Dec, 1915] Geometry of Translated Normal Curve 65 



In the region beneath the parabola and to the right from 

 the shaded area of Fig. I the curve is essentially of the shape 

 shown in Fig. II. This type includes the most common skew 

 ■curves and hence is of great importance in statistics. 



As the point (X, k) moves from the X-axis the crest rises 

 until the parabola is reached when the infinite ordinates appear 

 as two coincident lines, shown in Fig. III. 



After the parabola is passed, the infinite ordinates separate 

 and the curve apparently separates into three branches as in 

 Fig. IV. 



In crossing the K-axis to the left one asymptote rnoves o.'T to 

 infinity giving a curve of the type shown in Fig. V. 



Then the asymptote reappears giving a curve of the type 

 shown in Fig. VI. 



This general shape is preserved as the point moves toward 

 the X-axis and when the point reaches the discriminant curve 

 the middle branch is flattened at the minimum point. 



For points within the discriminant curve two minimum 

 points appear and the central branch now shows a maximum 

 with a minimum point on either side as in Fig. VII. 



The Tri-modai Curves. The curves corresponding to values 

 of (X, k) within the discriminant, because of the requirement 

 that an element of area under the translated curve must always 

 be equivalent to the corresponding element under the base or 

 generating curve, can be of statistical value only under the 

 following conditions. 



The area between the two ordinates corresponding to t = ±3 

 is 0.99998 of the total area under the curve, so that when 

 neither of the minimum points corresponds to points closer than 

 three units to the origin of the base curve the curve may be 

 practically valuable. A moment's consideration will show that 

 the abscissas of the two minimum points must be practically 

 the same as that of the corresponding infinite ordinates. The 

 roots of the quadratic 



3Xt- + 2Kt-fl=0 



are numerically greater than 3 for all pairs of values of (X, k) 

 lying above the line 



27X— 6k-M=0 



As statistically promising within the discriminant of the 

 ■cubic we then have the shaded area of the (X, k) plane. 



