THE THEOEY OF EOLLING CUEVES. 29 



Ex. 4. "When it is a parabola, the focus traces the directrix, and the vertex 

 traces the cissoid. 



Ex. 5. When it is the hyperbolic spiral, the traced curve is the tractory of 

 the circle. 



Ex. 6. When it is the polar catenary, the equation of the traced curve is 



a / 2a --i r 



6 = / -- 1 versm ' - . 

 V r a 



Ex. 7. When it is the curve whose equation is 



the equation of the traced curve is r = a(e 6 e~"). 



This paper commenced with an outline of the nature and history of the problem of rolling 

 curves, and it was shewn that the subject had been discussed previously, by several geometers, 

 amongst whom were De la Hire and Nicole in the Memoires de I'Academie, Euler, Professor 

 Willis, in his Principles of Mechanism, and the Rev. H. Holditch in the Cambridge Philosophical 

 Transactions. 



None of these authors, however, except the two last, had made any application of their 

 methods ; and the principal object of the present communication was to find how far the general 

 equations could be simplified in particular cases, and to apply the results to practice. 



Several problems were then worked out, of which some were applicable to the generation 

 of curves, and some to wheelwork ; while others were interesting as shewing the relations which 

 exist between different curves; and, finally, a collection of examples was added, as an illus- 

 tration of the fertility of the methods employed. 



