493]
ON EVOLUTES AND PARALLEL CURVES.
37
Hence for the curve in question the equation of the normal is
{(Jc - 1) a!6* - WO*-' + 7 '} {(/3X - aY) 0* + (aZ-yX) 0 + ( 7 F- /3Z)}
+ {(Jc -1) aO* - WO*-' + 7}. {W'X - a' Y) 6* + (a'Z- y'X) 0 + (y'Y- f3'Z)} = 0,
or, expanding and reducing, this equation is
0-* .(Jc- 1) {(a/3' + a'/3) X — 2aa' F}
+ &*-'. - Jc {‘2/3/3'X - (a/3' + ol'/3) Y}
+ 6* +1 . (Jc — 1) {2 aa' Z — (ay + ay) X]
+ 0*. {(Jc + 1) (£7' + ffy)X + (Jc- 2) (ya' + y'a) Y-(2Jc-l)(a/3' + af/3) Z}
+ 0*-'. - Jc {(/3y' + ¡3'y) Y - 2/3/3'Z]
+ 0 . i(7 a/ + 7 a ) Z — 2yy'X{
+ {277 F - (/3y' + /3'y) Z} = 0,
where & is a positive integer not less than 2; hence except in the case Jc = 2, all
the terms 0-*, 0-*~',...0, 0°, have different indices, and the coefficients Jc — 1, Jc, &c.
none of them vanish; if however Jc = 2, then the terms 6“*~', 0* +1 coalesce into a single
term, as do also the terms 6*~ 1 and 0; moreover the coefficient Jc — 2 is =0.
The evolute is the envelope of the line represented by the foregoing equation,
considering therein 0 as an arbitrary parameter; viz. the equation is obtained by
equating to zero the discriminant of the foregoing equation in 0. Hence in general
the class of the evolute is =2Jc, and its order is =2 (21c— 1); results which agree
with the formulae for n", m", since in the present case m + n, =Jc + Jc, = 21c, a — Sn + k,
= oJc + (Jc — 2), = YJc — 2. And moreover there are not any inflexions, t" = 0 as before.
The equation may however contain a factor in 6 independent of (X, F, Z), and
throwing out this factor, say its order is = s, the expression for the class is 2Jc — s,
= m + n — s, and that for the order is — 2 — 2s, = a — 2s. Moreover, in the original
equation or in the equation thus reduced, it may happen that the equation will on
writing therein il = 0 (il a linear function of X, Y, Z) acquire a factor of the order
(w, independent of (X, Y, Z); the line fl = 0 is in this case a stationary tangent,
= a) — 1 inflexions; and the discriminant contains the factor il" -1 , which may be thrown
out; that is we have here n" = 2Jc — s, 1" = co - 1, m" — Yk — 2 — 2s — (a> — 1); agreeing
with the relation m" — 2n" + 2 + l" = 0 which holds good in virtue of the evolute being
a unicursal curve. It is in this manner that the values of m", n", t" are obtained
in the several cases to be considered, viz. :
Ah Inflexion situate on IJ, which is not a tangent.
Bje Inflexion situate on IJ, which is a tangent.
G k Cusp situate on IJ, which is not a tangent.
D k Cusp situate on IJ, which is a tangent.
P k Inflexion at J, IJ not a tangent.
Q k Inflexion at J, IJ a tangent.
14 Cusp at J, IJ not a tangent.
S k Cusp at J, IJ a tangent.
