Dec, 1915] Geometry of Translated Normal Curve 



63 



Values of A and k for points on the discriminant give curves 

 with two modes coinciding. All points on one side of the dis- 

 criminant have three real and distinct modes, and all on the 

 other have one real and two imaginary modes. To determine 

 on which side the points giving three real modes lie we examine 

 a point inside the discriminant. When /c = the modal equation 

 becomes 



3Xt-'* + (l+6X)t = 



Hence the roots are t = and t 



1 + 6X. 



3X 



The quantity 



under the radical is positive for values of X between and -1/6, 

 Therefore, all points within the discriminant curve yield 

 tri-modal curves and all without uni-modal curves. 



The plane of \ and K 



K 



Fig.! 

 ( The horizontal 5cale js twice the vertical scale) 



The infinite values of dy/dx arise from zero values of the 

 quadratic, l+S/ct+SXt^. The greatest possible number of 

 modes for any one curve is therefore five, three from the cubic 

 and two from the quadratic. Since for infinite values of dy/dx 

 the corresponding ordinates are infinite, it is advisable to study 

 the location of the infinite points of the curve, rather to the 

 neglect of the idea of maximum values at such points. 



