ON CERTAIN CUBIC CURVES. 



BY F. H. LOUD. 



When a curve of the third order has three inflectioDal tan- 

 gents which converge at a point, the curve belongs always to 

 that non-singular genus called simplex or unipariite, and has 

 accordingly six more inflectional tangents, and these will also 

 converge by three to two other points. As only three of the 

 nine tangents are real, there will be two distinct cases, either 

 the three real tangents may converge, when there will be two 

 imaginary points, each the meeting-point of three imaginary 

 tangents, or two of the imaginary tangents may meet upon a 

 point of a real stationary tangent, and then each of the other 

 real inflectional tangents will have a point in which it is met by 

 two that are imaginary. The harmonic polars of three points of 

 inflection whose tangents converge, all pass through the point of 

 convergence of the three tangents. Through the same point 

 pass also the two lines which join by threes the points of con- 

 tact of the six remaining tangents. 



When two of the points of convergence are imaginary, the 

 special case may arise that these two should be the circular 

 points at infinity. The real inflectional tangents are then 

 asymptotes, and meet at equal angles, which are bisected by 

 the harmonic polars. The curve consists of three equal branches 

 lying alternately in three of the six angles formed by the asym- 

 ptotes, and within each branch is a focus, the foci forming ver- 

 tices of an equilateral triangle. In a system of tangential co- 

 ordinates, wherein the coordinates of a line are its perpendicu- 

 lar distances from these three foci, if we put the sum of these 

 coordinates I + 'z + ^ = 3w, and write as the equation of the 

 circular points IJ = 0, then the equation of the curve ia 

 81 (3--r;: + c^IJ)^ = 4TJ'. 



