those of a Bring–Jerrard quintic: The extra parameter this fourth-order transformation provides allowed Bring to decrease the degrees of the other parameters. This leads to a system of five equations in six unknowns, which then requires the solution of a cubic and a quadratic equation. This method was also discovered by Jerrard in 1852, but it is likely that he was unaware of Bring's previous work in this area. The full transformation may readily be accomplished using a computer algebra package such as Mathematica or Maple. As might be expected from the complexity of these transformations, the resulting expressions can be enormous, particularly when compared to the solutions in radicals for lower degree equations, taking many megabytes of storage for a general quintic with symbolic coefficients. Regarded as an algebraic function, the solutions to involve two variables, d1 and d0; however, the reduction is actually to an algebraic function of one variable, very much analogous to a solution in radicals, since we may further reduce the Bring–Jerrard form. If we for instance set then we reduce the equation to the form which involves z as an algebraic function of a single variable , where . This form is required by the Hermite–Kronecker–Brioschi method, Glasser's method, and the Cockle–Harley method of differential resolvents described below. An alternative form is obtained by setting so that where . This form is used to define the Bring radical below. Brioschi normal form There is another one-parameter normal form for the quintic equation, known as Brioschi normal form which can be derived by using the rational Tschirnhaus transformation to relate the roots of a general quintic to a Brioschi quintic. The values of the parameters and may be derived by using polyhedral functions on the Riemann sphere, and are related to the partition of an object of icosahedral symmetry into five objects of tetrahedral symmetry. This Tschirnhaus transformation is rather simpler than the difficult one used to transform a principal quintic into Bring–Jerrard form. This normal form is used by the Doyle–McMullen iteration method and the Kiepert method. Series representation A Taylor series for Bring radicals, as well as a representation in terms of hypergeometric functions can be derived as follows. The equation can be rewritten as By setting the desired solution is since is odd. The series for can then be obtained by reversion of the Taylor series for (which is simply ), giving