Chord theorem

Figure 7.3 Interactions between a spherical particle and a solid slab surface D is the distance between the particle and the surface of the plane solid RSph is the radius of the spherical particle fiRing is the radius of the ring RSiice is the radius of the slice in the spherical particle. Other terms are self-descriptive and [Rsiice = x(2/ Sph - x)l from the chord theorem.

Figure 7.4 Right-angled triangle, ABC, captive in a semi-circle. The drawing is used in the proof of the chord theorem of plane geometry, resulting in the expression [/i2 = x(2R - x)], which will be used in Hamaker constant calculations.

Two-Chord Power Theorem—RS and TU are chords of the same circle, intersecting at Q then... 

To round off this section we note a few unusual applications of Polya s Theorem an application to telecommunications network [CatK75], and one to the enumeration of Latin squares [JucA76]. In pure mathematics there is an application in number theory [ChaC82], and one to the study of quadratic forms [CraT80], being the enumeration of isomorphism types of Witt rings of fields. Finally, we note a perhaps unexpected, but quite natural, application in music theory to the enumeration of chords and tone rows for an n-note scale [ReiD85]. In the latter paper it is shown that for the usual chromatic scale of 12 semitones there are 80 essentially different 6-note chords, and 9,985,920 different tone rows. 

This follows from a theorem by Cauchy [18] according to which the mean chord length / in a convex site is given by 4F/5, where V and S are, respectively, the volume and surface area of the site. For a spherical site of diameter d ... 

It can be shown from the theorems of fractal geometry that the sets of intercepts created in a line search examination of a dispersed field of view constitutes a Cantorian set in one dimensional space. If limited to the points where the search tracks, enter and leave the profiles, revealed is a Cantorian dust that can be linked to the richness of the ore [31,32]. It can be shown that the fraction of the random lines falling within the profiles is the same as area fraction occupied by the dispersed fineparticles in the field of view. It can also be shown that the average size of the chord tracking across the dispersed fineparticles is a function of the average size of the dispersed fineparticle. These relationships were first developed by Rosiwal [33, 34, 35]-... 

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