III. On Rolling Curves. By Hamnett Holditch, M.A., Fellow of 

 Cuius College, and of the Cambridge Philosophical Society. 



[Read December 10, 1838.] 



In the fifth volume of the Acta Petropolitana, Euler referred to 

 a class of curves which, when caused to turn round fixed centres, 

 possessed the property of communicating motion to each other without 

 friction ; he deduced also their characteristic property, that the point of 

 contact remains always in the straight line joining their centres: he has 

 not however followed out the investigation so as to furnish actual forms 

 of curves, neither has this been done by any other writer that I am 

 aware of, and consequently no method exists by which such curves 

 can be obtained. But as they are practically employed in a manner 

 which I shall proceed to explain, and commonly found by a tentative 

 process, it appeared worth while to search for forms and rules for their 

 construction, independently of the analytical interest that may be sup- 

 posed to attach to such investigation. 



Let Anm, Bn i m l (Fig. 1.) be two curves capable of rolling together, 

 and having their centres of rotation A and B fixed at a distance equal 

 to the sum of their apsidal distances, Am being a long and Bn t a short 

 apsidal distance, then if nAm be caused to turn round in the direction 

 of the arrow, it will press against Bnm t and communicate a rotation to 

 it. This action will, however, cease when the point m has reached « ; 

 for beyond that point the radii of mAn will diminish, and its circum- 

 ference begin to recede from the other curve. 



