66 Mr HOLDITCH, ON ROLLING CURVES. 



, ab 2ab ,. 



and r = n = - rr rr the equation to an 



2 v , . , fa + b) + (a - b) . cos ? 



a cos 8 - + b sin 2 - v J v J 



2 2 



ellipse round the focus, which is known to be capable of rolling upon 

 another equal and similar ellipse. 



Hence 9 = ^% . tan" 1 \/j • \Z r -^ 

 Vab b a — r 



is the equation to the curve constructed in the ninth section of 

 Newton's Principia, which is therefore a self-rolling curve. 



In the equation found above, if r = b, 



n I 2k, k.(a + by . . .J 



so that the minor apsidal distances recur, the angular distances between 



them being = \—^= + l a + / - k Ja+ b)\ .tt. 

 Wab 2y/ab K j 



If r = a, 



and the major apsidal distances recur and bisect the angles between 

 the minor distances: and if that portion of the curve between two 

 minor distances, including as they do, a major distance between them, 

 be called a Lobe, the number of lobes in a revolution 



f 2k, k.ia + by . , ,J 

 Wab 2<Vab ) 



and in order that the curve may return into itself and so be capable 

 of successive revolutions, this must be an integer = n ; 



