6 



C. V. L. Chailier. 



Typo B. The frequency curve of the second form may be expressed with 

 tlie help of the auxiliary function 



where X is a parameter, and the general form of F (x) is tlien 

 F (x) = B, <^ + B, ^ [x] + B, ^' '\ (,x) + . . 



where 7?^, /i^, i^g, . . . are coefficients independent of x. 



Beyond these two forms no other frequency curves can occur, except those 

 obtained through a superposition (addition) of several curves of the types A and B. 



I will iu this memoir more fully discuss these two forms of the frequency 

 curve. 



As to the conditions for the rise of these two types, it inay for the present 

 suffice to observe that type B arises, if the probability of a deviation from the 

 »ideal» value of a character, caused by each single source of error is very small, 

 whereas those sources of error, that possess an equal or nearly equal probability 

 for such values of the character as lie in the neighbourhood of the »ideal» one 

 give rise to a frequency curve of the first type. 



By ideal value of the character here is meant such a value as would arise 

 if all sources of error that may influence on the character had their most probable 

 state. For the more precise formulation of the conditions for the two forms I refer 

 to the mathematical investigation in the memoirs cited. It must be remarked that 

 it is possible to pass continuously from one form to the other. 



sm 71 X 



II. Type A of frequency curves. 



Let X be the value of a character and F [x) dx the f requeue}' of those values 

 that lie between x and x -\- dx. The frequency F [x] is represented by means of 

 the equation 



(1) F(,:r) = A, 'f (,r) + A, (x) + A, 'f [xJ] + . . ., 



where 



1 9 



(1*) <p(,x) = — -=.e 



The quantities 6, a, A^, A^, A^, . . . are dependent on the form of the equa- 

 tion y = F(x). The formulae for determining these quantities have been given in 

 my treatise »Über die Darstellung willkürlicher Functionen» (»Meddelanden» 

 N:o 27). 



