22 ON THE THEORY OF THE SINGULAR SOLUTIONS OF [636
Suppose the coordinates of the given point are y = cos a, x = c + \ol — ^ sin 2a, where
a is a determinate quantity ; then, to find 6, we have
cos 6 — cos a, 26 — sin 26 = 2a - sin 2a.
The first equation gives 6 = 2nnr + a, and the second equation then is
4»i7r + 2a + sin 2a = 2a — sin 2a ;
viz. taking the upper signs, this is 4umr = 0, giving m — 0 and 6 = a ; and, taking the
lower signs, it is mir = a — sin a, which, a being given, is not in general satisfied ;
hence to the given point there corresponds only the value a of the parameter 6. If,
however, a is such that a — sin a is equal to a multiple of 7r, say rir, then the last-
mentioned equation is satisfied by the value m = r, so that to the given point of the
curve correspond the two values a and 2?*7r — a of the parameter ; these values are
in general unequal, and the point is then a node ; but they may be equal, viz. this
is so if a = T7T (the point on the curve being then y = cos rir, = ± 1, x=c-\-^rir), and
the point is then a cusp ; showing what was known, that there are on each of the
lines y — — 1, y = + 1, an infinite series of equidistant cusps.
More definitely, suppose a = rir ± /3, where /3 is a root of the equation 2/3 — sin 2/S = 0,
then
sin 2a = + sin 2/3, 2a — sin 2a = 2r7r ± (2/3 — sin 2/3) = 2r7T,
and to the given point on the curve correspond the two values a and 2?’7r — a of
the parameter. If /3 = 0, we have, as above, the cusps on the two lines y — -1-1,
y = — 1 respectively ; but if /3 be an imaginary root of the equation 2/3 — sin 2/3 = 0,
then we have an infinite series of nodes on the imaginary line y = cos rir cos /3 ; and
there are an infinite number of such lines corresponding to the different imaginary
roots of the equation 2/3 — sin 2/3 = 0.
From the form in which the equation of the curve is given, we cannot directly
form the equation of the envelope by equating to zero the discriminant in regard to
the constant c ; but we may determine the intersections of the curve by the con
secutive curve (corresponding to a value c + 8c of the constant), and thus determine
the locus of these intersections.
Consider for a moment the curves belonging to the constants c, c u and let 6, 6 X
be the values of the parameter 6 belonging to the points of intersection; we have
cosd = cos0!, 4c + 26 — sin 26 = 4^ + 26^ — sin 26 l ; we have 6 1 = 2rir + 6, but we cannot
thereby satisfy the second equation; or else 6 1 = 2rir — 6, giving
4c + 26 — sin 26 = 4Cj. + 4?"7r — 26 + sin 26,
that is, 26 — sin 26 = 2c x — 2c + 2rir ; and we have thus corresponding to any given value
of r a series of values of 6, viz. these are 6 = rir + /3, where /3 is any root of the equa
tion
636]
In j
any root
intersecti
on the
equation
what pre
than the
the node
thereof w
Ex.
that is,
I ha
the conic
X = COS (f>,
Equa
viz. the
of cusps.
2(3 — sin 2/3 = 2cj — 2c.
