MATHEMATICS: R. D. CARMICHAEL 
u{x) = u(a) + P(x) — N(x)y 
553 
where P(x) and N(x) are respectively the positive and the negative 
variation of u(x) on (ax); and by U(x) denote the total variation 
P(x) + N{x). Then P(x), N(x), U{x) are continuous at the points of 
continuity of u{x), 
A sufficient condition for the existence of the integral of f(x) as to 
u(x) from a to 6 is the existence of the integral of f(x) as to P{x) and 
as to N{x). Sufficient to this is the existence and equality of the upper 
and lower integrals of f(x) as to P(x) and as to N(x). In C3,seu{x) is 
continuous or f(x) is continuous at the discontinuities of u{x), these 
upper and lower integrals surely exist and a sufficient condition for 
their equality is that the sums 
n n 
2 (Mi-m,) {P(xd -P } and ^ (Mi-md {N(xd-N(x,.,) ] 
shall have the limit zero as 8 approaches zero. Hence a sufficient 
condition for the existence of the integral of f(x) as to u{x) from a to b 
is that 
n 
Hm y\ {Mi-m,) [ C/fe) - U{xi.,) } = 0. (D 
Bliss (1. c, p. 634, 11. 1-10) has shown that a necessary condition for 
the existence of the integral is that 
n 
From this necessary condition if follows readily that f{x) must be con- 
tinuous at the points of discontinuity of u{x) (see Bliss, 1. c, p. 636, 
11. 7-17). If we write u{x) = v(x) +j{x) (Bliss, p. 636, 11. 1-7), where 
j{x) is the function of 'jumps' of u(x), it may be shown (Bliss, p. 636, 
11. 18-36) that the integral of f(x) as toj(x) exists whenever that as to 
u(x) exists. In the same way it may be shown that the integral of 
f{x) as to J(x) must also exist, where J(x) is the total variation of j{x) 
on the interval (ax). Hence a necessary condition for the existence of 
the integral of f(x) as to u(x) is the existence of the integral of f(x) as 
to v(x). 
We propose to show next that the existence of the integral of f(x) as 
to u(x) impKes that of f(x) as to U(x). In view of the results of the 
