30 



O. A. Åkesson 



From the last of these formulée the following appears 

 ] :o. E = 0 if mj = ni^ ; Oj = . 

 2:o. E > 0 il nij = ; :^ g.-,. 



3:o. E^Oilm,^m,;.,^G,;l-^^lp>0 

 or what is the same 



N N 



E>0, if anyone of the quotients and is < 0.211. 



NN. 



E < 0, if the quotients ^ and ^ are within tlie limits 0.211 and 0.789. 



Hence it follows, that a positive excess always arises ivhen two normal curves 

 with the same medium and different dispersions are added. If we should thus be 

 able to show that our material of observation can be divided into two or more 

 groups, with different dispersions in each group, this would possibly be a means by 

 wliich one might be able to explain the positive excess. 



My first idea was to make a division of the material according to the spot- 

 area, for it seems, in all probability, that the motions of the great spots should have 

 a smaller dispersion than the small ones, since the dispersion might also be con- 

 sidered as a measure of the stability of the spots during the motion. 



In dividing the material according to the spot-area, it however showed that 

 the distribution obtained was of the same type which Chaklier in his Researches 

 into the Theory of Probability calls the B-type, the number of spots diminis- 

 hing with increasing spot-area. As the frequency-function of this distribution, from 

 a mathematical point of view, is considerably more complicated than the frequency- 

 function of the A-type, I introduced on trial log A instead of A and divided the spots 

 according to this new variable, which in the following I designate by eo. For 

 those spots the areas of which are smaller than 0.5 and therefore designated by 

 zero in the Ledgers, I have put A = 1. Thus zero will be the lower limit of co. 

 It then appeared, that the distribution according to w was almost normal. 



As an example of the correlation between x and w , I here give the table ot 

 correlation between these variables of the zone first examined (Table XII). The 

 provisory mean of (o is l.'JOO and the class-breadth is assumed at 0.200. From the 

 form of this table, it is evident that the dispersions of the motions in longitude 

 increase with diminishing spot-area. 



A more, minute investigation of the correlation between dispersion and spot- 

 area will be the subject of another chapter. Here we will only discuss this question 

 as far as it can serve as an explanation of the positive excess. Should the distri- 

 bution of the motions of spots having a certain spot-area prove normal, we have 

 previously shown that the distribution of all spots combined will have a positive 

 excess, the dispersions of the motions for spots of different sizes being different. 

 It now appear, however, that even after dividing the material according to the spot- 



