40 
ON EVOLUTES AND PARALLEL CURVES. 
[493 
Case Q k . 
equation is 
We have 7 = 0, /3' = 7' = 0, viz. /3'= 0 in the formulae of B k . 
6 2k . ( k — 1) a/3Y 
+ 6' 2k ~ l . — k (— a!BY) 
+ 6 k+1 . ( k — 1) 2aa'Z 
+ 6 k .-(2k- l)a'/3Z=0, 
e k x 
+ 6 k - x Y 
+ 0 z 
+ z=0, 
The 
so that n" = k. For Z = 0 there is the factor 6 k \ hence 1" = k — 2, m"— 2 (k — 1 )—(k - 2), = k. 
The process holds good for k = 2. 
Case R k . 
equation is 
We have /3 = 0, a! = 0, /3' — 0, viz. o! = 0 in the formulae of G k . 
0k+i . (& _ i) (_ cy'X) 
+ 6 k ,(k — 2) (ay'F) 
+ 6 . (a.<y'Z — 2yy'X) 
+ 277 Y = 0, 
e k+i x 
+ e k Y(k - 2) 
+ 6 (Z, X) 
+ F = 0, 
The 
so that n" — k + 1, 1" = 0, m" = 2k. But observe that in the particular case k = 2, the 
form is 6 3 X+6(Z, X) + F= 0, the term 6 k Y disappearing on account of the factor 
/ Q \2 
& — 2. Here on writing X = 0, there is a factor i 1 — — J (indicated by the reduction 
of order from 3 to 1), hence 1" = 1, n" = 3, m"=2.2 —1 = 3, agreeing with the column 
p 2 r 2 . 
Case S k . 
equation is 
We have a = 0, a' =/3'= 0, viz. y3' = 0 in the formulae of D k . The 
e k . (k +1) #/x 
+ -WF 
+ 0 — 2<yy'X 
e k x 
+ F 
+ e x 
+ 
277' F — Bi ^ 
+ 
F+^ = 0, 
so that n" = t" — 0, m" — 2k — 2. The process applies to the case k = 2. 
As to the formula for A 3 , B 3 ,...S 3 , there is nothing special in these; they are 
simply deduced from those for A k , B k ,... S k by writing therein k= 3. And we ha,ve 
thus the foregoing series of formulae, which will apply to the greater part of the 
cases which ordinarily arise. For instance suppose there is at / or J a triple 
point = cusp + 2 nodes; there is here an ordinary branch and a cuspidal (ordinary 
cuspidal) branch and according as IJ touches neither branch, _the ordinary branch, or 
the cuspidal branch, the corrections to m", n', t", k" are B 2 + R 3 , S 2 + R 3 > R 2 + S 3 
respectively. Observe moreover that A 2 C 2 is no speciality, B 2 D 2 is the speciality g = 1, 
P 2 R 2 the speciality /= 1. 
There is a remarkable case in which the fundamental assumption of the (1, 1) 
correspondence of the evolute with the original curve ceases to be correct. In fact,