62 The Ohio Journal of Science [Vol. XVI, No. 2, 



From the vanishing of the numerator of dy/dx there must 

 result either one or three real modes for each pair of values for 

 X and K, that is, for each translated curve. To determine what 

 values of X and k give uni-modal curves and what tri-modal 

 it is convenient to consider the plane of X and k. 



The discriminant of the equation 



3Xt=* +2Kt2 + (1 +6X)t +2/V = 

 is 



IB/c"— k2(1+66X + 117X2)+3X(1+6X)3 = 



This fourth degree curve crosses the horizontal or X-axis at 

 X = and at X=-l/6 and when X=0 its equation reduces to 

 16k^ — K- = or K' = ± 0, K= ^}4:- There is thus contact with the 

 vertical or K-axis at the origin and that axis is crossed at the 

 points (0, ±3^^}'). At the point (X=-l/6, k = 0) there is a cusp 

 with the X-axis for tangent. The other two intersections with 

 the line X=-l/6 are imaginary, indicating the presence of two 

 branches to the curve. 



The discriminant of the denominator of dy/dx is the 

 parabola (in X and k), 



/c2— 3X = 



The evident close geometrical connection between the two 

 discriminants suggests arranging the discriminant of the cubic 

 curve in the following form: 



(k^— 3X) (16k2— 117X2— 18X—1)—27XH1—24X) =0 



From the equation in this, the well known uv+kws = form, 

 numerous elementary geometrical facts can be derived. The 

 relations to the hyperbola, IG/c^ — IHX^ — ISX — 1=0, and to the 

 parabola, k'^ — 3X = 0, premit of the ready plotting of the curve 

 with sufficient accuracy. The general shape of the curve is 

 shown in Figure 1. 



It is to be noted that one branch of the curve is within the 

 parabola, almost coinciding with it, while the other crosses it at 

 X = l/24, From the original form of this equation it appears 

 that the two branches of this discriminant meet just inside the 

 parabola in the end points with approximate co-ordinates 

 (0.043, ±0.360). The geometry of the cusp and end-points on 

 the discriminant curve is suggestive of interesting development 

 in detail. 



