1868.] Prof. Cayley on the Roots of a Quartic or Quintic. 229 



January 9^ 1868. 



Lieut.-General SABINE, President, in the Chair. 



The following communications were read : — 

 I. On the Conditions for the existence of Three Equal Roots, or of 

 Two Pairs of Equal Roots of a Binary Quartic or Quintic.^' By 

 A. Cayley, F.R.S. Received November 26, 1867. 



(Abstract.) 



In considering the conditions for the existence of given systems of 

 equalities between the roots of an equation, we obtain some very interesting 

 examples of the composition of relations. A relation is either onefold, 

 expressed by a single equation U = 0, or it is, say, /<;-fold, expressed by a 

 system of k or more equations. Of course, as regards onefold relations, 

 the theory of the composition is well known : the relation UV = is a re- 

 lation compounded of the relations U=0, V = ; that is, it is a relation 

 satisfied if, and not satisfied unless one or the other of the two component 

 relations is satisfied. The like notion of composition applies to relations 

 in general ; viz., the compound relation is a relation satisfied if, and not 

 not satisfied unless one or the other of the two component relations is 

 satisfied. The author purposely refrains at present from any further dis- 

 cussion of the theory of composition. The conditions for the existence of 

 given systems of equalities between the roots of an equation furnish in- 

 stances of such composition ; in fact, if w^e express that the function 

 (#^^, ai^d its first-derived function in regard to x, or, what is the 

 same thing, the first-derived functions in regard to x, y respectively, have a 

 common quadric factor, we obtain between the coefficients a certain twofold 

 relation, which imphes either that the equation (^^a?, ?/)*^=0 has three 

 equal roots, or else that it has two pairs of equal roots ; that is, the rela- 

 tion in question is satisfied if, and it is not satisfied unless there is satisfied 

 either the relation for the existence of three equal roots, or else the relation 

 for the existence of two pairs of equal roots ; or the relation for the 

 quadric factor is compounded of the last-mentioned two relations. The 

 relation for the quadric factor, for any value whatever of n, is at once seen 

 to be expressible by means of an oblong matrix, giving rise to a series of 

 determinants which are each to be put =0; the relation for three equal 

 roots and that for two pairs of equal roots in the particular cases w=4 

 and w=5, are given in the author's "Memoir on the Conditions for the 

 existence of given Systems of Equalities between the roots of an Equation," 

 Phil. Trans, t. cxlvii. (1857), pp. 727-731 ; and he proposes in the present 

 Memoir to exhibit, for the cases in question w=4 andw = 5, the connexion 

 between the compound relation for the quadric factor and the component 

 relations for the three equal roots and for the two pairs of equal roots 

 respectively. 



VOL. XVJ. Y 



