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' are each integrable),
Next suppose that / ' and <£' are not integrable functions but A//' 2 + $'2 is a perfect differential, say ctt we can then prove that* as follows. * Lebesgue, Lemons sur VIntegration (1904), p. 61.
LENGTH AND LINEAR CONTENT
Taking any point P on the curve of parameter tlt we can, since / ' and <j>' both exist, de2d termine a small quantity dlt such r—^*ggr^> that for any point Q of the curve g whose parameter t2 lies between " ti+d*!, the length of the chord PQ differs from
V/'2 (Q + f^)
(«, - h)
by less than e (t2 — ^). Further we can determine a small quantity d2 such that for any point Q of the curve whose parameter £, lies between t, + d3, ^/' a (*i) +