Abstract
The necessary and sufficient condition that a curve should possess a length, this length being given by the usual integral formula, is well known. The curve being defined by the equations
x
=
x
(
u
),
y
=
y
(
u
), the condition is that
x
(
u
) and
y
(
u
) should be expressible as integrals with respect to
u
. It may seem scarcely credible that no corresponding theorem is known with regard to the area of a surface. Such is, however, the case. And what is more surprising, no one has hitherto succeeded in giving such a definition of the area of a curved surface as permits of a determination of a sufficient condition of a general nature that the surface should possess an area, this area being given by the integral formula known to hold in the simplest cases.
Cited by
4 articles.
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1. Peano on definition of surface area;Rendiconti Lincei - Matematica e Applicazioni;2016
2. On Geöcze’s problem for non-parametric surfaces;Transactions of the American Mathematical Society;1950
3. William Henry Young, 1863 - 1942;Obituary Notices of Fellows of the Royal Society;1943-11-30
4. Trasformazioni piane, superficie quadrabili, integrali di superficie;Rendiconti del Circolo Matematico di Palermo;1930-12