( 311 ) 



from which ensues, that the integral of this equation, called the 

 curve of the sums, is expressed by the form 



Z> «J> + D x <t> x -f Z),*, -f etc. ..... (5) 



where * means <I>(/ti . as is the case m what follows. 



From (4) it is immediately evident when regarded in connection 

 with (3) that the suggested analysis of the curve (called by Bruns 

 not frequency-curve but curve of distribution) shows a resemblance 

 in the development of a function in terms of a Fourier series. 



In the different </',, terms appear polynomial functions of order 

 /> — I ; the 'l' r curve -hows p maxima and minima and intersects 

 the .c axis in p— 1 points and alternately will lie found for x = 

 either an extreme value (order uneven) or a point of intersection 

 (order even). 



The constants I) are determined in the well-known way by 

 evaluating the moments of various orders with respect to the </-axi> 

 through the origin of coordinates; it' we take for this origin the 

 value of ./■ corresponding to the arithmetical mean and if we put: 



./' 



x n y dx z= n„ , 



we find evidently D = ^ on account of /<„ being equal to 1; further- 

 more f<! must be =0 on account of the choice of the origin, so 

 D x must he put equal to 0, whilst, if one defines the value of the 

 constant It in such a way that 



2 AV, = 1 , 



it is then easy to deduce that also Z) 2 must be 0. 



The expressions (4) and (5) can thus he simplified and they become 



y = h\\ <P X + ZV*\ + #,* 6 ... | ... . (4«) 

 and 



i *, + D,*, + i><*< (5") 



The constants Z),, D 4 , etc. can easily be calculated by means of 

 the formula (3) where, with a view to this, the above mentioned 

 form is given. 



To calculate 2D„ we have namely, to consider the form appearing 

 between square brackets in the expression for *„-(-i and to substitute 

 in it h"(i„ for x n . 



Our finding in this way '2D„ instead of D, t is due to the same 

 reason why D a must be put equal to I ; namely to the form of (1) 

 in which the number 2 stands as coefficient. 



