NATURE OF FUNCTIONAL EQUATIONS. 127 



To solve an equation of any kind, is to determine the un- 

 known operations by means of the known. If one symbol is 

 said to be a function of another, it is, in reality, the result of 

 an operation performed upon it. Thus the idea of functional 

 dependence pervades the whole science of symbols, and on this 

 idea the following remarks are based. 



In order to classify equations, we can make use of two 

 considerations : 1st. The nature of the operations which are 

 combined together; 2nd. The order in which they succeed 

 one another in the congeries of operations which is made equal 

 to zero. 



Let us illustrate these remarks by some examples. 



If we have an equation of the form 



x*+ax+t> = ........................ (1), 



the bases are a and b ; the operations are, first, the unknown 

 one denoted by x, and then certain known ones denoted by the 

 index, the coefficient, &c. All these are what are called alge- 

 braical operations. 



If again we have an equation of the form 



. 



the base is x\ the operations are, first, the unknown one 

 denoted by #, which is a function of x, then the operation 



-T-, and lastly, certain algebraical operations. From the pre- 



UX 



sence of the operation -y- , this is called a differential equa- 



tion. Equations (1) and (2) are discriminated by the nature 

 of the operations combined, on our first principle of classifi- 

 cation. 



But in one important point these equations agree. In 

 both, the unknown operation is performed immediately on the 

 bases ; the known are subsequent to the unknown : but in what 

 are called functional equations this is not so. Thus, in the 

 equation 



$(mx) + x = Q ........................ (3), 



the base is x, the unknown operation is <f>, which is performed, 

 not on x, but on the result of a previous operation. In the 



