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LXIX. Analysis of the Theory of Equations. Second and Con- 

 cluding Part. By James Cockle, Esq., M.A., Barrister- 

 at-Law. In a Letter to T. S. Davies, Esq., F.R.S. $c* 



[The First Part will be found at pp. 351-367 of vol. xxxii. S. 3.] 



2 Pump Court, Temple, 

 My DEAR Sir, October 8, 1850. 



1. "\TARIOUS considerations, which it is not necessary to 



▼ detail, render it desirable that, in resuming and con- 

 cluding the subject of a former letter of mine to you, I should 

 confine my remarks to rational equations. And I shall be 

 forgiven for reminding you, and those under whose notice 

 this letter may perhaps eventually come, of the existence in 

 print, not only of that former letter, but also of my Notes on 

 the Theory of Algebraic Equations, commenced in the forty- 

 sixth volume of the Mechanics' Magazine, and the Third and 

 Concluding Series of which is now in the course of publica- 

 tion in that Journal. I mention these Notes rather than other 

 papers of mine, because in them, as in this Analysis, I have 

 attempted, though from different points of view, to take some- 

 thing like a general survey of the whole subject. 



2. All those equations, the discussion of which forms the 

 primary object of the Theory before us, involve at least one 

 general symbol — that of the unknown quantity — and are, con- 

 sequently, algebraic in their form. But a little consideration 

 enables us to perceive that such equations may be advantage- 

 ously separated into two great classes, the characteristics of 

 which are respectively furnished by an observation of the 

 nature of the coefficients of the unknown quantity. When any 

 one or more of the coefficients of a given equation is a letter, 

 or general symbol, the equation may be termed a literal, 

 symbolic, or algebraic equation. When all its coefficients are 

 numbers, it may be called a numerical or arithmetical equation. 

 These two classes then, (a) the literal, and (b) the numerical, 

 include all equations algebraic in their form. 



3. But, if we should carry our views further than the mere 

 form of the equation, and look beyond that to the object we 

 have in view in discussing it, we must admit a third and hy- 

 brid class which partakes of the natures of the other two, being 

 literal in its form, but virtually numerical, inasmuch as all our 

 processes treat the literal coefficients, not as general symbols, 

 but as generalized numbers. For the sake of clear and accu- 

 rate classification, I shall separate the class of literal equations 

 into two others, the algebraic, and the quasi-algebraic. In 



* Communicated by T. S. Davies, Esq., F.R.S.L. & E., F.S.A. 



