ESSAYS 509 



Since the operations are of very different types, the analogy between the two 

 equations is not very close and is confined almost to the fact that the method 

 of iteration can be used to obtain the solution in certain cases, but the details of 

 the analysis are quite different. 



The use of symbols to denote operations was advocated by Leibniz, 1 who 

 showed that the operator D = d/dx can in some cases be treated as an algebraic 

 quantity, as for instance in the equation 



D m (D"y) = D"' + "y 



and in the well-known expression for the » th differential coefficient of the product 

 of two functions. 



Although this calculus of symbols was developed by Lagrange 3 (1772), the first 

 great step in the theory was made by Servois, 3 who showed that the analogies 

 between the calculus of symbols and ordinary algebra depend on the fact that the 

 various symbols A, djdx, of the differential calculus and calculus of finite differences 

 possess the commutative, distributive, and associative properties : the first two 

 terms were in fact introduced by Servois. Two symbols A, B, are said to obey 

 the commutative law if 



AB = BA; 



they are said to obey the associative law if 



A{BC) = (AB)C, 



where C is any other symbol of the same type. If the symbols A, B denote 

 operations it is convenient to interpret AB as the result of first operating with B 

 and then with A. If the two operations obey the commutative law, the order 

 in which they are performed is immaterial. 



In order to define the distributive law we must introduce the idea of objects 

 or functions on which the symbol or operation acts. If a, /3, . . . denote different 

 objects, an operation A is said to be distributive when 



A(a + /3 -I- ) = A(a) + A<£) + . . ., A(ca) = cA(a), 



c being an arbitrary constant. 



General analysis can be divided into two main branches according as the 

 operations which are considered are distributive or not. When we limit ourselves 

 to distributive operations we practically abandon the interesting branch of analysis 

 which is generally called " the calculus of functions." The analogies which exist 

 between this subject and the theory of distributive operations depend chiefly on 

 the idea of iteration and the commutative and associative laws. 



In the calculus of functions the commutative law suggests the interesting 

 functional equation 



/OK*)] - *[/(.r)], 



which may be regarded as an equation for <fi when f is given. In the theory 

 of integral equations the corresponding equation is 



f b f{s,x)${xj) dx =f 6 J>(s,x)f(x,t)dx, 



1 Berol. Miscell. 1710, 1, 160. Cf. the correspondence of Leibniz and John 

 Bernoulli in Leibniz's Mathematical Works (1), 3, 175. 



2 CEuvres, 3, 441. 



3 Gergonne's Ann. 1814,5,93. 



