Mr. J. Cockle on the Remarks of Mr. Jerrard. 197 



by the fact that, for such equations, u and v vanish, and the 

 reduite is not an Abelian sextic. 



2. The function designated by \/K is generally expressible 

 in rational terms of K. The suggestion that that function is an 

 irreducible surd cannot be reconciled with the fact that it is a 

 root of an equation of the sixth degree with rational coefficients. 

 Assume, however, for a moment, that it can. Then, asw r e learn 

 from a theorem of Abel, four other roots will be obtained by 

 multiplying \/K into the unreal fifth roots of unity. Hence, 

 denoting the remaining root by 6 and the absolute term by A, 

 we find 



Next, change the signs of the quadratic surds in K, or multiply 

 its cubic surds into unreal cube roots of unity, or perform both 

 operations simultaneously, and denote the result by K'. Then, 



by AbePs theorem, ^. f is a root of the sextic. Consequently, for 



some of the forms of K', we may establish the relation 



in other words \/K is not an irreducible surd, and the argument 

 of M. Hermite is unshaken. 



3. Mr. Jerrard' s paper in this Journal for May 1861 does not 

 disprove a rational commmunication between the resolvent pro- 

 duct and its fifth power. His equation (e\) is not illusory but 

 identical, and we may expunge it and substitute u° for v in the 

 second of the equations marked (e). The result of this substitu- 

 tion, reduced by means of (U), may be made to take the form 



X 5 + \u + X 3 w 2 + . . + \u b == 0, 



and the six linear, but not homogeneous, equations 



\ 5 = 0, \ 4 = 0,....X = 0, 



will determine the six quantities v s , v 4 , . . v . 



4. Mr. Jerrard says that it follows from what has been de- 

 monstrated in his ' Essay ' that the fifth power of the resolvent 

 product depends directly upon an Abelian equation. But the sup- 

 posed demonstration consists in applying to 5 the suffix /upon 

 no assigned grounds. The consequence of this arbitrary assump- 

 tion is a result apparently so inexplicable that Mr. Jerrard 

 deemed it necessary to verify it. The attempted verification in 

 this Journal for February 1852 is, to me at least, as inexplicable 

 as the result itself; and in his ' Essay ' Mr. Jerrard gives no 

 explanation whatever of the difficulty. Tbat ( Essay/ I am 



