514 On a Relation between two Ternary Cubic Forms. 



two cases of failure, viz. for 1= — £, the equations for the linear 

 transformations are 



X = Y=Z=— x—y— z; 



so that X, Y, Z are not only not independent, but they are con- 

 nected by two linear relations. And the formula of transforma- 

 tion becomes 



(X + Y + Z) 8 -27XYZ=0, 



which is, in fact, true in virtue of the equations X = Y = Z. 

 The two forms of equation, 



x 3 -\-y 8 + z 3 + 6lxyz = f 

 (x+y + z) 3 + 27kxyz=Q, 



represent each of them equally well a curve of the third order 

 without a double point. In the first form the three real points 

 of inflexion are given by (# = 0, y-\-z = 0), (y = 0, z + x=0), 

 (2=0, x+yz=0) j or what is the same thing, the points in 

 question are the intersections of the lines x=0, y = 0, z = with 

 the line x + y + z=0; or we have x+y+z=Q for the equation 

 of the line through the three points of inflexion. And the equa- 

 tions of the tangents at the points of inflexion are 



2lx—y—z=0, 2ly—z—x=0, 2lz-x—y=0. 



For the second form it is obvious that the points of inflexion are 

 the intersections of the lines x=0, y=0, z=0 with the line 

 x-t-y + z = 0; and, moreover, that the lines x = 0, y==0, z=0 

 are the tangents at the point of inflexion. 



The first of the above-mentioned forms, however, cannot repre- 

 sent a curve with a double point. In fact the condition for its 

 doing so would be 1+8/^=0; but when this condition is 

 satisfied, the left-hand side breaks up into linear factors, and 

 the equation represents, not a proper curve of the third order, 

 but a system of three lines. The second form can represent a 

 curve having a double point ; viz. if k= — 1, the curve will have 

 a conjugate or isolated point at the point x^y=z. It is clear 

 a priori that (#=0, y = 0, z = being real lines) neither of the 

 forms can represent a curve of the third order having a double 

 point with two real branches through it, since in this case the 

 curve has only one real point of inflexion. 



I have elsewhere used the word " node " to denote a double 

 point, and I take the opportunity of suggesting the employment 

 of the words " crunode M (crus) and " acnode n (acus) to denote 

 respectively a double point with two real branches through it, 

 and a conjugate or isolated point. 



2 Stone Buildings, W.C., 

 October 19, 1860. 



