MATHEMATICS: L. P. EISENHART 
555 
n 
Theorem V. In order that a bounded function f{x) shall be integrable 
from a to b as to a function u{x) of bounded variation, it is necessary and 
sufficient that the interval (ab) may be divided into partial intervals so that 
the total variation of u{x) in those in which the oscillation of f{x) is greater 
than an arbitrarily preassigned positive number 03 shall also be as small 
as one wishes. 
Theorem VI. // u{x) is of bounded variation andf(x) is bounded on 
the interval (ab), then a necessary and sufficient condition for the existence 
of the integral of f(x) as to u(x) from a to b is that the total variation of 
u(x) on the set D of discontinuities of f(x) shall be zero {Theorem of Bliss). 
Of the four preceding theorems the only one needing further proof 
is the last. [For the definition of the total variation of u{x) on a set 
of points, see Bliss, 1. c, p. 633, 11. 12-19.] Let €1, €2, €3, ... be a 
sequence of positive numbers decreasing monotonically toward zero, 
and let Z)i, D2, D3, ... be the closed set of points at which the oscilla- 
tion of f(x) is ^ €1, ^ €2, ^ €3, . . . Then the set D of discontinuities 
oif(x) is the limit of the set when n is indefinitely increased. Now 
if /(x) is integrable as to u(x) we have seen that the interval (ab) may 
be divided into partial intervals so that the total variation of u{x) on 
those in which the oscillation of/(x) is greater than an arbitrarily pre- 
assigned positive number shall be as small as one pleases; and this im- 
plies that the total variation of u{x) on D„, and hence on D, is zero. 
Again, if the total variation of u{x) on D is zero so is it on D„ for every 
n; and hence f(x) is integrable as to u(x) since it is such that the inter- 
val (ab) may be divided into partial intervals so that the sum of those 
in which the oscillation is greater than an arbitrarily preassigned 
positive number shall be as small as one pleases. 
TRANSFORMATIONS OF CYCLIC SYSTEMS OF CIRCLES 
By L. p. Eisenhart 
Department of Mathematics, Princeton University 
Communicated by E. H. Moore, October 9, 1919 
When two surfaces, S and S, are applicable, there is a unique con- 
jugate system on S which corresponds to a conjugate system on S. 
Denote these conjugate systems, or nets, by N and N respectively. 
