14 
as the differential resolvent of 
y % — 3y 2 + 2x = 0 (ii) 
The sinister of this resolvent is of the form of the sinister of 
(3) of my last paper, and the symbolical decomposition of (i) 
of this paper is (i denoting as usual ± J - 1) 
( 3 -<>-l <“) 
and the resolvent of this form of cubic, like that of the other, 
is soluble by change of the independent variable. 
The evanescence of the dexter of a resolvent does not 
necessarily indicate that all its algebraic coresolvents are 
wanting in their second term. Thus, for instance, 
y( l-^_^| + y =0 (iv) 
is the differential resolvent not only of 
y*-3y + 2x = 0 (v) 
but of every algebraical equation whose roots are homoge- 
neous linear functions of the roots of (v), that is to say of 
any two of them, for any root of (v) is such a function of the 
other two. Let a, j3, and y be roots of an equation whereof 
any one root is a linear and homogeneous function of any 
two other roots. Then we may put 
y = ma + nf3 (vi) 
and, from the identities 
( da d& \ ( ( dQ n da \ 
+ ) ~ [ ma + w/3 )di = T _ ) 
\ (jLJU WW / \ / tlt/s \ tit// l It// J 
and 
(ma + »/3)f - (mg. + »g)/3 = m(«g- - /3-|) 
we infer that 
and 
