Researc'lies into the tlieory of prul lability. 



5 



the coordinates of the stars are affected by small errors, the temperature, pressure 

 and other conditions of the atmosphere differ from one moment to another a. s. o. 

 Each error of observation therefore may be considered as the sum of a multitude of 

 small errors, derived from equally many "independent sources. The law according 

 to which the errors of each source varies may be different for each source and 

 must a priori be considered as unknown. 



In essentially the same manner we can declare the variation of the characters 

 in biology. Consider, for instance, the stature of a group of adult men. If all men 

 in the group be supposed to possess identically similar ancestors, if they have 

 enjoyed identically the same education, the same food, the same climatical in- 

 fluences, if all other circumstances that may have some influence on the stature 

 of the man were identically similar for all men in the group, we must conclude 

 that the length of the stature of all these men must be the same, as truly as the 

 effect is determined from the cause. The differences in ancestral heredity, in 

 education, in food a. s. o. for a group of men may be considered as different 

 sources of error as to the stature of these men. Each source of error may cause 

 a positive or negative »elementary error» in the length; and through the addition 

 of these small quantities the resulting deviation in the length of an individual 

 from the supposed ideal length is obtained. Obviously the number of the sources 

 of these elementary errors must be considered as very great, if not infinite. 



This manner considering things seems to be very plausible. Meanwhile a new 

 difficulty appears, a difficulty of a mathematical character, which seems to make 

 the problem almost uusoluble. The number of the sources of error that each give 

 elementary errors is supposed to be very great and each source has its own law of 

 error, which must be considered as unknown. How great is the sum of all these 

 elementary errors? The problem is very difficult, but it has been attacked and 

 in principle solved by Laplace in his great work »Theorie analytique des probabili- 

 tés» (1820). In two memoirs on the law of errors (Medilelanden från Lunds Obser- 

 vatorium N:ris 25 och 2t)) I have discussed the problem, and shown some conse- 

 quences that may be drawn from the results of Laplace. 



These consequences are the following ones. 



A frcquencii curve may possess one of the follounng two forms: 



Type A. If the frequency curve is defined by the equation >/ = F {x), 

 where x is the measure of the character in cjuestion, and y its frequency, and we put 



designating by h and a two parameters, which must be duly determined, we can re- 

 present the frequency curve of type A through the equation 



1 



2 c-' 



. r = - 



r= e 



2 TT 



■) + •••, 



