MATHEMATICS: C. A. FISCHER- 
641 
In the present note equation (1) is derived by a different method, 
and the function u (s, t) is proved to be regular in both arguments when 
determined by this method, and unique if regular. The theorem can 
then be extended by mathematical induction to w-linear functionals. 
The variation V^u of u {s, t) is defined by Frechet as the least upper 
bound of the expression 
where ei and ej are taken equal to plus or minus 1 in such a way as to 
make the summation as large as possible, and 
Aij u {s, /) = u {Si+ 1, ^^-^ i) - u (Si, tj^ i) - u (Sj^i, tj) + U (Si, tj). 
The double Stieltjes integrals considered here will all be of the form 
J W g W d,u {s, t) = limit 2/ (^i) ^ij ^ 0, 
where the region T is defined by the inequalities 
and s'i and tj are in the intervals (si, Si+i) and (tj, tj+i) respectively. 
Such an integral always exists if / and g are continuous and VzU is 
finite, and it must satisfy the inequality 
I J^/ (s) g (0 d,u is, t) I ^ mfmgV^u.' (2) 
A function u {s, t) will be called regular here if F2W is finite, u [a, t) = 
u{s, h) = 0, andw(j?, t) = u{s-\-0, t)=u(s, t-\-0) excepting on the bound- 
ary of T. This makes some of the work simpler than to assume with 
Frechet that 2u (s, t) = u(s — 0, t) + ti (s + 0, /). If ti (s, t) is regular, 
its variation in one variable, when the other is constant, cannot be 
greater than Wu. The double integral can then be expressed as an 
iterated integral by the equation^ 
Jy/ is) g (0 d,u is, t) = J]'/ (5) d, J^'' g it) d,u (s, t). 
The functional 
'^k (t);s) = j'^git) d,u is, t) 
can be proved regular 'n s by proving that its var'ation cannot be greater 
than mgV^u, and then proving that v{g;s-\-e) approaches v{g\s) when e 
