Using a Tschirnhausen transformation, the principal quintic form can be transformed to the one-parameter form
(1)
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named after Francesco Brioschi (1824-1897) and which is important to the Klein's solution of the general quintic in terms of hypergeometric functions (Doyle and McMullen). This can be attained by using the transformation,
(2)
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(Dickson 1959) and eliminating the variable between the two using resultants to form a new quintic
(3)
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where
(4)
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(5)
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(6)
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Equating coefficients with a generic principal quintic
(7)
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results in a system of three equations in the three unknowns , , and . Amazingly, this can be resolved to a single equation that is only a quadratic and given in the variable by
(8)
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(Dickson 1959).