Algebraic loci traced by certain mechanisms determination of order or class of loci of points, planes or lines with one or two degrees of freedom

The principal results presented in this thesis are Theorems 1 and 2 in ¶2.1, both of which are new. Theorem 1 gives the feather of a particular group of couplings with connectivity two, and Theorem 2 defines the feather of any coupling with connectivity one. The terms "coupling" and "connectivity" were defined by Davies and Umphrey in an unpublished paper. The term "feather" is new; it is defined and discussed in ¶1.2. All three terms are also defined in Appendix I. The conclusiions presented in Theorems 1 and 2 are always true provided only that the restriction of Bézout's theorem is complied with. In spite of this there are many apparent exceptions; a few examples of these are discussed in ¶2.2. As demonstrated there the apparent exceptions are all due to singular situations, e.g. part of the locus is at infinity or part of it is imaginary. Several of Theorems 1 and 2 are presented in Chapter 3. Some of the results are not ne but are included for illustrative purposed, but much nrw information is presented. Some of this new information would be very difficult to obtain by other methods. More important than specific results, however, is the ease with which they are obtained; tthis ease of application is one of the primary attributes of Theorems and 2. At the end of Chapter 3 there is some discussion of constant velocity shaft coupling; these applications are less directly related to Theorems 1 and 2 than earlier applications. These discussions of constant velcoty shaft couplings are applications of the synthetic methods used throughout the thesis. The secondary object of this thesis is to reintroduce synthetic geometry to kinematics. Synthetic geometry is discussed in ¶1.1 in general terms; Appendix V gives a short history of synthetic geometry and analytic geometry and a comparison of the two; in ¶1.2 and ¶2.1 the development of the ideas presented in this thesis is given in some detail. These discussions point out some of the weaknesses of synthetic approach and demonstrate some of its strengths. The principal results of the literature survey are presented in ¶1.3. In addition to these principl findings which relate to Theorems 1 and 2 several unkown or little know facts were discovered. First, there are Darboux's theorems presented here in Appendix II and slightly extended. Second, the relation between Darboux's theorems and a work by Chasles is discussed here in ¶1.3.3. Third, two theorems derived, more or less directly, from certain results obtained by Cayley are presented here in Appendix II. Fourth, some discussion of two remarks made by Samuel Roberts is presented in Appendix IV. Chapter 4 presents a few proposals for further research along the lines begun in this thesis. Appendix I defines the special terms used in the thesis. Most of the definitions are standard and may easily be found in othere books; these are include for convience. A few are new or would be difficult to find in other places.