200 Mr. J. Cockle on the Theory of Equations 
have inferred that it is (arts. 58, 59). The presumption is now 
rebutted. But since 3° replaces $ in the formulz, we have to 
inquire whether 3,° be a rational function of 3,°._ If it be, then, 
since every root of a rational equation is a rational function of its 
own square (for the equation may be written 2(2*) + (27) =0), 
we see that 0,5 must be a rational function of 0,°, and 6,)° of 6,'°. 
Consequently each of the expressions 
6,505 +0,°05+0,°0,° and 0,9 ,!°+ 0,10 6,39 4.8.1 6,10 
must be a symmetric (and indeed rational) function of the roots 
of the given quintic. Hence it is readily seen that the cubic 
whose roots are the above three values of 63, 6° will have all its 
coefficients symmetric in x, and therefore invariable under all 
interchanges of the 2’s. It would follow that 6;, 0; has only 
three values; and that for some one value (at least) of r and s 
we have 6=6', 
an inadmissible result. 
87. The same difficulty presents itself in another shape. 
Since all functions of the above form are invariable under inter- 
changes of the z’s, the doctrine of similar functions shows that 
the second coefficient of the cubic could only be determined by 
the solution of a quintic, even if the first were known. But 
masmuch as one of the most distinguished of writers_on the 
theory of equations has recently repeated the expression of a 
belief, formed many years ago, that the general quintic is soluble 
by means of an Abelian sextic, I shall add a few words upon the 
point. 
88. Let = =-V,, «= 40,9 0,5+00,40,44+..+60,0,4f, 
then, as* we know, 
V,,4+ Vo,6+ Vs,s=71(%s), 
where 7, denotes a rational function. Let the ratios of a, b,..,¢ 
* Recurring to arts 14 et seg., and grouping the 6’s thus, 
{P (at). LOR =. CG) rn ae)» C2) 
the omitted interchange (2 e) being equivalent to (4 q), and the inter- 
changes in each of the other groups being complementary, let every single 
interchange be applied. The order of the groups will or may be changed, 
but the members of each group will be inseparable. Consequently no pos- 
sible interchange can, save as to the order in which they are written, affect 
the prone, And since (see art. 15) the form of ¢ is arbitrary, and we may 
make 
p=2,+ a2, +bx,+cr3+dr,, 
we see that many of our ec-aclusions are true, whatever be the values of a, 
b,c, and d. It is when we seek a symmetric product that those quantities 
become unreal fifth roots of unity, and that ¢ becomes one of the func- 
tions of Lagrange and Vandermonde. 
