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equation to a trinomial form ; and at the close of a Paper on 
the history of that celebrated problem, published at pp. 38-47, 
of the 6th vol. of the Quarterly Journal of Pure and Applied 
Mathematics , I have given Bring’s solution in extenso. 
Since the publication of these Papers Mr. Samuel Bills, of 
Hawton, has communicated to me a very ingenious simplifi- 
cation of Bring’s method. Tu place of transforming the 
quadratic in d by means of the assumptions 
b = ad +% and c = d + <y, 
and making the resulting quadratic vanish identically, Mr. 
Bills, observing that the equation (E) may be written 
5(3 pc + 4 qd + br)b + lO/c 2 + 2 bred — 15 p l c - 3 p 2 d 2 - 23pqd 
- 2 r/ - 2C )rp = 0, 
assumes 
3 pc + iq d + 5r — 0 
or, what is the same thing, 
_ _ 4 qd + 5r 
and by substitution in (E) obtains a quadratic in d, tbe solution 
of which gives d in terms of/j, q, r ; c and a then also become 
“ known”; and substituting for a, c, d their several values in 
the coefficient of if, and equating the result to zero, there 
arises a cubic in l>. Mr. Bills also observes, that if tbe values 
of a, c, d thus found were substituted for these letters in the 
coefficient of y, in place of that of if, and the result were 
equated to zero, we should be conducted to a biquadratic in h. 
Hence it follows that the general quin tic equation may be 
deprived of its second, third, and fourth terms by the solution 
of auxiliary equations none of which rise above the third 
degree ; and of its second, third, and fifth terms by the solu- 
tion of equations none of which rise above the fourth degree. 
This method of solution may be extended, as I propose to 
show shortly elsewhere, to the corresponding problems Jo r 
any general algebraic equation of a degree higher than the 
fourth. 
