Mr. Harley’s paper on Quintics. 273 
the roots of the trinomial equation also must, if there be no error 
in the processes, admit of bemg expressed by means of radicals 
characterized by the symbols 7, ~/ only, that is to say, without 
the aid of any function of the form “z. And in effecting the 
reduction of the general equation to the trinomial one, no such 
function is introduced. 
The method, therefore, even should it be found to extend 
beyond the extreme case of #,=0, z,=0,..2;=0, cannot be 
applied except when 4/z (I mean an irreducible radical of that 
form) does not enter into the expressions for the roots. 
2. Postscript.—Since writing these lines, I have seen a paper 
by Mr. Cockle in the Philosophical Magazine for the present 
month. The objection which attaches to his method from the 
absence of quintic surds is not overlooked. But, instead of con- 
fining himself to the elucidation of the origin of an error, the 
existence of which is so obvious, he goes on to infer, from the 
failure of his method, the impossibility of effecting, except in a 
limited number of cases, the algebraical resolution of equations 
of the fifth degree,—a result in the accuracy of which I cannot 
concur. It is true that all methods of solution, if equally general, 
must ultimately coincide. The success of one involves the suc- 
cess of all*. The failure of one the failure of all. It is, how- 
ever, far otherwise with respect to methods subject to conditions 
not necessarily, or rather not universally, inherent in the subject. 
3. But I come to what Mr. Cockle brings forward as his 
weightiest objection to my method. He says, “The error of 
Mr. Jerrard inheres, in my opinion, in his mode of comparing 
the equations (ab) and (ac) at pages 80 and 81 of his most 
valuable ‘ Essay.’ His functions ,=, ,&, ,=, and ,= in art. 104 
are foreign to the question, mere instruments for eliminating 
radicalities. They lead to no other result than that to which the 
immediate comparison of (ac) and 
| =0 
would conduct us, viz. an expression for & into which Pree) 
enters irrationally.” Doubtless the result in both cases would 
be the same. On substituting the expression for = in (ac) we 
should find by the evanescence of Nz, No, N,, No that 
Nu) =HV6) ==p 
* See applications II. III. at pp. 84, 85 of my ‘ Essay on the Resolution 
ofEquations,’ published by Taylor and Francis, Red Lion Court, Fleet 
Street, London. 
