We would like to thank Professors Serguei Denissov and Alexander Kiselev for their constant support throughout the research and writing processes. We are grateful to them for suggesting the topic and providing us with the opportunity to apply the concepts we have learned throughout our mathematical careers. We thank them both for their excellent advice and encouragement.Abstract. For θ ∈ [0, 2π), consider the rotation matrix R θ and h = λ 0 0 0 , λ > 1.Let Wn(θ) denote the product of m R θ 's and n h's with the condition m ≤ [ǫn] (0 < ǫ < 1). We analyze the measure of the set of θ for which Wn(θ) ≥ λ δn (0 < δ < 1). This can be regarded as a model problem for the so-called Bochi-Fayad conjecture.

Tables of extended distances for PKP1, PKP2 and PcP are presented, for surface focus and for focal depths ranging from 0.00R to 0.12R by steps of 0.01R. The tables are based on the Jeffreys-Bullen travel-time curves for the equivalent phases. They are consistent with earlier tables giving extended distances for P, so that the several phases can be used in a single solution.

The paper presents tables of extended distances for several secondary phases which are occasionally useful in fault-plane solutions, and extends earlier tables for P to very short distances. The tables are given for surface focus and for focal depths ranging from O·00R to O·12R in steps of O·01R. They are consistent with earlier tables of extended distances so that the several phases can be used in a single solution.

Tables of extcnded distances, consistent with those already published for the P phases, are prosented for those S phases which are likely to be useful in fault-plane studies. As in earlier papers, the extended distances are given for surface focus, and for focal depths from O·00R to O·12R in steps of O·1R.

This paper is the third in a series extending Byerly's method of determining the direction of faulting in an earthquake to deep focus earthquakes and permitting the use of secondary P phases. Here tables of extended distances are presented for the reflected rays of PP and pP, for earthquakes of all focal depths down to 0.12 R. The tables are consistent with earlier ones for P, PKP, and PcP, so that the several phases can be used in a single solution.