39G ilEPOttTS ON TttE STATE OB 1 SCIENCE. 



l 



then (f>(y) = P cosec v(y - x)x(x)dx, 



J (Hardy) 



where P denotes that the principal value of the integral must be taken. 



18. Distributive Operations. 



It has been pointed out by Pincherle l that many of the theorems of 

 integral equations are simply illustrations of theorems belonging to the 

 general theory of distributive operations, and this is one of the leading 

 ideas in the applications to integral equations of the General Analysis 

 developed by E. H. Moore. 2 



Although the use of symbols to denote operations was first advocated 

 by Leibnitz 3 and his calculus of symbols was developed by Lagrange* 4 

 the first great step in the theory was made by Servois s Who showed that 

 the analogies between the calculus of symbols and ordinary algebra depend 



on the fact that the various symbols A, - - , 2, &c, possess the commuta- 



CltXs 



tive, 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 deilote 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 per- 

 formed 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, ft ... . 

 denote different objects, an operation A is said to be distributive when 



A (a + ft +....) = A(«) + A(/3) + . . . . A(ca) = cA(«). 



A set of functions which are chosen according to some law is called 

 afield of functions, a function a is said to be a condensation function 

 of the field when to every positive quantity e there is a function ft 

 belonging to the field and different from « and such that 



A field Which contains all its condensation functions is said to be 



1 Bend. Lined, 1906 ; Bologna Memoirs, set. 6», vol. iii. A good account of the 

 theory of distributive operation is given in the book by Pincherle and Amaldi, Le 

 Operazioni Distributive, Bologna (1901). 



3 Newhaven, Mathematical Colloquium • Yale Univ. Press 1910 

 „ * nwol-Miscell, 1710, i. p. 160. 'The Correspondence of Leibnitz and John 

 Bernoulli, Leibnitz's Mathematieal Works (1), iii , p 175 



• (Euvret, 1772, t. 3, p. 441. * Gergonne Ann., 1814, t. 6, p. 93. 



