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XXIII. On the Conditions for the existence of Three Equal Boots, or of Two Pairs of Equal 
Boots, of a Binary Quartic or Quintic. By A. Cayley, F.B.S. 
Received November 26, 1867, — Read January 9, 1868. 
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=Q is a relation 
compounded of the relations U = 0, V=0; 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 satisfied unless one or the other of the two component rela- 
tions is satisfied. I purposely refrain at present from any further discussion of the 
theory of composition. I say that the conditions for the existence of given systems of 
equalities between the roots of an equation furnish instances of such composition ; in 
fact, if we express that the function y ) n -> an< ^ first-derived function in regard to x, 
or, what is the same thing, that the first-derived functions in regard to x, y respectively, 
have a common quadric factor, we obtain between the coefficients a certain twofold rela- 
tion, which implies either that the equation (*fx, y) n — 0 has three equal roots, or else 
that it has two pairs of equal roots ; that is, the relation 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 existence of 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 deter- 
minants 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 n — 4 and n— 5, are given in my “ Memoir 
on the Conditions for the existence of given Systems of Equalities between the roots of 
an Equation,” Phil. Trans, vol. cxlvii. (1857), pp. 727-731 ; and I propose in the present 
Memoir to exhibit, for the cases in question n — 4 and n= 5, the connexion between the 
compound relation for the quadric factor with the component relations for the three equal 
roots and for the two pairs of equal roots respectively. 
Article Nos. 1 to 8, the Quartic. 
1. For the quartic function 
(a, h, c, d, ejx, y)\ 
4 k 2 
