31 °]
THE EQUATION OF THE FIFTH ORDER.
53
But there is a more simple form of the analytical expression of such a 12-valued
function; in fact, if [o/3y8e] be any function which is not altered by any one of the
above ten substitutions—if, for instance, [a/3] is a symmetrical function of x a , xp, and
[a/37§e] = [a/3] + [#7] + [78] + [Se] + [ea],
and therefore
[076/38] = [07] + [ 7 e] + [e/3] + [/38] + [8a],
then any unsymmetrical function of [a/3y8e] and [076/38] will be a 12-valued function
homotypical with P.
Mr Jerrard’s function W is the sum of two values of his function P; the sub
stitution by which the second is derived from the first can only be that which
interchanges the two functions [a/3y8e] and [076/38]; and hence any symmetrical function
of [0/3786] and [076/38] is a function homotypical with Mr Jerrard’s W; such symmetric
function is in fact a 6-valued function only. Indeed it is easy to see that the twelve
pentagons correspond together in pairs, either pentagon of a pair being derived from
the other one by stellation, and the six values of the function in question corresponding
to the six pairs of pentagons respectively.
Writing with Mr Cockle and Mr Harley,
T = X a Xp +' XpXy + XyXs + x$x e + x e x a ,
r = X a Xy + X y X e + X e Xp + XpXs +x s x a ,
then (t + r is a symmetrical function of all the roots, and it must be excluded; but)
(r — t) I 2 or tt' 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
fi = Xj + lx 2 + i 2 x 3 4- i 3 x 4 + i 4 x 5 ,
then the coefficients of the equation the roots whereof are (ft) 5 , (ft 2 ) 5 , (ft 3 ) 5 , (jd 4 ) 5 , and
in particular the last coefficient (ftft?ft?ft 4 ")*, are determined by an equation of the
sixth degree ; and this last coefficient is a perfect fifth power, and its fifth root, or
fi ft? ft? fi 4 , is the function just referred to as the Resolvent Product.
The conclusion from the foregoing remarks is that if the equation for W has the
above property of the rational expressibility of its roots, the equation of the sixth order
resulting from Lagrange’s general theory has the same property.
I take the opportunity of adding a simple remark on cubic equations. The
principle which furnishes what in a foregoing foot-note is called the a priori demon
stration of Lagrange’s theorem is that an equation need never contain extraneous roots;
a quantity which has only one value will, if the investigation is properly conducted,
be determined in the first instance by a linear equation; one which has two values
by a quadratic equation, and so on; there is always enough, and not more than enough,
to determine what is required.
