300 
[409 
409. 
ON THE CONDITIONS FOR THE EXISTENCE OF THREE EQUAL 
ROOTS, OR OF TWO PAIRS OF EQUAL ROOTS, OF A BINARY 
QUARTIC OR QUINTIC. 
[From the Philosophical Transactions of the Royal Society of London, vol. clviii. (for the 
year 1868), pp. 577—588. Received November 26, 1867,—Read January 9, 1868.] 
[It is remarked, Proc. R. Soc. vol. xvil. p. 314, that the above title is a misnomer: 
I had in fact in regard to the quintic considered not the twofold relations belonging 
to the root-systems 311 and 221 respectively, but the threefold relations belonging to 
the root-systems 41 and 32 respectively. The proper title would have been “ On the 
conditions for the existence of certain systems of equal roots of a binary quartic or 
quintic.”] 
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 
JJV — 0 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 relations 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 (*$#, y) n , 
and its first-derived function in regard to x, or, what is the same thing, that the 
first-derived functions in regard to x, y respectively, have a common quadric factor, 
we obtain between the coefficients a certain twofold relation, which implies either that 
the equation (*$#, 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