Colloquium publications.

110

THE CAMBRIDGE COLLOQUIUM.

where {vi(t)} is a sequence of functions
vi(t) = ailsil(t)pil(t) + ai2Ai2(t)ti2(t) + * * * + -aiini(t)-in(t),
the functions -ij and iij being arbitrary functions from 9X, and
the coefficients aij arbitrary numbers from t. The convergence,
as is indicated, is uniform over $, relatively to some scale function which is the product of two functions of 9PW. Similarly, the
significance of (13) is that St is the totality of functions K(st)
such that
(13') K(st) = lim vi(st),
i=00
where {vi(st) } is a sequence of functions
ni
vi(st)= Ejaijiij(s)ij(t).
The convergence is uniform over the range ($13) (i. e., the range
of the composite variable s, t) relatively to some function A(s)u(t),
as a scale function.
All of the description of the basis, just given, is implied by the
array (11). In order to develop the complete analysis of the
equation (G) it is necessary merely to specify certain postulates
which are to hold for the class of functions 9N, and the operation J.
As to the class of functions E9, the postulates must be four,
(L), (C), (Do) and (D) as follows:
(L) If 1 UL, '* *, un are functions of 92, and al, * *, an elements
of W, then alui + * * * + ann is a function of 9f.
(C) If {U,,} represents a sequence of functions of 9P, then the
limit
- = lim nn ($; 9T)
is a function of 9P, provided that the convergence is uniform
relatively to a scale function of 9W.
(Do) If,u is a function of 9), there must be some real-valued
nowhere-negative function /uo (which may vary with 1u) of 9X,
such that for every p of $3
Jl(p) I|~ o(p).