360 Mr. J. Cockle's Analysis of the Theory of Equations^ 



to assume an expression consisting o^ four terms for the pur- 

 pose of representing the root of the transformed equation. 

 Every diminution of the number of the terms of this expres- 

 sion very greatly facilitates the formation of the transformed 

 equation. 



A purely indirect method of transformation is that which I 

 sent through you to the Philosophical Magazine, and which 

 will be found at pp. SSS-SS* of the twenty-sixth volume of that 

 work. It proceeds by modifying the roots at once. 



3. On our knowledge of the number of the roots of a given 

 equation is founded much of the theory of algebraic equations ; 

 so that it becomes absolutely necessary that a demonstration 

 that the number of its roots equals the number of its dimen- 

 sions should be found in every work treating of the subject. 

 So far as the algebraic theory is concerned, the demonstration 

 of Cauchy, of the existence of a root of an equation, is perhaps 

 the preferable one. 



4. The number of the roots of an equation once known to 

 be in general the same as the index of its dimensions, the 

 combination of the theory o^ symmetric functions with that of 

 equations at once ensues. By means of this combination, the 

 elimination requisite in the transformation of algebraic equa- 

 tions is greatly facilitated. An instance of this occurs in the 

 improvements effected by Sir J. W. Lubbock in the processes 

 of Bezout, and in Mr. Jerrard's eliminations. The latter 

 mathematician has made an important advance in this branch 

 of science, by preserving the notation unaltered in the case 

 where different terms of a symmetric function become identical. 



We thus see that indirect methods of elimination may be 

 preferable to direct ones; and the same theory of symmetric 

 functions which enables us to employ them with effect, is 

 equally essential to the efficiencj' of the (indirect) general me- 

 thod of solving equations given by Lagrange. It is not my 

 intention to enter into the subject of the finite solution of equa- 

 tions of the fifth degree ; but, on the development of Lagrange's 

 method in such a case, there is much information to be ob- 

 tained from a paper by Sir W, R. Hamilton (Transactions 

 of the Royal Irish Academy, vol. xix. pp. 329-376) on the 

 Formulas of Professor Badano. 



5. Tlie rule of Descartes (or more properly perhaps of 

 Harriot) enables us to ascertain the greatest number of posi- 

 tive or of negative (real) roots that can possibly enter into a 

 given equation. The number of unreal roots is capable of 

 being infallibly ascertained by the criterion of Sturm ; but the 

 labour which its application requires is a serious objection {e). 



(e). This objection was, I believe, first made in the Phil. Mag. 



