Mr. J. Cockle on Transcendental Roots. 371 



the Appendix to ' The Ladies' Diary ' for 1839. CommentiDg 

 on such an expansion as 



1 x 



l—x + x 2 — ±x n ~ 1 + 



White observes (pp. 60, 61) that " in summing such finite 

 serieses, the remainders must never be neglected The in- 

 troduction of the remainders removes all paradox. Hence, how- 

 ever great n may be supposed, the remainders must never be 

 neglected in serieses generated by algebraic division," &c. And 

 I presume that White would have maintained that the equation 

 i 



1 — # + # 2 — ±x n ~ 1 + &c. in infin. 



1+x 



was always algebraically false, though it might be arithmetically 

 true. 



Other doubts (Ibid. p. 437), to which I sought to direct Mr. 

 Jerrard's* attention, he seems to have cleared up (S. 4. vol. hi. 

 p. 457). Butf, in leaving unassailed the theory of Abelians with 

 composite indices, he virtually leaves his own processes without 

 defence from the objections to which that theory gives rise. As 

 observed by M. Kronecker, all the distinct values of a cyclical 

 function of x , x v . . x 4 being given, the five roots x may be de- 

 duced from them rationally %. Now Mr. Jerrard's functions S, 



* Mr. Jerrard (Ibid. p. 459) has adverted to some researches of Legendre 

 on cubics,in connexion with which I would call attention to "a few remarks" 

 of James Lockhart " on the roots of the cubic a? 3 — 3a?— 1=0, which apply- 

 to all cubic equations," and which I communicated to the Mechanics' Ma- 

 gazine (vol. lv. p. 1J3). Lockhart's theorem respecting quintics (Ibid, 

 vol. liii. p. 449; et vide lv. pp. 172, 173) may be easily demonstrated by- 

 multiplying the quadratic into x, eliminating ar 3 and repeating this process 

 as far as may be necessary. 



t M. Kronecker states that every (algebraically) soluble equation of a 

 prime degree fx is an Abelian, if we regard as known a quantity which 

 itself is a root of an Abelian of the (fx. — l)th degree (see Serret, Cours, 1854, 

 p. 564). There is not, as I once suspected, a corresponding proposition 

 when fi is not prime. The solution of the problem for a composite number 

 n is obtained the moment that we have resolved it for the case in which 

 the degree of the Abelian is one of the powers (of a prime number) con- 

 tained in n (M. Kronecker, ibid. p. 566 ; M. Hermite, ib. pp. 569 et seq.). 



X See M. Hermite's Essay, Sur la Theorie des Equations Modulaires et 

 la Resolution de V equation du cinquieme degre (Paris, 1859), p. 27; consult 

 also M. Hermite's " Considerations sur la Resolution Algebrique de l'equa- 

 tion du 5° degre," at pp. 326-336 of vol. i. of the Nouvelles Annates de 

 Mathematiques, par. MM. Terquem et Gerono, 1842. I am not aware that 

 these important researches of M. Hermite have been completed. Mr. T. 

 T. Wilkinson informs me that the " suite " is not found in any subsequent 

 volume up to the present date. 



I may add that, if we had an irreducible non-cyclical reduite of the 5fith 



