In the 1850’s, Jerrard showed that a Tschirnhausen transformation could reduce *in radicals* the general quintic into one missing three terms,

In 1864, Bring independently would do the same. It is now known as the *Bring-Jerrard quintic*. Such a reduction is important because it proved that a formula for the general quintic *does* exist, albeit it went beyond radicals and used *elliptic functions*, as was first done by Hermite.

In 1885, Runge et al showed that *all* solvable quintics [1] with rational coefficients have the form,

A century later, Spearman and Williams gave their version,

So which is it? It turns out they are two sides of the same coin. Using the Spearman-Williams parametrization, let,

and eliminating *v* between them using resultants, one gets,

This is one of the simplest sextic resolvents for the quintic: given {*a, b*}, if one can solve for *z*, then [1] is solvable. Since [2] is only a quadratic for *b*, we can easily solve for it,

Still let,

and substituting it into the positive case of the square root yields the *b* of the 1885 version, while the negative case gives the *b* of the 1994 one, proving that they are indeed two sides of the same coin.

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