106 Mr. H. M. Jeffery. [June 15, 



3. A proof will be hence derived for two general theorems on 

 regular polygons given by Dr. Matthew . Stewart (Props. 39, 42), of 

 which many others are special cases. 



4. These propositions will first be elucidated by examples. 



In the case of the regular hypocycloids, we may consider them a& 

 referred to line-coordinates, the polygon of reference being formed by 

 joining the (11) cusps. 



If oi = 3, PilhPs=P =^(Pi+P2+Pzy° °r 1^+1^+^=0. 



n: 



Consequently by differentiating with respect to (the inclination 

 of a perpendicular on a tangent to the initial line) in the several cases- 

 on both sides (Besant, "On Roulettes," § 7) — 



If 11 = 3, cot a x + COt a 3 + COt a 3 = 3 COt a, 



n=4, 2 cot a ; «=4 cot a, 



11=5, 2cota ;; ;=S£ ^cota, 



p'~ — d 



where d is a constant, and « l9 a 2 , . . . a. denote the angles made by the- 

 tangent with the vectors from the cusps and centre. 



5. In the epicycloid the same polygon of reference is used. 



If n=l, ±0,-^ = ^, the unicuspid cardioid, 



n=2, lc^p lVi =^ 



n = 3, Ja^tft=-|gJp 6 -laV(^l)- 

 Consequently, if n = l, cot a x =3 cot a, 



n = 2, cot a 1 + C0t a 2 =4 cot a. 



The general theorems will now be established. 



6. Let these perpendiculars from the cusps be expressed by tan- 

 gential polar coordinates, when the initial line is drawn from the 

 centre to a cusp : — 



P1P2 - -Pn=(p— acos0)jp — cicos^— + • • |jp — a cos ^ n ~^ tt + jy 

 The artifice used by Gregory in a kindred-question (" Math. 



