MATHEMATICS: L. P. EISENHART 
637 
that the integral shall exist with v in place of u. For since u = v -\- j 
every sum 5 for u is the sum of two similar ones for v and j, and 
when / and u have no common discontinuity the last has a definite 
limit as d approaches zero, as has been shown. This condition is, how- 
ever, equivalent to that of the theorem. For the total variation of u 
on an interval is the sum of the total variations of v and j, and the fact 
that the set of points common to two denumerable sets of intervals is 
itself representable as such a denumerable set, makes it possible to 
show that the total variation of on a set of points D is also the 
sum of the similar variations for v and j. Furthermore the total varia- 
tion J{x) of j is the value obtained from the series (5) by replacing 
each term by its absolute value, and on a set of points D contain- 
ing no discontinuity of u it is zero. For if the are a set of points as 
described above, a denumerable set A of non-overlapping intervals can 
be selected approaching each as a limit on both sides and contain- 
ing all the points of ab except the ^kS. This set of intervals will also 
enclose D in its interior since D contains no ^k, and the sum of the total 
variations of j on A will surely be less than e. It is easy to see, con- 
versely, that when the variation of j is zero on D then the latter con- 
tains no discontinuity of u. Hence the existence of the integral with 
V in place of u and the conditions that / and u have no discontinuity 
in common imply that the total variation oi u on D is zero, and con- 
versely, which was to be proved. 
TRANSFORMATIONS OF APPLICABLE CONJUGATE NETS OF 
CURVES ON SURFACES 
By Luther Pfahler Eisenhart 
DEPARTMENT OF MATHEMATICS, PRINCETON UNIVERSITY 
Communicated by E, H. Moore, September 24, 1917 
When the rectangular point coordinates x, ;y, 2 of a surface satisfy 
an equation of the form 
= a — -\- b —, (1) 
budv bu bv 
the curves w = const., = const, form a conjugate system. We assume 
that the parametric system of curves is of this sort throughout this 
note, and we shall speak of it as a net. Equation (1) is the point equa- 
tion of the net. 
If N is such a net, the coordinates x', y' ^ z' of a second net N' are 
given by quadratures of the form 
