His most important work Mathematical Researches (1832-35) is on the theory of equations. Viète and Cardan had shown how to transform an equation of degree n so that it had no term in xn-1. This method had been generalised by Tschirnhaus to remove terms in xn-1 and xn-2. These methods were, to a large extent, motivated by attempts to solve equations algebraically. Abel and Ruffini showed this was impossible for general equations of degree greater than four.
In 1786 Bring reduced a general quintic to x5 + px + q = 0 while Jerrard generalised this to show that a transformation could be applied to an equation of degree n to remove the terms in xn-1, xn-2 and xn-3. Hermite used Jerrard's result saying that it was the most important step in studying the quintic equation since Abel's results. Hermite did not know of Bring's result and it is almost certain that Jerrard did not know of Bring's result either. Jerrard wrote a further two volume work on the algebraic solution of equations An essay on the resolution of equations (1858). He also wrote numerous articles which appear in the Philosophical Magazine and the journals of the Royal Society.
Jerrard did not accept that the algebraic solution of the quintic equation was impossible. Abel's proof of 1824 did not convince Jerrard and, one would have to add, many other mathematicians too. James Cockle was another mathematician who could not accept that Abel had proved the solution to be impossible, but Hamilton supported Abel and pointed out errors in Jerrard's work. In fact Jerrard had successfully shown that quintic equations could be solved but his error was to use methods which did not come under the precise definition of the 'method of radicals' which was required. After many years, Cockle came to accept that he was wrong. Bryce writes that :-
... in 1862, Cockle published a guarded surrender to Abel and Hamilton which must have been a blow to Jerrard: Cockle had been the best he had had by way of a mathematical supporter. Moreover, by the 1860s, Cockle was being forced to point out in print mistakes of Jerrard, and in this he was joined by Arthur Cayley. The exchange ended unpleasantly, with a good deal of asperity on either side.Jerrard died at his brother's house, the rectory at Long Stratton, Norfolk.
Article by: J J O'Connor and E F Robertson