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PROFESSOR J. CASEY ON A NEW 
Section IT . — The Hypocycloid. 
108. Having discussed at considerable length the properties of the epicycloid, we 
shall treat very briefly those of the hypocycloid. In fact, analytically, the latter curve 
differs from the former only in the sign of a parameter ; hence the properties of one 
curve are with slight modifications true of the other. The most interesting are those 
which are found by considering the curves in combination. 
109. In the equation <r=cLj0 of art. 96 let Q 
denote the angle which LP makes externally 
with the tangent to the director circle (which 
comes to the same thing as to consider 6 nega- 
tive. Now if Q change its sign, since we must 
regard <r as positive, must change sign ; in 
other words has changed direction). 
Also let denote the angle which LP makes 
with the tangent to the director circle at the 
origin ; then we have -^ = 0—$; 
<AJ/ dQ d<p 
da da da 
Hence, if denote the diameter of the circle LPE described about the infinitesimal 
triangle LL'P, we have 
'8 8 
-i § 
(231) 
and we find, as in art. 97, 
(e+'^)<p='^ ; 
that is, the arc AE=arc LP ; and making the arc AX = semicircle LPE, X will be a fixed 
.point, and we shall have the arc EP=arcEX. Hence the locus of P is the hypocycloid 
generated by the rolling of the circle EPL on the circle AEX. 
110. Since I=J_ — I, see equation (231), 
and see equation (222), 
8 o_, g 
I + 1 - 1 
8 ‘ '8 8 _ 
(232) 
Hence if an epicycloid and hypocycloid have the same director circle , and if the gene- 
rating circles of their involutes be equal to one another, the diameter of the generating 
circles of their involutes is a harmonic mean between the diameters of the generating 
circles of the curves themselves. 
111. From equation (231) we get 
g+W-V, 
? ?- 6 -i 
and from equation (222) 
