Derivative Points of Curves of the Third Degree. 117 

 degree in x, y, z reduced to the canonical form 

 gZ j^yij^^^ mxyz = 0, 



any solution x = a,y=b, z = c of this equation is of course one 

 of a group of six obtained by the permutations of the three let- 

 ters a, b, c, and having an obvious relation to one another fhi'ough 

 the medium of the points of inflexion. So, too, it is manifest if 

 we take the equation to the curve in its most general form, from 

 any given solution, a group of six, including the given one, may 

 be formed, the characteristics of each of which will be linear 

 functions of one another. For the piu-pose of the theorem about 

 to be enunciated, such a group of solutions will be treated as a 

 single solution ; and then we can affirm the proposition follow- 

 ing, in which a solvent system means a system of values of the 

 variables x, y, z satisfying the equation /(.r, y, 2')=0, and free 

 from any common factor. 



Let a, b, c he any solvent system to a cubic homogeneous equa- 

 tion in x, y, z ; then from a, b, c we may derive a new solvent 

 system, a', b', c', ivhere a', b', c' are each of them functions of the 

 fourth degree of a, b, c, and another system a", b", c" of the ninth 

 degree in a, b, c, and another a'", b'", c'" of the sixteenth degree, 

 and so in general a new solvent system of the degree n^ in a, h, c. 

 One such derivative system, and only one, of the degree n^ can be 

 formed, and none of any intermediate degree. 



Thus, for instance, the coordinates of the tangential (the name 

 adopted from me by ]\Ir. Cayley to express the point of intersec- 

 tion of a tangent to a cubic curve at any point with the curve) 

 being called a', U, c', these last letters are biquadratic functions 

 of a, b, c*. 



So again, as I also suggested to Mr. Cayley, the point in 

 which the conic of closest contact with a cubic curve cuts the 

 curve will necessarily have a derivative system of coordinates of 

 a square-numbered degree in respect of the original ones, which 

 by actual trial Mr, Cayley has found to be the 25th. Mr. Salmon, 

 I believe, has obtained in certain geometrical investigations deri- 

 vatives of the 49th degree. 



I am in possession of the equations by means of which the 

 successive systems of the fourth, fifth, &c. degrees, which I incline 

 to call the first or primary, the second, third, &c. derivative 

 systems, may be formed explicitly by successive derivation from 

 one another ; so that, for instance, as soon as I am informed 

 that the system investigated by Mr. Cayley is of the twenty-fifth 

 degree or fifth order, 1 can find them without any reference to 



* This derivative solution (though not as corresponding to the tangential) 

 was known also to rule for a particular case, as will be seen by reference to 

 hia ' Algebra.' 



