339 
of Edinburgh^ Session 186i-65. 
other things remaining the same, increases with the distance of the 
tracing-point from the centre. As that distance is augmented the 
curve at last becomes cusped as in the common cycloid traced by a 
nail in the tire of a cart wheel. If the distance of the tracing-point 
be still farther augmented, the cycloid becomes looped ; the loops 
become wider, and come each to touch that loop adjacent to it on 
either side. 
Mr Perigal’s problem is, to determine the conditions under which 
this contact of the loops takes place. 
If the arm of the revolving wheel be still farther lengthened, the 
loops come to touch those removed from them by two, by three, or 
by any number of steps, so that the problem may admit of many 
solutions. 
Mr Perigal has obtained the solution of this problem in a con- 
siderable variety of cases, by help of mechanical appliances, and 
has exhibited them in his beautiful series of machine-engraved epi- 
cycloids, or bi-circloids, as he calls them. 
The most comprehensive view of the genesis of epicycloidal 
curves is obtained by supposing two arms, OA and OB, to turn with 
uniform velocities on a com- 
mon centre 0, just as do the 
two hands of a watch, and 
by supposing the rhomboid 
OAPB completed at each 
instant; the point P then 
traces an epicycloid. 
If the angular velocities of 
the two arms be represented 
by a and while the lengths 
are A and B, the directions 
of these arms at any instant 
of time are represented by 
at and pt. It is shown that 
if T be the time correspond- 
ing to the contact of one loop with another, we must have the pro- 
portions 
a : /3 : : tan aT : tan 
A : B : : sin (3T : sin aT 
