366 Mr. J. Cockle's Analysis of the Theory of Equations. 



biaical theory itself is more fully developed than in any of the others 

 — and with quite as much skill and elegance. 



If I may be permitted to offer a few suggestions in connection 

 with future inquiries on equations, they would be of the following 

 kind : — 



1. That the algebraic investigation should be as little cramped as 

 possible by reference to the object of solving the numerical equa- 

 tions. The general properties of equations are of high interest and 

 furnish exercise for transcendent skill, independently of all col- 

 lateral practical applications of the results obtained. Any attempt 

 to keep in view simultaneously two distinct purposes is calculated 

 to frustrate the accomplishment of each. From the cornucopia (to 

 adopt a happy metaphor of Sir John Herschel on another subject) 

 of results obtained by the algebraical investigation, some property 

 may fall out, that would effect all that is required in the numerical 

 investigation. Numerical aims, however, only clog the wings of the 

 general algebraic investigator; and if numerical solution shall be 

 benefited by such means, it will be only accidentally so— like 

 Girard's spherical excess in the geodetical problem. 



2. That the attention of those who aim at the numerical solution 

 should be directed to obtaining more simple means (I mean less 

 labor'mis) of accomplishing the following purposes : — 



(a). Finding the equal roots. 



(/3). Finding the limits between which the several roots lie 

 without transforming the equation and without the substitution of 

 particular numbers in any equation, either the original or any one 

 derived from it by any process whatever. 



(y). An infallible criterion whether the roots lying within a given 

 interval be real or imaginary — -however closely the real ones may 

 approach to each other, and however small the value of /3 may be 

 in the expression a + /3^ZrY when they are imaginary. 



(iJ). The relations between the limiting values of an imaginary 

 pair, and the real and imaginary parts of that pair. " The real 

 part may lie in the positive region and yet the pair be indicated in 

 the negative region." — Young, Researches, p. 54. 



(g). Supposing the roots of the given equation to be represented 

 by a„, + /3,„, and when the equation is of an odd degree (-In+l) we 

 introduce the new root a„ — a.,„ and denote the corresponding odd 

 one by a„ + a„, we shall have the equation converted into one per- 

 haps better adapted to examination. If we can form two subsidiary 

 equations, one involving all the as and the other all the /3s ; and 

 if we can moreover discover any criterion by which the proper assort- 

 ment of as and /3s can be effected, the complete numerical solution 

 will be within our reach. It must be remarked that the /3s only 

 appear of even powers in the substitution, and hence we s houl d get 

 at once a knowledge whether they involved the symbol \/ — \. 



The method of Lagrange involves both the as and the /3s in one 

 expression, and possibly the proposal (e) may be supposed to be 

 mixed up in the method of that distinguished analyst : but to be 

 really effective, some method of separating the two parts of each 



