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Notes on the Apollonian Problem and the allied theory

Published online by Cambridge University Press:  20 January 2009

John Dougall
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Extract

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1. This paper contains a number of investigations, more or less connected, on the theory of systems of circles. In such a well-worn field one does not expect to have hit upon much that is absolutely new, but it may be hoped that there is sufficient freshness of treatment to give the paper some interest even where it deals with results already known.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 24 , February 1905 , pp. 78 - 119
Copyright
Copyright © Edinburgh Mathematical Society 1905

References

* See Art. 16 below.

* Conversely, if (3) is satisfied, the circles 1, 2, 3, 4 have a common tangent cirole. To prove this, take the circle 5 so that 152=252=352=0. The equation (1) is a quadratio for 452, one of the roots of which is zero, in virtue of (3). Hence one of the two circles touching 1, 2, 3 touches 4 also.