MATHEMATICS: E. H. MOORE 
629 
valued function, t{p) being the length of the curve p or the average 
value over its range of definition of the function p. However, the nature 
of the element p is not fully specified in Frechet's theory ^ (1906) of sets 
of elements ^ of a class ^ and of continuous functions on such sets. 
This general theory has as instances the theories for the linear con- 
tinuum ^ and for the w-dimensional space initiated by G. Cantor 
and for the class ^ of continuous curves due especially to Arzel^. 
Frechet conditions his class ^ by a. definite but undefined relation L 
(the concept of convergence of a sequence {pj - piy p2, • • • , 
pnj . . . , of elements p to an element po of ^ as a limit) possess- 
ing certain properties; these properties of the relation L are in the 
special theories immediate consequences of the current definitions of 
the relation L in those theories. 
Those theories of Analysis in which at least certain of the functions 
involved are on a range ^ of elements ^ of a nature not specified, or at 
least not fully specified, we may designate as theories of General Analysis. 
Thus, Frechet's theory with basis ( ^ ; L) is a theory of General Analysis, 
as is Hkewise my theory ^ (1910) of classes of functions on a general 
range ^. 
A general range ^ is an arbitrary particular range ^ with abstraction 
of its particular features, e.g., Frechet's range ^ with abstraction of 
the feature L. Properties of functions, classes of functions, etc., on or 
connected with a general range ^ (whose definitions accordingly involve 
no particular features of the range ^) are of 'general reference,' while 
others are of 'special reference.' Thus, relative to a linear interval 
^, the continuity of a function is of special reference, while the prop- 
erty of uniformity of convergence of a sequence of functions and the 
property of the class of all continuous functions that the Hmit of a 
uniformly convergent sequence of the class belongs to the class are 
properties of general reference. 
2. General Integral Analysis. A theory of General Analysis involving 
a numerically valued single-valued functional operation J (of the type 
of definite integration) on a class ^ of functions k whose range of 
definition involves a general range ^ we may designate as a theory of 
General Integral Analysis. Thus, my theory^ of linear integral equa- 
tions is a theory of General Integral Analysis. This general theory has 
as instances the classical theory due to Fredholm and Hilbert-Schmidt 
for the case of continuous functions, and other classical theories. • 
3. Definition of Limit in General Integral Analysis. In a subsequent 
note I shall indicate a general theory of linear integral equations having 
as instance Hilbert's theory of functions of infinitely many variables. 
