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In 1948 Harker and Kasper published their paper on inequality relationships, which actually opened the field of direct methods. They applied the Cauchy inequality:
In case then or in other words the sign of reflection 2H is positive whatsoever its |U2H| value is. Note that the sign of H may have both values. In practice does not often occur. However, when |U2H| is large, expression (13) requires the sign of 2H to be positive even if UH is somewhat smaller than . Moreover, when |UH| and |U2H| are reasonably large, but at the same time (13) is fulfilled for both signs of 2H, it is still more likely that S2H = + than that S2H = -. For example, for |UH| = 0.4 and |U2H| = 0.3, S2H = + leads in (13) to 0.16 0.5 + 0.3 which is certainly true, and S2H = - to which is also true. Then probability arguments indicate that still S2H = + is the more likely sign. The probability is a function of the magnitudes |UH| and |U2H| and in this example the probability of S2H = + being correct is .In conclusion the mathematical treatment leads to the same result as the graphic explanation from the preceding paragraph: the relationship.
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