168 
Proceedings of the Royal Society 
Fig. 7. 
in which it revolves, the hypotrochoid becomes an ellipse ; and if 
the point be on the circumference of the smaller circle, the ellipse 
degenerates into a straight line. 
Let the circle BPO (fig. 7), diameter BC = AC, revolve in 
the circle ABM, radius AC. If the initial position of the describ- 
ing circle be such that AC 
be its diameter along the 
axis of y, the point A will 
move along the diameter 
AM. When the circle has 
moved round to B, let P 
be the position of A ; then 
AB = BP. Join BKC and 
KP. Let ACB = 0; 
BKP = <p. The arc BP = 
AB .* . BK- <p = KC-6 = 
2 BK • 0. Hence <p = 2 6. 
Hence P must lie on AC. 
(Euc I. 32). 
The difficulty of constructing, and the inconvenience experienced 
in using an arrangement in which the annular wheel comes into 
action, are so great that it is seldom employed. The following is 
a simple method of obtaining the hypocycloidal motion without 
requiring the annular wheel. On the arm ACB (fig. 8), 
which is made to revolve 
in a circle MLP round A, 
attach two toothed wheels 
LNF and FEM, in gear with 
each other and with a fixed 
wheel EOH, whose radius 
AE is double that of NLF. 
The centre wheel EEM is 
used merely for the purpose 
of reversing the motion of 
NFL, hence it is immaterial 
how many teeth it has; the 
only postulate of the pro- 
Fig. 8. 
blem being that LFN revolve on its axis at double the speed of the 
