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X.— On the Contact of the Loops of Epicycloidal Curves. 



(Plates VI. to XII.) 



(Read 3d January 1865.) 



By Edward Sang, Esq. 



During the summer vacation, Mr Henry Perigal of London proposed to me 

 the following problem :— 



"To determine the proportions of an epicycloid of which the loops touch each 

 other." 



The solution of this problem contains some points of interest to the general 

 analyst, and exhibits relations between certain trigonometric formulae and alge- 

 braic equations. I, therefore, offer an outline of it to the attention of the Royal 

 Society. 



Mr Perigal had obtained the solution, in a considerable variety of cases by 

 the method of trial, aided by mechanical appliances, and has exhibited them in 

 his beautiful series of machine-engraved epicycloids. 



1. If we suppose two radii OA and OB to turn on a common centre with 

 uniform velocities, in the manner of the two hands of a 



watch, and if, at each instant, we complete the paral- 

 lelogram OAPB, the opposite corner P describes an 

 epicycloid. This curve may be obtained by causing the 

 line AP to turn on A as a centre, while A itself describes 

 a circle round ; or by causing the arm BP to turn on 

 B as a centre, while B moves round the fixed centre 0. 

 These are the ordinary arrangements by wheel- work. 



There are other arrangements by help of which 

 epicycloids may be produced, but they all result in 

 giving, for the equation of the curve referred to rect- 

 angular co-ordinates, the formulae 



x = A . cos at + B . cos fit 

 y — A . sin at 4- B . sin (3t 



in which A and B represent the length of the arms, a and fi their angular velo- 

 cities, and t the time elapsed since both arms were in the direction OX, so that 

 at and fit are the angles XOA and XOB respectively. 



2. fi being supposed to be the greater of the two angular velocities, if the arm 

 OB were minute as compared with OA, the curve described by P would be nearly 

 circular and slightly undulated ; as OB is augmented the waves become deeper, 

 as shown in figs. 13 and 14, and when OB reaches the magnitude determined 



VOL. XXIV. PART I. 2 L 



