results for three different apparatus, observations on the same vertical being made at the same time. The individual observations were taken thirty minutes apart Three or four means were deduced for the day. Apparatus I on the N.S. face of the pier fronting East has already been referred to in connection with figure I. Apparatus II was placed in a niche on the EW. wall of the pier fronting north, surrounding on all sides within 1 meter by the interior brick walls of the building. It thus receives secondary radiation only, and the graph in its details, departs utterly from curve I, particularly on clear days. If curves I and II were smoothed, however, they would show some resemblance.

=

Curve III are the results for the metal case (on the E.W. wall fronting south) with the needle kept in the partial vacuum, pressure p, marked on the curve. On the morning of September 12 and 13 at 39 and 36 cm. the two bodies M and m repelled each other. Even at lower pressures (p 8, 5, 3, etc.) the results seemed fluctuating. Hence after September 14, I observed for p < 1.5 c.m., only (numeral omitted), there being a slight leak in the apparatus so that a higher vacuum could not be held for a half hour. The astonishing feature of these high vacuum (III) results is that they agree very closely with the observations (I) made in a plenum; whereas if III had also been observed in a plenum, the results would necessarily be in total opposition to I, as the repulsions at the beginning of curve III indicate.

Now it may be shown by direct tests (Science, 1.c.) that the radiant forces of a hot body, M are repulsions for p < 4 cm. and attraction for higher pressures in the case. The exact pressure of radiant equilibrium resulting from this inversion is for incidental reasons difficult to specify; but one may estimate that in high vacua, y varies about 5 mm. per cm. of pressure p.

It follows from this that in the plenum apparatus (I) with eastern exposure, the attracting body M must be relatively warm in the morning and cold in the afternoon; while in the vacuum apparatus (III) with southern exposure, the body M is relatively cold in the morning and warm in the afternoon; for in such a case the radiant forces have the same sign. Hence the agreement in kind of the plenum graph, I, and the vacuum graph, III, in figure 2, after September 14, is a demonstration in a dark room, at practically constant temperature, of the rotation of the sun.

1 Advance note from a Report to the Carnegie Institute of Washington, D. C.

CONDITIONS NECESSARY AND SUFFICIENT FOR THE EXISTENCE OF A STIELTJES INTEGRAL

By R. D. CARMICHAEL

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF ILLINOIS

Communicated by E. H. Moore, October 8, 1919

The purpose of this note is to suggest a method for deriving a necessary and sufficient condition for the existence of the Stieltjes integral of f(x) as to u(x) from a to b in each of several forms generalizing those frequently employed (see Encyclopédie des sciences mathématiques, II1, pp. 171-174) in the special case of Cauchy-Riemann integration— where u(x) is of bounded variation and f(x) is bounded on the interval (ab), M≥f(x) ≥m. Bliss (PROCEEDINGS 3, 1917, pp. 633–637) has obtained one of these forms, perhaps the most satisfying of any; this note closes with his theorem, of which we give a new demonstration. Some of the other theorems stated (though here derived otherwise) are immediate consequences of the one due to Bliss or are otherwise intimately related to it, as the reader will readily see. Since it is believed that the present treatment will be found useful in connection with that of Bliss, our notation has been made to conform to his; moreover, reference to his paper is given for such isolated steps in the proof as may readily be supplied from it.

For a given partition π of (ab) due to the points x0 = a, x1,x2, Xn-1, Xn b, 0 < x; — x¡-1 < 8, form the sums

=

where ▲;u= u(x;) — u(x;−1), X; is any point of the interval (x;-1, x;), and M[m] is the least upper bound [the greatest lower bound] of f(x) on (x;-1, x;). If the limit of Sexists as d approaches zero, this limit is the Stieltjes integral of f(x) as to u(x) from a to b. In case u(x) is a monotonic non-decreasing function, the limit, if existing, of [], as & approaches zero, is here called the upper [lower] Stieltjes integral of f(x) as to u(x) from a to b.

