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- 8 - varies uniformly between 0 and it. According to J. SERRA (1982, pp. 330-331 , rel. V-21 et exercice X-b-10), we see that if is small and if ôX shows raddi of curvatures everywhere, we have : (4) f (X) - £■ X1- ' ~ 4 l 1 L(ôX) / i>x+ du ? (du o elementary arc on ôX, R = radius of curvature at, ôX+ part of ôX with positive radii of curvature). The same approach also applies for small traverses of Xe, i.e. of the zone above the boundary ôX (Pig. 7). This results in a function f il) Pig. 7 - Xe which satisfies a relatin symmetrical to (4). If we now sum up the two types of traverses, we obtain an average histogram f U) such that f(i) = £ [fxci) +1 an = {. i t / . § + f g] the quant ity ^ curvature = (5) f(^) fis X 8 lTôxT / ^ÔX du T R ÔX R 1 7 2 is nothing but the average of the square Therefore f U) | . E (_>,) =4e(C2) where E(*) means "mathematical expectation of . Gomment : this resuit is extremely instructive. It means that if the begining of the expérimental histogram f*(£>) is propor- tional to £ , then : i) the assumption that ôX has non zéro radii of curvature (almost) everywhere is acceptable. ii) the slope of f*(^) near the origin equals the average square curvature of the fracture, divided by eight.