A Treatise On the Theory of Determinants And Their Applications in Analysis And Geometry

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34—37.] APPLICATIONS TO GEOMETBT.
211 For two systems of five spheres we should get = 576 iv,r, + ...+ V5) «ft + . • • + v^Ps), tn- •«.a.
1 t.- ■• 1 1 .
.. 1 using the notation of Art. 31.
If t^ is the angle at which the plane of similitude of the first four spheres of the first system cuts each of these spheres, and (r^t^ the angle at which it cuts the fifth sphere, and similarly for the second system, we can reduce this to the form cos (p,T,)N tn- •«..
1 ts.- ■%.> 1 1 .
. 1 576v,r^v:p,(l- cos
...i.
COST.
Hence the determinant vanishes if one of the systems of five spheres has a common plane of similitude.
For two sets of four spheres, after some reduction we can prove that in- K.> 1 ^41 • • • *«' 1 1 ... 1 = 288..' (1--^^), \ cos t COS T/ where ^ is the angle between the planes of similitude of the two systems, and t, t the angles at which they cut their sets of spheres.
37. By compounding the arrays whose i'" rows are and - 2^„ - 2v„ - 2^„ 2p„ |/ + 7,/ + ?f - p^ 1, we get the homogeneous relation between the sets of tangents common to two sets of seven spheres in- i..


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