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On the Equations of Loci traced upon the Surface of the Sphere, 

 as expressed by Spherical Co-ordinates. By THOMAS STE- 

 PHENS DAVIES, Esq. F. R. S. LOND. & ED. F. R. A. S. 



(Read 1st April 1833;. 



the researches which I have made upon this subject 

 since the composition of my former paper, printed in the first 

 Part of this Volume, so many and so varied inquiries arising 

 out of it, and essentially connected with it, have presented 

 themselves, that I have been compelled to make a total change 

 in the plan I had then laid down for the completion of the 

 present section of my communication. I found that many sub- 

 jects to which I there alluded might, with propriety, be omit- 

 ted in the present case, as constituting little more, in reference 

 to principles, than illustrations, however interesting they might 

 be when viewed as properties of geometrical figures. On this 

 ground, therefore, I have cancelled a considerable number of pro- 

 perties of the Spherical Conic Sections, and retained only one or 

 two for the purpose of illustrating the method of discussing the 

 properties of those curves. The remarks I intended to make 

 upon the singular points of spherical curves, the geometrical sig- 

 nification of certain symbols, and other inquiries collateral to 

 these, have grown into systems of themselves, or been attached 

 to other dissertations (either wholly or partially completed), to 

 which they seemed to be as closely allied as even to the present 

 subject. From this paper, too, as originally written, I have ab- 

 stracted considerable portions, amongst which are three distinct 

 classes of spherical research ; and I have here confined myself to 

 the essential parts that belong to the subject of spherical co-or- 



VOL. XII. PART II. 3 C 



