( 800 ) 



give no clear indication of the part they play in the construction of 

 the curve, and it is not well possible to describe their function in a 

 simple manner either verbally or graphically. 



The object of this communication is to piopose a general and 

 simple method by which a curve may be found, which being inte- 

 grated between certain limits, defined by the distribution of the data, 

 will give the sums characteristic of this distribution, and that for 

 frequencies of different kinds, as far as this is possible owing to the 

 elements of uncertainty proceeding from the imperfection of the data 

 which, of course, always remain. 



This curve, representing the law which the phenomenon follows, 

 should be called the frequency-curve: the curve of the aggregate 

 values, obtained by grouping the original data within definite limits, 

 may then be called the curve of distribution according to Bruns. Its 

 form depends upon the degree of condensation of the original data 

 (Abrundung after Bruns), but approximates more to that of the 

 frequency curve as the condensation becomes less extensive and 

 consequently the number of observations is greater. 



Such a development of an arbitrary function can evidently be 

 made in an infinite number of ways ; it is therefoi-e necessary to 

 postulate some general principles. 



The following premises apply to the method of development selected : 



1. That the development takes place according to polynomia of 

 an ascending degree'. 



2. that for the determination of the constants, the calculation of 

 means of different orders is used, in relation to an origin favourably 

 selected according to the requirements of the various cases. 



The expression "moments" which is frequently employed, has been 

 avoided as an unnecessary analogy with mechanical problems. 



^ 2. Development between definite limits. 



a. No given values of the function at the limits. 



The polynomia, the degree of which is indicated by a suffix, are 

 represented by Qn, and the series by: 



« = ^„Q, + AQi+.^.Q. + • • • • ^tc (1) 



The simplest form which can be given to the polynomia is: 



Qn = X'' -f ajA-"— 1 + a^x''-- -}-... a„ 

 In this case the most practical choice for the origin of coordinates 

 is evidently the mean between the limits as then, on integrating 

 between the limits, all odd terms vanish ; hence a separation between . 



