whith wich Sx can be determined — the mean error 

 in S being 1.9325:]/ n — we see that, practically speaking, we might put ä,,, = 0 . 



I did not expect this originally. In the research made by Hiratama into 

 the systematic motions of the sun-spots, as stated in the introduction, page 11, the 

 author shows that if the material of observation is divided according to the rotation 

 period, a distribution is obtained, which, judging from the graphic representation 

 of Hirayama, would give a pronounced positive skewness, which should evidently 

 correspond to a negative skewness in the distribution of the motions. But such a 

 skewness does not exist. The discussion of this contradiction we shall relegate to 

 another section. 



Lastly, as regards the excess E,-, it is thoroughly positive, and the same cir- 

 cumstance is repeated in the motions in latitude. To find a plausible explanation 

 of this seems, however, to be attended with considerable difficulties. 



A positive excess may under certain circumstances arise by the addition of 

 two or more normal distributions. 



Let us take a zone with N observations of the motions in longitude. Suppose 

 these N observations to be divided into two groups with and N,^ observa- 

 tions in each group. In each of these groups we assume the distribution to be 

 normal. The dispersions we designate for a moment by a, and and the means 

 by and m^. If the mean of these two groups combined is designated by m, 

 the dispersion by a, the skewness by S and the excess by the following expres- 

 sions may be deduced : ^ 



W'a) (^i — ^2) 



NN\N 



C. V. L. Charliek: Studies in Stellar Statistics II. page 94. 



