FORM OF TANGENTIAL EQUATION. 
421 
Hence g+'&, f, § — & are in harmonical progression. Hence , with the same hypo- 
thesis as in the last article , the radius of the common director circle is a harmonic mean 
between the radii of their bases. 
112. Several propositions proved for the cycloid may with scarcely any modification of 
the demonstration be extended to epi- and hypocycloids. Thus : — 1 °. If from the point 
where the generating circle of an epi- or hypocycloid touches the base a perpendicular 
be let fall on the revolving radius , the envelope of the perpendicular is an epi- or hypo- 
cycloid. 2°. If the perpendicular be let fall from the point where the generating circle 
touches the director circle , the envelope is an epi- or hypocycloid. 3°. The envelope of 
the revolving radius is an epi- or hypocycloid. 4°. If two tangents , PT, PT', to an epi- 
or hypocycloid meeting the director circle in the points A, A! make a constant angle , 
the locus of the centre of the circle about the triangle APA' is a circle. 5°. The enve- 
lope of the diameter of this circle which passes through P is an epi- or hypocycloid. 
6°. The envelopes of the chords passing through P and through the highest or lowest 
points are epi- or hypocycloids. 
Section III. — Extern Epicycloids. 
113. In the same manner as we have called the curve described by a fixed point in 
the revolving radius of the generating circle of a cycloid an in- or extern cycloid, we shall 
call the curve described by a fixed point in the plane of the generating circle of an epi- 
cycloid an in- or extern epicycloid according as the point is inside or outside the circum- 
ference of the circle. Similarly we shall have an in- or extern hypocycloid ; so that the 
curve embraces four distinct species ; but as they differ only in the magnitude or sign 
of a parameter, their properties are virtually the same ; hence we shall discuss only the 
extern hypocycloid. 
114. Let P' be the point in the radius IP ; then, since B is the centre of instantaneous 
rotation, BP' will be a normal to the curve, and P'Z perpendicular to BP' will be a tangent. 
The curve will have points of inflection. This follows at 
once from a beautiful theorem of Professor Ball’s, Astro- 
nomer Poyal of Ireland : — “ That if a plane figure is mov- 
ing in a plane according to any law, there is always a circle 
of points rigidly connected with it, such that three conse- 
cutive positions of each point are in a right line” *. (See 
‘Proceedings of the Eoyal Irish Academy,’ December 11, 
1871.) Another proof will be given in the course of our 
investigations. 
Let the equation of the curve described by the point P 
he 
t T=nd ; 
* This circle is called the “circle of inflections.” The theorem in the test was originally given by Savary 
in his ‘ Legons des Machines.’ — November 1877. 
