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 
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any root 
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on the 
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the node 
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the conic 
X = COS (f>, 
Equa 
viz. the 
of cusps. 
2(3 — sin 2/3 = 2cj — 2c.