Mr. G. B. Jerrard on the Equations of the Fifth Order. 349 
(e,) may take the form 
Mn_s 
D >] 
M,_., D being expressive of whole functions, and D, which re- 
mains constant while M,,_, successively becomes M,_1, M,_»,..Mo, 
being such as not to vanish except when (U) has equal roots. 
We find in fact from the researches of Lagrange that 
HD Ply) E(u,).s Kah ‘ 
where F(u) =nu"-! + (n—1)a, u®-2 + (n— —2)agu"3+ aie 4c 
U), Uo)». Up denoting the n roots of the equation (U). 
Of the meaning of the analogous expression 
Naas 
Dp! 
which obtains in (e,) for v,_,, 1t is needless to speak. Indeed, 
having found one of the two equations (e), say (e), we may in 
general deduce the other, (e,), from it by the method of the 
highest common divisor. 
“Let us now examine the following extract from Mr. Cayley’s 
paper in the last Number (that for March) of the Philosophical 
Magazine. 
& Writing,” he says, “ with Mr. Cockle and Mr. Harley, 
TH Uylgt Uyly + Lyle + Lsle TL Ly, 
T= y2y +2, L Ag+ UUs t Leas 
then (7+7' is a symmetrical function of all the roots, and it : 
must be excluded; but) (r—7')? or r7' are each of them 6-valued 
functions of the form in question, and either of these functions 
is linearly connected with the Resolvent Product. In Lagrange’s 
general theory of the solution of equations, if 
fr=2,+tt,+ F034 82,4445, 
then ‘the coefficients of the equation the roots whereof are ( ft)®, 
(fu?)°?, (fe®)?, (fe*)®, and in particular the last coefficient 
(ft fe fe fe), are determimed by an equation of the sixth 
degree ; and this last coefficient is a perfect fifth power, and its 
fifth root, or fu fi? fr? fu*, is the function just referred to as the 
Resolvent Product. 
“The conclusion from the foregoing remarks 1s, that if the 
equation for W has the above property of the rational expresst- 
bility of its roots, the equation of the sixth order resulting from 
Lagrange’s general theory has the same property.” 
Here the question arises, Is it certain that fi fi? fi? fi4 can, by 
