493]
ON EVOLUTES AND PARALLEL CURVES.
39
where throwing out the factor 0 k ~\ the form is as shown on the right hand. Writing
Z = 0, we have a factor 8 k , whence i" = k-l, and then n” = k+ 1, m" = 2k- (k - 1) = k + 1,
agreeing with the table. The process holds good for k = 2.
Case G k . I write ft = ft = Q- } this brings as well the inflexion as the cusp upon
the line IJ \ but it has been seen (Case A k ) that there is not any reduction on
account of this position of the inflexion, hence the whole effect will be due to the
cusp. The equation is
0' 2k . (k — 1) {— 2aa'F}
+ 0 k+1 .(k — 1) {2aa!Z — (ay + ay) X}
+ 0 k .(k — 2) (ya' + y'a) F
+ 6 . |(ya' + r y' a ) 2yy'X}
+ 2yy' Y = 0,
ô- k Y
+ e k+i (z, x)
+ 6 k (k — 2) F
+ 6 (Z, X)
+ F= 0,
/0 y- 1
so that here n" = 2k. On writing F = 0, there is a factor il — — J thrown out
(indicated by the reduction of the order from 2k to k +1), whence
l" = k-2, m" = 2 (2k — l) — (k — 2), =3k.
The process holds good for k = 2.
Case I)*.
We may write a = a' = 0; the equation is
6 2k ~ l .-k. 2/3/3'X
+ 6 k . (k+l)(/3 y' + /3'y)X
+ 6 k ~ l . - k [(#/ + /3'y) Y - 2/3/3'Z]
+ 6 . — 2yy'X
+ 2yy'F — {¡3y + /3'y) Z =0,
0- k ~i X
+ e k x
+ e k ~ 1 (F, z)
+ e x
+ (Y,Z) = 0,
so that n" = 2k — 1. Writing X = 0, we have the factor il
0 \ k
— (indicated by the
reduction of order from 2& — 1 to k — 1), whence t" — k— 1, and then m' — (4k — 2) 2 (k 1),
= 3k — 3, agreeing with the table. The process holds good for k = 2.
Case P k . We have ft = y' = 0; the equation is
02k .(&_!) [a'/3X — 2aa'F}
+ 8 2k ~ l . — k {— a (3 Y\
+ 0 k+1 . (k — 1) {— a'yX}
+ 0 k . (k - 2) ya'F- (2k - 1) a!j3Z
+ 6 . <ya!Z=0,
0^-yX, Y)
+ 0*-* Y
+ e k x
+ 6 k ~yk- 2) Y+Z
+ Z=0,
so that here n" =2k — 1. Writing Z = 0, we have the factor 6 k 1 , whence i k . 2,
m" = 4tk-4-(k — 2) = 3k — 2, agreeing with the table. If, however, k = 2, then on writing
Z = 0 the equation (instead of the factor 6 k ~ l ) acquires the factoi 0 (— 0), so that
here n" = 3, t" = 1, m" = 3, agreeing with the column P 2 R, of the table.
