388 TRANSACTIONS OF SECTION A. 
the class ). Fundamentally important instances of such systems are the 
following :— 
I) P! is the class of a single element p. Ml) is the class of all functions p of 
—i.e, {M§! is the real number system A. ; 
II,,) 4, is the class of n elements p. MI)", is the class of all functions p» of 
—i.e. {IS", is the class of all n-partite real numbers. 
III) P"" is the denumerable infinitude of elements p=1, 2, 3, ..., m, ... 
@"™ is the class of all functions » of p for which the corresponding series, 
Su(p), is absolutely convergent. 
IV) P* is the linear interval : 0<p<1, of the real number system. JISV is 
the class of all continuous functions p of p. 
Properties of systems 5 common to the systems 31, .., 2"V are of fundamental 
importance. Such properties of = are essentially properties of {Ml}, since properties 
(eg. convergence, continuity) of individual functions » have usually refer- 
ence to special features (e.g. denumerability, possession of the operation — and of 
the relation <) of the class [f, the range of the variable p. Thus, there are the 
closure properties: the absolute value of a function of lS, or the product by a 
constant of a function of fl, or the sum or the product of two functions of Jl} is 
in every case a function of @M§. For {M—/iS'¥ there is furthermore the closure 
property that the limit of a uniformly convergent sequence of functions of ll) isa 
function of gM. This is not a property of (S—@S'". However, if we generalise 
uniform convergence to convergence uniform relative to a function taken as 
scale of convergence, replacing the e entering the final inequality of the definition 
of uniform convergence by e multiplied by the absolute value of the scale 
function, we secure a corresponding common property of the classes /S, . . fHS'Y, 
viz., the class /§ contains the limit of every sequence of functions of @f§ which 
converges uniformly relatively to some function of fM. There is further the 
dominance property D, that for every sequence of functions of ff} there exists a 
function of @ such that every function of the sequence is dominated by some 
numerical multiple of the function. 
A memoir, entitled ‘ Introduction to a Form of General Analysis,’ will appear 
shortly in a volume to be published by the Yale University Press. In the first part 
of this memoir I consider, especially with respect to these closure and dominance 
properties, the inter-relations of classes ¢M§ with a common range fp. In the 
second part I consider the composition of ranges [P and of classes Mf}, obtaining, 
by the introduction of the notion development of a range ff), and allied properties, 
somewhat akin to convergence and continuity, of classes /§, a very general 
theorem characterising functionally in terms of suitably conditioned classes 
M@, gi’ the functions of their *-composite class (JIN/IN’),; ¢g., for 
=A’, aS'=AN'", the *-composite class is the class of all functions of the 
two variables p, p’ continuous over the square: O<p, p’<1, and these functions are 
likewise the functions continuous in p’ for every p and uniformly continuous 
in p on P uniformly in p’ on fp’. 
In a paper: ‘On a Form of General Analysis with Application to Linear 
Differential and Integral Equations,! I defined a system as a system of General 
Analysis, and the corresponding theory as a theory of General Analysis in case 
they involve one or more variables about whose character and range of variation 
information is given only by the mediation of conditions involving one or more 
classes of functions of those and perhaps other variables. In connection with 
functional transformations of classes fff} into classes fff)’ there arises the question 
of generalising the fundamental existence theorems of Analysis. Indeed, the 
systematic study of the various analytic doctrines from the point of view of 
General Analysis will reveal types of properties and theorems of inter-relation of 
the highest novelty and importance. In a memoir to be presented to the London 
Mathematical Society I intend to take up in some detail the theory of systems 
of linear differential equations in General Analysis; the theory outlined in my 
Rome paper will be much simplified, 
1 Atti del IV Congresso Internationale dei Matematict, vol. ii. pp. 98-114. 
