Dec, 1915] Geometry of Translated Normal Curve 6l 



In order that the final curve may be written in terms of the 

 co-ordinates x and y the equation of the base or generating 

 normal probabiUty curve is written: 



1 ~ T^ 

 z = 



\/27r 

 where t denotes abscissas and z ordinates. 



Let the abscissas of the transformed curve be functions of 

 the corresponding abscissas of the base curve. Then it may be 

 assumed that x can be developed in powers of t, and hence we 

 may write on omitting fourth and higher powers, 



x = a(t + Kt2+xt3), 



where a, k and X are constants to be determined in "fitting" 

 the curve. 



Since x denotes the value of a measurement and y the 

 frequency of x, that is, the number of individuals possessing 

 that value of x, the magnitude of an element of area denotes the 

 number of individuals between two values of x.. Obviously, 

 therefore, if the transformation is to be of concrete value the 

 magnitude of an element of area must not be altered, though of 

 course the shape will be changed. Hence 



y dx = z dt, 

 and y = z dt/dx 



1 ~^ 1 



V27r a(l + 2/ct+3Xt-) 



The formulas of transformation are thus: 

 X = a(t + Kt' + Xt^), 



1 ~"2~ 1 



y = 



V27r a(l + 2«t+3Xt^) 



Maximum and Minimum Feints. Since only curves with 

 one maximum point or mode are practically useful it is desirable 

 to determine what values of the constants a, k and X give 

 unimodal curves. 



We have 



dy ^ dy d^ ^ l_ ~ T- (3Xt^+2Kt-+(l+6X)t + 2^) 



dx dt'dx -\/2^ ^ ' a(l+2/ct+3Xt2)== 



