Dec, 1915] Geometry of Translated Normal Curve 67' 



The general form for the moments about the median of the 

 area under the translated curve is 



1 c 



, ^L- r ° ^;f +f' +,\n: e " ^a(l + 2«t+3Xt')dt 

 -w^N^— a(l + 2/ct+3Xt') V r I y 



= ,1^ f " a-^Ct+Kt'+Xt^j^e ~ T~dt 



V 27r N ^ — 



On applying the two well known formulas : 



+ CC 



x^"+^e-"Mx= %^ x^"e--Mx, 



CO 2 •^ CO 



the determination of ju/, )U2', Ms' and jii is reduced to a matter of 

 algebraic detail. Then on transferring to the arithmetic mean 

 as origin the values of )U2, Ms, and m4 can be determined in terms. 

 of a, K and X. It is most convenient however, to make use of 

 the quantities /3i = m3Vm2"'' and /32 = M4/m2^ or rather jS = /5i/8 and 

 € = (/32 — 3)/12 and express the constants in terms of these 

 quantities. It is to be noted that both e and j8 are zero for a 

 normal distribution, that is, for X = /< = C. 



Omitting the detailed reduction* which is straightforward 

 and direct, we have 



(1) n' = aK 



(2) M2 = a2(l+6x+15x2+2/c2) 



2k2(2/c='+Q)2 



(3) 13 





^^^ ' (2k2+R)2 



where the symbols, S, R, Q and T are defined as follows: 



S =1 + 18X + 90X2, 

 R = 1+6X + 15X2, 

 Q = 1.5 + 18X + 135/2X2, 

 T =2X + 36X2+270X3+810X4. 



* Compare Edgeworth, "A Method of Representing Statistics by Analytical 

 Geometry," Proceedings Fifth International Congress of Mathematicians,. 

 Cambridge, 1912. 



