448 An easy Rule for Formalizing all Epicyclical Curves, 



&c. The f oral appearance of many of these curves induces 

 me to suggest the name of peialoids. They may possibly one 

 day lead to geometrical disclosures on the structure oi'Jlo'wers, 

 as Naumann and Moseley (Phil. Trans. 1838) have success- 

 fully shown in shells; each individual shell having its own 

 numerical parameter, which a verbal nomenclature would 

 vainly follow, as every additional digit increases the number 

 of varieties tenfold, three already denoting 999 varieties. 

 Mr. Perigal's finite spiroeids are very curious, especially the 



cos 

 retrogressive syphonoids(a?=acosg'<p, ?/ = 6 ^^^ p<p), the two- 

 branched syphonoid being the common conical parabola. {Vide 

 Mr. Sang's paper On the Vibration of Wires, Edin. Phil. 

 Journ. 1832, p. 317.) All these which terminate in points 

 Sive finite portions of generally infinite curves ; for in the latter 

 y'^ = ax, \f X be proportional to the periodic quantity cos^ A, 

 and ?/ proportional to cos X, 3/^ -r a; is still constant: the pa- 

 rabola is described, but only between the limits of + 1 = cos A, 

 towards which points the motion gradually slackens to zero, 

 as publicly shown by Mr. Perigal at Lord Northampton's 

 scientific smVe'^ in March 1846'. The Royal Society, Astro- 

 nomical Society and Royal Institution, possess three volumes 

 of various singular epicyclical curves executed by Mr. Perigal's 

 machinery, some of which are highly ornamental, and I think 

 might be useful for the arts, e. g. the drawing of volutes to 

 Ionic columns, &c. 



Kinematic Parabola. A retrogressive Syphonoid produced by compound 

 circular motion. 



7 Bedford Place, Hampstead Road, 

 September 1848. 



