traced on the Surface of the Sphere. 269 



duced by the perforation of the sphere by a particular cylinder. 

 The curve upon the earth to which the sun is vertical during his 

 annual path has also been assigned, and its identity with the most 

 general form of the equable spiral has been pointed out. I have 

 given also an investigation of LEXELL'S theorem, and another si- 

 milar locus, well known to geometers, the envelope of the base of 

 a triangle, the sum of whose sides and whose vertical are given. 

 They are solved, I believe for the first time, in a perfectly legiti- 

 mate manner. That is to say, in such a way as would have led 

 to the discovery of the locus itself, without any previous suspi- 

 cion being entertained as to what the locus was, by finding the 

 equations of those loci, and shewing they were the previously de- 

 termined equations of the circle. It would obviously have been 

 easy to extend this class of processes to all cases which have been 

 considered ; and I trust that there will be sufficient attention 

 given to the subject by mathematicians to supply a numerous 

 assemblage of elegant problems, for the use of the student just 

 entering upon the inquiry, a purpose for which they are well 

 adapted. 



The spherical epicycloid, a curve which had been considered by 

 HERMANN, JOHN BERNOULLI, MAUPERTIUS," NICOLE, and CLAI- 

 RAULT, is also brought under examination by this method, and in- 

 vestigated on the most general supposition, viz. when the tracing 

 point is not in the circumference of the rolling circle. I examine 

 the expression for its length in reference to the theorem of JOHN 

 BERNOULLI, that if the rolling circle be a great circle, and the 

 tracing point in its circumference, the curve traced out will be rec- 

 tifiable, and I shew that it is rectifiable in no other case. 



In discussing the loxodrome, I have given the three projec- 

 tions of the curve on the equator, and examined the course of 

 the curve itself on the surface of the sphere. As I had confined 

 my plan, in the present paper, to a discussion only of such pro- 

 perties of spherical curves as were deducible from the expres- 



VOL. XII. PART I. Mm 



