with Mr. T. S. Davies's Notes on some of the Topics. 353 



but it is often a very difficult matter to determine whether 

 what purports to be an equation in the strict sense of the term 

 be really so, or whether it be not an identity or a contradiction. 



In one point of view, then, the whole of algebraic science 

 may be said to be contained in the theory of equations. Even 

 when the latter term is used in its restricted meaning, the 

 theory is intimately connected with every part of the science, 

 and we find ourselves face to face with it at a very early period 

 of our progress into the domain of algebra. 



The vast and illimitable field comprehended under the 

 phrase theory of equations must undergo a minute subdivision 

 before we shall be in a condition, either to examine it with 

 anything like ease, or to arrive at a just appreciation of those 

 whose labours have advanced it to its present state, or to form 

 a proper estimate of what may be expected from future re- 

 searches. 



Corresponding to each of the following topics there is a 

 distinct department of the theory of equations, and this di- 

 stinctness we must keep steadily before us. The topics in 

 question are as follows : — 



I. The finite and rigorous solution of a given equation or 

 system of equations. II. The transformation of a given equa- 

 tion into another of a different form. III. The number of 

 solutions of a given equation or system of equations. IV. The 

 relations between the coefficients and roots of equations. V. 

 The natura of the roots. VI. The limits within which they 

 lie. VII, Their numerical values. 



We may also regard the theory of equations as consisting of 

 two great parts : the theory of algebraic equations, and the 

 theory of ?i«;«tv7c«^ equations. In the former part, accurate 

 results are exclusively aimed at; and the algebraic theory may 

 be defined as that which treats of the rigorous solution and 

 transformation of equations, and the number and properties 

 of their rools symbolically considered ; and we may here in- 

 clude the theory of symmetric functions. 



The aim of the numerical theory is to ascertain the nature 

 and limits of the roots of equations, and to arrive at values 

 wliich shall enable us to satisfy equations, either accurately or 

 to any required degree of approximation. The subject of 

 transformation plays an important part in this as well as in the 

 algebraic theory. 



A literal equation may be treated either as an algebraic or 

 a numerical one. In the latter case the letters stand for ge- 

 neralized numbers, and the processes applied to them are but 

 universal types or examples of those which are to be employed 

 when concrete are substituted for abstract numbers. It is only 



Phil. Mag. S. 3 . Vol . 32. No, 2 1 6. Mr/// 1 84.8 . 2 A 