Let us determine conditions under which the upper integral shall certainly exist when u(x) is monotonic non-decreasing. If we form a repartition of (ab) as to π by taking the points forming and certain additional points, it is clear that SS. Moreover, there is ob

π

viously a lower bound to S. Hence if the number of divisions of (ab) is increased by repartitions in such wise that & approaches zero, the sum S approaches a definite finite limit. Let us ask under what conditions we shall be led to a contradiction by supposing that two convergent sequences of sums σ for sequences of partitions (whether formed by repartitions or not) with norms & approaching 0 lead to different limits N1 and N2 where N1<N2. Let & and ʼn be two arbitrarily small positive numbers such that N1 + + n < N2. Let i be a partition of (ab) into s intervals belonging to the sequence by which N1 is defined and let s be so great that S, < N1+ §. Let π be a partition of (ab) into t intervals where t is an integer greater than s and subject to being made as large as one pleases. Let з be a partition formed by the points of 1 and 2, so that π is a repartition of both #1 and #2. Πι Then we have

η

Πι

where the intervals (x-1, *ia), obviously at most s — 1 in number, are all the intervals of T2 which are separated into parts in forming π3. Since t may be made large at our choice and since the sum in the second member of the foregoing relation never has more than s 1 terms, whatever the value of t, it is clear that t may be chosen so large that this sum is less than ʼn provided that u(x) is continuous at the discontinuities of f(x). Then we have S, ≤ST + n. But we have seen that S S S1<N1+§. Hence we have S„,<N1 + § + n. But, as t increases, ST approaches N2. Hence we have N2 ≤ N1 + § + n, contrary to the hypothesis N1++n<N2. Hence the upper integral of f(x) as to a monotonic non-decreasing function u(x) exists provided that u(x) is continuous at the points of discontinuity of f(x). The corresponding result may likewise be proved for the case of a non-decreasing function u(x); and also for the case of the lower integral. Hence, since every function of bounded variation u(x) may be expressed as the difference of two monotonic non-decreasing functions which are continuous at the points of continuity of u(x), we have the following theorem:

2

1

THEOREM I. If u(x) is of bounded variation and f(x) is bounded on (ab) and if f(x) is continuous at the points of discontinuity of u(x), then the limits as d approaches 0 of the sums S, σ of the preceding paragraph both exist.

Let us now write the function u(x) in the form

u(x) = u(a) + P(x) – N(x),

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 b 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 case u(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

(M;—m;) {P (x;) — P(x-1)} and (M;—m;){N(x;) — N (x;−1)}

shall have the limit zero as & 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

Bliss (1. c., p. 634, ll. 1-10) has shown that a necessary condition for the existence of the integral is that

lim
8-0

(M¡— m;) | u(x;) — u(x;_1) | = 0.

(2)

From this necessary condition if follows readily that f(x) must be continuous 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, ll. 1–7), where ¡(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 to j(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) implies that of f(x) as to U(x). In view of the results of the

preceding paragraph and of the fact that U(x) is obviously equal to V(x) + J(x), where V(x) is the total variation of v(x) on (ax), it is obviously sufficient to prove this for the case when u(x) is a continuous function. Now, when u(x) is continuous, we have (Vallée Poussin, Cours d'Analyse, vol. 1, 3rd edn, p. 73)

(3)

with a like relation when b is replaced by x and the interval (ab) by the interval (ax). Since no | u(x;) — u(x;_1) | is greater than the corresponding difference U(x;) — U(x-1) and since the sum of the latter differences, for i = 1, 2, . . ., n, is U(b), we see that

lim Σ[U (x;) – U (x; –1 ) − | u (x;) — u (x;–1) | ] = 0

8=0

(4)

Hence for every e there

n) of the sum in (4)

is less than e when 8<81. Hence, for such &, we have

(M¡—m;)[U(x;) — U (x;-1) — | u(x;) — u(x;_1) | ] < (M — m) e (5)

Hence, as d approaches zero the sum in (5) approaches zero as a limit.
This result and relation (2) imply (1). But the latter is sufficient to
the existence of the integral of f(x) as to U(x), and indeed as to u(x).
We are thus led to the following theorem:

THEOREM II. If u(x) is of bounded variation and f(x) is bounded on (ab), then a necessary and sufficient 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 U(x). Since (1) and (2) are identical when u(x) U(x) we now have readily the following theorems:

THEOREM III. A necessary and sufficient condition for the existence of the integral of the bounded function f(x) as to the function u(x) of bounded variation is that the upper and lower integrals of f(x) as to the total variation function U(x) of u(x) shall exist and be equal.

THEOREM IV. A necessary and sufficient condition for the existence of the integral from a to b of the bounded function f(x) as to the function u(x) of bounded variation is that the total oscillation of f(x) as to u(x) from a to b shall be zero, that is, that