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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:

(8) |

(9) |

(10) |

(11) |

(12) |

(13) |

In case then or in other words the sign
of reflection 2*H* is positive whatsoever its |*U*_{2H}| value is. Note that
the sign of *H* may have both values. In practice does
not often occur. However, when |*U*_{2H}| is large, expression (13) requires
the sign of 2*H* to be positive even if *U*_{H} is somewhat smaller than
. Moreover, when |*U*_{H}| and |*U*_{2H}| are reasonably large, but
at the same time (13) is fulfilled for both signs of 2*H*, it is still more
likely that *S*_{2H} = + than that *S*_{2H} = -. For example, for |*U*_{H}| = 0.4
and |*U*_{2H}| = 0.3, *S*_{2H} = + leads in (13) to 0.16 0.5 + 0.3 which
is certainly true, and *S*_{2H} = - to which is also true.
Then probability arguments indicate that still *S*_{2H} = + is the more likely
sign. The probability is a function of the magnitudes |*U*_{H}| and |*U*_{2H}|
and in this example the probability of *S*_{2H} = + being correct is .In conclusion the mathematical treatment leads to the same result as the graphic
explanation from the preceding paragraph: the relationship.

**Copyright © 1984, 1998 International Union of
Crystallography**