354 Mr. J. Cockle's Analysis of the Theory of Equations, 



in their possible application to particular numerical instances 

 that such processes are of any value whatever. 



On the other hand, a numerical equation (as we may call 

 an equation with numerical coefficients) may be treated as an 

 algebraical one, that is to say, by the algebraical theory. 

 We may obtain rigorous expressions for its roots, ascertain 

 their number, discuss their relations to the coefficients, or 

 transform the equation itself by rigorous processes and under 

 exact forms, as well when the coefficients are numbers as when 

 they are symbols. 



The different branches of the subject will thus be distin- 

 guished by bearing in mind their end and object, and not 

 by the circumstance of the equation under discussion being 

 numerical or not in its form {b). 



(b). This distinction is very important, and has been too much 

 overlooked by analysts. It does not, however, appear to be very 

 probable that the solution of numerical equations is ever likely to 

 receive much improvement from the most extended researches into 

 the algebraical theory. Even could we find "rigorous expressions 

 for the roots of an equation in terms of its coefficient?," it is proved 

 by the algebraical theory, and justified also by analogy, that these 

 " rigorous expressions " will contain radicals of a degree as high as 

 the index of the equation itself. The roots to be extracted would 

 thus become immensely numerous in equations of only a moderately 

 elevated degree. Now it is a remarkable fact, that by Horner's pro- 

 cess the gradual evolution of the figures of the root involves no 

 more labour than the similar evolution in a binomial equation of the 

 same degree. It hence follows that any system of solution which 

 should aim to present us with the radical expressions for the root, 

 would for all numerical purposes be utterly useless, since it would 

 add the most intolerable complication and labour to the practical 

 part of the process. This, too, would be the case, even Avere we to 

 bring into play the most powerful engines for extraction we possess, 

 viz. the methods of Horner and Weddle for the evolution of the 

 figures. The method proposed would be strikingly retrograde ; 

 and those who have given the closest attention to the question 

 under this aspect are, I believe, fully satisfied that this stage (evolu- 

 tion of the figures) has attained its final simplification. It is the 

 only one, however, of which this can be said. 



If by the order of an equation we denote the number of 

 unknown quantities which enter into it, an equation will be 

 completely defined when its degree and order are given. Now 

 equations of the first order present features so peculiar to 

 themselves, that it might almost be thought that the best 

 course would be to treat the theory of equations of that order 

 in the first instance and separately, and afterwards to engraft 

 on it such extensions as may be necessary for the purpose of 



