122 MR EDWARD SANG ON THE CONTACT OF THE 



by the proportion /3 2 : a 2 : : A : B, the curve becomes flat at certain points, that 

 is, the radius of curvature there becomes infinite ; this phase is exemplified in 

 figs. 15, 16. When OB increases beyond this value the curve becomes sinuous, 

 its concavity being turned inwards and outwards alternately, as is seen in figs. 

 17, 18 ; and when OB becomes so great as to satisfy the condition (3 : a : : A : B, 

 the curve becomes cusped, and assumes that form to which the name epicycloid 

 is sometimes restricted ; this form is exemplified in figs. 19, 20. When the arm 

 OB is made still longer the epicycloid is looped, the loops being arranged at 

 regular intervals, as in figs. 21, 22 ; and if OB be made sufficiently long, the loops 

 come to touch each other, as in figs. 23, 24. Mr Perigal's problem is to deter- 

 mine the conditions under which this contact of the loops takes place. The loop 

 may touch those adjacent to it on either side, or, if OB be made sufficiently long, 

 those separated from it by two, three, or more intervals ; so that the problem 

 may have more than one solution. 



3. From the very genesis of the curve it follows that the contact of the loops 

 must occur either on the major or on the minor radius-vector; now the angular 

 motions of the arms may be either in the same or in opposite directions, where- 

 fore there are four cases to be examined. 



4. When the arms turn in the same direction, and when the contact is to be on 

 the major radius-vector, we may use the formulae of Article 1, unchanged ; and 

 since, at the instant of contact, the curve must touch the radius- vector OX, we 



must have both #=0, and its derivative 757=0 ; hence, if T denote the time at 



which the tracing-point is in this position, we must have 



= A . sin aT + B . sin /3T 

 = aA . cos aT + /5B . cos /3T 



whence we obtain the two proportions 



a : (3 : : tan aT : tan /3T 

 A : - B : : sin /3T : sin aT 



5. When the arms turn in the same direction, and when the contact is to be 

 on the minor radius- vector, we have to change the sign of B in the preceding 

 formulae, which are thereby converted into 



= A . sin aT - B . sin /3T 

 = aA . sin aT - /SB . sin /3T 



whence 



a : 



:/3: 



: tan aT : 



tan/3T 



A 



: B : 



: sin /3T : 



sin aT 



