24 O. A. Åkesson 



The above equation may, howerer, be written in a form more convenient for nu- 

 merial purposes. We observe that 



where Es(x) is a whole rational function of x of the degree s. Thus, we hawe 



a-i?^ = — {x — x^), 



o^B^ = + — x^Y — 0^, 



o'R, = -~-{x-x,r-^3^^x-x,), 



gS/j-, = + {x — x,y — (3i V — + 3aS 



f'rom tliese and preceding expressions it follows that 



GZ{X), 0''f\x), 0^(2') 



are functions only of the variable ^ ^ ^" . 



On account of this fact we may conveniently write the expression for F{x) in 

 the form 



Fi^) = 'f N + + ß.^V'C^'), + ] 



or 



(2) = ^ ['f o(:^) + ß 3?3 (^) + T. (^) + ] 



In this equation N designates the number of observations. The deviation from the 

 normal distribution is indicated by and ß^. By a distribution according to the 

 usual law of errors, ßg and ß^ vanish. The arbitrary factor 5 is attached in order 

 to give the frequency-curve a suitable form. (See page 32). 



The characteristics we have to compute are the following: 



IXq = meat!, 

 a = dispersion, 

 S = Sßj = skewness, 

 I = 3ß^ = excess. 



These characteristics may be computed in a simple manner from the moments 

 of the function F[x). I designate the moment of the s'" order in reference to the 

 origin by [jl«. This quantity is determined by the formula 



+ =» 



(4) ^-8 = f F[x)xHlx. 



The corresponding relative moment is obtained from 



